text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
Operațiunea "Pilonul Apărării" (în ebraică: עַמּוּד עָנָן, Amúd Anán "Stâlpul de nori", după un episod din cartea Exodului din Biblie, unde se povestește cum evreii ieșiți din Egipt au fost apărați de un stâlp miraculos de nori) a fost o operațiune militară condusă de Armata Israeliană în Fâșia Gaza între 14 și 21 noiembrie 2012. Aceasta a început oficial odată cu uciderea lui Ahmed Jabari, șeful operațiunilor militare din Gaza ale organizației Hamas. Obiectivele declarate ale operațiunii au fost stoparea atacurilor cu rachete originare din Fâșia Gaza împotriva țintelor civile și distrugerea capacităților organizațiilor militante.
Conform Guvernului israelian, operațiunea a fost demarată ca răspuns la rachetele lansate din Fâșia Gaza și la atacurile împotriva militarilor israelieni de la granița dintre Israel și Fâșia Gaza. Grupările palestiniene au declarat că atacurile asupra civililor israelieni se justifică prin blocada impusă Fâșiei Gaza și ocuparea Cisiordaniei și Ierusalimului de Est.
În timpul operațiunii, armata israeliană a lansat peste 1.550 de focuri asupra unor ținte din Fâșia Gaza, printre care se numără lansatoare de rachete, depozite de arme, militanți individuali, numeroase clădiri guvernamentale, câmpuri, case și blocuri. Conform Centrului Palestinian pentru Drepturile Omului (CPDO), 158 de palestinieni au fost uciși până pe 22 noiembrie, dintre care 102 civili, 55 de militanți și un ofițer de poliție. Opt palestinieni au fost executați public de Brigăzile Izz ad-Din al-Qassam pentru presupusa colaborare cu Israelul. Ministerul Sănătății, condus de Hamas, a estimat că aproximativ 840 de palestinieni au fost răniți de la începutul operațiunii.
Declarând că asasinarea lui Jabari "a deschis porțile iadului", Brigăzile al-Qassam și Jihadul Islamic Palestinian și-au intensificat atacurile cu rachete asupra orașelor din Isarel. Această operațiune a fost numită Operațiunea Pietrele de lut ars (în arabă حجارة سجيل, ḥijārat sajīl) de către Brigăzile al-Qassam și Operațiunea Cerul albastru (în arabă السماء الزرقاء, as-samā' az-zarqā) de către membrii JIP.
Grupările palestiniene au lansat peste 1.456 de focuri (rachete Fajr-5, BM-21 Grad, Qassam și mortiere) în Rișon Le-Țion, Beer Șeva, Așdod, Așkelon și alte zone populate; Tel Aviv a fost bombardat pentru prima dată de la Războiul din Golf din 1991, rachetele fiind îndreptate și către Ierusalim. Rachetele au ucis patru civili israelieni – trei dintre ei într-un atac direct asupra unei case din Kiryat Malachi – doi soldați israelieni și mai mulți civili palestinieni. Până la 19 noiembrie, peste 252 de israelieni au fost răniți de atacurile cu rachete.
Iran, Egipt, Turcia și alte state arabe și musulmane au condamnat operațiunea condusă de Israel. Consiliul de Securitate al ONU a ținut o ședință de urgență pe această, dar nu a luat nici o decizie. Pe 21 noiembrie, Hamas și Israel au semnat un armistițiu mediat de Egipt, ambele părți revendicând victoria.
Context
Conflictul în forma sa actuală se află în desfășurare de când partidul islamic Hamas a câșigat alegerile legislative palestiniene din ianuarie 2006. În iunie 2007, după un conflict militar cu grupul său rival, Fatah, Hamas a preluat controlul Fâșiei Gaza. Drept răspuns, Israel și Egipt au închis granițele terestre cu Fâșia Gaza în aceeași lună, afectând situația economică și umanitară a acesteia. În timp ce Crucea Roșie consideră că blocada impusă de Israel este ilegală sub dreptul umanitar internațional, iar un raport al ONU a declarat că aceasta a fost de asemenea ilegală, o anchetă juridică a ONU a constatat că blocada a fost atât legală, cât și corespunzătoare. Cu toate că Israelul și-a retras în 2005 personalul militar și civil, SUA, ONU și Liga Arabă consideră acest stat ca fiind o putere ocupantă în teritoriu. Hamas, un grup islamist palestinian considerat organizație teroristă de SUA, Uniunea Europeană, Canada, Australia, și Japonia, a cerut distrugerea Israelului încă din 1988. Rusia, Turcia și Norvegia nu consideră că Hamas este o organizație teroristă.
Între Israel și Fâșia Gaza au continuat să apară tensiuni, ambele părți trecând periodic prin diferite conflicte. Unul dintre cele mai cunoscute este conflictul armat de la sfârșitul lui 2008 și începutul lui 2009, care a durat trei săptămâni. Israel a declarat că a avut drept scop oprirea atacului cu rachete din Fâșia Gaza, după ce aproximativ 2.378 de proiectile au fost lansate către Israel pe o perioadă de unsprezece luni. Conform organizației B'Tselem, forțele de securitate israeliene au ucis 271 de plaestinieni în Fâșia Gaza de la terminarea Războiului din Gaza și până la 30 septembrie 2012.
Conform oficialilor israelieni, Hamas, cu ajutorul experților tehnici din Iran și guvernului sudanez, a făcut contrabandă cu rachete iraniene de tip Fajr-5, pe care le-a plasat în zone extrem de populate din Israel. Cu toate acestea, comandantul Gărzilor Revoluționare din Iran, Generalul Mohammad Ali Jafari, a declarat: "Nu am trimis nici o armă în Gaza pentru că se află sub blocadă, dar suntem onorați să vă anunțăm că le-am dat tehnologia rachetelor Fajr-5." Între timp, Ali Larijani a declarat că Iranul a fost "onorat" să ajute Hamas cu "aspecte materiale și militare". În Gaza se află aproximativ 35.000 de luptători palestinieni, în timp ce Israelul are în armată 175.000 de recruți.
Cronologie
14 noiembrie
Operațiunea a început în jurul orei 16:00 (ora Israelului) cu un atac aerian care a vizat automobilul în care se afla Ahmed Jabari, liderul aripii militare a Hamas. Osama Hamdan, un reprezentant al Hamas în Liban, a susținut că și fiul lui Jabari a fost ucis, însă afirmația s-a dovedit a fi falsă. Armata israeliană a dat publicității înregistrarea cu raidul aerian.
Militanții din Gaza au continuat să lanseze rachete spre orașele israeliene Beer Șeva, Așdod, Ofakim și spre consiliile regionale Hanegev Shaar și Eshkol. Sistemul de apărare Cupola de fier a făcut 130 de interceptări. Aproximativ 55 de rachete au fost lansate în seara de 14 noiembrie, printre care și o rachetă BM-21 Grad îndreptată spre Centrul de Cercetare Nucleară Negev de lângă Dimona.
Extindere
Cisiordania
Conflictul armat a declanșat ample proteste în Cisiordania, care au dus la ciocniri între palestinieni și Armata Israeliană. Drumul dintre Ierusalim și Gush Etzion a fost închis ca urmare a protestelor violente.
Pe 18 noiembrie 2012, un palestinian de 31 de ani care participa la o demonstrație în localitatea Nabi Saleh a fost ucis de un foc israelian. Armata Israeliană, care a catalogat demonstrația drept "ilegal și violent", a lansat o anchetă asupra incidentului. O zi mai târziu, peste 50 de palestinieni au fost răniți în timpul protestelor de solidaritate organizate în Ierusalimul de Est, Ramala, Betleem, Beit Ummar și Qalandia. Mii de oameni au mărșăluit în semn de protest față de uciderea unui protestatar în ziua precedentă.
Mai multe proteste și ciocniri au avut loc în Cisiordania între 21 și 22 noiembrie. Mii de palestinieni au protestat față de moartea lui Rushdi al-Tamimi, al cărui proces a trecut prin Ramala și Universitatea Birzeit înainte de a se termina în orașul său natal, Nabi Saleh. Protestatarii care au participat la funeralii au aruncat cu pietre în trupele israeliene de la intrarea în sat, acestea răspunzând cu gaze lacrimogene și gloanțe de cauciuc. Sute de persoane îndoliate au participat la funeraliile palestinianului ucis în Hebron, la 20 noiembrie. După înmormântare, mai mulți tineri protestatari s-au apropiat de o așezare israeliană de lângă Piața Bab Al-Zawiya, ceea ce a dus la ciocniri cu forțele israeliene. Aproximativ 40 de palestinieni au fost răniți. În orașul Nablus, sute de protestatari au fluturat steagul Hamas. Intrarea în Bani Naim a fost închisă de armata israeliană, după mai multe ciocniri între aceasta și locuitorii orașului. Între timp, Al-Jalama, un sat din nordul Cisiordaniei, a fost declarat "zonă militară închisă", după ce sute de demonstranți palestinieni au protestat la punctul de control. Cinci palestinieni au fost arestați în raidurile armatei israeliene în Ya'bad și Tubas.
Liban
Tot pe 20 noiembrie, o patrulă a armatei libaneze a descoperit două rachete Grad de 107 mm, gata de lansare, între satele Halta și Mari, la aproximativ 2 mile de granița cu Israelul. Rachetele au fost dezamorsate, iar generalul israelian Yoav Mordechai a declarat că facțiuni palestiniene din Liban au fost probabil în spatele planului.
Pe 21 noiembrie, conform oficialilor de la Beirut, două rachete lansate din Liban către Israel s-au prăbușit în interiorul Libanului, iar pe 22 noiembrie, după semnarea armistițiului, armata libaneză a mai dezarmat o rachetă amplasată în localitatea Marjayoun, aflată la 10 km de granița cu Israelul.
Armistițiu
Israel și Hamas au refuzat să discute în mod direct, iar negocierile s-au desfășurat prin intermediari. Principalii jucători din negocierea armistițiului au fost oficiali din SUA și Egipt, care au îndeplinit rolul de facilitatori.
Tentative de armistițiu
Negocierile indirecte dintre Israel și Hamas au fost mediate Egipt. Președintele egiptean Mohamed Morsi a prezis că negocierie ar conduce la rezultate pozitive în curând. În schimb, Secretarul de Stat american Hillary Clinton, după o întâlnire cu Netaniahu, a declarat că procesul va avea loc "poimâine". Secretarul General al ONU, Ban Ki Moon, s-a întâlnit de asemenea cu Netaniahu pentru a încerca să pună capăt violenței. Ministrul de Externe turc și diplomații din Liga Arabă au fost trimiși în Gaza pentru a promova un armistițiu între părțile aflate în conflict.
Conform rapoartelor din Cairo, Israelul a făcut 6 cereri pentru un armistițiu:
Fără violență pe o perioadă mai mare de 15 ani.
Fără contrabandă sau transfer de arme în Gaza.
Încetarea tuturor atacurilor cu rachete asupra soldaților israelieni.
Israelul își rezervă dreptul de a ataca teroriștii în cazul unui atac sau a unui posibil atac.
Punctele de trecere a frontierei dintre Israel și Gaza vor rămâne închise.
Politicienii din Egipt trebuie să garanteze cererile de mai sus.
În schimbul unui armistițiu, cererile includ ridicarea blocadei navale din Gaza, garanțiile comunității internaționale pentru încetarea crimelor vizate, oprirea raidurilor transfrontaliere efectuate de Armata israeliană și încetarea atacului. Liderul Hamas Khaled Meshaal a cerut în plus "garanțiile internaționale" pentru ridicarea blocadei.
Victime
Victimele palestiniene
Pe 20 noiembrie 2012, oficialii din domeniul sănătății din Gaza au declarat că aproximativ 113 palestinieni au fost uciși de la începutul operațiunii, dintre care 53 au fost civili, 49 militanți și un polițist.
Reacții
Din perspectiva Israelului, operația a reușit deoarece a restabilit descurajarea.
Guvernele din Europa și America de Nord au declarat în timpul operațiunii că Israelul a avut dreptul de a se apăra împotriva atacurilor palestiniene originare din Fâșia Gaza asupra populației civile.
Guvernul islamist al Egiptului a condamnat operațiunea și a rechemat pe ambasadorul în Israel pentru consultări.
Note
Legături externe
Live Updates: IDF Targeting Terrorist Sites, Armata Israeliană (în engleză)
Q&A: Israel-Gaza violence la BBC News Online (în engleză)
Gaza Crisis la Al Jazeera English (în engleză)
Operațiuni militare ale Israelului
Conflictul dintre Israel și Fâșia Gaza din 2012
2012 în Israel
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,078
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<!-- NewPage -->
<html lang="en">
<head>
<!-- Generated by javadoc (1.8.0_111) on Wed Jan 04 22:31:28 EST 2017 -->
<title>Serialized Form</title>
<meta name="date" content="2017-01-04">
<link rel="stylesheet" type="text/css" href="stylesheet.css" title="Style">
<script type="text/javascript" src="script.js"></script>
</head>
<body>
<script type="text/javascript"><!--
try {
if (location.href.indexOf('is-external=true') == -1) {
parent.document.title="Serialized Form";
}
}
catch(err) {
}
//-->
</script>
<noscript>
<div>JavaScript is disabled on your browser.</div>
</noscript>
<!-- ========= START OF TOP NAVBAR ======= -->
<div class="topNav"><a name="navbar.top">
<!-- -->
</a>
<div class="skipNav"><a href="#skip.navbar.top" title="Skip navigation links">Skip navigation links</a></div>
<a name="navbar.top.firstrow">
<!-- -->
</a>
<ul class="navList" title="Navigation">
<li><a href="overview-summary.html">Overview</a></li>
<li>Package</li>
<li>Class</li>
<li>Use</li>
<li><a href="overview-tree.html">Tree</a></li>
<li><a href="deprecated-list.html">Deprecated</a></li>
<li><a href="index-files/index-1.html">Index</a></li>
<li><a href="help-doc.html">Help</a></li>
</ul>
</div>
<div class="subNav">
<ul class="navList">
<li>Prev</li>
<li>Next</li>
</ul>
<ul class="navList">
<li><a href="index.html?serialized-form.html" target="_top">Frames</a></li>
<li><a href="serialized-form.html" target="_top">No Frames</a></li>
</ul>
<ul class="navList" id="allclasses_navbar_top">
<li><a href="allclasses-noframe.html">All Classes</a></li>
</ul>
<div>
<script type="text/javascript"><!--
allClassesLink = document.getElementById("allclasses_navbar_top");
if(window==top) {
allClassesLink.style.display = "block";
}
else {
allClassesLink.style.display = "none";
}
//-->
</script>
</div>
<a name="skip.navbar.top">
<!-- -->
</a></div>
<!-- ========= END OF TOP NAVBAR ========= -->
<div class="header">
<h1 title="Serialized Form" class="title">Serialized Form</h1>
</div>
<div class="serializedFormContainer">
<ul class="blockList">
<li class="blockList">
<h2 title="Package">Package org.drip.analytics.support</h2>
<ul class="blockList">
<li class="blockList"><a name="org.drip.analytics.support.CaseInsensitiveHashMap">
<!-- -->
</a>
<h3>Class <a href="org/drip/analytics/support/CaseInsensitiveHashMap.html" title="class in org.drip.analytics.support">org.drip.analytics.support.CaseInsensitiveHashMap</a> extends java.util.HashMap<java.lang.String,<a href="org/drip/analytics/support/CaseInsensitiveHashMap.html" title="type parameter in CaseInsensitiveHashMap">V</a>> implements Serializable</h3>
</li>
<li class="blockList"><a name="org.drip.analytics.support.CaseInsensitiveTreeMap">
<!-- -->
</a>
<h3>Class <a href="org/drip/analytics/support/CaseInsensitiveTreeMap.html" title="class in org.drip.analytics.support">org.drip.analytics.support.CaseInsensitiveTreeMap</a> extends java.util.TreeMap<java.lang.String,<a href="org/drip/analytics/support/CaseInsensitiveTreeMap.html" title="type parameter in CaseInsensitiveTreeMap">V</a>> implements Serializable</h3>
</li>
</ul>
</li>
<li class="blockList">
<h2 title="Package">Package org.drip.json.parser</h2>
<ul class="blockList">
<li class="blockList"><a name="org.drip.json.parser.ParseException">
<!-- -->
</a>
<h3>Class <a href="org/drip/json/parser/ParseException.html" title="class in org.drip.json.parser">org.drip.json.parser.ParseException</a> extends java.lang.Exception implements Serializable</h3>
<dl class="nameValue">
<dt>serialVersionUID:</dt>
<dd>-7880698968187728548L</dd>
</dl>
<ul class="blockList">
<li class="blockList">
<h3>Serialized Fields</h3>
<ul class="blockList">
<li class="blockList">
<h4>errorType</h4>
<pre>int errorType</pre>
</li>
<li class="blockList">
<h4>unexpectedObject</h4>
<pre>java.lang.Object unexpectedObject</pre>
</li>
<li class="blockListLast">
<h4>position</h4>
<pre>int position</pre>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li class="blockList">
<h2 title="Package">Package org.drip.json.simple</h2>
<ul class="blockList">
<li class="blockList"><a name="org.drip.json.simple.JSONArray">
<!-- -->
</a>
<h3>Class <a href="org/drip/json/simple/JSONArray.html" title="class in org.drip.json.simple">org.drip.json.simple.JSONArray</a> extends java.util.ArrayList implements Serializable</h3>
<dl class="nameValue">
<dt>serialVersionUID:</dt>
<dd>3957988303675231981L</dd>
</dl>
</li>
<li class="blockList"><a name="org.drip.json.simple.JSONObject">
<!-- -->
</a>
<h3>Class <a href="org/drip/json/simple/JSONObject.html" title="class in org.drip.json.simple">org.drip.json.simple.JSONObject</a> extends java.util.HashMap implements Serializable</h3>
<dl class="nameValue">
<dt>serialVersionUID:</dt>
<dd>-503443796854799292L</dd>
</dl>
</li>
</ul>
</li>
</ul>
</div>
<!-- ======= START OF BOTTOM NAVBAR ====== -->
<div class="bottomNav"><a name="navbar.bottom">
<!-- -->
</a>
<div class="skipNav"><a href="#skip.navbar.bottom" title="Skip navigation links">Skip navigation links</a></div>
<a name="navbar.bottom.firstrow">
<!-- -->
</a>
<ul class="navList" title="Navigation">
<li><a href="overview-summary.html">Overview</a></li>
<li>Package</li>
<li>Class</li>
<li>Use</li>
<li><a href="overview-tree.html">Tree</a></li>
<li><a href="deprecated-list.html">Deprecated</a></li>
<li><a href="index-files/index-1.html">Index</a></li>
<li><a href="help-doc.html">Help</a></li>
</ul>
</div>
<div class="subNav">
<ul class="navList">
<li>Prev</li>
<li>Next</li>
</ul>
<ul class="navList">
<li><a href="index.html?serialized-form.html" target="_top">Frames</a></li>
<li><a href="serialized-form.html" target="_top">No Frames</a></li>
</ul>
<ul class="navList" id="allclasses_navbar_bottom">
<li><a href="allclasses-noframe.html">All Classes</a></li>
</ul>
<div>
<script type="text/javascript"><!--
allClassesLink = document.getElementById("allclasses_navbar_bottom");
if(window==top) {
allClassesLink.style.display = "block";
}
else {
allClassesLink.style.display = "none";
}
//-->
</script>
</div>
<a name="skip.navbar.bottom">
<!-- -->
</a></div>
<!-- ======== END OF BOTTOM NAVBAR ======= -->
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,538
|
Q: Oauth and scopes and its correct use May be I am missing something here but what the point of client requesting scope?
What I mean is that when a client request a token he/she can say "I want everything" i.e. read/write/delete/update etc.
In reality, user might only have read access. So what's the point of requesting these in scopes? or its not a good practice/idea to put these in scope? If not then what do you really use it for?
I have seen in some places where it's used for Email, Username while others its used for repo:read, repo:write...
Isn't Authorization server going decide what permission the user has?
Thanks
A: I'm not sure if I understand your question fully but here goes. In a typical OAuth scenario, there's the:
*
*OAuth provider - has the protected resource, issues access tokens to resource (e.g. Google)
*End user - owner of resource (e.g, Google user)
*Relay party - uses OAuth to access protected resource from provider on behalf of end user (e.g, some random app that does something with your google profile)
When you use the relay party's application, it needs to get your information from the OAuth provider. Most providers have different levels or "scopes" of access to the end user's information - it's usually not an all or nothing access. The relay party must define the scope of access, and ask the OAuth provider, the provider presents the scope of access request to the end user, and finally the end user can decide whether to grant the relay party that scope of access or deny it.
So without scope, how can a relay party request a granular level of access?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,584
|
Пло́щадь Па́вших Революционе́ров — городская площадь, расположенная в Центральном районе Челябинска, ограничена улицами: Российской, Красноармейской и Труда. До 20 февраля 1920 года площадь называлась Казарменной, позже — площадью Памяти Павших. Объект культурного наследия регионального значения.
История
Своё историческое название Казарменная (или в народе, Солдатская) площадь получила по расположенному на площади зданию Белых казарм (ныне ул. Российская, 151 и 149). Здание было возведено в 1878 году. Казармы на этой площади располагались с XVIII века, но сохранившаяся постройка — первое каменное здание казармы. На площади, близ здания казарм, произошло одно из самых трагичных событий в истории Челябинска. 3 июня 1918 года на Казарменной площади казаками были изрублены пять активных революционеров: Д. В. Колющенко (член городского комитета РСДРП(б)), М. А. Болейко (начальник революционного штаба охраны города); В. И. Могильников (зам. нач. штаба охраны города), П. Н. Тряскин (секр. правления профсоюза рабочих); Ш. И. Гозиосский (секр. исполкома Чел. Совета).
Очевидцы так описывали произошедшее:
20 февраля 1920 года, в честь погибших коммунистов, площадь получила название Памяти Павших. В 1957 году на здании Белых казарм (в то время в них уже располагался военный госпиталь) была установлена памятная доска, освещающая события 1918 года. В этом же году площадь получила своё настоящее имя — площадь Павших Революционеров. В середине 80-х годов XX века планировалось строительство памятника на площади, но на его месте заложен был только памятный камень.
В настоящее время площадь Павших Революционеров была значительно уменьшена из-за строительства крупной дорожной развязки в центре Челябинска. Но здание бывших казарм (ныне — Окружной военный клинический госпиталь) уцелело, так же как и небольшой пятачок земли, где и установлен памятный камень. Также на площади располагается действующая протестантская церковь.
Факты
В 1929 году в центре площади начали строительство деревянного цирка, который был открыт 5 сентября 1930 года и получил адрес улица Труда 48. С годами цирк обветшал, грибок источил несущие конструкции, и было принято решение его сжечь. Эта спецоперация была успешно проведена 15 марта 1973 года.
Примечания
Ссылки
Суть времени. Память Челябинска: площадь Павших революционеров.
Вечерний Челябинск. Площадь Павших революционеров.
Площади Челябинска
Объекты культурного наследия России в Челябинской области
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,606
|
\section{Introduction}
\label{sec-introduction}
Self-assembly is a term used to describe systems in which a small
number of simple components, each following local rules governing
their interaction with each other, automatically assemble to form a target
structure. Winfree \cite{Winf98} introduced the abstract
Tile Assembly Model (aTAM) -- based on a constructive version of Wang tiling
\cite{Wang61,Wang63} -- as a simplified mathematical model of
Seeman's work \cite{Seem82} in utilizing DNA to physically implement self-assembly at the molecular level. In
the aTAM, the fundamental components are un-rotatable, but
translatable square ``tile types'' whose sides are labeled with glue
``labels'' and ``strengths.'' Two tiles placed next to each
other \emph{interact} if the glue labels on their abutting sides
match, and a tile \emph{binds} to an assembly if the total strength on all of its interacting sides exceeds the ambient
``temperature,'' equal to 2 in this paper. The model is detailed more
formally in Section \ref{sec-tile-assembly-model}.
Winfree
\cite{Winf98} demonstrated the computational universality of the aTAM by showing how to simulate an arbitrary cellular automaton with a tile assembly system. Building on these connections to computability, Rothemund and Winfree \cite{RotWin00} investigated the minimum number of tile types needed to uniquely assemble an $n \times n$ square. Utilizing the theory of Kolmogorov complexity, they show that for any algorithmically random $n$, $\Omega\left(\frac{\log n}{\log \log n}\right)$ tile types are required to uniquely assemble an $n \times n$ square, and Adleman, Cheng, Goel, and Huang \cite{AdChGoHu01} exhibit a construction showing that this lower bound is asymptotically tight.
Real-life implementations of the aTAM involve (at the present time) creating tile types out of DNA double-crossover molecules \cite{RoPaWi04}, copies of which can be created at an exponential rate using the polymerase chain reaction (PCR) \cite{MolecularCloningLabManual}. PCR technology has advanced to the point where it is automated by machines, meaning that \emph{copies} of tiles are easy to supply, whereas the number of distinct tile \emph{types} is a precious resource, costing much more lab time to create. Therefore, effort has been put towards developing methods of ``programming'' tile sets through methods other than hard-coding the desired behavior into the tile types. Such methods include \emph{temperature programming} \cite{KS06,Sum09}, which involves changing the ambient temperature through the assembly process in order to alter which bonds are possible to break or create, and \emph{staged assembly} \cite{DDFIRSS07}, which involves preparing different assemblies in different test tubes, which are then mixed after reaching a terminal state. Each of these models allows a \emph{single} tile set to be reused for assembling different structures by programming different environmental conditions that affect the behavior of the tiles and therefore serve as an ``input'' to be processed by the tile set.
The ``input specification model'' used in this paper is known as \emph{tile concentration programming}. If the tile assembly system is nondeterministic -- if intermediate assemblies exist in which more than one tile type is capable of binding to the same position -- and if the solution is well-mixed, then the relative concentrations of these tile types determine the probability that each tile type will be the one to bind. Tile concentrations affect the expected time before an assembly is completed (such a model is considered in \cite{AdChGoHu01} and \cite{BeckerRR06}, for instance), but we ignore such running time considerations in the present paper. We instead focus on using the biased randomness of tile concentrations to guide a probabilistic shape-building algorithm, subject a certain kind of ``geometric \emph{space} bound''; namely, that the algorithm must be executed within the confines of the shape being assembled. This restriction follows from the monotone nature of the aTAM: once a tile attaches to an assembly, it never detaches.
We now describe related work. Chandran, Gopalkrishnan, and Reif \cite{ChaGopRei09} show that a one-dimensional line of expected length $n$ can be assembled using $\Theta\left( \log n \right)$ tile types, subject to the restriction that all tile concentrations are equal. Furthermore, they show that this bound is tight for \emph{all} $n$. Note that this is not tile concentration programming since the concentrations are forced to be equal. Nonetheless, they use the inherent randomness of binding competition to strictly improve the assembly capabilities of the aTAM; a simple pigeonhole argument shows that $n$ unique tile types are required to construct a line of length $n$ in the deterministic aTAM model. Two previous papers \cite{BeckerRR06,KaoSchS08} deal directly with the tile concentration programming model. Becker, Rapaport, and R\'emila \cite{BeckerRR06} show that there is a \emph{single} tile assembly system $\calT$ such that, for all $n\in\N$, setting the tile concentrations appropriately causes $\calT$ to assemble an $n' \times n'$ square, such that $n'$ has expected value $n$. However, $n'$ will have a large deviation from $n$ with non-negligible probability. Kao and Schweller \cite{KaoSchS08} improve this result by constructing, for each $\delta,\epsilon > 0$, a tile assembly system $\calT$ such that setting the tile concentrations appropriately causes $\calT$ to assemble an $n' \times n'$ square, where $(1-\epsilon) n \leq n' \leq (1+\epsilon) n$ with probability at least $1-\delta$, for sufficiently large $n\in\Z^+$ (depending on $\delta$ and $\epsilon$).
Kao and Schweller asked whether a constant-sized tile assembly system could be constructed that, through tile concentration programming, would assemble a square of dimensions \emph{exactly} $n \times n$, with high probability. We answer this question affirmatively, showing that, for each $\delta > 0$, there is a tile assembly system $\calT$ such that, for sufficiently large $n \in\Z^+$, there is an assignment of tile concentrations to $\calT$ such that $\calT$ assembles an $n \times n$ square with probability at least $1-\delta$. Therefore, with a constant number of tile types, any size square can be created entirely through the programming of tile concentrations.
The primary technique is a tile set that, through appropriate tile concentration programming, forms a thin structure of height $O(\log n)$ and length $O(n^{2/3})$ (and for arbitrarily small $\epsilon > 0$, the length can be made $O(n^{\epsilon})$) and encodes the value of $n$ in binary.
This binary string could be used to assemble useful structures other than squares, such as rectangles and other supersets of the sampling structure that are ``easily encoded'' in a binary string of length $O(\log n)$.
Kao and Schweller also asked whether arbitrary finite connected shapes, possibly scaled by factor $c\in\N$ (depending on the shape) by replacing each point in the shape with a $c \times c$ block of points, could be assembled from a constant tile set through concentration programming. Our construction answering the first question computes the binary expansion of $n$ with high probability in a self-assembled rectangle of height $O(\log n)$ and width $O(n^{2/3})$. By assembling this structure within the ``seed block'' of the construction of \cite{SolWin07}, our construction can easily be combined with that of \cite{SolWin07} to answer this question affirmatively as well, by replacing the number $n$ with a program that outputs a list of points in the shape, and using this as the ``seed block'' of the construction of \cite{SolWin07}.
Since it may be infeasible to specify tile concentrations with unlimited precision, we show how to generalize our construction to allow a smooth tradeoff between specifying the number $n$ through tile concentrations versus hard-coded tile types. Since $\log n$ bits are required to specify $n$ for almost all values of $n$, we show that for arbitrary $g$, it is possible to specify ``about'' $g$ of the bits through tile concentrations and the remaining ``about'' $\log n - g$ bits through the hard-coding of tile types; i.e., using a tile set that can be described with about $(\log n) - g + o(\log n)$ bits. The actual bound is complicated and is stated in Theorem \ref{thm-tradeoff}.
Finally, there are some unrealistic aspects of the concentration programming model, in addition to the assumption that concentrations can be specified to unlimited precision. Chiefly, the aTAM is itself an kinetically implausible model, but Winfree showed that the behavior of the aTAM can be approximated to arbitrary accuracy by the more realistic \emph{kinetic Tile Assembly Model} (kTAM) \cite{Winf98}. One of the assumptions Winfree employs to achieve this approximation is that all tile types have equal concentration, a condition clearly violated by our intentional setting of concentrations to be unequal. We will argue that our particular construction avoids the potential pitfalls of the concentration programming model, but leave open the task of defining a concentration programming model that is inherently immune to these pitfalls. We also show how to alter the construction to use only concentrations that are arbitrarily close to uniform, as a potential fix for the kinetic problems.
This paper is organized as follows. Section \ref{sec-tile-assembly-model}
provides background definitions and notation for the abstract TAM and tile concentration programming.
Section \ref{sec-construction} specifies and proves the correctness of the main construction, a tile set that can be used to assemble precisely-sized squares through concentration programming. This results of Section \ref{sec-construction} first appeared in \cite{RSAES}.
Section \ref{sec-shapes} specifies the construction of a tile set that assembles scaled versions of arbitrary finite shapes through concentration programming. This scaled shapes construction was announced in \cite{RSAES} but not demonstrated.
Section \ref{sec-tradeoff} discusses a relaxation of the model that allows a greater than constant number of tile types, while using fewer bits to specify the concentrations of each tile type, and shows how to achieve a smooth tradeoff between the resources of ``number of tile types'' and ``bits of precision of concentrations'', to assemble squares, while using an asymptotically optimal number of total bits of precision (i.e., the bits of precision of concentrations, plus the bits needed to describe the tile types, are $O(\log n)$ for an $n \times n$ square).
Section \ref{sec-unrealistic} discusses some unrealistic aspects of the concentration programming model, and argues that the constructions of this paper are resistant to the problems caused by those unrealistic assumptions, or can be fixed to alleviate these problems.
Section \ref{sec-conclusion} concludes the paper, discusses practical limitations of the construction and potential improvements and states open questions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,288
|
Q: How to prevent p tag causing gap (new line) I'm using the below <p> but it causes a gap between the tick the text after the </p>
<p class="tick">✓</p> Post messages to friends all over the world.
CSS:
.tick {
font-size: 12px;
color: green;
font-weight: bold;
line-height: normal
}
This is what it looks like:
✓
Post messages to friends all over the world.
but I want it all on the same line instead.
Any ideas?
A: display: inline; will do the trick.
.tick {
display: inline;
font-size: 12px;
color: green;
font-weight: bold;
line-height: normal;
}
But i think that better approach would be do this in unordered list with custom li bullets.
A: The answer of vegijtha works, but you shouldn't use the p-tag here. They have the semantic meaning of a paragraph of text - use a span-tag instead.
<span class="tick">✓</span> Post messages to friends all over the world.
A: Check this fiddle: http://jsfiddle.net/h4n2sstq/
The first example is obiously your original entry - not good.
In the second example i put the text inside the p tags so they align correctly.
<p class="tick">✓ Post messages to friends all over the world.</p>
If you don't want to put the text inside the p tags just add a css rule for the tick span with an inline-block or just inline display property.
display: inline-block;
Good luck!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 291
|
Serrata granum is a species of sea snail, a marine gastropod mollusk in the family Marginellidae, the margin snails.
Description
The size of the shell attains 4 mm.
Distribution
This marine species occurs off New Caledonia (depth range 381-469 m).
References
Boyer F. (2008) The genus Serrata Jousseaume, 1875 (Caenogastropoda: Marginellidae) in New Caledonia. In: V. Héros, R.H. Cowie & P. Bouchet (eds), Tropical Deep-Sea Benthos 25. Mémoires du Muséum National d'Histoire Naturelle 196: 389–436
External links
Marginellidae
Gastropods described in 2008
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 577
|
angular.module('modal', []);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,399
|
One call to The Painted Turtle Inn can be the easiest way to cross off the names on your Christmas List! For every $100 gift certificate you buy, you can choose to either get an additional $25 for YOU to use on your stay OR get an additional $25 in value on the card you are giving. Call today to take advantage of this awesome offer before time runs out! 269-982-9463 Offer must be requested at time of purchase. Cannot be combined with any other offers or third-party gift certificates.
Posted in | Comments Off on One-Stop Holiday Shopping!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,846
|
Vibrant and exuding a deep warmth, orange sapphires are a rare beauty to behold in their natural and untreated state.
A stunning radiant cut orange sapphire.
Orange sapphires are quite rare, and are some of the most difficult sapphires to find in a natural, untreated state. The color of most orange sapphires has a secondary tone of yellow as well. Intense orange sapphires that are untreated and naturally orange color can be expensive and difficult to find in a larger carat size.
Of all orange sapphires seen in the marketplaces of websites and jewelry stores, 99.999% will be treated with extreme heat to produce the vibrant orange color. These stones are not rare and are not expensive. A true natural untreated orange sapphire needs to have a reputable gem laboratory report if it is to be considered a true unheated rare orange sapphire.
An oval orange sapphire ring set in a flower diamond design mount.
Orange sapphires range from light pastel oranges to vivid orangish-reds due to their blend of red and yellow hues . These sapphires are colored by a combination of chromium (red) and iron (yellow) trace minerals , or by exposure to natural radiation. Orange sapphires whose color stems from natural radiation may fade on exposure to heat or intense daylight.
A princess cut orange sapphire with a lighter hue.
Most orange sapphires today are coming from Australia and Madagascar, with others from Sri Lanka and Tanzania. Sometimes orange sapphires have a secondary color tone of yellow or brown. This means the more pure the orange, the more expensive the sapphire. Large orange sapphires are almost non-existent. There is almost no substitute for an orange sapphire of natural color, as these are exceptionally unique sapphires.
An oval orange sapphire and diamond ring.
For orange heat treated sapphires, I won't consider buying them for personal use. The only way to get a heated sapphire orange is to cook it to the point where its color is getting close to brown; it's basically deep fried. I cannot see the value at that point. It doesn't speak to me after decades of seeing these stones in the marketplace.
A true untreated orange sapphire is something I absolutely admire and they are actually super rare. What I appreciate is an orange that is all-natural, that gives me a feeling of awe and beauty.
As we get closer to rounding out the unique and rare sapphire colors, we next explore Green Sapphires.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,341
|
Delimara Tower (), originally known as Torre della Limara, was a small watchtower on the Delimara Peninsula, in the limits of Marsaxlokk, Malta. It was built in 1659 as the tenth De Redin tower, and an artillery battery was later built nearby in 1793. Both the tower and the battery have been demolished.
History
Delimara Tower was built in 1659 at the tip of Delimara Point. It followed the standard design of the De Redin towers, having a square plan with two floors and a turret on the roof. A feature unique to Delimara Tower was that it had machicolations. It also had a buttress at the base, implying that it had some structural weaknesses. A similar buttress still exists at Triq il-Wiesgħa Tower.
Delimara Tower had Xrobb l-Għaġin Tower in its line of sight to the northeast, and Bengħisa Tower to the southwest. A mortar battery was built near the tower in 1793.
Both the tower and battery were demolished by the British to clear the line of fire of the nearby Fort Delimara.
References
De Redin towers
Batteries in Malta
Towers completed in 1659
Buildings and structures completed in 1793
Demolished buildings and structures in Malta
Marsaxlokk
Former towers
1659 establishments in Malta
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,607
|
Q: Changes references My cites had a format (1) and y want [1]
my code:
\usepackage[ pdftex, plainpages = false, pdfpagelabels,
pdfpagelayout = OneColumn, % display single page, advancing flips the page - Sasa Tomic
bookmarks,
bookmarksopen = true,
bookmarksnumbered = true,
breaklinks = true,
linktocpage,
pagebackref,
colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = red,
anchorcolor = green,
hyperindex = true,
hyperfigures
]{hyperref}
\usepackage[pdftex]{graphicx}
\DeclareGraphicsExtensions{.png, .jpg, .pdf}
\pdfcompresslevel=9
A: If your numeric-style citation call-outs have the form (1), (2), etc. it must because something in your code is inducing this non-default behavior; the LaTeX default is [1], [2], etc.
If you can't identify what's in your code that's causing the citation call-outs to be encased in round parentheses, do consider either one (but not both simultaneously!) of the following suggestions:
*
*If you're using the cite citation management package, be sure to issue the instructions
\renewcommand\citeleft{[}
\renewcommand\citeright{]}
after loading the cite package.
*If you're using the natbib citation management package, be sure to either load it with the options numbers and square or to issue the instruction
\setcitestyle{numbers,square}
after loading natbib.
Happy LaTeXing!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,012
|
Innovative Sustainable Building Material..
Union Minister Ramdas Athawale launches ..
FM Jaishankar launches ANI founder Prem ..
Union Minister Nitin Gadkari launches Pr..
Unique magnetic water treatment tech brings accolades to students of Sona College
Trans World Features | @twfindia | 16 Oct 2020
#SonaCollege, #MagneticWater
Salem: Recognizing the importance of water in agriculture, a group of students has developed farm-friendly system that reduces water usage and minimizes dependency on fertilizer.
The team from Sona College of Technology, Salem, has secured first place in the Smart India Hackathon 2020, in the Hardware category, for their solution that uses the principles of Internet of Things (IoT) and magnetisation.
Of the 343 problem statements released for all participants, across 40 nodal centres, solutions from four teams from across India were recognised in the finals.
The team consisted of six students – Manimoliselvan C: Dinesh Kumar B, Manikandan S, Lokeshwar S, all pursuing final year Civil Engineering and Kungumaswetha A, final year, Computer Science Engineering and Suvetha S, final year (Electronics and Communications Engineering).
Dr R Malathy, Dean (R&D) and Professor, Department of Civil Engineering, Sona College of Technology, who mentored the team shared, "Our project aims to help farmers by purifying their available borewell water significantly, with zero waste."
While borewell water is used for almost all farm applications, the nature of such water causes scaling. This in turn leads to non-uniform water supply to plants and poor mineral absorption. The limescale deposition also damages soil structure. Such hard water is absorbed by the plant cells with difficulty.
If through intervention, the water's minerals are broken down into smaller particles, they become more bio-available to these plant cells.
Sona College's solution achieves this by using a designated permanent rare earth magnet. The water is passed through a magnetic field and undergoes electrolysis and magnetisation. This breaks the larger water clusters into smaller, hexagonal-shaped clusters. Such magnetically treated, hexagonal-structured water molecules not only stop scaling, but also remove existing scaling.
The magnetic structuring breaks all minerals into smaller particles, and in the process, the salt in the soil is also broken down. As a result, the salt sinks deep into the soil and can be washed away easily. The desalinisation happens quickly over a season, creating much healthier plants and greater yields, and a better final product.
For even better effectiveness and easy monitoring, the inlet and outlet of the water pumping system developed by Sona College has been fitted with sensors that can measure water levels, weather and optical transducers to monitor nutrient absorption.
As per tests run by the team, the overall results in terms of plant growth proved to be dramatic. With minimal usage of water, the plants were hydrated well, were able to absorb maximum minerals. This resulted in greater yields, larger and better end product, earlier maturation, longer shelf life, and healthier plants.
Such a system allows a reduction in the water needs, as well as dependency on fertilizer and pesticides and can be a boon for farmers.
"This helps carry sufficient minerals in the standard composure, with water free from hardness for their transpiration and respiration, with zero maintenance cost. The IoT-enabled, user-friendly system makes our project unique," said Dr Malathy.
"This solution was developed in around six months, starting with selection of the right magnet that can be used for borewell water at all places, visual observation on potted plants and then moving to a peanut field for research," added Dr Malathy.
She shared that the team is also working on applying for a patent for their IoT-based solution.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,691
|
Die Burg Czorsztyn (deutsch Burg Czornsteyn oder auch Burg Zornstein) ist eine Festungsanlage in der Nähe von Czorsztyn in der Woiwodschaft Kleinpolen.
Geschichte
Der Beginn der Höhenburg reicht in das Ende des 13. Jahrhunderts zurück. Damals soll auf dem Felsen nahe dem Dunajec eine erste Befestigungsanlage, bestehend aus Erdwällen und eventuell verstärkt mit Steinen, entstanden sein. Im Übergang vom 13. zum 14. Jahrhundert entstand hier ein runder gemauerter Turm mit einem Durchmesser von 10 m und Mauerstärken bis zu 3 m. Die Anlage trug damals den Namen Wronin, der sich von der dunklen Farbe des Waldes, mit dem damals der Fels bewachsen war, ableitete. Die Burg wurde vermutlich von den Klarissen aus Stary Sącz (deutsch Alt Sandez) errichtet, möglicherweise aber bereits von der Herzogin Kunigunde von Polen, welche die Ländereien von Nowy Sącz und Pieninen (deutsch Kronenberge) besaß. Nach dem Tod ihres Gatten, Bolesław dem Keuschen trat sie in das von ihr gestiftete Klarissenkloster in Stary Sącz ein, das sie reich mit Gütern beschenkte.
Die Gründung der Burg steht in Zusammenhang mit Kolonisierungsbemühungen an den Ufern des Dunajec und der Auseinandersetzung zwischen Ungarn und Polen um diese Grenzgebiete. Hier verlief auch ein Verkehrsweg zwischen Buda und Krakau und eine Furt durch den Dunajec. Auf der ungarischen Seite lag die Burg Dunajec. Nach der Burg Czorsztyn gabelte sich der Weg, wobei eine Abzweigung nach Nowy Targ und der andere durch Krościenko nad Dunajcem nach Alt Sandez führte; diese Straße war auch ein Königsweg. Das älteste Dokument, in dem die Burg Wronin erwähnt wurde, ist eine Besiedelungsurkunde für das Dorf Kluszkowce, die 1320 von der Priorin Stronisława des Klosters in Alt Sandez ausgestellt wurde, wobei die Besiedelung bereits 1307 stattgefunden hat. Die erste Urkunde mit dem Namen Czorsztyn bezieht sich auf die Gründung von Krościenko im Jahre 1348 durch Kasimir den Großen. Unter Kasimir den Großen wird die Burg von dem König übernommen und zur Überwachung des Königsweges ausgebaut.
Die Starostei von Czorsztyn wurde während der Regierungszeit von Władysław Jagiełło gegründet. Zu dieser gehörten Krościenko nad Dunajcem, Sromowce Wyzne, Maniowy, Grywald, Hałuszowa, Tylmanova, Kluszkowce, Ochotnica, Szczawnica Wyzna, Szczawnica Nizna, Huba, Mizerna, Tylka und Krosnica. Auf der Burg residierte der jeweilige Starost, unter diesen auch Zawisza Czarny. Unter dem Starost Jan Baranowski wurde in der ersten Hälfte des 17. Jahrhunderts die Burg umgebaut und es wurden zwei neue Basteien hinzugefügt. In den zu der Starostei gehören Dörfern wurden neue Vorwerke angelegt (polnisch folwark). Er selbst gilt in der Überlieferung als sehr grausamer Despot; aufständische Bauern soll er auf Bäume, die auf Felsen gestanden haben, gebunden und getötet haben, indem diese von oben heruntergestürzt wurden. Der letzte Starost war der 1763 ernannte Josef Makary Potocki. In seiner Zeit wurde das bereits heruntergekommene Schloss 1790 von einem Blitz in Brand gesteckt und zerstört. In der Folge diente es für die Bevölkerung als Steinbruch.
Die Starostei bestand bis 1811. Dann wurde die nun unter österreichischer Herrschaft stehenden Besitzungen aufgeteilt und versteigert. 1819 wurden die Ruine und die umliegenden Gebiete von Jan Maksymilian Drohojowski gekauft. In der zweiten Hälfte des 19. Jahrhunderts ließ Marceli Drohojowski in der Nähe der Burg einen gemauerten Hof nach Entwürfen des galizischen Architekten Feliks Ksiezarski errichten. Der Nachfolger Konstatin Drohojowski ließ erste archäologische Untersuchungen und Sanierungsarbeiten an der Ruine durchführen. Er richtete 1921 um die Burg auch ein Naturreservat ein, das den Kern des späteren Pieniński Park Narodowny bildete. Der letzte Besitzer Marian Drohojowski führte vor dem Zweiten Weltkrieg Ausgrabungen am mittleren Schloss durch, bei denen Stein- und Keramikfunde gemacht wurden.
Nach 1945 wurde die Familie Drohojowski enteignet und der Besitz verstaatlicht. In den 1960er-Jahren wurden Ausgrabungen durchgeführt und Teile der Außenmauern um das untere Schloss, das Einfahrtsgebäude sowie Teile des Wirtschaftsgebäudes freigelegt. Auf Initiative des Pieniński Park Narodowny wurden großangelegte Sicherungsarbeiten am oberen und mittleren Schloss durchgeführt und das Bauwerk so vor dem weiteren Verfall bewahrt.
Historische Ereignisse in Czorsztyn
Auf der Burg hat sich mehrere Male Kazimierz der Große aufgehalten, ebenso zwei Mal Władysław Jagiełłlo; hier sollte die Rückgabe von 37 Tausend Schock Prager Groschen stattfinden, das diesem von Sigismund von Luxemburg gegen die Verpfändung von 13 Zipser Städten gewährt worden war. Der Betrag wurde aber nicht zurückerstattet. In des 1430er-Jahren zogen hier die Hussiten bei ihren Plünderungen von Zips vorbei.
1651 war die Burg Zeuge des Bauernaufstandes unter Führung von Aleksander Kostka Napierski. Mit Hilfe von gefälschten Königsbriefen begann er die örtliche Bevölkerung aufzuhetzen. Anfang 1651 sammelte er eine Gruppe bestehend aus zehn Bauern und Räubern und es gelang ihm, Czorsztyn einzunehmen, da der damalige Starost Platemberg gerade beim König in Beresteczko befand. Der Bischof Andrzej Gebicki schenkte den gefälschten Königsbriefen keinen Glauben und schickte Soldaten nach Czorsztyn. Diese griffen Napierski zuerst erfolglos an, die Verstärkung belagerte die Burg und am 24. Juni 1651 ergaben sich die Belagerten und lieferten die Anführer aus. Napierski wurde verurteilt und gepfählt.
Auf der Burg soll Jan Kazimierz auf der Flucht vor den Schweden dem Jerzy Lubomirski den polnischen Kronschatz und das Archiv zum Aufbewahren in seinem Landgut Lubowla übergeben haben.
Ruine Czorsztyn heute
Der frühere Fluss Dunajec wurde zwischen 1969 und 1996 zu dem Jezioro Czorsztyńskie aufgestaut. Dadurch ist die alte Ortschaft Czorsztyn ebenso wie die Unterburg von Czorsztyn in den Wasserfluten untergegangen. Ebenso liegt der alte Königsweg zwischen Czorsztyn und Nidzica nun unter Wasser.
Nach den Restaurierungsarbeiten ist die Burg 1996 erstmals der Öffentlichkeit zugänglich gemacht worden. Heute können das obere und mittlere Schloss sowie die Baranowski-Bastei besichtigt werden. In einem Lapidarium werden Steinfragmente aus früheren Bauepochen gezeigt. Im Hof des oberen Schlosses befindet sich ein in den Felsen gemeißeltes 6 m großes Loch, das einst die Zisterne war, deren Grund bis zum unteren Schloss reichte.
Literatur
Andrzej Weglarz: Zamek Czorsztyn. Rynek, Agencja Wydawnica WiT s.c., ISBN 978-83-89580-45-0.
Weblinks
Czorsztyn Burg
Dieter Motschmann: Die Burg Czorsztyn in den Pieninen
Czorsztyn
Gegründet im 13. Jahrhundert
Gmina Czorsztyn
Bauwerk in der Woiwodschaft Kleinpolen
Czorsztyn
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,068
|
{"url":"https:\/\/codereview.stackexchange.com\/questions\/247324\/closest-pair-heuristics-graph-with-adjacency-matrix-in-c17","text":"# Closest Pair heuristics, graph with adjacency matrix in C++17\n\nI was trying to solve a problem that was briefly mentioned at the beginning of the \"The Algorithm Design Manual\" by Steven Skiena (Ch 1, Problem 26).\n\nIt took me some time to build a working program from the pseudocode, and I think I've got it pretty close to the described idea. However my C++ knowledge is lacking, and I'm pretty sure there must exist much easier way to achieve the goal. There is a lot of things that I doubt about, specifically:\n\n\u2022 I have two versions of DFS-search, which seems excessive\n\u2022 Four nested loops to get the pairs, is there a way to make it more human-readable? Is the complexity of that block still O(n^2)? Will I be correct if I say, that complexity of the entire solution is also O(n^2), where n - number of input points, or it's actually worse than that?\n\u2022 Are there any obvious ways to make my code more clean, concise, better structured logically? Are they some well-known C++ constructions that I'm missing?\n\u2022 I'm specifically interested in help, when it's possible to save lines of code without sacrificing clarity (I know it's subjective, but if there's a way to rewrite a while loop into a for loop, such that it looks clearer and takes less space, I would like to know.\n\nI would like someone to review my code with the full rigor, and help me to improve upon it, as if my goal would be to provide a perfect C++ solution to a given problem.\n\nThe problem goes as follow:\n\nSolution that I come up with:\n\n#include <iostream>\n#include <vector>\n#include <string>\n#include <cmath>\n\ntypedef std::pair<double, double> pt_t;\ntypedef std::vector<pt_t> pts_t;\ntypedef std::vector<std::vector<int>> matrix_t;\n\nvoid print_point(pt_t pt) {\nstd::cout << \"(\" << pt.first << \", \" << pt.second << \")\" << '\\n';\n}\n\nvoid print_points(std::string headline, pts_t points) {\nstd::for_each(points.begin(), points.end(), print_point);\nstd::cout << \"---\\n\";\n}\n\nvoid print_matrix(std::string headline, matrix_t matrix) {\n\nfor (auto& row: matrix) {\nfor (auto& item : row) {\nstd::cout << item << ' ';\n}\nstd::cout << '\\n';\n}\n\nstd::cout << \"---\\n\";\n}\n\nvoid print_endpoint_pairs(std::vector<pt_t>& pairs) {\nfor (auto pair : pairs) {\nstd::cout << \"Pair: \" << pair.first << ' ' << pair.second << '\\n';\n}\nstd::cout << \"---\\n\";\n}\n\ndouble compute_distance(const pt_t& pt1, const pt_t& pt2) {\nreturn std::sqrt(\nstd::pow((pt1.first - pt2.first), 2) +\nstd::pow((pt1.second - pt2.second), 2)\n);\n}\n\nvoid dfs(matrix_t& matrix, std::vector<bool>& visited, std::vector<int>& path, int v) {\nvisited[v] = 1;\n\npath.push_back(v);\n\nfor (int i = 0; i < matrix.size(); i++) {\nif (matrix[v][i] == 1 && !visited[i]) {\ndfs(matrix, visited, path, i);\n}\n}\n}\n\nvoid dfs_ep(matrix_t& matrix, std::vector<bool>& visited, std::vector<int>& path, int v) {\nvisited[v] = 1;\n\nint connections = 0;\n\nfor (int i = 0; i < matrix.size(); i++) {\nif (matrix[v][i] == 1) {\nconnections++;\n}\n}\n\n\/\/ exclude points that have max number of connections\nif (connections <= 1) {\npath.push_back(v);\n}\n\nfor (int i = 0; i < matrix.size(); i++) {\nif (matrix[v][i] == 1 && !visited[i]) {\ndfs_ep(matrix, visited, path, i);\n}\n}\n}\n\nclass PlaneVector {\npublic:\npts_t points{};\nmatrix_t matrix;\n\nPlaneVector(pts_t points) :\npoints(points),\nmatrix(points.size(), std::vector<int>(points.size(), 0))\n{}\n\nmatrix_t get_vertex_endpoints() {\nmatrix_t chains;\nstd::vector<int> chain;\nstd::vector<bool> visited(points.size(), 0);\n\n\/\/ print_matrix(\"Matrix: \", matrix);\n\nfor (int i = 0; i < points.size(); i++) {\nif (visited[i]) {\ncontinue;\n}\n\nchain.clear();\n\ndfs_ep(matrix, visited, chain, i);\n\nchains.push_back(chain);\n}\n\nreturn chains;\n}\n\npts_t get_path() {\nstd::vector<bool> visited(points.size(), 0);\nstd::vector<int> path;\npts_t path_points;\n\ndfs(matrix, visited, path, 0);\n\nfor (int i = 0; i < path.size(); i++) {\npt_t pt = points[path[i]];\npath_points.push_back(pt);\n}\n\npath_points.push_back(path_points[0]);\n\nreturn path_points;\n}\n\nvoid add_edge(int m, int n) {\n\/\/ std::cout << \"Add edge: \" << m << ' ' << n << '\\n';\nmatrix[m][n] = 1;\nmatrix[n][m] = 1;\n}\n};\n\nstd::vector<pt_t> get_distinct_pairs(PlaneVector& vec) {\nstd::vector<pt_t> pairs{};\n\nmatrix_t chains = vec.get_vertex_endpoints();\n\/\/ print_matrix(\"Endpoints: \", chains);\n\n\/\/ generate pairs from vertex chains endpoints\nfor (int i = 0; i < chains.size() - 1; i++) {\nfor (int j = i + 1; j < chains.size(); j++) {\nfor (int n = 0; n < chains[i].size(); n++) {\nfor (int k = 0; k < chains[j].size(); k++) {\npairs.push_back(std::make_pair(chains[i][n], chains[j][k]));\n}\n}\n}\n}\n\nreturn pairs;\n}\n\npts_t closest_pair(PlaneVector& vec) {\nstd::vector<pt_t> pairs = get_distinct_pairs(vec);\n\nwhile (!pairs.empty()) {\n\/\/ print_endpoint_pairs(pairs);\n\ndouble distance = std::numeric_limits<double>::max();\nint min_i = 0;\nint min_j = 0;\n\nfor (auto pair : pairs) {\ndouble curr_distance = compute_distance(\nvec.points[pair.first],\nvec.points[pair.second]\n);\n\nif (curr_distance < distance) {\nmin_i = pair.first;\nmin_j = pair.second;\ndistance = curr_distance;\n}\n}\n\npairs = get_distinct_pairs(vec);\n}\n\n\/\/ connect two last endpoints to form a cycle\n\/\/ matrix_t chains = vec.get_vertex_endpoints();\n\nreturn vec.get_path();\n}\n\nint main() {\n\/\/ PlaneVector vec{{\n\/\/ {-2, -2},\n\/\/ {-2, 1},\n\/\/ {1, 0},\n\/\/ {2, -2},\n\/\/ {2, 1},\n\/\/ {5, 5},\n\/\/ }};\n\nPlaneVector vec{{\n{0.3, 0.2},\n{0.3, 0.4},\n{0.501, 0.4},\n{0.501, 0.2},\n{0.702, 0.4},\n{0.702, 0.2}\n}};\n\npts_t path = closest_pair(vec);\n\nprint_points(\"Points: \", vec.points);\nprint_points(\"Path: \", path);\n\nreturn 0;\n}\n\n\u2022 Currently, it is not quite clear what your code aims to do and why. You should make your question more clear. And what are these vertex chains? Aug 1 '20 at 11:23\n\u2022 It's a variance of Traveling Salesman Problem. Vertex chains are paths, that connect multiple points together. Endpoints are the ends of vertex chains (excluding the points in the middle). Aug 1 '20 at 14:01\n\n# Generalizing your graph search function\n\nThe reason you had to write two versions of the graph search algorithm is that you merged the search operation with the action you want to perform on each node. You have to separate the two.\n\nThere are various approaches you could use. One is to create an iterator class that can be used to iterate over the graph in the desired order, so that you could just write something like:\n\nfor (auto v: dfs(matrix)) {\npath.push_back(v);\n}\n\n\nAlternatively, you can write a function that takes a function object as a parameter, and applies it on each node that it finds in the desired order. You also want to avoid having to pass visited and v as a parameter to dfs(), since those variables are just internal details of the DFS algorithm, you should not expose that.\n\nstatic void dfs_impl(const matrix_t &matrix, std::function<void(int)> &func, static void dfs_impl(const matrix_t &matrix, const std::function<void(int)> &func, std::vector<bool> &visited, int v) {\nvisited[v] = true;\n\nfunc(v);\n\nfor (int i = 0; i < matrix.size(); ++i) {\nif (matrix[v][i] && !visited[i]) {\ndfs_impl(matrix, func, visited, i);\n}\n}\n}\n\nvoid dfs2(const matrix_t &matrix, int root, const std::function<void(int)> &func) {\nstd::vector<bool> visited(matrix.size());\ndfs_impl(matrix, func, visited, root);\n}\n\n\nNow you can call it like so:\n\npts_t get_path() const {\npts_t path_points;\n\ndfs(matrix, 0, [&](int v){ path_points.push_back(points[v]); });\n\npath_points.push_back(path_points.front());\nreturn path_points;\n}\n\n\nAnd instead of calling dfs_ep(), you can write the following:\n\nmatrix_t get_vertex_endpoints() const {\nmatrix_t chains;\nstd::vector<bool> visited(points.size());\n\nfor (int i = 0; i < points.size(); i++) {\nif (visited[i]) {\ncontinue;\n}\n\nstd::vector<int> chain;\n\ndfs(matrix, i, [&](int v){\nvisited[v] = true;\n\nif (std::count(matrix[v].begin, matrix[v].end, 1) <= 1) {\nchain.push_back(v);\n}\n});\n\nchains.push_back(chain);\n}\n\nreturn chains;\n}\n\n\nNote that here we had to keep a local vector visited. You could make it so you still pass a reference to visited to the function dfs(), but I find this is not as clean. Another approach is to have dfs() return an iterator to the next unvisited node:\n\nint dfs(...) {\nstd::vector<bool> visited(matrix.size());\ndfs_impl(matrix, func, visited, root);\nreturn std::find(visited.begin() + root, visited.end(), false) - visited.begin();\n}\n\n\nIn that case, you can rewrite get_vertex_endpoints() like so:\n\nmatrix_t get_vertex_endpoints() const {\n...\nfor (int i = 0; i < points.size();) {\n...\ni = dfs(matrix, i, [&](int v){\n...\n\n\n# Nesting for-loops\n\nIt is probably possible to make the four nested for-loops in get_distinct_pairs() look better. You could make a class that allows iteration over pairs, and use some kind of Cartesian product iterator from existing libraries, and use C++17 structured bindings to make the for-loops look approximately like this:\n\nfor (auto [chain1, chain2]: pairs(chains)) {\nfor (auto [vertex1, vertex2]: cartesian_product(chain1, chain2) {\npairs.push_back({vertex1, vertex2});\n}\n}\n\n\nHowever, those functions are not in the standard library, so to be portable you'd have to implement them yourself. I don't think four nested loops is bad here, the comment explains what you are going to do.\n\nThe complexity is still just O(n^2).\n\n# Other ways to make the code more readable\n\nThere are lots of functions in the standard library that can help you. I already shown a few example above, where I used std::count() and std::find() to remove manual loops. Not only does it make the code shorter, it also expresses intent explicitly.\n\nThere's also some places where you can use auto, structured bindings and so on to reduce the amount of code without hurting readability. I'll mention some more specific things that can be improved below.\n\n# Use std::hypot()\n\nTo compute the distance between two 2D points, you can make use of std::hypot():\n\ndouble compute_distance(const pt_t& pt1, const pt_t& pt2) {\nreturn std::hypot(pt1.first - pt2.first, pt1.second - pt2.second);\n}\n\n\n# Write std::ostream formatters instead of print() functions\n\nInstead of writing print_point(pt), wouldn't it be nicer to be able to write std::cout << pt << '\\n'? You can do this by converting your printing functions to overload the <<-operator of std::ostream, like so:\n\nstd::ostream &operator<<(std::ostream &o, const pt_t &pt) {\nreturn o << \"(\" << pt.first << \", \" << pt.second << \")\";\n}\n\n\nApart from printing your own objects in a more idiomatic way, it's now also much more generic, and allows you to print to files, stringstreams, and everything else that is a std::ostream.\n\n# Use const where appropariate\n\nAny time a function takes a pointer or reference parameter, and does not modify it, you should mark it as const, so the compiler can better optimize your code, and can give an error if you accidentily do modify it.\n\nAlso, class member functions that do not modify any of the member variables should also be marked const.","date":"2021-09-29 02:08:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.28770747780799866, \"perplexity\": 6525.098046578358}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780061350.42\/warc\/CC-MAIN-20210929004757-20210929034757-00136.warc.gz\"}"}
| null | null |
all: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
%.html: %.Rmd
Rscript -e "rmarkdown::render(\"$<\")"
0: basic_mcmc/nimble_basic_mcmc.html
1: build_a_model/nimble_build_a_model.html
2: CAR/CAR.html
3: customized_mcmc/nimble_customizing_mcmc.html
4: Ecology_Examples/Ecology_Examples.html
5: gaussian_process/gaussian_process.html
6: irt_models_example/IRT_example.html
7: linear_predictors/linpred.html
8: logistic_regression/nimble_logistic_regression.html
9: MLE/MLE.html
10: Partial_Pooling/PartialPooling.html
11: pumpMCEM/pumpMCEM.html
12: RJMCMC_example/RJMCMC_example.html
13: simulation_from_model/simulation_from_model.html
14: stochastic_volatility/stochastic_volatility.html
15: zero_inflated_poisson/zero_inflated_poisson.html
16: parallelization/parallelizing_NIMBLE.html
17: bnp/bnp_density.html
18: bnp/bnp_raneff.html
19: bnp/bnp_multivariate.html
20: converting_to_nimble/converting_to_nimble.html
21: posterior_predictive/posterior_predictive.html
22: restart_mcmc/restart_mcmc.html
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,118
|
Prudential Hong Kong Tennis Open 2016 – tenisowy turniej WTA kategorii WTA International Series w sezonie 2016 z cyklu Hong Kong Tennis Open rozgrywany w dniach 10–16 października 2016 roku w Hongkongu na kortach twardych o puli nagród wynoszącej 250 000 dolarów amerykańskich.
Gra pojedyncza
Zawodniczki rozstawione
Drabinka
Faza finałowa
Faza początkowa
Pula nagród
Gra podwójna
Zawodniczki rozstawione
Drabinka
Pula nagród
Uwagi
Bibliografia
Linki zewnętrzne
2016 w tenisie ziemnym
2016
2016 w Azji
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,768
|
\section*{Author Summary}
\section*{Introduction}
Gun violence has been an ongoing problem in the United States of
America, with an incidence of gun-related homicides that is
significantly higher than in most developed nations \cite{hemenway2000firearm}. While the pros and cons of gun control have been debated in the past, e.g. \cite{jacobs2002can}, this issue has recently gained new momentum. On December 14,
2012, the country witnessed one of the worst school shootings in the
history of the US, where 20 children and 6 adults were killed in Sandy
Hook Elementary School in Newton, CT. This has sparked an intense
debate among politicians, interests groups, and media
personalities. On the one hand, this tragedy has resulted in a call
for tougher gun control laws. On the other hand, there is the
suggestion to arm the population in order to protect them against
offenders. This debate cannot be settled satisfactorily by verbal
arguments alone, since these can be simply considered opinions without
a solid scientific backing. What is under debate is essentially a
population dynamic problem: how do different gun control strategies
affect the rate at which people become killed by attackers,
and how can this rate be minimized?
This question can be addressed with mathematical models that describe
the interaction between a criminal shooter and one or more people that
are the target of the shooter. The gun policy is defined as the
fraction of the population that can legally and readily obtain
firearms. On the one hand, the availability of firearms for a large
fraction of the population facilitates the acquisition of such weapons
by criminals, and this can increase the rate of attack on people. On the other hand, a relatively high prevalence of firearms in
the population can increase the chances of people to
successfully defend themselves against an attack, thus lowering the
death rate \cite{kleck1995armed, lott2010more}. The mathematical
models described in this study aim to analyze this tradeoff and to
suggest which type of gun policy minimizes firearm-related deaths under different assumptions. Calculations are performed for two
scenarios: the assault by a shooter of a single potentially armed
victim (what we call a one-on-one attack), or the assault of a crowd
of people that can be potentially armed (a one-against-many
attack). Note that the former scenario has been documented to be the
most prevalent cause of gun-related homicides \cite{cook1981effect,
maxfield1989circumstances, fox2000homicide}. The latter scenario
corresponds to incidents such as movie theater or shopping mall
shootings and requires a more complicated model. Although such
one-against-many attacks are responsible for a small minority of
gun-related homicides, they are an important focus of public attention
and are widely discussed in the press.
According to our models, both sides of the gun control argument could
in principle work, depending on parameter values. Gun-related deaths can
be minimized either by the ban of private firearms possession, or by a
policy that allows the general public to obtain guns. The following crucial
parameters determine the optimal gun control policy: (i) the fraction
of offenders that cannot obtain a gun legally but possess one
illegally, (ii) the relative degree of protection against death
during an attack, conferred by gun ownership, and (iii) the fraction of people who take up their legal right to own a gun and carry it with them when attacked. These parameters can be estimated from published statistical data. In the context of the parameter estimates, the model suggests that if gun control laws
are enforced at a level similar to that in the United Kingdom, gun-related deaths can only be minimized by a ban of private firearm possession. If this policy is impractical for cultural or constitutional reasons, the parameterized model suggests that a partial reduction of firearm availability lowers the gun-induced death rate, even though it does not minimize it. Most importantly, the model identifies the crucial parameters that decide which policy reduces gun-induced deaths, providing a guide for what needs to be measured statistically in more detail.
\section*{Results}
To calculate the effect of different gun control policies on the
gun-induced death rate of people, we turn to the following
mathematical framework. We consider the correlates of the total rate
(per year, per capita) at which people are killed as a result of
shootings. We introduce the variable $g$ to describe the gun control
policy. This quantity denotes the fraction of the population owning a
gun. A ban of private firearm possession is described by $g=0$, while a "gun availability to all"
strategy is given by $g=1$. We assume that a certain small fraction of
the population is violent, and that an encounter with an armed
attacker may result in death. The number of offenders that own
firearms is a function of the gun control policy $g$ and is denoted by
$z(g)$. The probability of a person to die during an attack
is also a function of the gun control policy $g$ because this
determines whether the person and any other people also present at the
place of the attack are armed and can defend themselves. This
probability is denoted by $F(z)$. The overall risk of being killed by
a violent attacker as a result of shooting is thus proportional to
\beq
\label{main}
{\cal F}(g)=z(g)F(g).
\eeq
An important aspect of this model is the form of the dependency of
these two quantities on the gun control policy, $g$. The number of
armed attackers, $z(g)$, is a growing function of $g$, i.e. $z'>0$. Note,
however, that even if offenders are not allowed to legally obtain
firearms, there is a probability $h$ to obtain them illegally. Hence,
the value of $z$ is non-zero for $g=0$. One example of such a function
is given by the following linear dependence,
\beq
\label{z}
z(g)=g+h(1-g)
\eeq
with $0<h<1$. The probability
$F$ for a regular person to die in an attack (once he or she is at an
attack spot) is a decaying function of $g$, $F'<0$. One example again
is a linear function (see the following section). More generally,
again we could assume $F''>0$, see the one-against-many attack scenario below.
We start by examining the case where the quantities $z$ and $F$ are
linear functions of the fraction of people that can legally obtain
firearms, $g$. This corresponds to the situation when a shooter attacks
a single individual in an isolated setting, i.e. no other people are
around to help defend against the attack. It could also correspond to
a classroom setting where a shooter attacks the entire class, but only
one person (the teacher) can be potentially armed for
protection. Subsequently, the more complex situation is examined where a
shooter attacks a group of people, each of which can be potentially
armed and contribute to defense. This would correspond to shootings in
movie theaters, malls, or other public places.
\subsection*{One-against-one attack}
Here we consider the situation where an attacker faces a single
individual who can be potentially armed. The fraction of people owning guns in the
population is defined by the legal possibility and ease at which guns
can be acquired ($g$), as well as the personal choice to acquire a
gun. Moreover, people who own a gun might not necessarily carry the firearm when attacked. Therefore, we will assume that the fraction of people armed with a gun when attacked is given by $cg$, where the parameter $c$ describes the fraction of people who take up their legal right of gun ownership and have the firearm in possession when attacked ($0\le c\le 1$). We will model the probability to be shot in an attack as
\beq
\label{Flin}
F(g)=\beta_1(1-cg)+\beta_2 cg,
\eeq
a linear function of $g$, where $\beta_1$ is the probability for an unarmed person to die in an
attack, and $\beta_2$ is the probability for an armed person to die in
an attack, with $\beta_1>\beta_2$. The number of attacks is
proportional to $z(g)$ given by equation (\ref{z}). The aim is to find the value of $g$ that minimizes the death rate, ${\cal F}$, given by equation (\ref{main}).
\paragraph{Optimization results.} The first important result is that the killing rate can only be
minimized for the extreme strategies $g=0$ and $g=1$, and that
intermediate strategies are always suboptimal (this is because ${\cal F}''<0$ for all $g$). In words, either a complete ban of private firearm possession, or a "gun availability to all" strategy minimizes gun-induced deaths.
Further, we can provide simple conditions on which of the two
extreme strategies minimizes death. Let us first assume that
\beq
\label{hc}
h<1-c;
\eeq
this case is illustrated in figure \ref{fig:one}(a) and interpreted below. Now, a ban of private firearm possession always minimizes gun-induced deaths. This can be seen in figure \ref{fig:one}(a), where we plot the shooting death rate, ${\cal F}$, as a function of the gun policy, $g$, for several values of the quantity $\beta_2/\beta_1$. For all of these functions, the minimum is achieved at $g=0$.
\begin{figure}
\centering \includegraphics[scale=0.4]{one-one.eps}
\vspace{\baselineskip}
\caption{\footnotesize {\bf The rate of death caused by shooting in
an one-against-one attack}, as a function of the gun control
policy, $g$, where $g=0$ corresponds to a ban of private firearm
possession, and $g=1$ to the ``gun availability to all''
policy. (a) The fraction of people who possess the gun and have
it with them when attacked is relatively low, $c=0.6<1-h$ with
$h=0.2$. The different lines correspond to different values of
$\beta_2/\beta_1$. For all values of $\beta_2/\beta_1$, the
shooting death rate is minimal for $g=0$. (b) The fraction of
people who possess the gun and have it with them when attacked is
relatively high, $c=0.9>1-h$ with $h=0.4$. As long as condition
(\ref{in1}) holds, the shooting death rate is minimal for $g=0$
(ban of private firearm possession, solid lines). If condition
(\ref{in1}) is violated, then the shooting death rate is
minimized for $g=1$ ("gun availability to all", dashed lines). }
\label{fig:one}
\end{figure}
Next, let us suppose that condition (\ref{hc}) is violated, that is, $h>1-c$, see figure \ref{fig:one}(b). In this case, a ban of private firearm possession minimizes death if the following additional condition holds,
\beq
\label{in1}
h<\frac{\beta_2}{\beta_1}c+1-c,
\eeq
Condition (\ref{in1}) defines a threshold value for $h$, the fraction of
offenders that cannot legally obtain a gun but possess one
illegally. If $h$ is smaller than the threshold value, then the policy
of choice is a ban of private firearm possession. The threshold value provided by inequality (\ref{in1}) depends on the
degree to which gun ownership reduces the probability for the attacked
person to die, $\beta_2/\beta_1$ (the smaller the quantity
$\beta_2/\beta_1$, the higher the gun-mediated protection), and on the fraction $c$ of people who take up their right of legal gun ownership and carry the gun with them when attacked. The right hand side of inequality (\ref{in1}) decays with $c$. When $c=1$ (everybody who has a right to a gun, carries a gun), condition (\ref{in1}) takes a particularly simple form:
\beq
\label{in2}
h<\frac{\beta_2}{\beta_1}.
\eeq
The case where inequality (\ref{hc}) is violated is illustrated in figure \ref{fig:one}(b). Larger values of $\beta_2/\beta_1$ satisfy condition (\ref{in1}), see the solid curves in figure \ref{fig:one}(b). For those curves, $g=0$ (the total firearms ban) corresponds to the minimum of the shooting death rate. If condition (\ref{in1}) is not satisfied (the dashed curves in the figure), then the $g=1$ (``gun availability to all'')
policy is the optimum.
Note that the key condition (\ref{hc}) relates two quantities, which in some sense are the
opposites of each other: The first quantity is $h$, the probability that a potential attacker who
cannot legally possess a gun will obtain it illegally and have it at
the time of the attack. This can be a measure of law enforcement, with lower values of $h$ corresponding to stricter law enforcement. The other quantity is $1-c$, the probability that a person who
can legally have a gun will not have it available when attacked. To make the ban of private firearm possession work (that is, to make sure that
it is indeed the optimal strategy), one would have to make an effort
to enforce the law and fight illegal firearm possession to decrease
$h$. To make the ``gun availability to all'' policy work, one would have to
increase $c$, for example by encouraging the general population to have
firearms available at all times. \\
\paragraph{Partial restriction of gun-ownership.} An important question is as follows. Let us suppose that the total gun
ban is impossible due to e.g. constitutional or cultural constraints. Would a partial restriction of gun ownership help
reduce the firearm-related homicide rate? It follows that if
\beq
\label{2nd}
c<\frac{1-h}{(1-\beta_2/\beta_1)(2-h)},
\eeq
then any decrease in $g$ will reduce the gun-related homicide rate. If
the value of $c$ is in the interval
$$\frac{1-h}{(1-\beta_2/\beta_1)(2-h)}<c<\frac{1-h}{1-\beta_2/\beta_1},$$
then the maximum death rate corresponds to an intermediate value of $g$
(while the minimum is at $g=0$, the total ban). This means that if the
current state is $g=1$, then a partial reduction in $g$ may actually
increase the gun-related homicide rate. The reduction must be
significant, that is, $g$ has to be lowered below a threshold, in order
to see a decrease in the gun-related death rate. Finally, if
$$c>\frac{1-h}{1-\beta_2/\beta_1},$$
which is the opposite of condition (\ref{in1}), then depending on
$\beta_2/\beta_1$, the minimum of ${\cal F}$ may correspond to $g=1$.
\paragraph{A more general model of the victim population.} Equation (\ref{Flin})
defines the death probability of a person involved in an
attack. Comparing this equation with equation (\ref{z}) we can see
that in this model, we clearly separate the population of attackers
and the population of victims. The attackers carry a gun with
probability $P^{attacker}_{gun}=g+(1-g)h$, which assumes that the
attackers will obtain a gun if it is legally available, and have a
probability to also obtain it illegally. The victims carry a gun with
probability $P^{victim}_{gun}=cg$, which implies that they never
obtain a gun illegally, and even if it is legally available, they may
not have a firearm with them at the site of the attack. This could be
an appropriate model for homicide description in suburban and
low-crime areas. Below we will refer to this model as the ``suburban'' model.
It is however possible that the population of victims is similar to
the population of the attackers in the context of gun ownership,
especially if we model the situation in different socio-economic
conditions, such as inner cities. The following model is more
appropriate for such situations: $P^{victim}_{gun}=cg+c_1(1-g)h_1$. It
states that a victim who is entitled to a legal weapon will have the
gun available at the time of the attack with probability $c$. Also,
victims who cannot possess a gun legally will acquire it illegally
with probability $h_1$ and have it with them with probability
$c_1$. This model reduces to model (\ref{Flin}) if $h_1=0$ (no illegal
gun possession among the victims). If on the other hand we set
$h_1=h$, then the population of victims is the same as the population
of attackers, apart from the fact that the attackers have a gun with
them with certainty (otherwise, there would be no attack), and the
victims may not be carrying a gun with them ($c,c_1<1$). We will refer
to this model as the ``inner-city'' model. In the follwoing text we
explore how our conclusions are modified under this more general
model.
The $g=0$ policy is the optimal as long as
\beq
\label{in3}
\frac{\beta_2}{\beta_1}>1-\frac{1-h}{c-c_1hh_1}.
\eeq
Note that if $h_1=0$, we have
$\frac{\beta_2}{\beta_1}>1-\frac{1-h}{c}$, which is the same as
condition (\ref{in1}). If $h_1=h$ and $c_1=c$, we have
$\frac{\beta_2}{\beta_1}>1-\frac{1}{c(1+h)}$, which is a weaker
condition than condition (\ref{in1}). In general, increasing $c_1$ and
$h_1$ makes condition (\ref{in3}) easier to fulfill. Therefore, we can
safely say that if condition (\ref{in1}) is fulfilled for the
``suburban'' model, then it will be fulfilled for the ``inner-city''
model.
Furthermore, partial measures to reduce $g$ from $g=1$ will lead to a decrease in the death toll as long as
$$\frac{\beta_2}{\beta_1}>1-\frac{1-h}{c(2-h)-c_1h_1}.$$
As before, with $h_1=0$ we recover condition (\ref{2nd}), and with an
increase in $h_1$ and $c_1$ lead to a weaker condition. Again, if
partial reduction of the gun ownership improves the death rate in the
``suburban'' model, it will also work in the ``inner-city'' model.
\subsection*{One-against-many attack}
Here, we consider a situation where a shooter attacks a crowd of people,
such as in a movie theater or mall shooting. The difference compared
to the previous scenario is that multiple people can potentially be
armed and contribute to stopping the attacker. We suppose that there
are $n$ people within the range of a gun shot of the attacker, and
$k$ of them are armed. We envisage the following discrete time Markov
process. At each time-step, the state of the system is characterized
by an ordered triplet of numbers, $(\alpha,i,j)$, where $\alpha\in
\{0,1\}$ tells us whether the attacker has been shot down ($\alpha=0$)
or is alive ($\alpha=1$), $0\le i\le k$ is the number of armed people
in the crowd, and $0\le j\le n-k$ is the number of unarmed people. The
initial state is $(1,k,n-k)$.
At each time-step, the attacker shoots at one person in the crowd
(with the probability to kill $0\le d\le 1$), and all the armed people
in the crowd try to shoot the attacker, each with the probability to kill
$0\le p\le 1$. The following transitions are possible from the state
$(1,i,j)$ with $0\le i\le k$, $0\le j\le n-k$ (below we use the convention that
expressions of type $i/(i+j)$ take the value $0$ for $i=j=0$):
\begin{itemize}
\item $(1,i-1,j)$: one armed person is shot, the attacker is not shot, with probability $d\frac{i}{i+j}(1-p)^{i-1}$;
\item $(1,i,j-1)$: an unarmed person is shot, the attacker is not shot, with probability $d\frac{j}{i+j}(1-p)^{i}$;
\item $(0,i-1,j)$: one armed person is shot, the attacker is shot, with probability $d\frac{i}{i+j}[1-(1-p)^{i-1}]$;
\item $(0,i,j-1)$: an unarmed person is shot, the attacker is shot, with probability $d\frac{j}{i+j}[1-(1-p)^{i}]$;
\item $(0,i,j)$: no potential victims are shot, the attacker is shot, with probability $(1-d)[1-(1-p)^{i}]$;
\item $(1,i,j)$: no one is shot, with probability $(1-d)(1-p)^{i}$.
\end{itemize}
This model is considered in detail in the Analysis section. For $n>1$, $F(g)$ is a decaying function of $g$, with $F''> 0$ for all $n>1$. The following empirical model mimics the key properties of the overall risk of being shot in a one-against-many attack:
\beq
\label{mod}
{\cal F}=[g+(1-g)h]e^{-\beta cg+\gamma (cg)^2},
\eeq
where parameter $c$ is again the fraction of all the people who take up their legal right of gun ownership and carry the gun with them when attacked. The parameter $\beta$ measures the effectiveness of the protection
received from the guns, and parameter $0<\gamma<\beta/(2c)$ is used to better
describe the curvature of function $F(g)$ obtained from the exact
model. The empirical model (\ref{mod}) has the advantage of simplicity,
which allows for a straightforward analysis.
The optimal strategy that
minimizes the gun-induced death rate of people again depends on the
degree of law enforcement (i.e. the probability for offenders to
obtain firearms illegally). More precisely, we have to evaluate the inequality
\beq
\label{ineq2}
h<e^{-c\beta}.
\eeq
Again, the limiting case of this inequality corresponds to the case of $c=1$:
\beq
\label{ineq}
h<e^{-\beta}.
\eeq
There are two cases:
\begin{itemize}
\item If inequality (\ref{ineq2}) holds (tight law reinforcement and/or gun protection ineffective), then the ``ban of private firearm possession'' policy ($g=0$) is optimal.
\item If $h>e^{-c\beta}$ (lax law reinforcement and/or gun protection highly effective), then depending on the value of $\gamma$ we may have different outcomes. Namely, if $\gamma<\frac{\beta}{2c}-\frac{1-h}{2c^2}$, then the $g=1$ (gun availability to all) policy is optimal. Otherwise, the optimal policy corresponds to an intermediate value of $g$:
$$g=\frac{\beta(1-h)-2c\gamma h+\sqrt{(\beta(1-h)-2c\gamma h)^2-8\gamma(1-h)(1-h-\beta ch)}}{4c\gamma(1-h)}<1.$$
\end{itemize}
\section*{Discussion}
We analyzed mathematical models in order to calculate the gun-induced
death rate of people depending on different gun control
strategies. The gun control strategies were expressed by a parameter
that describes the fraction of the population that can legally own
firearms. The strategies can range from a ban of private firearm possession to a
"gun availability to all" strategy. We first investigated a situation in which
one shooter is faced by only a single person that could potentially
own a gun and that could fight back against the shooter. This can
correspond to a one-on-one attack, such as a robbery, or a school
shooting where the only person in the classroom that could carry a gun
is the teacher. Subsequently, we examined a different scenario where a
shooter faces a crowd of people, all of which could potentially own a
gun and fight back against the attacker. This corresponds to shootings
in public places such as movie theaters and malls. The predictions of the model are similar for all scenarios. An
important parameter is the degree of law enforcement relative to the
amount of protection that gun ownership offers. If the
law is enforced strictly enough, a ban of private firearm possession minimizes the gun-induced death
rate of people. \\
The question arises how strict the law has to be enforced for the a ban of private firearm possession to minimize the gun-induced death of people. According to our results, this depends on the degree to which gun ownership protects potential victims during an attack and on the fraction of people who take up their legal right of gun ownership and carry the gun with them when attacked. These parameters in turn are likely to vary depending on the scenario of the attack and are discussed as follows. \\
\paragraph{One-against-one scenario.} The most prevalent use of guns is a one-against-one scenario and largely involves handguns \cite{maxfield1989circumstances, cook1981effect}. For this case, model predictions are relatively simple. Only one of the two extreme strategies can minimize gun-induced deaths, i.e. a ban of private firearm possession or a "gun availability to all" strategy. Intermediate gun policies lead to sub-optimal outcomes. Which strategy minimizes death depends on conditions that are easily interpreted. Gun-induced deaths are always minimized by a firearm ban if $h<1-c$. That is, we have to compare $h$, the fraction of offenders that illegally own a gun, with $1-c$, the fraction of the general population that could legally own a firearm but does not have it in possession when attacked. If the condition above is not fulfilled, gun-induced deaths can be minimized by either strategy, depending on the fraction of offenders who illegally obtain firearms relative to the level of protection offered by gun-ownership during an attack. If $c=1$ (all people take up their right of gun ownership and carry it when attacked), the conditions is simplest, and a ban of private firearm possession minimizes gun-related deaths if $h<\beta_2/\beta_1$, where $\beta_2/\beta_1$ is inversely correlated with the degree of protection offered by gun ownership to a victim during an attack, with $\beta_2/\beta_1=0$ meaning total protection, and $\beta_2/\beta_1=1$ corresponding to no protection associated with gun ownership. All these variables can be estimated from available statistical data, and the implications are discussed as follows. \\
In order to examine the fraction of offenders that cannot legally obtain a gun but own one illegally, $h$, we have to turn to a country with tough gun control laws. If a majority of people can legally own a gun, those that have to obtain one illegally is a negligible fraction. England and Wales have one of the strictest gun control laws since the 1997 Firearms Act, banning private possession of firearms almost entirely with the exception of some special circumstances \cite{firearmsact1997}. Estimating the fraction of potential offenders that illegally carry firearms is a difficult task. Most statistics quantify gun uses during the acts of offense, not among potential offenders. One study tried to fill this gap of knowledge by interviewing a pool of offenders that passed through prison \cite{bennett2004possession}. This was done in the context of the New English and Welsh Arrestee Drug Abuse Monitoring Programme (NEW-ADAM), covering a three year period between 1999-2002, and involving 3,135 interviewees. Among these offenders, 23\% indicated that they had illegally possessed a gun at some point in their life. However, only 8\% indicated illegal gun ownership within the last 12 months, which we consider a better measure of gun possession associated with committing crimes. More detailed questions revealed that only 21\% of people who owned a gun did so for the purpose of an offense. Similarly, among the 8\% of people who illegally owned a gun within the previous 12 months, only 23\% had taken the gun with them on an offense. Thus, as an estimate for the parameter $h$, we can say that 23\% of the 8\% constitutes people who illegally owned a gun which was also present during the offense, and hence $h=0.018$. \\
The fraction of people who legally own a firearm and have it in possession when attacked, $c$, can be partially estimated. Statistical data are available about the fraction of people who personally own a gun in the United States, but no data are available that quantify the probability that these gun owners have the weapon with then when attacked. Approximately $30\%$ of all adult Americans own a gun \cite{miller2007household, gallup}. Because not all of them will have the firearm with them when attacked, we can say that $c<0.3$. In this scenario, a "gun availability to all" policy can minimize firearm-related deaths if $c>1-h$, i.e. if $0.3>0.982$. This condition is clearly violated, and so the model predicts that gun-related deaths are minimized by a ban of private firearm possession. It is possible that the fraction of offenders that illegally carry a firearm derived from the study by \cite{bennett2004possession} is an underestimate. This study analyzed gun ownership among a prison population. It is conceivable that among those offenders, only a certain proportion has sufficient violent intention, and that among those people the frequency of gun ownership is higher. According to our calculations, however, the "gun availability to all" policy can only minimize gun-induced deaths if more than $70\%$ of potential offenders illegally owned a firearm, i.e. if $h=0.7$ or higher, depending on how many of the legal gun owners are likely to carry the firearm when attacked. This is unlikely to be the case given the results obtained by \cite{bennett2004possession}, but needs to be investigated statistically in more detail. \\
For the sake of the argument, let us consider the extreme scenario where all people who can legally own a gun do so and carry it with them at the time of an attack. This would require an effort by the government to persuade people to purchase firearms and carry them around at all times. As mentioned above, gun-related death is now minimized if $h<\beta_2/\beta_1$. The inverse relative protection that gun ownership provides during an attack ($\beta_2/\beta_1$) has also been statistically investigated \cite{branas2009investigating}. This is best done in a setting where a large fraction of the general population carries firearms, such as in the USA \cite{hemenway2000firearm}, and this study has been performed in Philadelphia. A total of 677 individuals assaulted with a gun were investigated and the study involved a variety of situations, including long range attacks where the victim did not have a chance to resist, and direct, short range attacks where the victim had a chance to resist. The study found that overall, gun ownership by potential victims did not protect against being fatally shot during an attack. In fact, individuals who carried a gun were more likely to be fatally shot than those who did not carry a gun. This also applies to situations where the armed victims had a chance to resist the attacker, and in this case, carrying a gun increased the chance to die in the attack about 5-fold. The authors provided several reasons for this. Possession of a gun might induce overconfidence in the victim's ability to fight off the attacker, resulting in a gun fight rather than a retreat. In addition, the element of surprise involved in an attack immediately puts the victim in a disadvantageous position, limiting their ability to gain the upper hand. If the victim produces a gun in this process rather than retreat, this could escalate the attacker's assault. These data would indicate that $\beta_2/\beta_1>1$, which in turn would again mean that a ban of private firearm possession is the only possible strategy that can minimize the gun-related death of people (the probability $h$ is by definition less than one). The results of this study have, however, been criticized on statistical grounds and it is currently unclear whether $\beta_2/\beta_1$ is indeed greater than one \cite{wintemute2010flaws, shah2010shah}. Results are also likely to depend on the geographical location. This study was conducted in a metropolitan area, and results might differ in smaller cities or more rural areas. However, the general notion that gun ownership does not lead to significant protection is also underlined by other studies that discussed the effectiveness of using guns as a defense against attacks \cite{kellermann1993gun, kleck1998risks, hemenway2000relative, hemenway2011risks}, especially in a home setting, although parameter estimates cannot be derived from these studies. In a literature review, no evidence was found that gun ownership in a home significantly reduces the chances of injury or death during an intrusion \cite{hemenway2011risks}. Even if this estimate of $\beta_2/\beta_1$ is somewhat uncertain, and even if its true value is less than one, the measured parameter $h=0.018$ means that firearm possession during an attack must reduce the chances of being fatally shot more than 50-fold for the "gun availability to all" policy to minimize the firearm-related deaths of people, i.e. the value of $\beta_2/\beta_1$ must be less than 1/50. This is unlikely in the light of the above discussed studies and again points to a firearms ban for the general population to be the correct strategy to minimize gun-induced deaths. \\
\begin{figure}
\centering \includegraphics[scale=0.4]{threshold.eps}
\vspace{\baselineskip}
\caption{\footnotesize {\bf One-against-many attacks: when is a ban of private firearm possession the optimal policy?} Presented are the contour-plots of the threshold value $\log_{10}(e^{-c\beta})$ with $c=0.3$, as a function of $\log_{10}p$ and $\log_{10}d$ for four different values of $n$. Darker colors indicate smaller values, and the contour values are marked. For each pair of probabilities $p$ and $d$, the plots show the highest possible value of $\log_{10}h$ still compatible with the ban of private firearm possession being the optimal policy. The black dashed lines on the bottom two plots indicate the approximate location of the contour corresponding to $h=0.018$; above those lines the ban of private firearm possession is the optimal solution. These lines are drawn according to the following relationship between $d$ and $p$: $d=4p$ for $n=20$, and $d=30p$ for $n=40$. For $n=5$ and $n=10$, the inequality $h<e^{-c\beta}$ holds for any values of $p$ and $d$, and the ban of private firearm possession is the optimal solution in the whole parameter space. }
\label{fig:thr}
\end{figure}
It is important to note that while according to the measured parameters, a ban of private firearm possession minimizes the gun-induced death rate, the implementation of such a policy might be impractical in some countries like the USA, because of constitutional and cultural constraints. It might only be practical to consider the option of partially restricting firearm access. In general, the model suggests that this might either decrease of increase the gun-induced death rate, depending whether condition (\ref{2nd}) is fulfilled, which in turn depends on the measured parameter values. To interpret condition (\ref{2nd}), let us suppose that $h\ll 1$, as measured by \cite{bennett2004possession}. Then we have the condition
$$c<\frac{1}{2(1-\beta_2/\beta_1)}$$
that guarantees that any reduction in $g$ provides an improvement in safety. In the most extreme scenario where the gun-assisted protection is the highest ($\beta_2/\beta_1=0$), the threshold value of $c$ is given by
$$c<\frac{1}{2}.$$
That is, as long as less than 50\% of people have the gun with them at
the time of an attack, a decrease in gun-ownership would decrease
gun-related homicides. With the estimate of $c\approx 0.3$ provided by statistical data, this condition holds. This means that even in the case
of very efficient weapons and high training of those who use them
($\beta_2/\beta_1=0$) a reduction in $g$ would be beneficial for the
society. In reality, the gun-related protection is not as high (which
corresponds to higher values of $\beta_2/\beta_1$), and the threshold
value of $c$ is most likely to be higher than $1/2$, which means that
most certainly a partial restriction on gun ownership would provide benefit
for reducing gun-related homicide rate.
\bigskip
\paragraph{One-against-many scenario.} Next, we discuss the one-against-many scenario. Here, two different gun control policies can again potentially minimize firearm-induced deaths of people: either a ban of private firearm possession, or arming the general population. However, in the latter case, not necessarily the entire population should carry firearms, but a certain fraction of the population, which is defined by model parameters. As in the one-against-one scenario, which policy minimizes gun-induced fatalities depends on the fraction of offenders that cannot legally obtain a gun but carry one illegally, the degree of gun-induced protection of victims during an attack, and the fraction of people who take up their right of gun ownership and carry the gun with them when attacked. In contrast to the one-against-one scenario, however, this dependence is more complicated here. In the empirical model described above, the degree of gun-mediated protection against an attack is given by the parameter $\beta$. This is a growing function of $n$ (the number of people involved in the attack) and $p$ (the probability for a victim to shoot and kill the attacker with one shot). Further, $\beta$ is a decaying function of $d$, the probability for the attacker to kill a victim with one shot. For a ban of private firearm possession to minimize gun-related deaths, the fraction of violent people that cannot obtain a gun legally but obtain one illegally must lie below the threshold given by condition (\ref{ineq}), i.e. $h<e^{-c\beta}$. Let us again assume that $30\%$ of the general population owns a gun, and for simplicity that they all carry the firearm with them when attacked ($c=0.3$). The
dependence of the function $e^{-c\beta}$ on the parameters $p$, $d$,
and $n$ is studied numerically in figure \ref{fig:thr}. Parameter $d$,
the probability for the attacker to kill a person with one shot,
varies between $10^{-2}$ and $1$. Parameter $p$, the probability for
an armed person to kill the attacker with one shot, varies between
$10^{-4}$ and $1$. The $\log_{10}$ of the right hand side of
inequality (\ref{ineq}) is represented by the shading, the lighter
colors corresponding to higher values. The dependence on parameters $p$ and $d$ is explored for different numbers of people in the crowd that is being attacked ($n=5, 10, 20, 40$). For each case, we ask what values of $p$ and $d$ fulfill the inequality $h<e^{-c\beta}$, assuming the estimated value $h=0.018$. In the parameter regions where this inequality holds, a ban of private firearm possession will minimize deaths, and outside those ranges, it is advisable that a fraction of the population is armed. For $n=40$ we find that a ban of private firearm possession requires $d>30p$. i.e. the attacker needs to be at least 30 more efficient at killing a victim than a single victim is at killing the attacker. For $n=20$ it requires $d>4p$. For $n=10$ and $n=5$, a firearms ban minimizes gun-related deaths for any value of $p$ and $d$. Thus, for smaller crowds, condition (\ref{ineq}) is easily satisfied and a ban of private firearm possession would minimize deaths. For larger crowds, a ban of private firearm possession would only make sense if the attackers were significantly more efficient compared to the victims to deliver a fatal shot. The meaning of these numbers further depends on the weapon carried by the attacker. Strictly speaking, the model considered here was
designed for attackers and victims with similar weapons. The victims would typically possess hand guns. If the attacker also uses a hand gun, it can be questioned whether the attacker is 30 times more efficient at fatally shooting someone than a victim, even if the attacker is better trained and has more experience. Therefore, if most gun attacks in the country involved a one-against-many scenario with a crowd of about forty people or larger being assaulted by a non-automatic weapon, firearm-induced deaths would be minimized by a policy that allows a certain fraction of people to carry guns. For smaller crowds, a firearms ban is likely to minimize deaths. In many situations, and certainly in the last string of mass shootings in the USA, automatic weapons were used to assault crowds, where hundreds of rounds per minute can be fired. The victims typically will not possess such powerful
weapons. Therefore, their ability to shoot is significantly lower than
that of the attacker. We can interpret the results of the model for a situation where the
attacker fires a machine gun and the victims respond with
non-automatic weapons. In this case we must assume that the
probability of victims to fire and shoot the killer is significantly
(perhaps 2 orders of magnitude) lower than that of the attacker. In this case, a ban of private firearm possession minimizes gun-related deaths even if most cases of gun violence involve the assault of relatively large crowds (such as $n=40$).
Now, for the sake of the argument assume that $c=1$, i.e. that everybody who can legally own a gun does so and has it in possession when attacked. The calculations yield the following results. For $n=40$, a firearms ban requires that $d>125p$, i.e. the attacker needs to be at least 125 more efficient at killing a victim than a single victim is at killing the attacker. For $n=20$, $n=10$, and $n=5$, the conditions are $d>30p$, $d>6p$, and $d>0.6p$. While the numbers are now shifted in favor of a "gun availability to all" policy, the general conclusions hold. If most attacks in the country occur in a one-against-many setting involving large crowds where the attacker does not use an automatic weapon, firearm-related deaths are probably minimized by the "gun availability to all" policy. If the crowds under attack are relatively small or if the attacker uses an automatic weapon, gun-induced deaths are minimized by banning private possession of firearms. \\
Having discussed the one-against-many scenario in some detail, it has to be pointed out that while assaults on crowds generate the most dramatic outcomes (many people shot at once), the great majority of gun-related deaths occur in a one-against-one setting \cite{cook1981effect, maxfield1989circumstances, fox2000homicide}, which generates less press attention. Therefore, it is likely that the results from the simpler one-against-one scenario are the ones that should dictate policies when the aim is to minimize the overall gun-related homicides across the country. \\
\paragraph{Further complexities.} We have discussed the prediction of the model in the context of the parameter estimates using what we called the suburban model. That is, we separate attacker and victim populations. We have shown, however, that in the context of the inner city model, the condition for a ban of private firearm possession becomes easier to fulfill. This model does not separate the attacker and victim populations but instead describes a scenario where a large fraction of the population can have criminal tendencies, and where a person may either be an attacker or a victim depending on the situation, e.g. two armed people getting into a fight, drug related crimes, etc. Because the suburban model indicates that gun-related homicides are likely to be minimized by a ban of private firearm possession, the same conclusion would be reached with the inner city model. It has to be kept in mind though that parameter estimates could be different depending on the setting, although there is currently no information available about this in the literature. Related to this issue, it is clear that crime is not uniform with respect to spatial locations. There are areas with adverse socio-economic conditions which are characterized by high homicide rates, and there are areas of relative safety with very few gun crimes. While our model does not take space into account explicitly, it takes into account different scenarios (such as the suburbal model or the inner-city model). The optimization problem solved here does not explicitly depend on the spatial distribution of different crime conditions. Further questions about crime management can however be asked if one utilized a spatial extension of this model. \\
Another assumption that could be changed in the model is the dependency of $z(g)$ on the policy parameter $g$. In the model analyzed here, we assume that the fraction of offenders that are not entitled to have a legal gun but get it illegally, $h$, does not depend on $g$. Let us suppose that it does. In other words, as the prevalence of firearms in the general population increases, the fraction of criminals acquiring an illegal gun increases too. In this case, our general conclusions hold even stronger. As $g$ increases, the frequency of crime will increase even more, and the probability of death during an attack will remain the same as in the original model.\\
An issue that we have ignored in our discussion so far are possible deterring effects of gun ownership, i.e. the notion that non-homicide crimes, for example burglaries, could occur less often if those offenders are deterred by the presence of guns in households. Our analysis was concerned with minimizing gun-related homicides, and not crime in general, which is a different topic and should be the subject of future work. If a gun is present in households, and the burglar would consequently carry a gun during the offense, however, the number of gun-related deaths is likely to increase, even if perhaps the total number of burglaries might decrease. This applies especially if guns in the household are unlikely to protect against injury or death, as indicated in the literature \cite{hemenway2011risks}. \\
Finally, it is important to note that this paper only takes account of factors related to the gun control policy, and assumes a constant socio-economic background. Of course in the real world a reduction in gun-related (and other) homicides would require improvement of the living and work conditions and education of underprivileged populations. Here we do not consider these issues. It is important to emphasize that cultural and socio-economic differences exist between different countries, making it impossible to draw direct comparisons. A lower or higher rate of gun-related deaths is not only a function of gun control policies, but also of those other factors.
Comparisons in the context of this model are only possible within the same cultural and socio-economic space. We single out the direct effects of gun-control policies and investigate those under fixed cultural and socio-economic circumstances.
\section*{Conclusions}
To conclude, this paper addresses the vigorous debate going on about the future of gun policies. One side argues that because guns are the reason for the deaths, stricter gun control policies should be introduced. The other side argues that guns also protect the life of people, and that an increased prevalence of guns in the general population could lead to fewer deaths. Rather than relying on opinions, we investigated this debate scientifically, using mathematical, population dynamic models to calculate the firearm-induced death rate of people as a function of the gun control policy. The results show that in principle, both arguments can be correct, depending on the parameters. Based on parameters that could be estimated from previously published data, our model suggests that in the context of one-against-one shootings, a ban of private firearm possession minimizes gun-related deaths if the gun control law can be enforced at least as effectively as in England/Wales. However, if a private firearm ban is not practical or possible due to constitutional and cultural constraints, the model and parameter measurements suggest that a partial reduction in firearms availability could lower gun-induced death rates. In a one-against-many scenario, the situation is a little less clear. While in the context of assaults with automatic or semi-automatic weapons, gun ownership by the general population is unlikely to minimize firearm-induced deaths, it could be the right strategy when an attacker assaults a large crowd with a non-automatic weapon. The one-against-many scenario is, however, less important than the one-against-one scenario if the aim is to reduce the number of gun-related homicides in the country, because most gun-related homicides involve the attack of single individuals \cite{cook1981effect, maxfield1989circumstances, fox2000homicide}. The next step in this investigation is to perform further statistical studies to estimate the crucial parameters of the model that we have discussed here. These parameters are clearly identified by the model, which is perhaps the most important contribution of this analysis. Detailed statistical measurements would provide data that could further inform the design of gun control policies.
\section*{Analysis}
The absorbing states of the stochastic system are (i) all the states $(0,i,j)$, where the attacker has been shot, and (ii) $(1,0,0)$, where all the people have been shot. The state $(0,0,m)$ is unreachable from $(1,i,j)$. Let us denote by $h_{i,j\to l,m}$ the probability to be absorbed in state $(0,l,m)$ starting from state $(1,i,j)$. The goal is to calculate the function $F(g)$, which is proportional to the probability for an individual involved in an attack to die.
We have $h_{i,j\to l,m}=0$ as long as $l>i$ or $m>j$. Further, $h_{i,j\to 0,m}=0$ for all $(i,j,m)$. Let us denote by $\lambda_{i,j}$ the probability to be absorbed in state $(1,0,0)$ starting from state $(1,i,j)$. We must have
\beq
\label{check1}
\sum_{l=1}^i\sum_{m=0}^jh_{i,j\to l,m}+\lambda_{i,j}=1.
\eeq
The following equations can be written down for these variables, by using a one-step analysis:
\bbar
h_{i,j\to i,j}&=&(1-d)[1-(1-p)^i]+(1-d)(1-p)^ih_{i,j\to i,j},\nonumber \\
h_{i,j\to i-1,j}&=&\frac{di}{i+j}[1-(1-p)^{i-1}]+\frac{di}{i+j}(1-p)^{i-1}h_{i-1,j\to i-1,j}+(1-d)(1-p)^ih_{i,j\to i-1,j},\nonumber\\
h_{i,j\to i,j-1}&=&\frac{dj}{i+j}[1-(1-p)^i]+\frac{dj}{i+j}(1-p)^ih_{i,j-1\to i,j-1}+(1-d)(1-p)^ih_{i,j\to i,j-1},\nonumber\\
h_{i,j\to l,j}&=&\frac{di}{i+j}(1-p)^{i-1}h_{i-1,j\to l,j}+(1-d)(1-p)^ih_{i,j\to l,j},\quad l<i-1,\nonumber\\
h_{i,j\to i,m}&=&\frac{dj}{i+j}(1-p)^ih_{i,j-1\to i,m}+(1-d)(1-p)^ih_{i,j\to i,m},\quad m<j-1,\nonumber\\
h_{i,j\to l,m}&=&\frac{di}{i+j}(1-p)^{i-1}h_{i-1,j\to l,m}+\frac{dj}{i+j}(1-p)^ih_{i,j-1\to l,m}+(1-d)(1-p)^ih_{i,j\to l,m},\nonumber\\
&& \quad l<i,\quad m<j.\nonumber
\eear
The values for all $h_{ij\to lm}$ can be calculated recursively from
this system. For completeness, we also write down the equation for the
variable $\lambda_{i,j}$:
$$\lambda_{i,j}=\frac{di}{i+j}(1-p)^{i-1}\lambda_{i-1,j}+\frac{dj}{i+j}(1-p)^i\lambda_{i,j-1}+(1-d)(1-p)^i\lambda_{i,j}.$$
One can check that equation (\ref{check1}) holds.
Given the probability to carry a gun $cg$, the probability to have $i$ individuals out of $n$ armed is given by
$$\frac{n!}{i!(n-i)!}(cg)^i(1-cg)^{n-i},$$
and the probability to have $l$ armed and $m$ unarmed individuals still alive after an attack is given by
$$\sum_{i=1}^n \frac{n!}{i!(n-i)!}(cg)^i(1-cg)^{n-i} h_{i,n-i\to l,m}.$$
Therefore, the probability to have $k$ people out of $n$ to survive the attack is
$$P_k=\sum_{l=1}^k\sum_{i=1}^n \frac{n!}{i!(n-i)!}(cg)^i(1-cg)^{n-i} h_{i,n-i\to l,k-l}.$$
Examples of this probability distribution for three different values of $g$ are shown in figure \ref{fig:res}(a). The expected number of people that survive is given by
$$\sum_{k=1}^nP_kk.$$
Therefore, the function $F(g)$, proportional to the probability to be killed in an attack, is given by
\beq
\label{exact}
F(g)=1-\frac{1}{n}\sum_{k=1}^nP_kk.
\eeq
\begin{figure}
\centering \includegraphics[scale=0.22]{results.eps}
\vspace{\baselineskip}
\caption{\footnotesize {\bf Results of the stochastic model of the one-against-many attack.} (a) A typical shape of the probability distribution of the number of people who survive an attack, plotted for three different values of $g$. (b) The probability to survive an attack, $F(g)$, as a function of the gun control (please note the logarithmic scale). The function $F(g)$ obtained in formula (\ref{exact}) is plotted by a thick gray line. The solid black line corresponds to approximation $e^{-\beta g}$ with $\beta$ given by formula (\ref{beta}). The dashed line corresponds to approximation (\ref{Fform}) with $\gamma=1.6$. Other parameters are $n=10$, $d=0.1$, $p=0.02$, and $c=1$. }
\label{fig:res}
\end{figure}
First of all, we can apply the one-against-many attack model for the case $n=1$. As expected, $F(g)$ for $n=1$ is a linear function of $g$, which can be written as equation (\ref{Flin}), with
parameters $\beta_1$ and $\beta_2$ giving rise to the threshold value of $h$,
$$\frac{\beta_2}{\beta_1}=\frac{d}{d(1-p)+p}.$$
We can see that the stochastic model informs our previous simple model by relating the quantity $\beta_2/\beta_1$ to the probability of the attacker to kill a victim with one shot, $d$, and the probability of a victim to shoot the attacker, $p$. As expected, the quantity $\beta_2/\beta_1$ grows with $d$ and decays with $p$. In other words, the gun-mediated protection decays with $d$ and it grows with $p$.
The expression for $F$ for $n>1$ is complicated. A typical shape of this function is shown in figure \ref{fig:res}(b), the thick gray line. We can
calculate the approximation for this function for small values of
$g$, by setting
$$F(g)\approx 1-\beta cg,$$
where
\bbar
\beta&=&\frac{p}{n(d(1-p)+p)^n}\left(\sum_{j=1}^{n-1}j^2d^{n-j}(1-p)^{n-j-1}(d(1-p)+p)^{j-1}\right.\nonumber \\
&+&\left. n^2(1-d)(d(1-p)+p)^{n-1}\right)\nonumber \\
&=&\frac{1}{np^2}\left(2d^2+dp-2d^2p-2dnp+n^2p^2\right.\nonumber \\
\label{beta}
&-&\left.(2d(1-p)+p)d^{n+1}(1-p)^n(d(1-p)+p)^{-n}\right).
\eear
Instead of working with the particular model described above, let us design a simpler model, which would retain some of the properties of the stochastic model considered, but be easier to analyze. First we notice that $F'<0$ and $F''>0$. Consider the following approximation of this function which satisfies $F'<0$ and $F''>0$:
\beq
\label{appr}
F(g)\approx e^{-\beta cg}.
\eeq
Figure \ref{fig:res}(b) shows that while expression (\ref{appr}) is
a good approximation of the function $F(g)$ for small values of $g$,
it deviates from the function $F$ as $g$ approaches $1$, see the solid
black line. The function $F$ given by exact formula (\ref{exact})
has a higher curvature for larger values of $g$ (the thick gray line
in figure \ref{fig:res}(b)). To mimic this trend, we will set
\beq
\label{Fform}
F(g)=e^{-\beta cg+\gamma (cg)^2},
\eeq
where $\beta>0$ and $0<\gamma<\beta/(2c)$, such that
$F'<0$ for $0\le g\le 1$. This approximation is shown in figure
\ref{fig:res}(b), the dashed line. In equation (\ref{Fform}),
$\beta$ is given by expression (\ref{beta}).
If $\gamma=0$, we have ${\cal F}''<0$, and the optimal strategies are
the same as in the one-against-one attack: only the two extreme
strategies can minimize the gun-induced death rate of people, i.e. either
a "ban of private firearm possession" strategy ($g=1$) or the "gun availability to all" strategy
($g=1$). Conditions (\ref{ineq2}) (or (\ref{ineq}) if $c=1$) help
separate the two cases. If we assume the existence of a nonzero
correction $\gamma>0$ in the expression for $F(g)$, it follows that
inequality (\ref{ineq2}) still plays a key role in separating two
different cases, as described in the Results section.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,610
|
\section{Introduction}
Estimating relevant quantities is an underlying key issue of many engineering problems, e.g. estimating a vehicle's position or velocity for tracking control in driver assistance systems. Since its introduction by \cite{Kalman.1960}, the Kalman filter has become one of the most prevalent estimators due to its optimality for linear systems. Naturally, numerous filter derivations for nonlinear dynamics were introduced thereafter, e.g. the extended Kalman filter (EKF), already used for the Apollo missions to the moon, and the unscented Kalman filter (UKF) by \cite{Julier.1997}.
However, an accurate model is crucial for the estimation performance regardless of which model-based filter is deployed. Recalling complex plants or systems, the process of modeling does not become difficult or time-consuming only due to parameter identification but due to complicated partial dynamics, often containing highly nonlinear parts and being challenging to capture.
Though many powerful filter techniques exist in control engineering, they strongly rely on the quality of the plant's model and will provide faulty estimates if it is inaccurate.
Therefore, a filter that is able to cope with a low-quality model and its uncertainty while also returning a small estimation error and a comprehensive model would be desirable.
Regarding parameters uncertainty, concepts like joint and dual estimation are sufficiently known, e.g. \cite{Wan.1999}, but only few studies on filter approaches dealing with complex model uncertainties and their detection have been carried out so far. Recently, only \cite{Khajenejad.2021}, \cite{BuissonFenet.2021} and \cite{Kullberg.2021b} investigated situations when just a partial model of the system is known. \cite{Khajenejad.2021} utilize a complex geometrical approach to estimate an upper and lower barrier for the approximated state to lie in between. The states and unknown partial dynamics are estimated very closely, but the approach does not enable an interpretable representation of the unknown dynamics.
In \cite{BuissonFenet.2021}, a high gain observer is deployed to estimate states that are then utilized to infer a Gaussian process model of the plant. Subsequently, the model is used within the observer and vice versa, affecting and correlating with each other. Only \cite{Kullberg.2021b} strive for a similar strategy as we do: They propose linear combinations of radial basis functions (RBFs) with compact support to approximate the unknown partial dynamics. This is carried out by extending the state with the linear combinations' parameters to infer dominant RBFs. Established within an EKF, the authors focus on an efficient computation by altering the EKF formulas rather than identifying and interpreting the unknown partial dynamics while simultaneously estimating the states.
In this work, we start from a problem formulation similar to \cite{Khajenejad.2021}, \cite{BuissonFenet.2021} and \cite{Kullberg.2021b} and define a joint state, that additionally contains parameters of the linear combinations for the partial dynamics to be approximated. However, we use a square-root UKF (SQ-UKF) combined with a physics-motivated library for the linear combinations, simplifying the inference of physical coherences. Finally, arguing similar to \cite{Brunton.2016} that most systems in nature and technology can be characterized by a small amount of dynamical terms, we identify a method from compressed sensing (CS), see e.g. \cite{Carmi.2010}, for our field of research and adjust it to estimate the full state and an interpretable representation of the system's unknown partial dynamics. In particular, we deploy the idea of pseudo measurements within a SQ-UKF to enforce the linear combinations' parameters being sparse. In contrast to the method in \cite{Brunton.2016} and related approaches such as LASSO (\cite{Tibshirani.1996}), the model selection is carried out online while estimating the system's states.
Therefore, our main contributions are the following:
\begin{itemize}
\item Joint estimation of states and model uncertainties within a SQ-UKF, using physics-based and experts' knowledge,
\item a sparse strategy to promote interpretable models by methods of CS,
\item demonstration and discussion on relevant application systems.
\end{itemize}
The paper proceeds as follows: In Sec. \ref{sec:problem} the challenge of estimating states while only a partial model of the considered system being available is discussed. Hence, the strategy of a sparse, joint SQ-UKF, that estimates states and model uncertainties simultaneously, is outlined and clarified by notions of CS in Sec. \ref{sec:sparseUKF}. The proposed filter is then demonstrated for experiments in Sec. \ref{sec:Simulations}. A short discussion and outlook in Sec. \ref{sec:conclusion} concludes this work.
\textit{Notation:} $\bld{\tilde{\bullet}}$ denotes the joint state vector or relating variables, such as the covariance matrix. A subscript $\bullet_{k\vert k-1}$ represents the variable at time step $k$ with observations up to time step $k-1$. Bold variables refer to vectors or matrices.
\section{Problem Formulation}\label{sec:problem}
For state estimation we wish to obtain the state $\bld{x}_k\in\mathbb{R}^{n_{\bld{x}}}$ at time step $k$ of a real system, whose dynamics are modeled by the following discrete system with control input $u_k\in\mathbb{R}$
\begin{align}\label{eq:model}
\begin{split}
\bld{x}_{k+1}&=\bld{f}(\bld{x}_k,u_k)+\bld{w}_k,\\
\bld{y}_k&= \bld{h}(\bld{x}_k,u_k)+\bld{v}_k.
\end{split}
\end{align}
The real system is assumed to be observable and can be observed by measurements $\bld{y}_k\in\mathbb{R}^m$.
Here, $\bld{f}$ and $\bld{h}$ denote the dynamical and observational models, respectively, with additive process and measurement noise $\bld{w}_k\sim\mathcal{N}(\bld{0},\bld{Q})$ and $\bld{v}_k\sim\mathcal{N}(\bld{0},\bld{R})$ assumed to be Gaussian distributed.
However, if the underlying model \eqref{eq:model} lacks of relevant quantities, the estimator based upon that model will not be able to capture the non-measurable states closely or will diverge.
Nonetheless, we could assume the unknown partial dynamics $\bld{g}(\bld{x},u)$ as a mapping that can be approximated by linear combinations of suitable functions $\Psi_i$, stored in a library $\bld{\Psi}$, and evaluated by parameters $\bld{\theta}\in\mathbb{R}^{n_{\bld{\theta}}}$. These functions contain prior knowledge, resulting from observational experience or experts' hypotheses, so that a close approximation by $\bld{g}(\bld{x},u)\approx\bld{\theta}^T\cdot\bld{\Psi}(\bld{x},u)$ can be expected. Presuming that the real system's order and its states can be defined, $\bld{\Psi}$'s minimal number of elements is $n_{\bld{x}}+2$, as it contains at least constants, the states and the control input. However, $\bld{\theta}$'s values are initially unknown and need to be calculated. Regarding parameter identification the joint estimation scheme is sufficiently known for estimating states and parameters simultaneously during the filter iterations, see e.g. \cite{Wan.1999} and \cite{vanderMerwe.2001}. Therefore, we extend the model \eqref{eq:model} to estimate unknown partial dynamics by defining the parameters $\bld{\theta}$ within a joint state vector $\tilde{\bld{x}}_k^T = \left(\bld{x}_k^T,\bld{\theta}_k^T\right)^T\in\mathbb{R}^{\tilde{n}}$ with $\tilde{n}=n_{\bld{x}}+n_{\bld{\theta}}$:
\begin{align}\label{eq:JEmodel}
\begin{split}
\tilde{\bld{x}}_{k+1}&=\begin{pmatrix}
\bld{f}(\bld{x}_k,u_k,\bld{g}(\bld{x}_k,u_k))+\bld{w}_k^{\bld{x}}\\
\bld{\theta}_k+\bld{w}_k^{\bld{\theta}}
\end{pmatrix}\\
&=\begin{pmatrix}
\bld{f}(\bld{x}_k,u_k,\bld{\theta}_k^T\bld{\Psi}(\bld{x}_k,u_k))+\bld{w}_k^{\bld{x}}\\
\bld{\theta}_k+\bld{w}_k^{\bld{\theta}}
\end{pmatrix},\\
\bld{y}_k&=\bld{h}(\bld{x}_k,u_k)+\bld{v}_k.
\end{split}
\end{align}
Remark that $\bld{\theta}$'s evolution is modeled by stationary dynamics and additive Gaussian noise $\bld{w}_k^{\bld{\theta}}\sim\mathcal{N}(\bld{0},\bld{Q}_{\bld{\theta}})$, whereas it holds $\bld{w}_k^{\bld{x}}\sim\mathcal{N}(\bld{0},\bld{Q}_{\bld{x}})$. Hence, the process covariance matrix of the joint state is a block matrix $\bld{\tilde{Q}}= \text{blkdiag}(\bld{Q}_{\bld{x}},\bld{Q}_{\bld{\theta}})$ with zeros elsewhere. Note within the scope of this paper the unknown partial dynamics are considered as one dimensional, namely $g(\bld{x},u)$, and $\bld{h}$ is linear. To receive a small estimation error $\bld{e}_k =\bld{\hat{\tilde{x}}}_k-\bld{\tilde{x}}_k$, any Kalman filter relies on the minimum-mean-squared-error to estimate $\bld{\hat{\tilde{x}}}_k$ as stated e.g. in \cite{Gibbs.2011}:
\begin{equation}\label{eq:filterProblem}
\argmin_{\bld{\hat{\tilde{x}}}} \frac{1}{2}\mathbb{E}[(\bld{\hat{\tilde{x}}}-\bld{\tilde{x}})^T(\bld{\hat{\tilde{x}}}-\bld{\tilde{x}})].
\end{equation}
Thus, the lift towards a higher dimensional state $\bld{\tilde{x}}$, including the unknown partial dynamics identifying parameters $\bld{\theta}$, affects the optimization problem \eqref{eq:filterProblem} by the cost of increased model complexity and computational burden.
\section{Joint Estimation by sparsity}\label{sec:sparseUKF}
After the model's reformulation to estimate states and unknown partial dynamics simultaneously, this section copes with the model's application within an SQ-UKF and presents a sparse solution as remedy towards the computational burden and the degrees of freedom due to the high dimensional state.
\subsection{Square-root unscented Kalman filter}
The unscented Kalman filter (UKF), established by \cite{Julier.1997, Julier.2002}, is a powerful nonlinear filter design method. In contrast to the extended Kalman filter (EKF), which linearizes nonlinear dynamics along a single sample point by the first order Tailor series, the UKF seeks to estimate the moments of the nonlinear probability distribution by multiple sample points. Therefore, it is independent of the analytical differentiation calculations the EKF requires. This enables a higher estimation accuracy when systems with strong nonlinearities are considered. However, the calculation of the covariance matrix in every filter iteration for $2\tilde{n}+1$ samples, called sigma points, is computational expensive. But, as presented in \cite{vanderMerwe.2001,vanderMerwe.2004}, the UKF's efficiency and its numerical stability can be improved by using the square-root covariance matrix $\bld{S}_k$ instead of $\bld{P}_k=\bld{S}_k\bld{S}_k^T$, leading to the square-root UKF (SQ-UKF) algorithm.
In Algo. \ref{algo:UKF} the SQ-UKF's procedure is outlined.
To ensure a symmetric distribution during the unscented transformation, the UKF's parameters $\lambda=\alpha^2(\tilde{n}+\kappa)-\tilde{n}$ and $\eta=\sqrt{\tilde{n}+\lambda}$ define the weights $\bld{W}^{(c)}$ and $\bld{W}^{(m)}$:
\begin{align}\label{eq:UKFparameters}
\begin{split}
W_0^{(m)}&=\frac{\lambda}{\lambda+\tilde{n}},\;
W_0^{(c)}=\frac{\lambda}{\lambda+\tilde{n}}+1-\alpha^2+\beta,\\
W_i^{(m)}&=W_i^{(c)}=\frac{1}{2(\tilde{n}+\kappa)}\quad\text{for}\,i=1,\dots,2\tilde{n}.
\end{split}
\end{align}
The parameters $\alpha, \beta$ and $\kappa$ need to be chosen beforehand, since they reflect how the nonlinear distribution is covered. In \cite{Nielsen.2021} it has been shown that the choice of these is problem-dependent and has a significant influence on the filter performance. However, for the scope of this paper, we choose $\alpha=10^{-3}, \beta=2$ and $\kappa=0$.
\begin{algorithm}
\caption{SQ-UKF}\label{algo:UKF}
\begin{algorithmic}[1]
\STATE \textbf{Initialize:} $\alpha, \beta, \kappa,\bld{\hat{\tilde{x}}}_0=\mathbb{E}[\bld{\tilde{x}}_0],$\\
\qquad\quad\quad\, $\bld{S}_0 = \text{chol}(\mathbb{E}[(\bld{\tilde{x}}_0-\bld{\hat{\tilde{x}}}_0)^T(\bld{\tilde{x}}_0-\bld{\hat{\tilde{x}}}_0)])$
\item[]
\STATE \textbf{for} {$k\in \{1,\dots,\infty\}$}\\
\quad Calculation of sigma points:
\STATE $\quad\mathcal{X}_{k-1}=[\bld{\tilde{x}}_{k-1}\quad\bld{\tilde{x}}_{k-1}+\eta\bld{S}_{k-1}\quad\bld{\tilde{x}}_{k-1}-\eta\bld{S}_{k-1}]$\\
\quad Prediction:
\STATE $\quad\mathcal{X}_{k\vert k-1}=\bld{f}(\mathcal{X}_{k-1},u_{k-1})$
\STATE $\quad\bld{\hat{\tilde{x}}}_k^- =\sum_{i=0}^{2\tilde{n}} W_i^{(m)}\mathcal{X}_{i,k\vert k-1}$
\STATE $\quad\bld{S}_k^-=\text{qr}\left(\left[\sqrt{W_1^{(c)}}(\mathcal{X}_{1:2\tilde{n},k\vert k-1}-\bld{\hat{\tilde{x}}}_k^-)\quad\sqrt{\bld{\tilde{Q}}}\right]\right)$
\STATE $\quad\bld{S}_k^-=\text{cholupdate}(\bld{S}_k^-,\mathcal{X}_{0,k}-\bld{\hat{\tilde{x}}}_k^-,W_0^{(c)})$
\STATE $\quad\mathcal{Y}_{k\vert k-1}=\bld{h}(\mathcal{X}_{k\vert k-1},u_{k-1})$
\STATE $\quad\bld{\hat{{y}}}_k^- =\sum_{i=0}^{2\tilde{n}}W_i^{(m)}\mathcal{Y}_{i,k\vert k-1}$\\
\quad Correction:
\STATE $\quad\bld{S}_{y_k}=\text{qr}\left(\left[\sqrt{W_1^{(c)}}(\mathcal{Y}_{1:2\tilde{n},k\vert k-1}-\bld{\hat{\tilde{x}}}_k^-)\quad\sqrt{\bld{R}}\right]\right)$
\STATE $\quad\bld{S}_{y_k}=\text{cholupdate}(\bld{S}_{y_k},\mathcal{Y}_{0,k}-\bld{\hat{{y}}}_k^-,W_0^{(c)})$
\STATE $\quad\bld{P}_{xy}=\sum_{i=0}^{2\tilde{n}}W_i^{(c)}[\mathcal{X}_{i,k\vert k-1}-\bld{\hat{\tilde{x}}}_k^-][\mathcal{Y}_{i,k\vert k-1}-\bld{\hat{{y}}}_k^-]^T$
\STATE $\quad\bld{K}_k =\bld{P}_{xy}\bld{P}_{yy}^{-1}$
\STATE $\quad\bld{\hat{\tilde{x}}}_k=\bld{\hat{\tilde{x}}}_k^- +\bld{K}_k(\bld{y}_k-\bld{\hat{{y}}}_k^-)$
\STATE $\quad\bld{U}=\bld{K}_k\bld{S}_{y_k}$
\STATE $\quad\bld{S}_k=\text{cholupdate}(\bld{S}_k^-, \bld{U}, -1)$
\STATE \textbf{end}
\end{algorithmic}
\end{algorithm}
If the SQ-UKF is utilized in its original form (Algo. \ref{algo:UKF}) without any modifications for the additional degrees of freedom resulting from $\bld{\theta}$, it has been observed that the filter tends to become locally unstable or divergent, which may lead to faulty state estimates. Depending on the number of design degrees $n_{\bld{\theta}}$, the calculation of $\bld{\theta}$ comes close to an underdetermined optimization problem. Hence, a strategy to choose only a selection of $\bld{\theta}$ while retaining the design degrees is required. Therefore, notions from the field of compressed sensing are adopted.
\subsection{Compressed sensing}\label{sec:cs}
In the domain signal processing methods of compressed sensing cope with the reconstruction of signals $\bld{x} =\bld{\Omega}\bld{s}$ with $\bld{\Omega}\in\mathbb{R}^{n\times n}$ a suitable basis matrix and $\bld{s}\in\mathbb{R}^n$ the sparse representation of the signal $\bld{x}\in\mathbb{R}^n$. Based on early works of \cite{Donoho.2006} and \cite{Candes.2006}, the linear underdetermined problem with the measurement $\bld{y}=\bld{H}\bld{x}$ and its measurement matrix $\bld{H}$
\begin{equation}\label{eq:l0}
\argmin_{\bld{s}} \vert\vert \bld{H}\bld{\Omega}\bld{s}-\bld{y}\vert\vert_2+\tilde{\lambda}\vert\vert\bld{s}\vert\vert_0,
\end{equation}
which seeks for a sparse solution $\bld{s}$ and is non-convex due to the $\ell_0$-norm leading to a high combinatorial effort, can be transformed into
\begin{equation}\label{eq:l1}
\argmin_{\bld{s}} \vert\vert \bld{H}\bld{\Omega}\bld{s}-\bld{y}\vert\vert_2+\tilde{\lambda}\vert\vert\bld{s}\vert\vert_1.
\end{equation}
Here, $0<\tilde{\lambda}\ll1$ denotes the sparsity balancing factor. The minimization problem \eqref{eq:l1} is now convex and provides a solution if sufficiently many measurements $\bld{y}$ are given and the rows of $\bld{H}$ are independent of $\bld{\Omega}$'s columns, for details see \cite{Carmi.2010,Brunton.2019}. Further, the problem can be reformulated into a constrained optimization problem
\begin{align}\label{eq:l1-relax}
\begin{split}
&\argmin_{\bld{s}} \vert\vert \bld{H}\bld{\Omega}\bld{s}-\bld{y}\vert\vert_2,\\
s.t.\,&\vert\vert\bld{s}\vert\vert_1\leq\epsilon,
\end{split}
\end{align}
with $0<\epsilon\ll1$.
\subsection{Promoting sparsity by pseudo measurements}
Considering the degrees of freedom introduced by the parameters $\bld{\theta}$, an accurate estimation by the filter problem \eqref{eq:filterProblem} can not be guaranteed any more as discussed in the previous sections.
However, using the same argument as in \cite{Brunton.2016}, most phenomena in nature or technology can be represented rather by a small amount of dominating terms. Therefore, only a selection of $\bld{\Psi}$ is needed to capture the system's behavior. To enforce this assumption, the filter problem \eqref{eq:filterProblem} is reformulated by adding a term that encourages a sparse vector $\bld{\theta}$. Adopting the notions \eqref{eq:l0} and \eqref{eq:l1} from Sec. \ref{sec:cs} to efficiently propagate a sparse $\bld{\theta}$ within the iterative filter Algo. \ref{algo:UKF}, the minimization problem can be formulated by
\begin{equation}\label{eq:filterProblemSparse}
\argmin_{\bld{\hat{\tilde{x}}}} \frac{1}{2}\mathbb{E}[(\bld{\hat{\tilde{x}}}-\bld{\tilde{x}})^T(\bld{\hat{\tilde{x}}}-\bld{\tilde{x}})] + \tilde{\lambda} \vert\vert\bld{\hat{\tilde{x}}}_{n+1:\tilde{n}}\vert\vert_1
\end{equation}
with $\tilde{\lambda}\in\mathbb{R}$ as the sparsity promoting term.
However, how to incorporate this approach within the iterative structure of the SQ-UKF is not straight-forward. Recalling the previous section and adjusting \eqref{eq:l1-relax} towards the problem \eqref{eq:filterProblemSparse} yields a sparse mean-squared-error for joint estimation
\begin{align}\label{eq:CS}
\begin{split}
&\argmin_{\bld{\hat{\tilde{x}}}} \frac{1}{2}\mathbb{E}[(\bld{\tilde{x}}-\bld{\hat{\tilde{x}}})^T(\bld{\tilde{x}}-\bld{\hat{\tilde{x}}})] \\
s.t. \,&\vert\vert\bld{\hat{\tilde{x}}}_{n+1:\tilde{n}}\vert\vert_1 \leq \epsilon.
\end{split}
\end{align}
By interpreting the soft constraint as a fictitious, additional measurement $0= \vert\vert\bld{\hat{\tilde{x}}}_{n+1:\tilde{n}}\vert\vert_1-\epsilon$ it can be utilized during estimation schemes, see e.g. \cite{Julier.2007}, \cite{Carmi.2010}, \cite{Hage.2020b}. Different approaches, e.g. as shown in \cite{Julier.2007}, exist how to incorporate the pseudo measurement within a filter iteration. For this paper, we define a pseudo measurement function $\tilde{h}(\bld{\tilde{x}})$ by
\begin{equation}\label{eq:pseudoh}
\tilde{h}(\bld{\tilde{x}}) = \vert\vert\bld{\hat{\tilde{x}}}_{n+1:\tilde{n}}\vert\vert_1-\epsilon,
\end{equation}
where $\epsilon\sim\mathcal{N}(0,R_{pm})$ now represents the fictitious, additive measurement noise and controls the tightness of the constraint. Since the SQ-UKF is based on the unscented transformation, the nonlinear observational model \eqref{eq:pseudoh} can be easily applied without any further adaption in contrast to its use within an EKF, e.g. \cite{Julier.2007}.
In Algo. \ref{algo:sparseUKF} the procedure of the proposed sparsity promoting, joint SQ-UKF (J-SQ-UKF) is depicted. Within every filter iteration $k$ additional updates for the estimated state $\bld{\hat{\tilde{x}}}_k$ and its square-root covariance $\bld{S}_k$ might be enclosed depending on the sparsity of $\bld{\theta}_k$. If the number $n_{\bld{\theta},act}$ of non-sparse elements, that are greater than the sparsity barrier $\tilde{\lambda}$, is exceeded (line 18), a filter update is calculated again but now using the pseudo measurement \eqref{eq:pseudoh} as fictitious observational model instead of the actual observational model $\bld{h}$ (line 19). This step is followed by additional checks and updates. The sparsity promoting loop terminates either when the maximal iteration of pseudo measurements $j=N$ is reached or the number of allowed non-sparse elements is less or equal $n_{\bld{\theta},act}$. In contrast to Sec. \ref{sec:cs} online estimation is possible because of the evolving covariance that contains information about past measurements.
Since with every filter iteration $k$ the dominant parameters of $\bld{\theta}$ could completely change due to the sparsity promoting loop and the degrees of freedom at the beginning of every iteration $k$, the stability of the filter algorithm may be easily affected as well as the model's interpretability might get lost. Hence, to avoid local instabilities or divergent behavior, a soft switching method for $\bld{\theta}$ is enforced afterwards (line 23) by balancing the state estimate before and after the pseudo measurement correction.
This guarantees a more continuous estimation of the dominant parameters, resulting in a more continuous state estimation itself.
Different methods of soft switching could be applied, e.g. batch updates, but herein a simple weighted approach is employed by a factor $0<\gamma<1$. It is recommended to enforce a higher emphasis on the newly corrected estimate after the pseudo measurement loop.
\begin{algorithm}
\caption{J-SQ-UKF}\label{algo:sparseUKF}
\begin{algorithmic}[1]
\setcounter{ALC@line}{14}
\STATE \quad$\vdots$
\STATE $\quad\bld{S}_k=\text{cholupdate}(\bld{S}_k^-, \bld{U}, -1)$\\
\item[]
\quad Sparsity promoting update:
\STATE \quad\textbf{Initialize:} ${\tilde{h}}, N, n_{\bld{\theta},act},\gamma,j = 1$\\
$\quad\quad\quad\quad\quad\quad\bld{S}_{pm,0}= \bld{S}_k, \bld{\hat{\tilde{x}}}_{pm,0}=\bld{\hat{\tilde{x}}}_k$
\item[]
\STATE $\quad$\textbf{while} {$\#\{\theta_j\vert\theta_j>\tilde{\lambda}\}>n_{\bld{\theta},act}\;\textbf{and}\; j<N$}
\STATE $\quad\quad \bld{\hat{\tilde{x}}}_{pm,j},\bld{S}_{pm,j}\leftarrow\text{SQ-UKF}\; \text{with}$\\
$\quad\quad\quad\quad\quad\quad\quad\quad\quad(\bld{\hat{\tilde{x}}}_{pm,j-1},\bld{S}_{pm,j-1},\bld{f},\tilde{h},\bld{\tilde{Q}},R_{pm})$
\STATE $\quad\quad j = j+1$\\
\STATE $\quad$\textbf{end}
\item[]
\STATE $\quad\bld{S}_{k,final}=\bld{S}_{pm,j}$
\STATE $\quad[\bld{\hat{\tilde{x}}}_{k,final}]_{1:n_{\bld{x}}}=[\bld{\hat{\tilde{x}}}_k]_{1:n_{\bld{x}}},$\\ $\quad[\bld{\hat{\tilde{x}}}_{k,final}]_{n_{\bld{x}}+1:\tilde{n}}=(1-\gamma)[\bld{\hat{\tilde{x}}}_{pm,j}]_{n_{\bld{x}}+1:\tilde{n}}+\gamma[\bld{\hat{\tilde{x}}}_k]_{n_{\bld{x}}+1:\tilde{n}}$
\STATE \textbf{end}
\end{algorithmic}
\end{algorithm}
\subsection{Curse of dimension}
As generally known, any Kalman filter's performance depends strongly on the choice of its covariance matrices. While the measurement covariance $\bld{R}\in\mathbb{R}^{m\times m}$ often is easier to determine by taking a closer look at the plant's measurements, the initial square-root covariance $\bld{S}_0$ and the process covariance $\bld{\tilde{Q}}\in\mathbb{R}^{\tilde{n}\times\tilde{n}}$ remain a major challenge in filter design, see e.g. \cite{Chen.2021, Nielsen.2021}. When promoting sparsity within the SQ-UKF even an additional covariance $R_{pm}\in\mathbb{R}$ comes along which determines the pseudo measurements' performance. With increasing numbers of states $n_{\bld{x}}$ and possibly uncertain partial dynamics, expressed by $n_{\bld{\theta}}$ numerous parameters $\bld{\theta}$, not only the curse of dimension but also the difficulty to initialize the performance parameters become obvious. Common approaches in filter design include manual tuning, grid or random search, which are unsatisfying in terms of effort and benefit while possibly resulting in local minima. Lately, Bayesian optimization becomes the state-of-the-art method for hyperparameter optimization and could be applied herein when offline estimation is pursued. For the scope of this paper, we focus on online estimation and therefore set the parameters as $\bld{P}_0=\bld{\tilde{Q}}=\text{blkdiag}(10^{-6}\bld{I}_{n_{\bld{x}}},10^{-4}\bld{I}_{n_{\bld{\theta}}}), \bld{R}=10^{-4}\bld{I}_m$, and $R_{pm}=1$.
\section{Results}\label{sec:Simulations}
The proposed filter algorithm is now demonstrated for two application examples. Compared to a traditional SQ-UKF, the advantages of the sparse, joint SQ-UKF filter become very clear. Within each application example both filters start with a corrupted initial state $\bld{\hat{\tilde{x}}}_0\neq \bld{\tilde{x}}_0$. For the J-SQ-UKF the parameters $\bld{\theta}_0$ are initialized equally with small values. Within all experiments discussed in this paper it holds $n_{\bld{\theta},act}=3$ and $\tilde{\lambda}=0.1$.
\subsection{Duffing oscillator}
The Duffing oscillator is a common example for nonlinear dynamical systems. Its dynamics are described in the following with parameters $\bld{p}=(-1,3,0.1)^T$:
\begin{align}\label{eq:Duffing}
\begin{split}
\bld{\dot{x}}&=\begin{pmatrix}
x_2\\ -p_3x_2-p_1x_1-p_2x_1^3+u
\end{pmatrix},\\
y&=x_1.
\end{split}
\end{align}
Here, we assume the nonlinear term $g(\bld{x},u)=p_2x_1^3$ as unknown and formulate the joint model, lacking of the mentioned term, analogously to the description in Sec. \ref{sec:problem}. Formulating \eqref{eq:Duffing} as a discrete model and using the explicit Euler scheme and the library $\bld{\Psi}_1=(1,x_1,x_2,x_2^2,\sin(x_2),x_1^3,x_1x_2,\cos(x_1),u)^T$ to evaluate the model, we can deploy it within the J-SQ-UKF.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{Fig1.pdf}
\caption{Estimation of the Duffing oscillator's states with sine excitation comparing various filter performances}\label{fig:duffingJE}
\end{figure}
As a result, we see through the red dashed line in Fig. \ref{fig:duffingJE} that the J-SQ-UKF is able to cope with the faulty model and to deliver a correct estimate. This is due to the selection of the true dynamics, which is illustrated in the top plot in Fig. \ref{fig:duffingJEtheta} by $\bld{\theta}$'s magnitude over time. Here, the line, that represents $\theta_6$ and therefore indicates that $\Psi_6(\bld{x},u)=x_1^3$ is mainly present, is of the highest magnitude. Further, it is able to deal with incorrect initial states and indicates a fast transient behavior towards the true states. Conversely, the traditional SQ-UKF is not able to handle the model uncertainty and even provides incorrect estimates regarding the initial state as depicted in Fig. \ref{fig:duffingJE} by the blue dotted line.
When utilizing a library without the true term $g$, e.g. $ \bld{\Psi}_2=(1,x_1,x_2,x_2^2,\sin(x_2),x_1x_2,\cos(x_1),u)^T$ or $\bld{\Psi}_3=(1,x_1,x_2,x_2^2,\sin(x_2),x_1^2,x_1x_2,\cos(x_1),u)^T$, the J-SQ-UKF is yet able to estimate the true dynamics. As illustrated in Fig. \ref{fig:duffingJE} and \ref{fig:duffingJEtheta}, the filter clearly extracts other physical terms that are close to the true missing term $g$ and still delivers an accurate state estimate, though a small degradation compared to the J-SQ-UKF with $\bld{\Psi}_1$ can be observed. Therefore, the proposed method appears to be confident in detecting either the correct partial dynamics or finding alternatives to interpret the system's behavior, namely $\Psi_2(\bld{x},u)=x_1$ or $\Psi_6(\bld{x},u)=x_1^2$. These interpretations can then be utilized for further model usage or refinement.
\begin{figure}
\centering
\subfloat[Dominant term $\theta_6$: $\Psi_6(\bld{x},u)=x_1^3$]{
\includegraphics[width=0.9\columnwidth]{Fig2a.pdf}}\\
\subfloat[Dominant term $\theta_2$: $\Psi_2(\bld{x},u)=x_1$]{
\includegraphics[width=0.9\columnwidth]{Fig2b.pdf}}\\
\subfloat[Dominant term $\theta_6$: $\Psi_6(\bld{x},u)=x_1^2$]{
\includegraphics[width=0.9\columnwidth]{Fig2c.pdf}}
\caption{Course of $\bld{\theta}$ corresponding to different libraries $\bld{\Psi}_1,\bld{\Psi}_2$ and $\bld{\Psi}_3$ from top to bottom: After a transient phase the dominant $\theta_i$ indicates which term is best for $g$ to be approximated.}\label{fig:duffingJEtheta}
\end{figure}
\subsection{Golf robot}
The golf robot, shown in Fig. \ref{fig:golfrobot}, is a test bench at our laboratory to investigate the capability of machine learning methods on real world problems. After extensive modeling effort the robot's dynamics are described by
\begin{align}\label{eq:golfrobot}
\begin{split}
\bld{\dot{x}}
&= \begin{pmatrix}x_2\\
J^{-1}\left(-m \tilde{g} a\sin(x_1)- M_F+4u\right)
\end{pmatrix},\\
M_{F} &= dx_2+2r\mu \,\text{arctan}(10^3 x_2)\pi^{-1} \left| m x_2^2a + m \tilde{g}\cos\left(x_1\right)\right|,\\
y &= x_1,
\end{split}
\end{align}
with $\tilde{g}$ as gravity constant and the same parameters $\bld{p}=(m, a, d, J, r, \mu)^T$ utilized as in \cite{Gotte.2022, Schon.2022}. Here, the joint model \eqref{eq:JEmodel} is the model that completely lacks of the term $M_F$. Since the modeling effort of \eqref{eq:golfrobot} lies mainly in modeling the friction terms, this assumption is reasonable and close to real world situations. Further, the library is defined as $\bld{\Psi}(\bld{x},u)=(1,x_1,x_2,x_2^2,x_1^3,\sin(x_2),\cos(x_1),u)^T$ and the model is again discretized and evaluated with the explicit Euler scheme.
\begin{wrapfigure}[]{r}{0.25\columnwidth}
\centering
\includegraphics[width=0.25\columnwidth]{Fig3.png}\caption{Golf robot}\label{fig:golfrobot}
\end{wrapfigure}
As illustrated in Fig. \ref{fig:golfstates} the J-SQ-UKF provides a close estimation of the robot's states and even shows a faster transient behavior than the traditional filter that delivers faulty angular velocities. The corresponding dominant parameters $\bld{\theta}$ are depicted in Fig. \ref{fig:golftheta} and reveal that more than one term of the library is relevant to approximate $g$, in fact these are $\Psi_1, \Psi_2$ and $\Psi_4$. Further, it is observed that these terms seem to vary in time which is reasonable since friction forces are dependent on velocity and are often highly nonlinear. It is also noteworthy that the dominant terms are close to the ones which have been identified so far by extensive modeling effort in model \eqref{eq:golfrobot} or correlate to those by factor, e.g. $x_1\approx\cos(x_1)$ when $x_1$ is small. However, extracting a more accurate representation for those dynamics requires a follow-up step like a modal transformation when there is more than one identified dominant term $\Psi_i$.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{Fig4.pdf}
\caption{Estimation of the golf robot's states with sine excitation comparing various filter performances}\label{fig:golfstates}
\end{figure}
\begin{figure}[h]\centering
\includegraphics[width=\columnwidth]{Fig5.pdf}\caption{Course of $\bld{\theta}$ corresponding to library $\bld{\Psi}$: Dominant $\theta_i$ indicate that $\Psi_1(\bld{x},u)=1, \Psi_2(\bld{x},u)=x_1$ and $\Psi_4(\bld{x},u)=x_2^2$ are the best terms to approximate $g$. The barrier due to $\tilde{\lambda}=0.1$ is illustrated by the transparent gray rectangle.}\label{fig:golftheta}
\end{figure}
\section{Discussion and outlook}\label{sec:conclusion}
The proposed method has shown for several application examples its power to jointly estimate states and model uncertainties. We could observe two cases: Either the library $\bld{\Psi}$ contains the unknown partial dynamics $g$ and the algorithm extracts the correct term or close approximations of it, or it provides a suggestion for $g$ based on its library terms.
Further, it clearly outperforms the traditional filter when only an incomplete model of the system is available. However, the interpretability has not been automated so far and needs the user to interpret and extract the dominant $\theta_i$ and its corresponding terms by sight.
Additionally, parameters like the number of allowed sparse elements $n_{\bld{\theta},act}$, the sparsity barrier $\tilde{\lambda}$ and the terms of $\bld{\Psi}$ currently have to be set by the user. Correlations between those parameters are to be expected. Moreover, it has not yet been investigated if there exists a dependency of $n_{\bld{\theta}}$ and $n_{\bld{x}}$, limiting possibly the number of $\bld{\Psi}$'s elements. Nonetheless, the sparse, joint SQ-UKF allows a deeper insight into the system's dynamics, while simultaneously providing sufficient estimates. This insight could be then concretized offline by a modal transformation method, e.g. proper orthogonal decomposition, to automate the model interpretations and utilize an updated model within the filter. However, experiments have shown to be real-time capable and encourage the usage within online estimation.
Therefore, future research focuses on the relation of $n_{\bld{\theta}}$ and $n_{\bld{x}}$ and on a higher dimensional $\bld{g}$.
\section*{Acknowledgment}
This work was developed in the junior research group DART (Daten\-ge\-trie\-be\-ne Methoden in der Regelungstechnik), University Paderborn, and funded by the Federal Ministry of Education and Research of Germany (BMBF - Bundesministerium für Bildung und Forschung) under the funding code 01IS20052. The responsibility for the content of this publication lies with the authors.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 510
|
The Art in Hospitality: Brent Harris and Andrew Browne
Words by Eliza Ackland
Photography by Jana Langhorst
From food stems the rituals and idiosyncrasies that make up the fabric of our being. The Art in Hospitality uses food as the stimulus to explore the minds of artists and creatives—because from food comes everything: necessity, nourishment, culture, ritual, connection and delight.
We revel in the opportunity to be guests in an artist's home and realise the potential of the art in hospitality. As we fill our minds and bellies, we learn about the artist and their practice, discover their relationship to the rituals of preparing and sharing meals and talk about the philosophies embedded in the ceremony of cooking, eating and hosting.
When visiting as guests, we believe you should only knock on the host's door with one hand, reserving the other to carry something to share. On this autumn Sunday in Fitzroy, Melbourne, we knock on Brent Harris and Andrew Browne's door armed with a plate of devilled eggs and a bottle of wine. The devilled eggs were inspired by a visit to Brent and Andrew's studio in Collingwood a few days prior. Brent and Andrew have shared the studio space for 15 years, with Brent occupying the second story of the warehouse space and Andrew downstairs.
Brent's space upstairs is relatively tidy, considering the complex themes that inform his work his process appears steady and deliberate. Brushes, books and tuna cans doubling as paint pots take up the tabletops. Several WIPs hang on the walls—a pansy, an elephant, a swamp, sperm, his mother, his father portrayed as a dark mask, a fried egg.
Brent is preparing for a show at Tolarno Galleries and pings around his space like the arcade game Pong, showing one work, pulling a book out from a shelf for reference, showing an earlier iteration. It's as if we're witnessing him flick through the archives of the filing cabinet that is his mind. In the corner a golden yolk cracks onto a pink canvas. When asked about the symbolism of the egg, Brent explains that eggs often appear as symbols of origin, fertility and life. But for him, and speaking on behalf of the world, life is pretty fried at the moment, hence the fried egg.
So it's devilled eggs, prepared by the chef at Congress that we bring to share with Brent and Andrew as they prepare lunch for a group of friends on a Sunday afternoon. Tucked under the same arm is a bottle of white Garganega, 2021 vintage. A recent collaboration with Mac Forbes and Congress, Brent's work To The Forest, 1999 adorns the wine label. The black silhouette of the print mimics the drip of the black wax that seals the bottle.
It's a few minutes after noon as Brent and Andrew welcome us into their home, so we agree it's appropriate to open the wine. We sit in the living room amongst the company of art adorned walls while Andrew potters in the kitchen prepping a simple pear cake he says never fails him.
Andrew is the head chef in this household. He'd once described Brent's cooking as slop but admits Brent does make good curries and a nice bolognese. Brent says his secret to a good bolognese is milk. We discuss bolognese as a particularly personal dish that is most often inherited from one's family. Brent explains his mother was a terrible cook. Her only redeeming dish was made from the rabbits they used to shoot and bring home as kids. She would put the rabbit on a roasting tray with onions and other veg—garlic was too fancy at the time. She'd top the rabbit with slices of white bread and fill the dish with milk and bake.
Brent grew up in New Zealand and moved to Melbourne in 1981. Over his career Brent has produced work that refers back to his challenging family dynamic in New Zealand. Through his work Brent confronts his father, comforts his mother and simultaneously acknowledges universal themes such as intimacy, desire, spirituality and mortality.
Andrew was born in Melbourne and works across painting, photography and graphic mediums including drawing, photogravure, intaglio and lithography. His works are often influenced by photographs of the urban environment, bringing a sense of realism that is coupled with a sense of abstraction and ambiguity.
Eliza Ackland
Let's start by talking about your daily routines. I'm interested in how you live and work together and the rituals that either bring you together or give you space. What do your days look like?
Brent Harris: One of us has to get up and make a coffee.
Andrew Browne: Yeah, that's pretty standard. I wasn't drinking coffee for about two years. I just started again. I don't know what it is. And then sometimes I'll eat breakfast, sometimes I'll skip breakfast. Brent usually cooks himself a large breakfast.
BH: What do you mean "large"?
AB: Like leftovers from the night before. Like, if he's got peas from last night he'll do eggs with them and when we're eating bread, (which we're not at the moment), toast. Beans… he likes baked beans.
BH: He's telling you what I have!
AB: What do I have, Brent? You can tell them what I have.
BH: When we were eating toast you'd have avocado, tomato, and a little olive.
AB: Brent usually asks at breakfast: "what do you think we should have for dinner?" I go, "I haven't even had breakfast. I can't even think about it!"
I noticed the tuna cans in the studio. Tuna in chilli oil. Is that a lunch favourite? And then they become the paint pots?
AB: Yeah he mixes a lot of paint in those to get particular consistencies.
I was thinking about the layout of your studio as well. Having two levels and one of you downstairs and one of you upstairs. Do you ever go downstairs and bug the other one about what they're up to?
BH: Yes, sometimes.
AB: If I'm starting a new body of work and I'm feeling hypersensitive, I've been known to ban Brent from my studio for up to a month. Because Brent's very blunt. I'm blunt when I go up to his studio. He likes the bluntness when I'm in his studio. Brent's actually one of those people who likes people going into his studio offering opinions.
BH: Yeah, I like to know, "should this eyeball be this big? Or that big? Or a bit to the left?"
AB: When Brent comes down to mine he just says "no" and it's something I've invested a bit of time and energy in. So I get sensitive about that. But other times we can be more agreeable.
Our conversation is interrupted by the timer going off for the cake. Our noses knew the cake was close to ready as the room was filled with sweet cinnamon. Andrew checks the centre and it's not quite set.
BH: I seek opinions about my work from maybe three or four people.
AB (as he checks the cake in the oven): I used to get more opinions from a couple of friends of mine. I think it's just the way you are psychologically in yourself. One of the friends who's coming for lunch today is really great in the studio. And it's nice to have people you've known for a period of time and they've seen different series of works go through.
Are there any crossovers between how you work in the kitchen to how you work in the studio?
AB: Well, I'm the neat one.
BH: We've got quite a big library upstairs but the books start there and then they slowly migrate into my studio and they end up being stacked up in the corner. I really use my library. The books are a clean mess though.
AB: A lot of Brent's work refers to art history. An anecdote: Fred Williams used to keep massive amounts of art postcards and if he was having a problem with a painting he'd just flick through the postcards and he'd use something there as a prompt. Brent often does that with the books, hence the large number strewn around. Whereas my practice is more an engagement with the urban environment - that becomes the source material, and I abstract from there. I've read a lot of art history, kinda processed it and now I guess I apply that to subjects from the real world. Which is different from Brent.
BH: You don't use many recipe books either.
AB: We used to buy recipe books, generally I think a lot of people find this, you buy a recipe book and you use one or two recipes. It's like the New York Times one, it's a fat recipe book with 1000s of recipes, yet I've only extracted a couple of recipes from it!
BH: We use a recipe for that lamb dish from a Mario Batali cookbook 'Molto Italiano'.
AB: Abbacchio alla romana, which is lamb shoulder with vinegar. It's slow cooked.
That sounds like good comfort food. What are your favourite comfort foods?
BH: Pasta. That's why we're fat.
And happy!
BH: Haha, generally happy.
AB: My comfort food is spaghetti with anchovies, garlic chilli and radicchio. Or pesto. I make my own pesto. And Brent might be catching up with his friend Kevin on a Wednesday night, during that season of basil being great, I'll come home by myself, make some pesto and watch some TV, that's my comfort.
We've been trying to cut down. We try not to have pasta at night anymore. It's part of our low carb diet. Which is a pity because it is quite comforting. And I find having fish and salad, or meat and salad, or meat and broccoli…
Life's too short for meat and broccoli.
BH: So in the kitchen I normally prep. And then he comes and puts it together. He says, "I'm not coming to the kitchen until you get out!"
AB: And then I go, "how many anchovies have you put in this?" Because he'll start doing the anchovies.
Do you think the way you approach preparing meals, particularly for other people, is something that's informed by how meals were prepared for you when you were growing up?
BH: Nope.
So you've learnt that somewhere else?
AB: My father was a produce merchant. The person that brings in all the food from Queensland or Tasmania and sells it to the fruiterers. So he worked very hard for 40 years getting up at 1 in the morning to go to the wholesale markets. I used to work there a bit when I was a student. He specialised in potato, onion, Swede, pumpkin and watermelon. So we always had mashed potatoes. As a result, I avoid mash these days!
BH: My mum would throw all the vegetables into the pressure cooker. So she's put in Brussel sprouts. And everything would come out grey. But they were quite delicious.
What about any memorable desserts or birthday cakes?
AB: We had pavlovas and things like that. My mother used to make trifles. She was very good at trifles actually.
Trifle is divisive, you either love it or hate it.
BH: We've moved from trifle to tiramisu.
AB: I'll tell you, Lupino in the city on Little Collins street, does a very good Tiramisu.
Our conversation is interrupted by the cake timer again. The centre around the pears has firmed up now. We decide it's time to try the deviled eggs.
AB: Down the hatch!
I wanted to ask you about what makes a good dinner party. I was looking into Immanuel Kant's Guide to a Good Dinner Party. Kant believed the number of dinner guests should follow Chesterfield's rule: no fewer than the Graces i.e., three, no more than the Muses i.e., nine. What's one of your rules for a successful dinner party?
AB: We used to do eight people often in the middle room. Usually it's a mix of people we know well alongside a newbie or two, but Covid has changed a lot, and we tend to have only a few people around the table these days. The recipe for a good dinner party for me is to spend half a day getting ready, doing all the prep so I'm not flustered at all.
BH: You'll see him flustered later.
Stefanie Breschi
What about elements or things that you appreciate about a dinner party?
BH: Where we had dinner last night, an old friend, he sets the most amazing tables. It's quite formal. He buys lots of stuff at auction houses like old silverware, quite fancy. We borrowed these napkins from him last night!
AB: I like simple but nice glassware. Just simple.
Can you give us a tour of the kitchen, what are your MVP items in here? Or your favourite items?
AB: These beautiful salad servers, which were made in pewter by a close friend of Brent's, Kevin Maritz, they're very beautifully hammered servers.
If your house was burning down and you had to grab something from the kitchen, what would you grab?
BH: Nothing!
You'd have your hands full of art.
BH: We've got about 18 Fred Williams…
AB: We ended up with some fabulous ones, including the circular one that's there on the back wall above the couches. That's the actual impression image of the print that's featured in his landmark book published in the 1970s. So we're very lucky to have that, that's what I'd grab!
As we finish the bottle of wine and polish off the devilled eggs, we talk about different artworks in the living room while Andrew continues to prepare lunch. He sets a table in the backyard that has a lemon tree twisting above it like a well trained bonsai. An orange tree pokes its head over the fence, as if it's a neighbour saying hi. Sam, their cat is stuck on the wall outside. Andrew gets a ladder to help him down. He's nine years old and looks like a ginger biscuit. Andrew puts olives on the table and prepares a plate of prosciutto di San Daniele and bresaola.
I notice how our conversations ping from art to food, but always back to art again. For Brent and Andrew, art is their comfort zone and food and wine is there to complement it.
Brent Harris will be showing 'Monkey Business', an exhibition of new work at Tolarno Galleries in August.
The Art in Hospitality: Claudia Lau and Kevin Cheung
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,118
|
{"url":"https:\/\/bitbucket.org\/spookylukey\/django-anonymizer\/src\/2fc675271dea\/docs\/anonymizers.rst?at=default","text":"# Writing Anonymizers\n\nFor each model, you need a subclass of :class:anonymizer.base.Anonymizer. They can be automatically generated using the :ref:create-anonymizers-command command.\n\nThe main attributes that must be set are model and attributes. You can also override other methods to customise the process.","date":"2015-09-05 02:44:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3320522606372833, \"perplexity\": 3193.058300211595}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440645371566.90\/warc\/CC-MAIN-20150827031611-00026-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
| null | null |
The Parterre Archives
La Cieca ci guarda
Trove Thursday
Cher Public
La Fille du Régiment
Truth or illusion
Les Contes d'Hoffmann
Musical comedy girl
Canta che ti passa
You don't need analyzing
I'm so happy I can't stop crying
Curtain up! Light the lights!
Tragic mike
Things go better with woke
"On stage, I am in the dark"
"I am grateful that my horizons were not narrowed at the outset"
Broadcast: Last Night at the Proms
Slapstick Tragedienne
"The most essential blog in opera!" (New York Times) Where opera is king and you, the readers, are queens.
questo e quello, video
She asks me why I'm just a hairy guy
The Royal Opera House's production of La bohéme (featuring Matthew Polenzani as Rodolfo) goes live on YouTube at 2:15 this afternoon.
By La Cieca on June 26, 2018 at 12:46 PM
The Royal Opera House's production of La bohéme (featuring Matthew Polenzani as Rodolfo) goes live on YouTube at 2:15 this afternoon.
Topics: livestream, royal opera house, youtube
Latest on Parterre
Everybody is nothing until you love them
Miss me blonde
Advertise on parterre box!
parterre box, "the most essential blog in opera" (New York Times), is now booking display advertising for 2021. Join Carnegie Hall, Lincoln Center, BAM, New York City Opera and many others in reaching your target audience through parterre box.
From Sponsored Content
Copyright © 2021 parterre box. All rights reserved. Registration or use of this site constitutes acceptance of our Terms of Service and Privacy Policy.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,025
|
Phragmiticola rhopalospermum är en svampart som först beskrevs av Wilhelm Kirschstein, och fick sitt nu gällande namn av Sherwood 1987. Phragmiticola rhopalospermum ingår i släktet Phragmiticola, ordningen disksvampar, klassen Leotiomycetes, divisionen sporsäcksvampar och riket svampar. Inga underarter finns listade i Catalogue of Life.
Källor
Disksvampar
rhopalospermum
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,013
|
WKVR (88.9 FM) is a radio station broadcasting a contemporary Christian format. Licensed to Flint, Michigan, it is a member of the K-Love radio network.
History
The station signed on in September 1997 as WGRI with an urban gospel format. After Gospel Radio International sold the station to the Educational Media Foundation, WGRI joined K-Love on October 1, 2001, and changed its call letters to WAKL on October 29. The station became WKMF on May 31, 2019, after EMF moved the WAKL call sign to a station it had acquired in the Atlanta market.
The station changed its call sign to WKVR on December 6, 2022.
Translators
References
External links
KVR
Contemporary Christian radio stations in the United States
K-Love radio stations
Radio stations established in 1998
1998 establishments in Michigan
Educational Media Foundation radio stations
KVR
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 952
|
{"url":"https:\/\/www.physicsforums.com\/threads\/differential-equation-bernoullis.709593\/","text":"# Homework Help: Differential equation, Bernoulli's\n\n1. Sep 10, 2013\n\n### Lorenc\n\n1. The problem statement, all variables and given\/known data\n\nHello everybody :) Now, I have a differential equation to solve. Its a Bernoulli's type of eq.\n\n2. Relevant equations\n\n(2x2lny-x)y' = y\n\n3. The attempt at a solution\n\nI tried of putting it in this way: y'\/y = 1\/2x2lny-x and then considerin z=1\/y, but it doesnt seem to solve the problem, because the lny sort of gets in the way. Any help would be appreciated. Thanks :)\n\n2. Sep 10, 2013\n\n### vanhees71\n\nSubstitute $z=\\ln y$ into the equation. Hint: Evaluate the derivative $z'$ first!\n\n3. Sep 10, 2013\n\n### Lorenc\n\nThe only thing I hadn't tried :D Thanks a lot :D","date":"2018-05-25 05:17:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6687719821929932, \"perplexity\": 1675.618433758762}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794867041.69\/warc\/CC-MAIN-20180525043910-20180525063910-00192.warc.gz\"}"}
| null | null |
Q: How do I show this function is measurable? Let $h:[0,\infty)\to\mathbb{R}$ be a monotone function, with $\int_0^\infty |h(x)|x^2\,dx<\infty$.
And let $f:\mathbb{R}^3\to\mathbb{R}$ with $f(x)=h(|x|)$ for all $x$.
Prove that $f$ is (Lebesgue) measurable on $\mathbb{R}^3$.
I tried several techniques but did not manage to prove.
Try 1) $h$ is monotone function, thus continuous almost everywhere. $g(x)=|x|$ is a continuous function. However $f=hg$ is not necessarily continuous almost everywhere.
Try 2) I know that if $g$ is continuous and $h$ measurable, then $gh$ is measurable. Unfortunately, the order is wrong, we need $hg$ measurable.
Thanks for any help!
Another thing is that $g(x)=|x|$ is Lipschitz, but again that doesn't seem to help as we need $g^{-1}$ Lipschitz instead.
A: With $g(x)=|x|$ , we have that $g$ is continuous, hence measurable. $h$ is monotone, therefore $h$ is measurable. It follows that $f = h \circ g$ is measurable.
In this case, what saves this argument is that both $h$ and $g$ are Borel measurable, so the composition is also Borel measurable. – Sangchul Lee
A: This can be proved "by hand" in a couple of ways: One would be to notice that since $f$ is monotone, $f^{-1}((\alpha,\infty))$ is an interval. Once we have that, it's simple to finish. Another way would be to recall that $f$ has at most a countable number of discontinuities. It follows that $f\circ a$ is continuous on $\mathbb R^n$ minus a countable number of spheres. It's easy to finish from there.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,662
|
Minimum wage: What I expect from Nigerian workers – Saraki
President of the Senate, Dr Bukola Saraki, says he hopes that the new minimum wage will spur Nigerians to work harder.
Saraki in a goodwill message to mark the 2019 Workers' Day celebration, also commended the leadership and members of the organised labour for their patriotism in mostly choosing dialogue over industrial action in resolving trade disputes.
He expressed hope that workers would be encouraged to always put in their best to uplift and sustain the nation's economy, "in view of the recent signing into law of a new minimum wage of N30,000.
"No nation can develop without a virile and agile workforce.
"It is trite to say that workers are the mainstay of our nation's economy, since no policy of government, no matter how remote, will succeed without the commitment and collaboration of workers saddled with implementation.
"Having interacted with Nigerian workers and their leadership times without number, I can say without fear of contradiction that the nation's workforce is among the best in the continent.
"All that is left is to adequately harness their abundant talents and spirit of patriotism to further improve on the nation's economic and political development through timely and adequate motivation, training and retraining," he said.
Saraki advised the public sector, "to work to ensure that the country realized her potentials.
"It should eliminate tardiness, increase the level of discipline and strive to provide enabling environment for the private sector to thrive with the resultant broadening of the scope of national prosperity."
He called on the leadership of organised labour to continue to discharge their responsibilities in the overall interest of the country.
(Daily Post)
Related Topics:2019Bukola sarakilabourMinimum wageNigeriaWorkers
Osun Guber: What court decided on Senator Adeleke's case on Thursday
Crisis looms in Osun as Redeemer's University, Muslim community clash over land
Abass Sulaiman Adegoke, well known as Adegoke is a student of Federal Polytechnic Ede studying Civil Engineering, He is a media enthusiast, loves traveling, and has a special interest in personal development.
Funke Olakunrin's killing: Herdsmen want to set Nigeria on fire – Yoruba elders tell Buhari
ADULTS ONLY! HOW TO ENJOY S***X DURING RAINY SEASON
Adeleke vs Oyetola: APC replies Davido father's claim on Osun guber
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,698
|
{"url":"http:\/\/crypto.stackexchange.com\/questions\/10517\/extracting-only-the-entropy\/10716","text":"# Extracting only the entropy\n\n(I cleaned up the question a bit, It was previously titled \"extracting entropy without hashing\" .. thus some of the comments \/ answers)\n\nI'd like to extract the entropy of data without injecting \"pseudo\" entropy. For example, let's say I have a series of bytes, but only the low bit is changing (or something is changing that I'm not aware of beforehand) - how can I just output that low bit?\n\nIs there a known way to extract the random part of the data without doing custom programming for each different type of data series?\n\nedit: To clarify, I'm looking to extract only the random part. Eg, assuming F is the magical function, I want:\n\nLength (F(X)) = # of bits of min-entropy in X.\n\nAs has been pointed out elsewhere, this is probably impossible to do automagically with any reasonably assured level of confidence in its real accuracy.\n\n-\nWhy do you want to avoid hashes? \u2013\u00a0 CodesInChaos Sep 23 '13 at 8:49\nBecause it injects pseudo entropy. I'm looking for an alg which just extracts the entropy. \u2013\u00a0 Blaze Sep 23 '13 at 11:02\nIf you manage to estimate entropy, then you can still simply truncate the hash to the estimated entropy. Entropy estimation and extraction are mostly independent. \u2013\u00a0 CodesInChaos Sep 23 '13 at 12:33\nOk, so now we just need an alg to estimate the entropy. \u2013\u00a0 Blaze Sep 23 '13 at 19:42\nI don't know what you mean by \"inject pseudo entropy\". A hash does not inject entropy. I'm not familiar with \"pseudo entropy\" as a technical term. Can you tell us more about what your real problem is? You might want to edit the question to provide more context and to explain the particular application setting and what you're trying to achieve or what problem you're trying to fix. \u2013\u00a0 D.W. Oct 2 '13 at 6:09\n\nFor randomness extraction, in some cases, you could use alternatives to hash functions. However, mostly hash (or hmac) is preferable, because hash and hmac are very good in extracting randomness.\n\nRFC 5869 describes HKDF, HMAC-based extract-and-expand key derivation function, with randomness extraction and expansion phase. NIST has made equivalent standard from HKDF, NIST SP 800-56C, which also allows AES-CMAC as alternative to HMAC. Thus, NIST SP 800-56C compliant randomness extraction can be done without hash functions by using AES-CMAC instead.\n\nHowever, NIST SP 800-56C is only usable when there is a good reason to expect that input contains sufficient entropy to meet the intended entropy of the output.\n\nAnother obvious approach is that you could create NIST SP 800-90A Deterministic Random Bit Generator using AES-CTR algorithm with derivation function. The randomness you want to extract could be input entropy to the algorithm.\n\n## Estimate entropy\n\nHowever, for both of uses described above, you need good means to estimate how much entropy you have within the input. If you use something like NIST SP 800-56C and you did not have enough entropy in input, you don't have lot of entropy in output either. For this reason, it is critical you can correctly estimate how much entropy you have.\n\nUsual compression functions, for example, like CodesInChaos mentioned above are not good enough, for determining the amount of entropy. They can give pointers. Just never expect a compression function to produce full entropy.\n\nLinux kernel's \/dev\/random uses various mathematical means to estimate amount of entropy present in the events in information theoretic sense. This is in fact something pretty close to compression. However, it does not actually compress anything, but instead selects entropy estimate which is strictly smaller than any compression approach it could have used. A lot of information about Linux \/dev\/random is in this analysis. The analysis is old, and the issues found are largely fixed, but the basic structure remains the same.\n\nFor estimating the amount of entropy, it is necessary to understand what kind of input materials you have. Software like ent are useful to make estimate, but it is not at all hard to find materials where ent will overestimate entropy. For instance, try estimate entropy of AES-CTR(128 x 0 bit, 1024 x 0 bit). This input has around zero bits of input, but ent will estimate it to have nearly 1024.\n\nI would almost say that if you cannot indicate what the input material is (unfortunately commonly the case), and you feel ent or compression are good enough, you're very likely to end up with system that is not very strong (because you most likely will overestimate entropy).\n\n-\nYeah, though perhaps there is some middle ground. Wish there were more tools I could use to crunch the data. \u2013\u00a0 Blaze Oct 2 '13 at 20:07\n\nThe best answer is almost certainly to use a cryptographic hash.\n\nYour reason for avoiding a cryptographic hash makes no sense to me. Your problem does not explain the motivation for your question, but I suspect you've fallen prey to the XY problem (see also here).\n\nYou haven't told us what you're ultimately trying to accomplish, but I suspect the right answer in practice is ultimately going to be:\n\n1. Use \/dev\/urandom to generate cryptographic-quality pseudorandom numbers.\n\n2. If you have a source of entropy, feed it into \/dev\/urandom (e.g., cat somekindarandombits > \/dev\/urandom), then see step 1 above.\n\nInternally, this effectively uses a hash.\n\nI also recommend you read the following:\n\n-\nThe answer I suspect is what codes in chaos said above in his comments. What I'm really looking for is a way to estimate entropy. And, what I've come to realize, is that there is no \"ent\" utility you can run that magically does this, but rather it's a combination of heuristic + statistical analysis. Painful, but necessary because of the hidden patterns in seemingly random data that can't be pried out by mere software. Automated entropy estimation simply fails (which is why dev\/random is no good) \u2013\u00a0 Blaze Oct 2 '13 at 9:23\nAs for the reason I want to do this, I'm under NDA sadly! Still, the problem of entropy estimation I think is something everyone might be interested in. \u2013\u00a0 Blaze Oct 2 '13 at 9:35\n@Blaze could you edit your question to contain the actual problem, instead of hiding it in a comment? \u2013\u00a0 Pa\u016dlo Ebermann Oct 2 '13 at 18:50\n@Blaze, careful there. There is no good way to do accurate entropy estimation (at least not in general, without knowing anything about the source). But be careful about what inferences you draw. That doesn't mean \/dev\/random is no good. In general, there's a lot of work on this sort of topic; I encourage you to read up on it before drawing too many conclusions. \u2013\u00a0 D.W. Oct 2 '13 at 19:27\nI have read up on it, and the entropy estimator is often the root of its problems. It's why Fortuna, Barak and Halevi, etc have been advocated. \u2013\u00a0 Blaze Oct 2 '13 at 19:47\nYou need to assume something about the data. $\\:$ Would you prefer to assume that the data is divided $\\hspace{.12 in}$ into independent blocks, or even iid blocks, rather than a lower bound on its smoothed min-entropy? $\\hspace{.27 in}$ \u2013\u00a0 Ricky Demer Sep 23 '13 at 0:29","date":"2014-04-20 03:39:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6513059735298157, \"perplexity\": 1045.0185508696288}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-15\/segments\/1397609537864.21\/warc\/CC-MAIN-20140416005217-00540-ip-10-147-4-33.ec2.internal.warc.gz\"}"}
| null | null |
This isn't a low cost solution; it's a "no-cost" decorator idea that will make your presentation look like a magazine in 6-simple steps. And believe it or not, you most likely have all the materials needed already inside and outside your home.
Step 1: Put a table cover over your buffet table.
Step 2: Place 3-boxes on the table, a short, medium and taller box, that are wide enough to hold a platter.
Step 3: Cover your boxes with a contrasting color of fabric. Could be drapery fabric, or scrap fabric.
Step 4: Place your third color fabric somewhat twisted-on the table, slightly tucking it into the boxes.
Step 5: Greens, gourds, candles. Place any greens from your yard alongside gourds and then pepper in some safe candles in glass containers.
Step 6: Food. Place veggies, cheese and cracker plates at the front and then the savory Dream Dinners chilled steak or chicken entrees presented as appetizers.
Next, watch the video "How to Turn Dream Dinners into Appetizers" to see how we added food to the table.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,043
|
<?xml version="1.0" encoding="iso-8859-1"?>
<!DOCTYPE html
PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html>
<head>
<title>adapter_name (::Oracle)</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
<link rel="stylesheet" href="../.././rdoc-style.css" type="text/css" media="screen" />
</head>
<body class="standalone-code">
<pre><span class="ruby-comment cmt"># File lib/jdbc_adapter/jdbc_oracle.rb, line 112</span>
<span class="ruby-keyword kw">def</span> <span class="ruby-identifier">adapter_name</span>
<span class="ruby-value str">'Oracle'</span>
<span class="ruby-keyword kw">end</span></pre>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,862
|
Landesstraßen (singular: Landesstraße) are roads in Germany and Austria that are, as a rule, the responsibility of the respective German or Austrian federal state. The term may therefore be translated as "state road". They are roads that cross the boundary of a rural or urban district (Landkreis or Kreisfreie Stadt). A Landesstraße is thus less important than a Bundesstraße or federal road, but more significant than a Kreisstraße or district road. The classification of a road as a Landesstraße is a legal matter (Widmung). In the free states of Bavaria and Saxony – but not, however, in the Free State of Thuringia – Landesstraßen are known as Staatsstraßen.
Designation
The abbreviation for a Landesstraße consists of a prefixed capital letter L and a serial number (e. g. L 1, L 83, L 262 or L 3190). Staatsstraßen in Saxony are similarly abbreviated using a capital S (e. g. S 190) and the Staatsstraßen in Bavaria are prefixed with the letters St (e. g. St 2108).
The kilometrage is shown on white signs by the roadside with black letters, known as location signs (Stationszeichen), that replace the former kilometre stones. The beginning and end of a Landesstraße is specified using so-called hub (Netzknoten) numbers. That makes its location unambiguous, which is important for rapid assistance when there is an accident, for example. The hub numbers are displayed on the upper part of the sign and also indicate their direction. In the example in the photograph, therefore No. 6608 039 is left of the sign 6608 023 to the right. In the bottom right-hand corner of the sign can be seen the so-called Stationierungsrichtung or direction of signage. It runs in the example from right to left and indicates in which direction the road kilometres are counted.
In Lower Saxony, this new system has been in place since 2007 and divides the Landestraßen into sections numbered 10, 20, etc. The location signs (Stationszeichen) comprise two panels. The location panel (Stationierungstafel) displays the name of the state (Niedersachsen) and county letters (e.g. Kreis WF) at the top, and the road letter and number below (e.g. L 627). The classification panel (Klassifizierungstafel) shows the section number, kilometrage and direction of the start hub (e.g. 10 Abschnitt and 2,0 → 0,0). The letters OD indicate a location post within a town or village and may be displayed in places other than on a white post. By the end of 2008, almost all the 8,000 kilometre posts on Lower Saxony's Landesstraßen had been replaced.
Properties
In terms of their construction, Landesstraßen tend to be built to a lesser standard than Bundesstraßen and their cross-section is generally smaller. In individual cases, however, the standard of construction may vary depending on when it was built and its importance as a route. However, Landesstraßen can be built as limited-access dual carriageways in densely populated areas.
Due to the division of funding, the federal states usually try to get the more substantial Landesstraßen officially designated as Bundesstraßen, so that their subsequent improvement and maintenance is funded from the Federal budget. The Bundesstraßen are, however, intended as links between cities (major centres) and radiate from them. It is not usually possible to have concentric roads, which link the satellite towns with one another, designated as Bundesstraßen. Similarly, it is difficult to transfer responsibility for the short stub roads running from cities to nearby motorways to the Federal Authorities.
Situation in the new federal states
Following German reunification the Bezirksstraßen of the GDR (also called Category 1 Landstraßen) were generally classified as Landesstraßen without consideration for their condition. This leads to a wide range of road types falling within this category. On the one hand, there are inter-city roads which have been modernised to a good quality standard. On the other hand, due to the austere design of the country road network in the GDR (e. g. occasionally one of two parallel roads or, where they were of low traffic importance, the links between them were simply allowed to fall into disrepair) there are today in the new federal states several unpaved roads and dirt tracks that are formally Landesstraßen (e. g. the L 208 between Burkersroda and Balgstädt in Saxony-Anhalt or the L 1062 between Wittersroda and Lengefeld in Thuringia). The upgrade of these roads is unlikely in view of the lack of funding and their low importance; in most cases attempts are being made to have their status downgraded.
Situation in Austria
In Austria today all important roads, apart from autobahns and Schnellstraßen (limited-access roads) managed by the publicly owned ASFiNAG corporation, are called Landesstraßen. Since 2002, even the former Bundesstraßen national highways are Landesstraßen, because they were placed under the responsibility of the federal states.
Before 2002 there were two types of Bundesstraße:
A white number on a blue square sign identified the more common type that are at the same time priority roads (Vorrangstraße). Their vehicle users had the right of way already by the blue sign.
A black number (typically greater than 100) on a yellow circular sign marks state roads that are not per se priority roads.
Except in Vorarlberg, the former Bundesstraßen continue to be designated with the prefix B. The remaining Landesstraßen are prefixed with the letter L. On traffic signs the prefixes are usually not used, unlike the A (for Autobahn) and S (Schnellstraße). Roads numbered with fewer digits are generally of more importance in the road network. The former designation of more important Landesstraßen in several states as Landeshauptstraßen (LH, in Vienna: Hauptstraßen A) is only occasionally seen now on road and street maps.
See also
Autobahn
Bundesstraße
Gemeindestraße
Kreisstraße
References
Roads in Germany
Roads in Austria
Roads by type
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,540
|
WATCH: Paraplegic Veteran Stuns Bride With Wedding Dance
Jeffrey Rindskopf
An Afghanistan veteran confined to a wheelchair wanted to make his wedding day special, and it's safe to say he succeeded. Sgt. Joey Johnson surprised his wife at their wedding reception by rigging himself up so the two could enjoy their first dance, wheelchair-free.
The video destroyed us.
Johnson, 27, suffered a spinal injury in a motorcycle accident just four months after meeting his now-wife Michelle. Johnson, who rode motorcycles after returning from combat to cope with his post-traumatic stress disorder, became a paraplegic.
Michelle stayed with him, and the two were married on June 28, 2014. Johnson and his friends worked to prepare a system that could help him stand and dance up to the big day.
"If you know Joey, he cannot keep a secret. But he made my dreams come true and I never knew how special our day would really be," she told ABC News.
The day of the wedding, Michelle took a break in her bridal suite after cutting the cake at the reception. When she returned for the first dance, she saw her husband standing in wait for her. See how the first dance went above.
http://www.firsttoknow.com
WATCH: Dog Gets a Christmas Present of a Lifetime!
'We can't have anything nice!' – Mom, Chaos, and Christmas
Friendships & Personal Relationships
5 Things I Learned from My Dying Mother — And How I Said Goodbye
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,613
|
{"url":"https:\/\/www.calc-medic.com\/precalc-unit-9-day-3","text":"Unit 9 - Day 3\n\nUnit 9Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 9Day 10Day 11Day 12Day 13Day 14Day 15Day 16Day 17Day 18Day 19Day 20Day 21All Units\n\u200bLearning Objectives\u200b\n\u2022 Understand that limits turn an estimate of the instantaneous rate of change into the exact value of the slope\n\n\u2022 Set-up and evaluate a limit expression that gives the slope at a single point\n\n\u2022 Write the equation of a tangent line to a curve at a given point\n\nLesson Handout\n\nExperience First\n\nBased on the common myth about a penny dropped from the Empire State Building being able to kill someone (and the subsequent Mythbuster episode to disprove it), students explore the actual speed of a penny in freefall. On page 1, students calculate the penny\u2019s average speed over a 1.5 second interval and then estimate the penny\u2019s actual speed after 1.5 seconds. The too-low and too-high estimates garnered great discussion. While some used way too high and way too low estimates, many groups calculated slopes between t-values just to the left and just to the right of t=1.5 and concluded that the true slope would be somewhere in between. Many were able to estimate the penny\u2019s speed as being 48 ft per second based on these measurements! As you are monitoring students, record what strategies you see being used, and determine how you want to sequence these strategies in the debrief.\n\nOn page 2, students actually calculate the slope on various intervals that get smaller and smaller. We introduce the notation of \u201ca\u201d and \u201ch\u201d and ask students to interpret what h means. Students realize that as the two x-values get closer and closer together, the slope calculated in the last column becomes closer and closer to the penny\u2019s true speed at t=1.5. The formal limit notation and evaluation is done as a class in the debrief.\n\nFormalize Later\n\nNote that we always use \u201ca\u201d when calculating slope at a specific point and \u201cx\u201d when we are finding the slope at any point, i.e. the derivative function. We have found that this alleviates much of the confusion around the notation. The biggest hang-up for students tends to be the shift away from the (y2-y1)\/(x2-x1) definition of slope to the (f(a+h)-f(a))\/h definition. The debrief of the table given in question 4 should make this connection clear for students. Ask the students why the denominator of the slope column doesn\u2019t have a difference in it. Students should be able to explain that h already represents the difference between the two x-values. If students are still confused by the idea of h, ask them how they would calculate x2-x1 in their table. It should become evident that a+h - a=h. We make this explicit again in the important ideas, where we mark the interval between the two x-values as h.\n\nTo find the equation of the tangent line, we prefer using point-slope form, even though students continue to favor slope-intercept. We allow students to write the equation in either form (all equivalent answers are accepted) but we hope that over time they will see the value in not having to solve for the y-intercept!\n\nAlgebra errors abound when students evaluate the limit definition of slope at a point. Proper use of parentheses and a review of how to evaluate functions at expressions like \u201ca+h\u201d is helpful in setting students up to tackle the Check Your Understanding questions. Keep in mind that while the algebraic skills are important, we shouldn\u2019t let students lose sight of what this limit expression represents. More purposeful practice will take place tomorrow!","date":"2021-06-21 10:22:22","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8539317846298218, \"perplexity\": 953.7310605940697}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488269939.53\/warc\/CC-MAIN-20210621085922-20210621115922-00449.warc.gz\"}"}
| null | null |
'''
Created on May 17, 2011
@author: frank
'''
from OvmCommonModule import *
class OvmVifDecoder(json.JSONDecoder):
def decode(self, jStr):
deDict = asciiLoads(jStr)
vif = OvmVif()
vif.mac = deDict['mac']
vif.bridge = deDict['bridge']
return vif
class OvmVifEncoder(json.JSONEncoder):
def default(self, obj):
if not isinstance(obj, OvmVif): raise Exception("%s is not instance of OvmVif"%type(obj))
dct = {}
safeDictSet(obj, dct, 'mac')
safeDictSet(obj, dct, 'bridge')
safeDictSet(obj, dct, 'type')
safeDictSet(obj, dct, 'name')
return dct
def fromOvmVif(vif):
return normalizeToGson(json.dumps(vif, cls=OvmVifEncoder))
def fromOvmVifList(vifList):
return [fromOvmVif(v) for v in vifList]
def toOvmVif(jStr):
return json.loads(jStr, cls=OvmVifDecoder)
def toOvmVifList(jStr):
vifs = []
for i in jStr:
vif = toOvmVif(i)
vifs.append(vif)
return vifs
class OvmVif(OvmObject):
name = ''
mac = ''
bridge = ''
type = ''
mode = ''
def toXenString(self):
return "%s,%s,%s"%(self.mac, self.bridge, self.type)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,072
|
<?php
namespace Users\Model\Table;
use Cake\ORM\Query;
use Cake\ORM\ResultSet;
use Cake\ORM\RulesChecker;
use Cake\ORM\Table;
use Cake\ORM\TableRegistry;
use Cake\Validation\Validator;
use Users\Model\Entity\User;
/**
* Users Model
*
* @property \Cake\ORM\Association\HasMany $Aros
*/
class UsersTable extends Table
{
/**
* Initialize method
*
* @param array $config The configuration for the Table.
* @return void
*/
public function initialize(array $config)
{
parent::initialize($config);
$this->table('users');
$this->displayField('email');
$this->primaryKey('id');
$this->addBehavior('Timestamp');
}
/**
* Default validation rules.
*
* @param \Cake\Validation\Validator $validator Validator instance.
* @return \Cake\Validation\Validator
*/
public function validationDefault(Validator $validator)
{
$validator
->add('id', 'valid', ['rule' => 'numeric'])
->allowEmpty('id', 'create');
$validator
->add('email', 'valid', ['rule' => 'email'])
->requirePresence('email', 'create')
->notEmpty('email');
$validator
->requirePresence('password', 'create')
->notEmpty('password');
$validator
->requirePresence('first', 'create')
->notEmpty('first');
$validator
->requirePresence('last', 'create')
->notEmpty('last');
return $validator;
}
/**
* Returns a rules checker object that will be used for validating
* application integrity.
*
* @param \Cake\ORM\RulesChecker $rules The rules object to be modified.
* @return \Cake\ORM\RulesChecker
*/
public function buildRules(RulesChecker $rules)
{
$rules->add($rules->isUnique(['email']));
return $rules;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,640
|
Q: assign one key of function output array to variable in tight away? I need to make following in one line, i am looking for method of making similar code one line not to solve this particular example.
$file_path = pathinfo($_SERVER["SCRIPT_NAME"]);
$file_name = $file_path["filename"];
e.g.
$file_name = pathinfo($_SERVER["SCRIPT_NAME"])["filename"];
A: Array dereferencing is only possible from php 5.5. If you have an earlier version, sadly, it will not be possible.
However you could try using list, which will assign all the array elements in pathinfo to individual variables. If you order your variables correctly, $file_name will have what you need.
A: Unfortunately PHP's syntax does not support this for versions lower than PHP5.5. As of PHP5.5 you can just write:
echo pathinfo($path)['filename'];
If you are working with a version < 5.5, I would suggest to write a custom function:
function array_get($key, $array) {
if(!array_kex_exists($key, $array)) {
throw new Exception('$array has no index: ' . $key);
}
return $array[$key];
}
Then use it:
$filename = array_get('filename', pathinfo($path));
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,112
|
\section{Introduction}
Resonant states play a central role
in the quantum description of decaying nuclear states. In the ordinary
formulation of Quantum Mechanics these states appear as complex energy
poles of the $S$ scattering matrix. In turn, these states can be
defined as solutions of the time-independent Schr\"{o}dinger equation
with purely out-going waves at large distances \cite{tp4}.
Several attempts have been performed to handle adequately
resonant states, the main obstacle being the divergent behaviour
of the corresponding wave functions at large distances, which makes
it impossible to normalize them in an infinite volume with the
conventional mathematical tools.
The first succesful attempt to handle resonant wave functions
has been made by Tore Bergreen
\cite{tp5} using a regularization method first suggested by Zel'dovich.
In his work Bergreen has shown that at least for finite range potentials
it is possible to define an ortogonality criteria among bound and
resonant states, and also a pseudonorm can be evaluated
using the general analysis of Newton \cite{tp4}.
A proper inclusion of resonant states within the general framework of
Quantum Mechanics has been done through the Rigged Hilbert Space (RHS) or
Gelfand's Triplet (GT) formulation \cite{tp2}. Resonant states are
described, within the RHS, as generalized complex energy solutions
of a self-adjoint Hamiltonian.
The structure of the RHS guarantees that
any matrix element involving resonant states is a well defined quantity,
provided the topology in the GT has been properly choosen to handle
the exponential growing of Gamow States at large distances.
The literature concerning the aplication of RHS to resonant states is
extensive \cite{tp9}-\cite{tp12}.
Among these works we shall mention, for instance, those of Bohm
\cite{tp9}, Gadella \cite{tp10} and also reference \cite{tp11},
where resonant states are introduced using a RHS of entire
Hardy-class functions defined in a half complex energy-plane. This allows
to extend analyticaly the concept of a resonant state
as an antilinear complex functional over the intersection of
Schwartz test functions with Hardy class.
A more general theory of resonant states follows if the RHS is built up
on tempered ultradistributions \cite{tp14}. In this case resonant states
arise as continuous linear functionals over rapidly decreasing entire
analytical test functions.
This can be obtained by using the Dirac's formula, which
allows a more direct determination of these states.
Another advantage of using tempered ultradistributions is that only
the physical spectrum appears in the definition of complex-energy states
\cite{tp14}.
In the present paper we want to show that
it is possible to define a complex pseudonorm for
resonant states in the sense of Bergreen using tempered ultradistributions.
With this pseudonorm we generalize
the Bergreen's result \cite{tp5}, and hence it can be
considered as the proper analytical extension of a pseudoscalar
product for resonant states.
We give an introduction of tempered ultradistributions and Gelfand
Triplet in section 2. In section 3 we define resonant states starting
from the Schr\"{o}dinger equation, and then we focus our atention
on the calculus of the pseudonorm of a complex-energy state.
We apply in section 4 the results of the previous section to the case
of a square well potential. We give a resume in section 5.
\section{The Tempered Ultradistributions}
\setcounter{equation}{0}
\subsection{The Triplet $(H,{\cal H},{\Lambda}_{\infty})$}
We define the space $H$ of test functions $\phi(x)$ such that
$e^{p|x|}|D^q\phi(x)|$ is bounded for any $p$ and $q$ by means of the
set of countably norms (ref.\cite{tp1}):
\begin{equation}
{\|\hat{\phi}\|}_p^{''}=\sup_{0\leq q\leq p,\,x}
e^{p|x|} \left|D^q \hat{\phi} (x)\right|\;\;\;;\;\;\;p=0,1,2,...
\end{equation}
According to the ref.\cite{tp2} $H$ is a space ${\cal K}\{M_p\}$
with:
\begin{equation}
M_p(x)=e^{(p-1)|x|}\;\;\;;\;\;\; p=1,2,...
\end{equation}
\begin{equation}
{\|\hat{\phi}\|}_p=\sup_{0\leq q \leq p}
M_p(x)\left|D^q \hat{\phi}(x) \right|
\end{equation}
${\cal K}\{e^{(p-1)|x|}\}$ satisfies condition $({\cal N})$ of Guelfand
( ref.\cite{tp3} ). Then if we define:
\begin{equation}
{<\hat{\phi}, \hat{\psi}>}_p = \int\limits_{-\infty}^{\infty}
e^{2(p-1)|x|} \sum\limits_{q=0}^p D^q \overline{\hat{\phi}} (x) D^q
\hat{\psi} (x)\;dx \;\;\;;\;\;\;p=1,2,...
\end{equation}
\begin{equation}
{\|\hat{\phi}\|}_p^{'}=\sqrt{{<\hat{\phi}, \hat{\phi}>}_p}
\end{equation}
${\cal K}\{e^{(p-1)|x|}\}$ is a countable Hilbert and nuclear space.
\begin{equation}
{\cal K}\{e^{(p-1)|x|}\} = H = \bigcap\limits_{p=1}^{\infty} H_p
\end{equation}
where $H_p$ is the completed of $H$ by the norm (2.1.5).
Let
\begin{equation}
<\hat{\phi}, \hat{\psi}> = \int\limits_{-\infty}^{\infty}
\overline{\hat{\phi}}(x) \hat{\psi}(x)
\end{equation}
Then, the completed of $H$ by (2.1.7) is ${\cal H}$, the Hilbert space
of square integrable functions. Now
\begin{equation}
<\hat{\phi}, \hat{\psi}> \leq C\,{\|\hat{\phi}\|}_1^{'}
\,{\|\hat{\psi}\|}_1^{'}
\end{equation}
and according to ref.\cite{tp3} the triplet
\begin{equation}
\left(H,{\cal H},{\Lambda}_{\infty}\right)
\end{equation}
is a Rigged Hilbert space or Guelfand's Triplet. Here
${\Lambda}_{\infty}$ is the dual of $H$ and it consist of
distributions of exponential type $T$ ( ref.\cite{tp1} ):
\begin{equation}
T=D^p\left[e^{p|x|} f(x) \right] \;\;\;;\;\;\;p=0,1,2...
\end{equation}
where $f(x)$ is bounded continuous.
\subsection{The Triplet $(h,{\cal H},{\cal U})$}
\setcounter{equation}{0}
The space $h={\cal F}\{H\}$ ( ${\cal F}$= Fourier transform )
consist of entire analytic rapidly decreasing test functions given
by the countable set of norms :
\begin{equation}
{\|\phi\|}_{pn} = \sup_{|Im(z)|\leq n} {\left(1+|z|\right)}^p
|\phi (z)|
\end{equation}
Then $h$ is a ${\cal Z}\{M_p\}$ space, complete and countable
normed ( Frechet ) with:
\begin{equation}
M_p(z)= (1+|z|)^p
\end{equation}
If we define:
\begin{equation}
{<\phi (z), \psi (z) >}_p={<\hat{\phi}(x), \hat{\psi}(x)>}_p
\end{equation}
then, ${\cal Z}\{(1+|z|)^p\}$ is a countable Hilbert and nuclear
space. Let be:
\begin{equation}
\psi (z) = \int\limits_{-\infty}^{\infty} e^{izx}
\hat{\psi} (x) dx
\end{equation}
\begin{equation}
\phi (z) = \int\limits_{-\infty}^{\infty} e^{izx}
\hat{\phi} (x) dx
\end{equation}
\begin{equation}
{\phi}_1(z)=\frac {1} {2\pi} \int\limits_{-\infty}^{\infty}
e^{-izx} \overline{\hat{\phi}}(x) dx
\end{equation}
Then we define:
\[<\phi (z), \psi (z)>=\int\limits_{-\infty}^{\infty}
{\phi}_1(z) \psi (z) dz =
\int\limits_{-\infty}^{\infty} \overline{\hat{\phi}}(x)
\hat{\psi}(x) dx \]
\begin{equation}
=<\hat{\phi}(x), \hat{\psi}(x)>
\end{equation}
The completed of $h$ by this last scalar product is the Hilbert
space ${\cal H}$ of square integrable functions
and the dual of $h$ is the space ${\cal U}$ of tempered
ultradistributions ( ref.\cite{tp1} ). Then $(h,{\cal H},{\cal U})$ is
a Guelfand's triplet.
The space ${\cal U}$ can be characterized as follow (ref.\cite{tp1}).
Let be ${\cal A}_{\omega}$ the space of all functions $F(z)$ such that:
(i)$F(z)$ is analytic in $\{z\in {\cal C} : |Im(z)|>p\}$.
(ii)$F(z)/z^p$ is bounded continuous in $\{z\in {\cal C} :
|Im(z)|\geq p\}$ where $p$ depends of $F(z)$. Here $p=0,1,2,...$
Let $\Pi$ be the set of all z-dependent polinomials $P(z)$, $z\in {\cal C}$.
Then ${\cal U}$ is the quotient space:
\begin{equation}
{\cal U}= \frac {{\cal A}_{\omega}} {\Pi}
\end{equation}
Due to these properties any ultradistribution can be represented as a
linear functional where $F(z)\in {\cal U}$ is the indicatrix of this
functional (ref.\cite{tp1}):
\begin{equation}
F(\phi)=<F(z), \phi(z)>=\oint\limits_{\Gamma} F(z) \phi(z) dz
\end{equation}
where the path $\Gamma$ runs parallel to the real axis from
$-\infty$ to $\infty$ for $Im(z)>\rho$, $\rho>p$
and back from $\infty$ to $-\infty$ for $Im(z)<-\rho$, $-\rho<-p$
( $\Gamma$ lies outside a horizontal band of width $2p$ that contain
all the singularities of $F(z)$ ).
Formula (2.2.9) will be our fundamental representation for a tempered
ultradistribution. An interesting property, according to ``Dirac formula''
for ultradistributions (ref.\cite{tp8}),
\begin{equation}
F(z)=\frac {1} {2\pi i}\int\limits_{-\infty}^{\infty} dt\;
\frac {f(t)} {t-z}
\end{equation}
is that the indicatrix $f(t)$ satisfies :
\begin{equation}
\oint\limits_{\Gamma} dz\; F(z) \phi(z) =
\int\limits_{-\infty}^{\infty} dt\; f(t) \phi(t)
\end{equation}
While $F(z)$ is analytic on $\Gamma$, the density $f(t)$ is in
general singular, so that the r.h.s. of (2.2.11) should be interpreted
in the sense of distribution theory.
The representation (2.2.9) makes evident that the addition of a
polinomial $P(z)$ to $F(z)$ do not alter the ultradistribution:
\[\oint\limits_{\Gamma}dz\;\{F(z)+P(z)\}\phi(z)=
\oint\limits_{\Gamma}dz\;F(z)\phi(z)+\oint\limits_{\Gamma}dz\;
P(z)\phi(z)\]
But:
\[\oint\limits_{\Gamma}dz\;P(z)\phi(z)=0\]
as $P(z)\phi(z)$ is entire analytic ( and rapidly decreasing ),
\begin{equation}
.{}^..\;\;\;\;\oint\limits_{\Gamma}dz\;\{F(z)+P(z)\}\phi(z)=
\oint\limits_{\Gamma}dz\;F(z)\phi(z)
\end{equation}
In the Rigged Hilbert spaces $(\Phi,{\cal H},{\Phi}^{\ast})$
is valid the following very important property:
Every symmetric operator $A$ acting on $\Phi$, that admit a self-adjoint
prolongation operating on ${\cal H}$, has in ${\Phi}^{\ast}$
a complete set of generalized eigenvectors ( or proper distributions )
that correspond to real eigenvalues(ref.\cite{tp3}).
This property is then valid in $(H,{\cal H},{\Lambda}_{\infty})$
and in $(h,{\cal H},{\cal U})$.
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{The pseudonorm of eigenstates of short range potentials}
\setcounter{equation}{0}
In this paragraph we describe the main properties of the solutions
of the Schr\"{o}dinger equation for a central short range potential
ref.\cite{tp4}. According to this reference the regular
( ${\phi}_l(k,r)$ ) and irregular ( $f_l(k,r)$ )
solutions for this equation satisfy, respectively, the following
boundary conditions
\begin{equation}
\lim_{r\rightarrow 0}\;(2l+1)!!\;r^{-l-1}{\phi}_l(k,r)=1
\end{equation}
\begin{equation}
\lim_{r\rightarrow \infty}\;e^{ikr}f_l(k,r)=i^l
\end{equation}
Both solutions are related by
\begin{equation}
{\phi}_l(k,r)=\frac {1} {2} ik^{-l-1}\left[f_l(-k)f_l(k,r)-
(-1)^lf_l(k)f_l(-k,r)\right]
\end{equation}
In (3.3) $f_l(k)$ is the Jost function defined by:
\begin{equation}
f_l(k)=k^l{\cal W}\left[f_l(k,r), {\phi}_l(k,r)\right]
\end{equation}
where ${\cal W}[f,\phi]$ is the Wronskian of the two solutions.
The zeros of the Jost function $f_l(k)$ are the bound
( $Re(k)=0, Im(k)\leq 0$ ), virtual ( $Re(k)=0, Im(k)>0$ ) and resonant
states ( $Re(k)\neq 0, Im(k)>0$ ) ( refs.\cite{tp4,tp5} ).
With these definitions we are now in position to calculate the pseudonorm
of the above states.
According to ref.\cite{tp4} the derivative of the Jost function with
respect to the variable $k$, ${\dot{f}}_l(k)$, satisfies
\[{\dot{f}}_l(k)=l k^{l-1} {\cal W}\left[f_l(k,r), {\phi}_l(k,r)\right]
+ k^l {\cal W}\left[{\dot{f}}_l(k,r), {\phi}_l(k,r)\right] \]
\begin{equation}
+ k^l {\cal W}\left[f_l(k,r), {\dot{\phi}}_l(k,r)\right]
\end{equation}
In particular, when $k_0$ is a zero of the Jost function then (3.5) takes
the form:
\begin{equation}
{\dot{f}}_l(k_0)= k_0^l {\cal W}\left[ {\dot{f}}_l(k_0,r),
{\phi}_l(k_0,r)
\right] + k_0^l {\cal W}\left[ f_l(k_0,r), {\dot{\phi}}_l(k_0,r)
\right]
\end{equation}
Due to eq.(3.3) at $k=k_0$ we have the equality:
\begin{equation}
f_l(k_0,r)=C(k_0){\phi}_l(k_0,r)\;\;\;;\;\;\;
C(k_0)=\frac {-2ik_0^{l+1}} {f_l(-k_0)}
\end{equation}
and following the procedure of ref.\cite{tp4} we get:
\[{\dot{f}}_l(k_0)={k_0}^l \lim_{\beta \rightarrow \infty} \left\{
{\cal W}\left[{\dot{f}}_l(k_0,\beta), {\phi}_l(k_0,\beta)\right]
-\right.\]
\begin{equation}
\left.2k_0 C(k_0) \int\limits_0^{\beta} {{\phi}_l}^2(k_0,r)\;dr \right\}
\end{equation}
From (3.8) we deduce immediately:
\[\lim_{\beta \rightarrow \infty} \int\limits_0^{\beta}
{{\phi}_l}^2(k_0,r)\;dr=
- \lim_{\beta \rightarrow \infty}
\frac {f_l(-k_0)} {4ik_0^{l+2}} {\cal W}\left[{\dot{f}}_l(k_0,\beta),
{\phi}_l(k_0,\beta)\right]\]
\begin{equation}
+\frac {{\dot{f}}_l(k_0) f_l(-k_0)} {4ik_0^{2l+2}}
\end{equation}
Now, we want to show that: i) the integral appearing in (3.9) can be defined
as an ultradistribution in the variable $k_0$ and ii) in the limit
$\beta \rightarrow \infty$, as an ultradistribution in $k_0$,
the Wronskian ${\cal W}$ vanishes.
With this purpose and according to ref.\cite{tp4} we note that
$k^l f_l(k_0,r)$=$h_l(k_0,r)$ is an entire analytic function of the
variable $k_0$ and therefore $k^{l+1} f_l(k_0,r)$ is also too.
Hence $k^{l+1} {\dot{f}}_l(k_0,r)=g_l(k_0,r)$ is an entire
analytic function of $k_0$. Moreover it has been shown in ref.\cite{tp4}
that $h_l(0,r)=C{\phi}_l(0,r)$.
And as a consequence we have $h_l(0,0)=g_l(0,0)=0$ because
${\phi}_l$ has the property ${\phi}_l(0,0)=0$.
We can write now (3.8) in terms of $g_l(k,r)$ as:
\[{\dot{f}}_l(k_0)=\lim_{\beta \rightarrow \infty}\left\{
\frac {{\cal W}\left[g_l(k_0,\beta), {\phi}_l(k_0,\beta)\right]}
{k_0}\right.\]
\begin{equation}
\left.-2k_0^{l+1} C(k_0) \int\limits_0^{\beta}
{{\phi}_l}^2(k_0,r)\;dr\right\}
\end{equation}
But:
\[\lim_{\beta \rightarrow \infty}\oint\limits_{\Gamma}
\frac {{\cal W}\left[g_l(k_0,\beta), {\phi}_l(k_0,\beta)\right]}
{k_0}\phi(k_0)\;dk_0=\]
\begin{equation}
\lim_{\beta \rightarrow \infty}
{\cal W}\left[g_l(0,\beta), {\phi}_l(0,\beta)\right]
\phi(0)
\end{equation}
where $\phi(k_0)\in h$ is an entire analytic test function and the
path $\Gamma$ runs parallel to the real axis from
$-\infty$ to $\infty$ for $Im(k_0)>\rho$, $\rho>0$
and back from $\infty$ to $-\infty$ for $Im(k_0)<-\rho$, $-\rho<0$
( $\Gamma$ lies outside a horizontal band that contains
the singularity in the origin ).
Taking into account that $f_l$ satisfies:
\begin{equation}
\frac {d} {dr} {\cal W}\left[{\dot{f}}_l(k,r), f_l(k,r)\right]=
2kf_l^2(k,r)
\end{equation}
it is easy to show that:
\begin{equation}
\frac {d} {dr} {\cal W}\left[g_l(k,r), h_l(k,r)\right]=
2k^2h_l^2(k,r)
\end{equation}
and then
\begin{equation}
\frac {d} {dr} {\cal W}\left[g_l(0,r), h_l(0,r)\right]= 0
\end{equation}
Eq.(3.14) implies that:
\begin{equation}
{\cal W}\left[g_l(0,r), h_l(0,r)\right]= constant
\end{equation}
and from $h_l(0,0)=g_l(0,0)=0$ we obtain:
\begin{equation}
{\cal W}\left[g_l(0,r), h_l(0,r)\right]=0
\end{equation}
This implies that
\begin{equation}
{\cal W}\left[g_l(0,r), {\phi}_l(0,r)\right]=0
\end{equation}
and then we have:
\begin{equation}
\lim_{\beta \rightarrow \infty}\oint\limits_{\Gamma}
\frac {{\cal W}\left[g_l(k_0,\beta), {\phi}_l(k_0,\beta)\right]}
{k_0}\phi(k_0)\;dk_0=0
\end{equation}
As a consequence of (3.18) it results that:
\begin{equation}
\lim_{\beta \rightarrow \infty}
\frac {{\cal W}\left[g_l(k_0,\beta), {\phi}_l(k_0,\beta)\right]}
{k_0}= P(k_0)
\end{equation}
where $P(k_0)$ is an arbitrary polynomial in the variable $k_0$.
Now we have the freedom to select $P(k_0)\equiv 0$, and in this
case (3.9) takes the form:
\begin{equation}
\lim_{\beta \rightarrow \infty} \int\limits_0^{\beta}
{{\phi}_l}^2(k_0,r)\;dr=
\frac {{\dot{f}}_l(k_0) f_l(-k_0)} {4ik_0^{2l+2}}
\end{equation}
where the limit is taken in the sense of ultradistributions.
By definition the pseudonormalized state is:
\begin{equation}
{\psi}_l(k_0,r)={\left[\frac {4i k_0^{2l+2}}
{{\dot{f}}_l(k_0)f_l(k_0)}\right]}^{1/2}{\phi}_l(k_0,r)
\end{equation}
and it can be thought as a tempered ultradistribution in the
variable $k_0$.
\section{The square well potential}
\setcounter{equation}{0}
We start from the Schr\"{o}dinger equation for the radial component
${\cal R}_l(r)$ (ref.\cite{tp6,tp7}):
\begin{equation}
{\cal R}_l^{''}(r)+\frac {2} {r} {\cal R}_l^{'}(r)+\left[q^2-
\frac {l(l+1)} {r^2}\right]{\cal R}_l(r)=0
\end{equation}
($'$ denotes the derivative $d/dr$) and with
\begin{equation}
q^2=\frac {2m} {{\hbar}^2}\left[E-{\cal V}(r)\right]=
k^2-\frac {2m} {{\hbar}^2} {\cal V}(r)
\end{equation}
where
\begin{equation}
{\cal V}(r)=\left\{\begin{array}{ll}
\;0 & for\;\;r>a \\
-{\cal V}_0 & for\;\;r\leq a
\end{array}\right.
\end{equation}
and
\begin{equation}
k^2=\frac {2mE} {{\hbar}^2}
\end{equation}
The regular solution is:
\begin{equation}
{\phi}_l(k,r)=\left\{\begin{array}{ll}
q^{-l}r\;j_l(qr) & for\;\;r<a \\
r\left[A_l\;j_l(kr)+B_l\;n_l(kr)\right]& for\;\;r>a
\end{array}\right.
\end{equation}
where $j_l$ and $n_l$ are,respectively, the spherical Bessel and Newmann
functions. The constants $A_l$ and $B_l$ in (4.5) are:
\[A_l= ka^2q^{-l}\left[k\;j_l(qa)\;n^{'}_l(ka)-q\;j^{'}_l(qa)\;n_l(ka)
\right]\]
\begin{equation}
B_l=ka^2q^{-l}\left[q\;j_l(ka)\;j^{'}_l(qa)-k\;j^{'}_l(ka)\;j_l(qa)
\right]
\end{equation}
The irregular solution $f_l(k,r)$ is given by:
\begin{equation}
f_l(k,r)=\left\{\begin{array}{ll}
r\left[C_l\;j_l(qr)+D_l\;n_l(qr)\right] & for\;\;r<a \\
-ikr\;h_l^{-}(kr) & for\;\;r>a
\end{array}\right.
\end{equation}
where $h_l^{-}=j_l-in_l$ is the spherical Hankel function and the constants
$C_l$ and $D_l$ are given by
\[C_l=-ikqa^2\left[q\;h_l^-(ka)\;n_l^{'}(qa)-k\;h_l^{-'}(ka)\;
n_l(qa)\right]\]
\begin{equation}
D_l=ikqa^2\left[q\;h_l^-(ka)\;j_l^{'}(qa)-k\;h_l^{-'}(ka)\;
j_l(qa)\right]
\end{equation}
Using eqs.(3.4),(4.5) and (4.7) we can evaluate the corresponding Jost
function $f_l(k)$:
\begin{equation}
f_l(k)={\left(\frac {k} {q}\right)}^lika^2\left[k\;j_l(qa)\;h_l^{-'}(ka)
-q\;j_l^{'}(qa)\;h_l^{-}(ka)\right]
\end{equation}
We wish to calculate eq.(3.20) for this example in the case $l=0$.
With this purpose we need the expressions of
$f_0(-k_0)$ and ${\dot{f}}_0(k_0)$. For this pourpose
we take into account that $f_0(k_0)=0$.
From (4.9) we obtain for $l=0$ :
\begin{equation}
f_0(k_0)=e^{-ik_0a} \left(ik_0\frac {\sin q_0a} {q_0}+\cos q_0a\right)=0
\end{equation}
and
\begin{equation}
{\dot{f}}_0(k_0)=i\frac {q_0^2-k_0^2} {q_0^3}e^{-ik_0a}
\left(\sin q_0a-q_0a\cos q_0a\right)
\end{equation}
where
\[q_0^2=k_0^2+ \frac {2m} {{\hbar}^2} {\cal V}_0\]
Therefore we deduce from (4.10) and (4.11) that :
\begin{equation}
f_0(-k_0)=-\frac {2ik_0} {q_0} e^{ik_0a} \sin q_0a
\end{equation}
\begin{equation}
{\dot{f}}_0(k_0)=i\frac {q_0^2-k_0^2} {q_0^3}\left(1+ik_0a\right)
e^{-ik_0a}\sin q_0a
\end{equation}
If we replace eqs.(4.12) and (4.13) into eq.(3.20) we obtain finally:
\begin{equation}
\int\limits_0^{\infty}{\phi}_0^2(k_0,r) dr = \frac {1+ik_0a} {2ik_0}\;
\frac {q_0^2-k_0^2} {q_0^4}\;{\sin}^2 q_0a
\end{equation}
It should be noted that when $k_0$ corresponds to a bound
state the integral (4.14) is real and positive. When $k_0$ corresponds
to a virtual state or a resonant state ($Re k_0\neq 0$,
$Im k_0 > 0$) the integral (4.14) is in general a complex number. It is
not surprising since (4.14) is an analytical extension in the sense of
ultradistributions of the habitual Lebesgue integral. In fact for a
bound state ($k_0=-i{\kappa}_0, {\kappa}_0>0$) we have
\begin{equation}
\int\limits_0^{\infty}{\phi}_0^2(k_0,r) dr = \frac {1+{\kappa}_0a}
{2{\kappa}_0}\;
\frac {q_0^2+{\kappa}_0^2} {q_0^4}\;{\sin}^2 q_0a
\end{equation}
which is the well-known norm of the $l=0$ bound state of the
square well. For the $l=0$ virtual state
($k_0=i{\kappa}_0, {\kappa}_0>0$) we have:
\begin{equation}
\int\limits_0^{\infty}{\phi}_0^2(k_0,r) dr = \frac {{\kappa}_0a-1}
{2{\kappa}_0}\;
\frac {q_0^2+{\kappa}_0^2} {q_0^4}\;{\sin}^2 q_0a
\end{equation}
and in this case the integral is real.
\section{Discussion}
We have shown here that tempered ultradistributions allow to perform a
general treatment of complex-energy states, incorporating
in a natural way bound and continuum states as well as resonant and
virtual states together, within a more general framework of Quantum
Mechanics, based on the Rigged Hilbert Space formulation.
In this work we have applied this formulation to the specific evaluation
of the complex pseudonorm, showing that the results come out in a more
transparent way, since they are free from regularization schemes.
As an example of the goodness of the procedure introduced in this
paper we give the evaluation of the pseudonorm of virtual and
resonant s-states for the square-well potential.
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 513
|
Wayne Jett: TRUMPING THE FED What Will It Mean?
TRUMPING THE FED: What Will It Mean?
A new report has been posted on the site of Classical Capital discussing President Trump's appointment of all but one of the Federal Reserve Board's governors as he proceeds towards what has been called a currency reset, which almost surely means ending the Federal Reserve itself. The Fed's fiat currency is being rejected by much of the world, including BRICS and more than 100 other allied nations. If the U. S. Military is no longer willing to enforce adherence to the dollar, as appears to be the case under President Trump, then America itself must seize the opportunity to return to sound money for the benefit of its people. This is a major element of the Trump Plan, and here we discuss some of the basic considerations of what to expect in coming weeks and months.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,465
|
Q: React Native - modals overlay Modal 1 -- Component 1
|
Component 2
|
Modal 2 -- Component 3
I have a scenario like the 'drawing' above, where component 3 is initialized in component 2 which is initialized in component 1.
Component 1 and 3 each holds a component with a modal.
If component 3's modal is open. The modal from component 1 is hidden behind it if it's opened.
I however want it to be on top of the stack and in the front view.
I tried with ordering things in the code, and i tried with zIndex with position either absolute or relative.
{position: 'relative/absolute', overlay: {zIndex: 99}};
but neither seems to work.
I've looked into similar posts but was unable to find an answer that work's.
How can i force modal 1 to always be visible if other modals are opened from a 'higher' component in the stack
A: You could use Modal component or some other module.
But if you want to create your own modal you can follow this pattern.
in your main component App, have 2 child components
<App>
<Home /> //All components/navigation are here
{
isModalOpen && <Modal />
}
</App>
Set Modal style to something like position:absolute, left: 0, right: 0, top: 0, bottom: 0
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,833
|
Q: How to compare if a timestamptz date equal a timestamptz date in Javascript/Postgresql I got a date in this format when i get my data in my postgresql database like that : 2022-02-20 00:37:11.337937+00
But when i get this from my javascript code i get this result instead : 2022-02-20T00:37:15.279Z
How can i compare this date since 2022-02-20T00:37:15.279Z is cut with a 'Z' at the end.
I try to compare strictly exazctly this date : 2022-02-20 00:37:11.337937+00 in my JS backend code.
EDIT:
In fact, i can have an array of waht i want, but i can't get value message, and others value in my database.
This is the request i have, but i can't get the message property:
SELECT ID, MAX(lastmsg) FROM
( SELECT m.receiver AS ID, MAX(m.created_at) as lastmsg FROM message m
WHERE m.sender = '1'
GROUP BY m.receiver
UNION
SELECT m.sender AS ID, MAX(m.created_at) as lastmsg FROM message m
WHERE m.receiver = '1'
GROUP BY m.sender
) as table2 GROUP BY ID ORDER BY max LIMIT 10 OFFSET 0
I found this solution from another post, but i don't know how to get message field, if i change that i have to add it to the GROUP bY but then my result is not the result expected
I tried something like that, but it doesn't work:
SELECT *
FROM (
SELECT DISTINCT ON (sender, receiver)
sender, receiver, message AS msg, created_at AS created_at
FROM message
WHERE sender = '1' OR receiver = '1'
ORDER BY sender, receiver, created_at
) f
I got duplicated value with that
Edit 2:
I'm running version 12.2 of Postgresql on a Debian
A: You cannot compare the two dates with timestamps. First, you should convert the timestamp into a date format (DD/MM/YY) before executing your query.
You can also use the below method as well.
datetime: {
[Op.gte]: moment(req.body.datetime).startOf('day'),
[Op.lte]: moment(req.body.datetime).endOf('day'),
},
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,516
|
Q: Django bicrypt password different from ruby I have ruby app with mysql database which has password stored.I don't know ruby but from internet what i found is that ruby stores password in bicrypt format.
I created one user with password : Password123.
What i printed in console was its encrypted password.
Tasks:
Now i am creating a django app which needs to be connected to same database.I need to verify password from same database.That is i used bicrypt algorithm in django dummy app and created user with same password : Password123.
But encrypted text from ruby app and django app are different.It needs to be same for verification from django app.
How to do this? Why bicrypt output of both language different.
A: I am going to assume you mean bcrypt. I don't know what bicrypt is. The 'encrypted' password is probably salted to protect against rainbow attacks, which is why they have different 'encrypted' values.
Why do you need the 'encrypted' password to be the same on both systems? Does using 'Password123' on either system not work?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,808
|
At dusk, a child is woken by strange and ethereal music surrounding their home. Running to the front door with their parents, the music can be heard coming from down the end of the street. A shoal of twinkling lights is seen in the distance getting closer. Only as the mass of illuminated bikes pass their house will they realise that the music is coming from speakers attached to dozens of decorated bikes producing the most ambient and serene music.
Lullaby is a gift to a city. A surround sound illuminated artwork delivered to the public's door at dusk.
Last night, this happened. As darkness fell, A was drifting off to sleep with the window open, and Ben and I sat in the living room, working. Suddenly we heard the sound of birdsong and music coming from the top of the street, and raced to the window to watch. Twenty or so bikes, covered in lights, meandered by in the dusky evening, the music getting louder and then quieter again as they passed the house. Ben, outside with his camera, looked up to the bedroom window to see A, a look of wonder on his face, pressed up against the glass.
Bristol: you're so very good at this sort of thing.
Lullaby by Luke Jerram with Sustrans took place on Monday 26 August 2013.
Midnight feasts and pillow fights. Giant sandcastle competitions and stories round the campfire. Fairy houses in the trees and acrobats in the branches. Lantern parades and a wild rumpus. Pirate training and lasso lessons. Sing-a-long Jungle Book and a 70s disco. Gruffalos in the forest and a dancing dragon.
Oh my goodness me, it sounds like something from an Enid Blyton book, doesn't it? (Well, maybe not the 70s disco.) But I'm just describing our weekend at Just So festival. And what a weekend it was. I hope you can see from the photographs just what an incredibly thoughtful and well-put-together event this was; every single tiny detail was just perfect, just so.
The organisers of the festival are dreamers of dreams, able to imagine their way into the head of a child and to create a combination of everything that's magical about being a kid. Six tribes roamed the site, and festival-goers were invited to choose from frogs, lions, fish, foxes, owls, and stags (the best-costumed earning points for their tribe). We hung out with friends old (my bezzie and her sweet daughter) and new (Kat and Heather), and fell in love with a brass band (Perhaps Contraption) and a dazzling dance show (Shiny).
Arthur and Ted had the times of their little lives, loving every second of the fun, the music, the dancing, the stories, the camping, the laughing, the exploring, the eating, the hay fights and the sheer joy of it all. They will not stop talking about it, and already they're asking when we can go back. Real life feels very dull in comparison.
We were kindly invited to the festival by the organisers of Just So, and we're incredibly grateful. Thank you Rowan and Sarah - you are AMAZING!
Likewise, we were each sent a pair of beautiful Hunter boots to take with us - we love them (Hunter was one of the festival sponsors). I chose Original Shorts in aubergine, Ben went for Original Adjustables in navy, Arthur opted for the Glow in the Dark Original Kids' (how cool are they?!), and Ted looked fetching in red Original Kids' Gloss. Thank you Hunter.
You might remember that I went to the Raven and the Writing Desk back in early summer, an 'immersive dining experience' based on Alice in Wonderland. Well, the same clever artists and performers behind this show have come up with another extraordinary event - Jack Rabbit's Derby - which promises an evening of gourmet diner food, '50s dancing, music and theatre at a mystery location in Bristol.
This is just the sort of thing Bristol is brilliant at: interesting, innovative creatives putting on thrilling events. It's the stuff of dreams and just one reason why I love this city.
It's this weekend, and tickets for Saturday and Sunday are still available from the website, and cost £25 which includes all entertainment, a cocktail on arrival and a two-course meal devised by the excellent pop-up, Mi Casa.
To get you in the mood, I've put together a Fifties board full of glorious images from the decade of diners and dancing.
This is Your Kingdom, one of my regular writing haunts, invited its south west contributors for a day out at Yeo Valley. I confess I'd been thinking it'd be all wellies and cow pats, but Yeo Valley is one of the loveliest places I've seen in a very long time; more boutique hotel than barn.
After a demo from head chef, Jaime (we saw butter churned from cream, and a fragrant, basil ice cream being made), we took a tour of their beautiful building before returning to the cafe area for lunch and silly amounts of chat. Then, we headed down to the tea room and garden, both of which are open to the public (go!). As you can imagine from a bloggy meet-up, the sound of camera shutters was loud in the air.
That was on Friday, since when I've eaten a LOT of yoghurt (we were given a very generous goodie bag and recipe book). Yeo've convinced me, Yeo Valley (sorry). When can I come back?
My kids have always been up with the lark. Most of the time it's a drag, but there are a few occasions when it's actually useful. Like today.
6.10am said my clock when I heard those familiar heavy footsteps stomping around in the bedroom next to ours. My usual routine is to have a quick few minutes of email/social media while the kids are happily occupied, and this morning I checked my Twitter account. Bristol Balloon Fiesta takes place this weekend, and at the top of my feed was a tweet saying the mass ascent would be going ahead with a hundred balloons taking to the sky, any minute.
I threw clothes on us all, grabbed a few snacks and a blanket, and hopped in the car to the nearest hilly point. We ate a picnic breakfast while watching the beautiful balloons take to the sky. You always know it's going to be a good day in Bristol when you hear the furry hum of a hot air balloon floating overhead, and we are clearly very spoilt this weekend (as I write these words I can hear the same sound - there's a 6pm mass ascent as well as a 6am red-eye job). I proper loves Bristol.
We were more than a little bit heartbroken to miss our friends' micro-festival last weekend. Ted got sick, then it spread through the house. But by Saturday, we all felt well enough for a day trip, so we headed to Glastonbury.
After a potter about in the magic shops of the town itself and a very lentil-y lunch, we donned wellies and waterproofs and headed up the hill to the Tor. It's a steep old climb to the top, but such an adventure.
As we climbed, the clouds darkened and rumbles of thunder echoed across the valley. The wind picked up and flashes of lightning blazed through the sky but, miraculously, we stayed dry. The children were thrilled by everything, emerging at the top triumphant.
Among the crowds, we found a patch of grass to sit and break open the Fruit Pastilles. We people-watched, our favourite Tor-ist being the goth reading aloud and applying copious amounts of factor 50 to her neck. And we told tales of Avalon; when you're six years old and your name is Arthur, legends of ancient kings are about the most exciting stories in the world.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,523
|
'''
Created on Apr 30, 2015
@author: selly
'''
import ConfigParser
class GreenbaseConfig(object):
'''
Pulls all the configuratoin data for Greenbase TwitterPuller
'''
def __init__(self):
'''
Pull in the config file or set defaults and throw a warning
'''
config = ConfigParser.RawConfigParser()
config.read('greenbase.cfg')
self.mysqlHost = config.get("mysql", "host")
self.mysqlDb = config.get("mysql", "db")
self.mysqlUser = config.get("mysql", "user")
self.mysqlPassword = config.get("mysql", "password")
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,710
|
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/intermediate-algebra-connecting-concepts-through-application\/chapter-5-exponential-functions-chapter-review-exercises-page-476\/18","text":"## Intermediate Algebra: Connecting Concepts through Application\n\nPublished by Brooks Cole\n\n# Chapter 5 - Exponential Functions - Chapter Review Exercises: 18\n\n#### Answer\n\n$x=-5$\n\n#### Work Step by Step\n\nIsolate the exponential term on one side. Then make the base of the right side equal to the base of the left side. Then set the exponents equal to each other. $(\\frac{1}{4})^x=1024$ $(\\frac{1}{4})^x=(\\frac{1}{4})^{-5}$ $x=-5$\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2018-08-15 08:16:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6900449991226196, \"perplexity\": 1265.031688466437}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221209980.13\/warc\/CC-MAIN-20180815063517-20180815083517-00564.warc.gz\"}"}
| null | null |
{"url":"https:\/\/ccssmathanswers.com\/convert-a-fraction-to-an-equivalent-fraction\/","text":"Convert a Fraction to an Equivalent Fraction | How to Turn a Fraction into an Equivalent Fraction with Examples?\n\nDo you want to know how to convert a fraction to an equivalent fraction? You will find all the information regarding Equivalent Fractions such as the Definition, Procedure on How to convert a fraction to an equivalent fraction, etc. You can also check the solved examples of converting a fraction to an equivalent fraction explained step by step so that you can understand them easily.\n\nEquivalent Fraction \u2013 Definition\n\nIf two or more fractions on simplification results in the same value then they are said to be Equivalent Fractions. Consider an example $$\\frac{2} {4}$$, $$\\frac{3} {6}$$. These two on simplification results in the fraction value $$\\frac{1} {2}$$. As both of them have the same fractional value on simplification they are said to be Equivalent Fractions.\n\nHow to Verify if Two Fractions are Equivalent or Not?\n\nTo verify whether the two fractions are equivalent or not, multiply the numerator of one fraction by the denominator of another fraction. Also, multiply the denominator of one fraction by the numerator of another fraction. If the products are the same after the multiplication then the fractions are said to be equivalent.\n\nHow do you Turn a Fraction into an Equivalent Fraction?\n\nFollow the simple measures listed below in order to change a fraction into an equivalent fraction. They are along the lines\n\n\u2022 You can make Equivalent Fractions either by multiplying or Dividing the Numerator and Denominators with the Same Amount.\n\u2022 Doing so, will not alter the value of the fraction. However, we can\u2019t Perform Addition and subtraction as they may alter the value of the fractions.\n\u2022 Simplify as much you can till both the Numerator and Denominator stay Whole Numbers.\n\nHow to Convert a Fraction to an Equivalent Fraction with a Large Denominator?\n\nIf we multiply the numerator and the denominator of the fraction with the same number the value of the fraction doesn\u2019t change and we get the equivalent fraction.\n\nExample 1:\nConsider the fraction $$\\frac { 1 }{ 4 }$$ and find the equivalent fraction.\nSolution:\nMultiply with 2 to get equivalent fraction.\n$$\\frac { 1\u00d72 }{ 4\u00d72 }$$ =$$\\frac { 2 }{ 8 }$$\nThe equivalent fraction of $$\\frac { 1 }{ 4 }$$ is $$\\frac { 2 }{ 8 }$$\n\nExample 2:\nconsider the fraction $$\\frac { 3 }{ 8 }$$\nSolution:\nMultiply with 5 to get an equivalent fraction.\n$$\\frac { 3\u00d75 }{ 8\u00d75 }$$=$$\\frac { 15 }{ 40 }$$\nThe equivalent fraction of $$\\frac { 3 }{ 8 }$$ is $$\\frac { 15 }{ 40 }$$\n\nHow to Convert a fraction to an equivalent fraction with the smaller denominator\n\nIf the numerator and the denominator of the fraction are divided by the same number then the value of the fraction doesn\u2019t change and we will get an equivalent fraction.\n\nExamples on Converting a Fraction to an Equivalent Fraction with the Smaller Denominator\n\nExample 1:\nConsider the fraction $$\\frac { 15 }{ 75 }$$ find the equivalent fraction.\nSolution:\nDivide with 5 to get the equivalent fraction.\n$$\\frac { 15\u00f75 }{ 75\u00f75 }$$=$$\\frac { 3 }{ 15 }$$\nThe equivalent fraction of $$\\frac { 15 }{ 75 }$$ is $$\\frac { 3 }{ 15 }$$.\n\nExample 2:\nConsider the fraction $$\\frac { 10 }{ 28 }$$ find the equivalent fraction.\nSolution:\nDivide with 2 to get the equivalent fraction.\n$$\\frac { 10\u00f72 }{ 28\u00f72}$$=$$\\frac { 5 }{ 14 }$$\nThe equivalent fraction of $$\\frac { 10 }{ 28 }$$ is $$\\frac { 5 }{ 14 }$$.\n\nScroll to Top","date":"2022-09-24 20:17:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8881055116653442, \"perplexity\": 335.25047000225226}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030333455.97\/warc\/CC-MAIN-20220924182740-20220924212740-00062.warc.gz\"}"}
| null | null |
Scala for OpenShift - Docker images
========================================
This repository contains the source for building various versions of
the Scala application as a reproducible Docker image using
[source-to-image](https://github.com/openshift/source-to-image).
The resulting image can be run using [Docker](http://docker.io).
Versions
---------------
Scala versions currently provided are:
* SBT 0.13
Java versions currently provided are:
* Java 1.8.0
CentOS versions currently supported are:
* CentOS 7
Installation
---------------
* **CentOS based image**
To build a Scala image from scratch run:
```
$ git clone https://github.com/ticketfly/sti-scala.git
$ cd sti-scala/sbt-0.13-java-8
$ make build
```
Usage
---------------------
To build a simple [Scala-sample-app](https://github.com/pat2man/play-originv3-test) application
using standalone [STI](https://github.com/openshift/source-to-image) and then run the
resulting image with [Docker](http://docker.io) execute:
* **For CentOS based image**
```
$ sti build https://github.com/ticketfly/sti-scala.git --context-dir=sbt-0.13-java-8/test/test-app/ ticketfly/scala-0.13-java-8-centos7 scala-test-app
$ docker run -p 9000:9000 scala-test-app
```
**Accessing the application:**
```
$ curl 127.0.0.1:9000
```
Test
---------------------
This repository also provides a [S2I](https://github.com/openshift/source-to-image) test framework,
which launches tests to check functionality of a simple Scala application built on top of the sti-Scala image.
Users can choose between testing a Scala test application based on a RHEL or CentOS image.
* **CentOS based image**
```
$ cd sti-scala/sbt-0.13-java-8
$ make test
```
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,299
|
Q: how can i make an iframe (reddit embed post) responsive with width 100% and auto height I am trying to display some reddit posts (as an iframe) in my website.
I want to make them responsive. with 100% width and height that will display all the content without overflow scroll.
when i'm trying to do so - the width is working fine but the height is shorten then what it should. and the post gets cutoff.
See this jsfiddle i created:
https://jsfiddle.net/qr1tzo3g/1/
html:
<div id="container">
<div id="body">
<iframe id="reddit-embed" src="https://www.redditmedia.com/r/WhitePeopleTwitter/comments/ymji0q/oooooffff/?ref_source=embed&ref=share&embed=true" sandbox="allow-scripts allow-same-origin allow-popups" style="border: none; width:100%; height:auto" scrolling="yes"></iframe>
<iframe id="reddit-embed" src="https://www.redditmedia.com/r/funny/comments/ymkbo8/mountain_lion_wins_his_favourite_toy_in_the_shell/?ref_source=embed&ref=share&embed=true" sandbox="allow-scripts allow-same-origin allow-popups" style="border: none; width:100%; height:auto" scrolling="yes"></iframe>
</div>
</div>
css:
#container{
display: flex;
justify-content: center;
width: 100%;
}
#body{
width: 100%;
max-width: 700px;
display: flex;
justify-content: center;
flex-direction: column;
align-items: center;
border: 1px solid black;
}
Any ideas how to solve this issue?
Thanks you!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,173
|
{"url":"https:\/\/blogs.helsinki.fi\/kulikov\/category\/mathematics\/wednesday-problem\/","text":"# Category Archives: Wednesday Problem\n\n## (Ir)rational Behavior of Calculation\n\nWe had the following set of exercises with a group of secondary school students (in Ressun lukio). Assume that a and b are positive real numbers. Then assume one of the following: (i) a and b are both rational (ii) \u2026 Continue reading\n\nPosted in Algebra, Calculus, Education, Mathematics, Wednesday Problem | 1 Comment\n\n## The Product of Topological Spaces Does Not Obey Cancellation\n\nExercise 1. Find metric topological spaces $$A,B,C$$ such that $$A$$ is not homeomorphic to $$B$$, but $$A\\times C$$ is homeomorphic to $$B\\times C$$. Exercise 2. Find path connected metric topological spaces $$A,B,C$$ such that $$A$$ is not homeomorphic to $$B$$, \u2026 Continue reading\n\nPosted in Mathematics, Topology, Wednesday Problem | Leave a comment\n\n## Class-metric\n\nIn two weeks I start lecturing Topology I at our University. I am excited. This course was my favourite among the basic undergraduate courses. The course concentrates on metric topology and its goal is to prove simple results about complete \u2026 Continue reading\n\n## Time Limit: One Minute!\n\nI found this lovely geometrical riddle in Martin Gardner\u2019s book \u201cMy Best Mathematical and Logic Puzzles\u201d. The most lovely thing about it is that Gardner gives a time limit: one minute! So, now draw out your stopwatch and read the \u2026 Continue reading\n\n## Walks on Planets\n\nMy gps-equipped radiowave controllable robot Lori is somewhere on the surface of Earth. I order her to drive 100 miles South and she obeys. Then I order her to drive 100 miles West and she obeys. Then I order her \u2026 Continue reading\n\n## Prisoners\u2019 problem 6\n\nThree prisoners meet a guard. The guard says: I have hats with me, two of which are black and the others are white. The prisoners ask: \u201cHow many hats there are all together?\u201d He says: \u201cIt\u2019s a secret!\u201d. The guard \u2026 Continue reading\n\nPosted in Combinatorics, Recreation, Wednesday Problem | Tagged | 2 Comments\n\n## Splitting A Rectangle\n\nFind all ways to cut a rectangle into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different dimensions (side lengths) than the orgininal one. I already know countably many ways. \u2026 Continue reading\n\nPosted in Geometry, Mathematics, Recreation, Wednesday Problem | 2 Comments\n\n## Sphere\n\nPick randomly three points on the unit sphere. The geodesic triangle with these points as vertices divides the sphere into two parts. What is the probability that the north pole is contained in the smaller of these parts? For instance \u2026 Continue reading\n\nPosted in Geometry, Mathematics, Probability, Wednesday Problem | Leave a comment\n\n## A Year Problem\n\nThere are more year problems than years. But since I have been pondering on this particular one, I will present it here. You are allowed to use +, -, \/ and * (plus, minus, division and multiplication) signs and bracketing. \u2026 Continue reading\n\n## More Chessboard\n\nYou are free to comment your solutions, questions and remarks.. You have two kinds of allowed moves. One move is to jump with a piece over another piece in a horizontal or vertical direction like this: and to jump with \u2026 Continue reading\n\n| Tagged | 3 Comments\n\n## Incomplete Chess Board\n\nPlease comment your solutions, questions and remarks.. I learned this puzzle from Juha Oikkonen, but it is probably quite famous anyway. You have left your chess board on the table in the summer house and when you came back you \u2026 Continue reading\n\n| Tagged | 2 Comments\n\n## Poisoned Bunnies\n\nPlease comment your solutions, questions and remarks.. Imagine that you have one thousand bottles in front of you. You know that one of them is filled with poison and others with water, but you do not know which one is \u2026 Continue reading\n\n## Irrational Rectangles\n\nPlease comment your solutions, questions and remarks.. This riddle is not easy. I solved it only after I got a hint. I got the hint without wanting it though, and I was already thinking in the right direction; so with \u2026 Continue reading\n\nPosted in Geometry, Mathematics, Wednesday Problem | 9 Comments\n\n## Ambient Isotopy\n\nNote: One is able to do puzzles 2 and 3 without reading or understanding the text before them. In knot theory, people define equivalence of knots using the concept of ambient isotopy. Two knots (embeddings of the unit circle into \u2026 Continue reading\n\n## Euler\u2019s Riddle\n\nPlease comment your solutions, questions and remarks.. I found in a recreational book the following riddle attributed to Leonard Euler. The book gives a clumsy solution involving solutions of quadratic equations, a lot of fractions and half a page of \u2026 Continue reading\n\nPosted in Algebra, Mathematics, Recreation, Wednesday Problem | 2 Comments\n\n## How To Find Your Way Out of Woods\n\nPlease comment your solutions, questions and remarks.. A lumberjack got lost in a forest. He knows that the area of the forest is S and that there are no meadows in the forest. Show that he can get out from \u2026 Continue reading\n\nPosted in Geometry, Mathematics, Recreation, Wednesday Problem | 3 Comments\n\n## If You Beg For Money, You Might End Up Doing Math\n\nYesterday I was walking around with my friend in a market hall. Two girls of age 15 asked us whether we could give them 40 euro cents. It is about \\$0.6. We asked them what could we gain in response. \u2026 Continue reading\n\nPosted in Education, Meta, Recreation, Wednesday Problem | 3 Comments\n\n## Prisoners\u2019 problem 4\n\nThe rules are as in the previous problem, but this time there are only 100 prisoners. How should they do so that 50 of them certainly survives? 3\/5\n\n## Prisoner\u2019s problem 3\n\nPlease comment your solutions, questions and remarks.. There are infinitely many prisoners in a prison. The prisoners are taken to the yard and everyone is given a hat which is either black or white. No one knows the colour of \u2026 Continue reading\n\n| Tagged | 2 Comments\n\n## Prisoner\u2019s problem 2\n\nPlease comment your solutions, questions and remarks.. Consider a prison with several millions prisoners. The head of the prison arranges the following challenge to the prisoners. One prisoner at a time will be called to a room with a lamp. \u2026 Continue reading\n\n| Tagged | 2 Comments\n\n## Prisoner\u2019s problem 1\n\nPlease comment your solutions, questions and remarks.. We are going to have several prisoner problems! Better solve them before you end up in jail. There are 2010 prisoners in the prison. One day the chair of the prison gathers the \u2026 Continue reading\n\n| Tagged | 5 Comments\n\n## How Many Good Colorings?\n\nPlease comment your solutions, questions and remarks.. Suppose G is a graph in which each vertex has three neighbours, for example: Suppose we have three colors, red green and blue, with which we want to color the edges of the \u2026 Continue reading\n\nPosted in Algebra, Combinatorics, Mathematics, Wednesday Problem | Leave a comment\n\n## Nasty Finite Combinatorics\n\nPlease comment your solutions, questions and remarks.. I was blamed for having too easy Wednesday problems. So beware! The level is coming up! Suppose that K is a set of n elements, $$n\\in\\mathbb{N}$$. Suppose that $$K_1,\\dots,K_n,K_{n+1}$$ are subsets of K. \u2026 Continue reading\n\n## Back To College\n\nPlease comment your solutions, questions and remarks.. Show that the following inequality holds for all $$k>0$$ and $$x\\in \\mathbb{R}$$: $$e^x>kx-k\\ln k.$$ Level 1\/5\n\nPosted in Calculus, Mathematics, Wednesday Problem | 2 Comments\n\n## How Many Are True?\n\nPlease comment your solutions, questions and remarks.. At most 1 statement of this post is true. At most 2 statements of this post are true. At most 3 statements of this post are true. At most 4 statements of this \u2026 Continue reading\n\nPosted in Logic, Mathematics, Recreation, Wednesday Problem | 1 Comment\n\n## Intermediate Value Functions\n\nPlease comment your solutions, questions and remarks.. A continuous function $$f\\colon\\mathbb{R}\\to\\mathbb{R}$$ on the real numbers has the following property: Given any two reals a<b, the function f gets all possible values between f(a) and f(b) on the interval [a,b] (i.e. \u2026 Continue reading\n\nPosted in Calculus, Mathematics, Wednesday Problem | 1 Comment\n\n## Sorry I Am Late\n\nPlease comment your solutions, questions and remarks.. How many of the statements below the line are true? 2\/5 \u2014\u2014\u2014\u2014 This post was not published on Wednesday 24. February 2010 There are at least 2 true statements in this post (after \u2026 Continue reading\n\nPosted in Logic, Recreation, Wednesday Problem | 3 Comments\n\n## More Tricks With Triangulations\n\nPlease comment your solutions, questions and remarks.. I learned this riddle from Sergei Chmutov. It is also a problem concerning triangulation and sharp angled triangles. Suppose we have a triangulated regular polygon with odd number of edges. The vertices of \u2026 Continue reading\n\nPosted in Geometry, Mathematics, Recreation, Wednesday Problem | Leave a comment\n\n## Triangulation\n\nPlease comment your solutions, questions and remarks.. In the last week\u2019s many in one post I explained what is a triangulation. Can you triangulate a square using only sharp angled triangles (i.e. triangles whose all angles are <$$\\pi\/2$$)? For example \u2026 Continue reading\n\nPosted in Geometry, Mathematics, Recreation, Wednesday Problem | Leave a comment\n\n## Brouwer\u2019s Fixed Point Theorem: Many in One Post\n\nIn this post I will (1) give a simple proof of Brouwer Fixed Point Theorem (2) fulfill the promise given here (3) present the Wednesday Problem in the form fill in the details in the below text Theorem (Brouwer\u2019s Fixed \u2026 Continue reading\n\n| | 2 Comments\n\n## The Coin Placing Game.\n\nSee the Wednesday Problem\u2019s vague rules here. Some games again. According to Sam, from whom I heard this riddle, it divides people into cathegories i) those who realize the answer immidiately and ii) those who think quite long about it. \u2026 Continue reading\n\nPosted in Games, Recreation, Wednesday Problem | 6 Comments\n\n## How Old Are The Kids?\n\nPlease comment your solutions, questions and remarks.. This is quite famous. But maybe you haven\u2019t heard it yet: A math student goes to a party organized by her supervisor. The student asks: How many daughters do you have? And the \u2026 Continue reading\n\nPosted in Combinatorics, Recreation, Wednesday Problem | 2 Comments\n\n## The Monk\n\nPlease comment your solutions, questions and remarks.. The posts have been a bit advanced lately. Let us lighten the atmosphere by this riddle which admits a simple solution, though mathematicians tend to use calculus in solving it: An Indian monk \u2026 Continue reading\n\nPosted in Calculus, Recreation, Wednesday Problem | 2 Comments\n\nPlease comment your solutions, questions and remarks.. Maximal Almost Disjoint families. This is not so much of a riddle than just a theorem, but the solution is fun, so I would like to place it here. This is like a \u2026 Continue reading\n\n## Happy Transition 2009-2010!\n\nPlease comment your solutions, questions and remarks.. Which one is bigger: $$\\sqrt{2009 + \\sqrt{2010}} + \\sqrt{2010 + \\sqrt{2009}}$$ or $$\\sqrt{2009 + \\sqrt{2009}} + \\sqrt{2010 + \\sqrt{2010}}$$? (If you use a calculator, show that it does not lie) 2\/5 \u2026 Continue reading\n\n## Joulupukki Is Fair: Your Christmas Riddle\n\nPlease comment your solutions, questions and remarks.. Joulupukki came to a kindergarten. He had some number of candies to give to the children. He saw that there are more boys than girls and that he could divide the candies evenly \u2026 Continue reading\n\nPosted in Mathematics, Recreation, Wednesday Problem | Leave a comment\n\n## Handshaking Lemma\n\nPlease comment your solutions, questions and remarks.. This one I learned from Sam Hardwick. Show that the number of those, who have shaken their hands with others an odd number of times, is even. Level 1\/5 P.S. This lemma has \u2026 Continue reading\n\nPosted in Combinatorics, Recreation, Wednesday Problem | 2 Comments\n\n## Ants\n\nPlease comment your solutions, questions and remarks.. This funny riddle I heard from Marcin Sabok. There are n ants on the unit interval, [0,1]. Each has a direction, left or right. In the picture above there are three ants and \u2026 Continue reading\n\nPosted in Mathematics, Recreation, Wednesday Problem | Leave a comment\n\n## The Chocolate Game\n\nPlease comment your solutions, questions and remarks.. I heard this from Lauri Hella, but he doesn\u2019t remember from whom he heard this. The game is played between two players as follows. There is an $$n\\times m$$ bar of chocolate on \u2026 Continue reading","date":"2021-09-18 20:10:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.365201473236084, \"perplexity\": 2013.5659914088458}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056572.96\/warc\/CC-MAIN-20210918184640-20210918214640-00491.warc.gz\"}"}
| null | null |
Trapiche peut faire référence à :
Trapiche, commune Malaga, en Espagne ;
Trapiche, commune de San Luis, en Argentine ;
Trapiche, caractéristique d'une gemme ;
Trapiche, vin argentin ;
Trapiche, moulin pour la canne à sucre ou l'extraction du minerai.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,320
|
We're just back from a nine-day family vacation. Nine days. And my teenagers actually wanted to go.
While the trip started out a bit rocky. Rocky, as in I vomited all the way from the TSA line at National Airport through a layover in Milwaukee and onward to San Francisco. Rocky, as in the flight attendant on the second flight asked Andrew if I was a nervous flyer. He said, "No, she's just sick." Bet that made her day. And rocky, as in I missed the first 36 hours of our San Fran visit.
But Andrew and the boys kept on ticking. I had a great and comfortable place to rest (go VRBO!) and they saw the Golden Gate Bridge and Sausalito (which they fondly call Sausagelito). And they saw the SF Museum of Modern Art. And rode cable cars. And such.
I totally rallied for Alcatraz.
It was so interesting and a beautiful day. We were so happy to be together. It was, for me, the start of a wonderful vacation.
Did you know that there are still three Alcatraz prisoners alive? All are still in prison somewhere. Andrew is really into the show Alcatraz. I think it's just creepy. But, I digress.
Dinner that night? So fun. We went to Range. And yes, my sons were the only non-adults there. But no one seemed to mind. They're as tall as the adults, anyway. And they know their way around a nice restaurant. It was all good.
The next 4 days were incredible. Monterey. Yosemite. (We stayed in a Caboose and had fabulous Cajun food. Not kidding.) Sequoia.
And Death Valley. Very fun "resort" in Death Valley. We met the nicest people there. And honestly, I think that I slept better there than any other night. It was just so peaceful and we were so darn happy. After dinner, there was nothing to do. I mean nothing. No Internet. No TV. No cell service.
We bought a 40 at the convenience store (for the adults, obviously) and some snacks for all. And we played poker. All of us in one room, laughing and playing cards. So fun.
And the next morning, we headed into the park and saw the sites. We hiked. We took pix. (Okay, that was me.) and we went to the lowest place in North America.
After a full and exhausting day, we headed east to Las Vegas.
There are a million details I left out. Like the Artichoke Capital of the World. Like the funny Mexican restaurant in Laughlin. Or Laughlin, in general.
I can't express how happy I am that we had a week of laughs. A week of talking. A week of experiencing. A week I'll never forget.
My boys are growing up so fast. Davis will be applying to colleges soon. And as of next fall, they'll all be in high school. No more middle school. No more elementary school. No more preschool.
< If you're wondering where we are.
I loved this post. So nice of you to share you amazing vacation adventures with us. And, uhm…so good to have you back!
Thanks, Candace. Glad to be back, but I sure enjoyed the time off!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,021
|
Q: Using library installed in one virtual environment in another virtual environment Can we use library installed in one virtual environment from another virtual environment?
For eg:
Library installed in venv from another virtual environment.
A: Short answer: No
Virtual environment's purpose is to isolate different python instalation, including their libraries. If you ever find a way to do this, it must be considered a hack and should be reported to virtual environment as a bug.
A: It should be possible, you just have to import the module by giving the full path to the module present in the source virtualenv. How to import a module given the full path?
But it'll NOT WORK as dependencies to the module will not be present in your target virtualenv. You can try importing all dependencies by using full path it'll work that way.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 108
|
Oddział pułkownika Czerniecowa (ros. Отряд полковника Чернецова) - ochotniczy oddział partyzancki Białych podczas wojny domowej w Rosji
W listopadzie 1917 r. antybolszewickie oddziały wojskowe nad Donem organizował ataman Kozaków dońskich gen. Aleksiej M. Kaledin. 30 listopada w Nowoczerkasku oddział partyzancki utworzył esauł Wasilij M. Czerniecow. Składał się z kadetów, gimnazjalistów, seminarzystów i uczniów szkół realnych. Początkowo stanowił jedyną realną siłę bojową gen. A. M. Kaledina. Po wyjściu z miasta wyruszył w kierunku północnym na Woroneż. Działał tam półtora miesiąca, po czym powrócił na terytorium obwodu dońskiego. W tym czasie oddział rozrósł się liczebnie. W jego skład wchodził 50-osobowy pluton pieszy, pododdział karabinów maszynowych z 2 karabinami maszynowymi, pododdział saperów i telefonistów oraz pluton junkierskiej baterii (2 działa) ppłk. Dmitrija T. Mionczinskiego przydzielony z 1 Samodzielnego Lekkiego Dywozjonu Artylerii. 15 stycznia 1918 r. partyzanci esauła W. M. Czerniecowa otrzymali rozkaz rozgonienia zjazdu bolszewickiego, jaki odbywał się w stanicy Kamienskaja. Wkrótce zajęli stację kolejową Zwieriewo, a następnie stanicę Lichaja, stanowiącą ważny węzeł kolejowy. Oddziały bolszewickie zaatakowały Zwieriewo, próbując odgrodzić czerniecowców od Nowoczerkaska. W celu obrony stacji kolejowej został skierowany pododdział oficerski, zaś główne siły esauła W. M. Czerniecowa zajęły pozycje obronne w stanicy Lichaja. Były to sotnia por. Kuroczkina, sotnia esauła Bryłkina i sotnia sztabsrtm. Inoziemcewa. Esauł W. M. Czerniecow postanowił zająć stanicę Kamienskaja, leżącą na północ od Lichoj, aby odciągnąć oddziały bolszewickie. W połowie drogi doszło do starcia z przeciwnikiem. Oddziały Kozaków dońskich, znajdujące się w składzie wojsk bolszewickich, powstrzymały się od walki, co doprowadziło bolszewików do odwrotu. Następnego dnia partyzanci bez walki zajęli stanicę Kamienskaja. Ochotnicy utworzyli kolejną sotnię, zaś b. oficerowie kozaccy sformowali drużynę. Rozkazem atamana gen. A. M. Kaledina esauł W. M. Czerniecow został awansowany do stopnia pułkownika. Kolejny bój miał miejsce po drodze do stanicy Głubokaja. Bolszewicy zaatakowali partyzantów od tył, ale zostali rozgromieni. Stracili ponad 100 zabitych, głównie z 3 Moskiewskiego Pułku Czerwonych. Partyzanci zdobyli wagon kolejowy z pociskami artyleryjskimi i 12 karabinami maszynowymi. Ranny został por. Kuroczkin, pełniący faktycznie funkcję zastępcy płk. W. M. Czerniecowa. Po dojściu pod Głubokoj 21 stycznia doszło do walki, w wyniku której partyzanci zostali zmuszeni do odwrotu w kierunku Kamienskoj. Dopadła ich jednak kozacka konnica starsziny wojskowego Gołubowa, rozbijąjąc oddział płk. W. M. Czerniecowa. Plutonowi artylerii ppłk. D. T. Mionczinskiego udało się przedostać do stanicy Kamienskaja. Ranny w nogi płk. W. M. Czerniecow dostał się do niewoli. Podczas przenoszenia go do stanicy Głubokaja zaatakowały ocalałe resztki oddziału partyzanckiego, próbując bezskutecznie odbić swojego dowódcę. W dalszej drodze płk W. M. Czerniecow próbował zastrzelić dowódcę konwojującego go oddziału kozackiego, ale sam został zabity. 9 lutego ocaleli partyzanci weszli w skład Pułku Partyzanckiego Armii Ochotniczej.
Niektóre epizody z walk oddziału partyzanckiego płk. W. M. Czerniecowa zostały przedstawione w książce Michaiła A. Szołochowa pt. "Cichy Don".
Linki zewnętrzne
Działania oddziału partyzanckiego płk. Wasilija M. Czerniecowa (jęz. rosyjski)
Bibliografia
Anton I. Denikin, Очерки Русской смуты, 2006
W. J. Szambarow, Белогвардейщина, 2007
Partyzanckie formacje Białej Armii
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,190
|
\section*{Conclusions}
In this work, we have developed an algorithmic approach to construct dynamical systems with prescribed statistical properties. We demonstrate that our approach can be used to design chaotic dynamical systems that reproduce paintings and photographs. Akin to a human painter, the dynamical system first captures the large scale features and then fills in the finer details. The given picture is decomposed into its color components, thus yielding distributions of red, green, and blue (or equivalently cyan, magenta, yellow, and key) colors. The algorithm then constructs a separate dynamical system for each color that optimally samples the corresponding color distribution. In a robotic system, these dynamical systems will provide the instructions for each paintbrush with associated colors. These dynamical systems statistically sample the prescribed distributions, consequently, the results are independent of initial conditions. The resulting equations for chaotic sampling are shown (in the supplementary material) to have three positive Lyapunov exponents, implying sensitive dependence to initial conditions, a key property of chaotic systems~\cite{Cit:Strogatz}. The paintings are the ``attractors'' for this dynamical system.
Additionally, in our supplementary material, we investigate the utility of the chaotic sampling approach for machine learning and Bayesian inference in big data settings. In particular, we demonstrate significant gains (in accuracy and convergence time) over traditional MCMC~\cite{Cit:MCMC3} methods. We compare chaotic sampling to slice sampling, Hamiltonian MCMC, and Metropolis-Hastings on a multi-modal test example in two dimensions. Our approach has the advantage over Metropolis-Hastings that it does not require the construction of proposal distributions. The construction of these distributions can be challenging. Moreover, given that the problem of designing systems with prescribed statistical properties arises in numerous applications such as 3-D printing~\cite{Cit:3dprinting}, biological systems~\cite{cit:DNA,Cit:DNA2} and microfluidics~\cite{cit:Microfluidics}, we anticipate our approach will also be valuable in these scenarios.
\begin{figure}[htb!]
\begin{center}
\subfigure[Time = $0.016$ sec]{\includegraphics[width=0.51\textwidth]{figures/Mona_Lisa_Fast16_v2.png}}
\subfigure[Time = $0.046$ sec]{\includegraphics[width=0.51\textwidth]{figures/Mona_Lisa_Fast46_v2.png}}
\subfigure[Time = $0.076$ sec]{\includegraphics[width=0.51\textwidth]{figures/Mona_Lisa_Fast76_v2.png}}
\subfigure[Time = $0.106$ sec]{\includegraphics[width=0.51\textwidth]{figures/Mona_Lisa_Fast106_v2.png}}
\subfigure[Time = $0.151$ sec]{\includegraphics[width=0.55\textwidth]{figures/Mona_Lisa_Fast151_v2.png}}
\end{center}
\vspace*{-0.33in}
\caption{Evolving reproduction of the Mona Lisa as recreated by chaotic sampling. The first frame is the superposition of the red, green, and blue frames. Note that the red, green, and blue frames are composed of a single trajectory for each color evolving over time.}
\label{Fig:ergodicmonalisa}
\end{figure}
\begin{figure}[htb!]
\begin{center}
\subfigure[Original Big Sur Photograph]{\includegraphics[width=0.4\textwidth]{figures/BigSurCoast.jpg}}\hspace{0.5in}
\subfigure[Big Sur reproduced using chaotic sampling]{\includegraphics[width=0.4\textwidth]{figures/BigSur_Simulation.png}}\\
\subfigure[Original Pines Switz Photograph]{\includegraphics[width=0.4\textwidth]{figures/Pines_Switz.jpg}}\hspace{0.5in}
\subfigure[Pines Switz reproduced using chaotic sampling]{\includegraphics[width=0.4\textwidth]{figures/Pines_Switz_Simulation.png}}\\
\subfigure[Original Starry Night by Van Gogh]{\includegraphics[width=0.4\textwidth]{figures/VanGoghStarryNight.jpg}}\hspace{0.5in}
\subfigure[Starry Night reproduced using chaotic sampling]{\includegraphics[width=0.4\textwidth]{figures/StarryNight.png}}\\
\end{center}
\caption{Various pictures and paintings generated by chaotic sampling.}
\label{Fig:ergodicpaintings}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,155
|
\section{Introduction}
The ability to perform an arbitrary operation on a quantum system is a crucial prerequisite for advanced quantum information processing and quantum computing \petr{\cite{Braun05}}. In optical implementations, quantum states of light are manipulated mainly with passive and active linear optical elements such as beam splitters and squeezers. The resulting state transformations preserve the Gaussian form of the Wigner function and are thus referred to as Gaussian operations. It is readily apparent that such operations alone are not sufficient for universal continuous-variable (CV) quantum computation \cite{Lloyd99,Bartlett02a,Bartlett02b} and must be supplemented by access to some other resources such as nonlinear dynamics \cite{Lloyd99}, single-photon detectors \cite{Knill01,Gu09}, or non-Gaussian states \cite{Gottesman01,Bartlett03}. While several schemes for generation of highly nonclassical states of light and implementation of various non-Gaussian operations have been suggested \cite{CKW,Fiurasek03,Fiurasek05}, a systematic study of usefulness of non-Gaussian states for universal quantum state manipulation and engineering has been missing.
\petr{In the present paper we focus on implementation of quantum gates using off-line generated ancilla states $|\psi\rangle$ and Gaussian measurements and operations \cite{Bartlett03,Filip05,Yoshikawa08}. The ancilla states represent the only non-Gaussian ingredient and can thus be seen as a resource that is converted into a non-Gaussian CV quantum gate. It is our aim to investigate what non-Gaussian ancilla states are sufficient for realization of arbitrary CV quantum gate within this approach. We shall prove that arbitrary pure single-mode non-Gaussian state $|\psi\rangle$ possessing finite expansion in Fock-state basis is sufficient for (probabilistic) implementation of any $n$-mode quantum gate on Hilbert space $\mathcal{H}_N^{\otimes n }$, where $\mathcal{H}_N$ is spanned by the first $N+1$ Fock states and both $N$ and $n$ are finite but otherwise arbitrary. The formulation in terms of truncated finite-dimensional Hilbert spaces is necessary in order to ensure that a scheme with finite number of components can be constructed that (conditionally) implements the requested gate. }
The core of our argument is the reduction of the problem to generation of single-photon Fock states $|1\rangle$ from the resource state $|\psi\rangle$. We provide explicit scheme for this latter task and assess its performance. For the sake of presentation clarity we explain the protocol on the example of traveling light modes, but the scheme is applicable also to other physical platforms such as atomic ensembles
or optomechanical systems.
\begin{figure}[!b!]
\centerline{\psfig{figure=fig1.eps,width=0.85\linewidth}}
\caption{(color online) (a) Setup for projective measurement on a single-photon state. D - Homodyne detectors, BS - balanced beam splitter.
(b) Setup for approximate photon subtraction. D - homodyne detector, BS - balanced beam splitter, $t_{a,b}$ - beam splitter with with transmittance $t_{a,b}$, $D(\xi)$ - displacement driven by detected value $\xi$.} \label{fig_singlephotondet}
\end{figure}
\section{Sufficiency of single-photon states}
We start by demonstrating that only single-photon states, apart from Gaussian operations and measurements, are required for probabilistic implementation of arbitrary quantum operation on $\mathcal{H}_N^{\otimes n}$.
A crucial observation is that the projection on a single-photon state can be performed with help of an ancillary single-photon state, a balanced beam splitter and a pair of homodyne detectors
measuring amplitude quadrature $x_1$ and phase quadrature $p_2$, respectively, c.f. Fig.~\ref{fig_singlephotondet}(a). Successful projection is heralded by outcomes $x_1=0$ and $p_2=0$. In this case, the two input modes impinging on the balanced beam splitter are projected on the maximally entangled
EPR state $|\Psi_{\mathrm{EPR}}\rangle=\sum_{n=0}^{\infty} |n,n\rangle$. This in conjunction with the ancillary single-photon state implements the probabilistic projection on a single-photon state,
$_{12}\langle \Psi_{\mathrm{EPR}}| 1\rangle_1=\, _2\langle 1|.$
To achieve a nonzero success probability,
a finite acceptance window for the measurement outcomes $x_1$ and $p_2$ has to be introduced, which reduces the fidelity of the projection and leads to trade-off between operation quality and its success probability. This is an unavoidable feature of our protocol arising from involvement of only Gaussian measurements.
Single-photon states and single-photon measurements combined with Gaussian operations are sufficient for probabilistic preparation of arbitrary multimode quantum state \cite{Fiurasek03} and implementation of arbitrary transformation on $\mathcal{H}_N^{\otimes n}$, e.g. by exploiting the scheme described in Ref. \cite{CKW} or simply by quantum teleportation \cite{Gottesman99}.
The whole question about nature of non-Gaussian resources sufficient for universal quantum state manipulation is thereby reduced to finding a class of states from which a single-photon state can be generated with help of only Gaussian operations and measurements. We are going to show that any collection of non-Gaussian pure states possessing finite expansion in the Fock-state basis is sufficient for this.
\section{Generalized photon subtraction}
Let us consider a steady supply of states of the form
\begin{equation}
|\psi_N\rangle = \sum_{k=0}^{N} c_k|k\rangle.
\label{psiN}
\end{equation}
An essential ingredient of our protocol is the setup depicted in Fig.~\ref{fig_singlephotondet}(b)
which employs one auxiliary state $|\psi_N\rangle$ and Gaussian operations
to remove the highest Fock state $|M\rangle$ from the input state $|\chi_M\rangle=\sum_{m=0}^M b_m |m\rangle$. This produces a state
$|\chi_{M-1}\rangle=\sum_{m=0}^{M-1} b_m' |m\rangle$ and this operation can be thus seen as a version of approximative photon subtraction.
First part of the process lies
in a deterministic transformation of $|\psi_N\rangle$ into a state
\begin{equation}\label{1step02}
|\phi_{\overline{0}}\rangle = \sum_{k=1}^{\infty} d_k |k\rangle,\quad \sum_{k=1}^{\infty}|d_k|^2 = 1,
\end{equation}
with $d_1 \neq 0$ and missing vacuum term, $d_0=0$. This can be achieved by coherent displacement of the state $|\psi_N\rangle$ if the displacement amplitude $\alpha$ satisfies
\begin{equation}
\langle 0|D(\alpha)|\psi_N\rangle =e^{-|\alpha|^2/2} \sum_{k=0}^{N} c_k \frac{(-\alpha^\ast)^k}{\sqrt{k!}} = 0.
\label{alpharoot}
\end{equation}
Such $\alpha$ exists for all finite $N$.
However, for a particular set of scenarios, \emph{e.g.} when $|\psi_N\rangle = |N\rangle$, this approach does not work as the required displacement is $\alpha = 0$ corresponding to no action
at all and the scheme in Fig.~\ref{fig_singlephotondet}(b) would produce vacuum state from input $|\psi_N\rangle$. This problem can be fortunately circumvented using an ancillary vacuum mode, a beam splitter, a single homodyne detection and feed-forward, see Fig.~\ref{fig_singlephotondet}(b). After passing through the beam splitter with transmittance $t_b$, the homodyne detection of the amplitude quadrature $\hat{x}_3$ yielding a value $x$, and the displacement $\alpha$, the state $|N\rangle$ transforms into
\begin{equation}
| \phi\rangle= D(\alpha) \sum_{k=0}^N \sqrt{N\choose k}
t_b^k r_b^{N-k}\langle x |N-k\rangle |k\rangle,
\end{equation}
where $r_j = \sqrt{1-t_j^2}$ for any $j$. By employing the relation for an overlap of a quadrature eigenstate and a Fock state,
\begin{equation}\label{fockXoverlap}
\langle x|n\rangle = \frac{H_n(x)}{\pi^{1/4}\sqrt{n!2^n}} e^{-x^2/2},
\end{equation}
where $H_n(x)$ stands for the Hermite polynomial, we can see that to arrive at the form (\ref{1step02}) with $d_0=0$ and $d_1 \neq 0$ for an arbitrary measured value $x$, the real displacement $\alpha$ must satisfy
\begin{equation}
H_N(\tilde{x}) = 0, \qquad
N t_a \sqrt{2} H_{N-1} (\tilde{x}) \neq \alpha r_a
H_N(\tilde{x}),
\end{equation}
where $\tilde{x}=x-\alpha t_a/(\sqrt{2}r_a)$.
The first condition can be for all values of $x$ fulfilled by a suitable choice of $\alpha$, while the second condition is in these cases satisfied automatically, as Hermite polynomials of unequal orders have different roots.
To summarize, the universal setup for deterministic generation of a state (\ref{1step02}) from a completely arbitrary state $|\psi_N\rangle$ consists of a beam splitter, homodyne detection, and a suitable displacement operation, where the specific values of parameters have to be adjusted according to the state employed.
Also note that the displacement operation could be replaced by a suitable post-selection - allowing only states for which no displacement is necessary and discarding the rest. Thus, experimental feasibility can be gained at the cost of a reduced success rate.
To perform the approximate photon subtraction on the input state $|\chi_M\rangle$, this state in mode 1 is combined with vacuum in mode 2 on a beam splitter with transmittance $t_a$
yielding a two-mode entangled state at the output. A balanced beam splitter and a pair of homodyne detectors are then used to project the mode $2$ and the mode $4$ prepared in
auxiliary state $|\phi_{\overline{0}}\rangle$ onto the EPR state $|\Psi_{\mathrm{EPR}}\rangle$, c.f Fig.~1(b).
This conditionally prepares the remaining output mode $1$ in the state $|\chi_{M-1}\rangle=\sum_{m=0}^{M-1} b_m' |m\rangle$, where
\begin{equation}
b_m' = \sum_{k=m+1}^M d_{k-m} b_k \sqrt{ {k \choose m}}
t_a^m r_a^{k-m}.
\end{equation}
\begin{figure}[!t!]
\centerline{\psfig{figure=fig2.eps,width=0.6\linewidth}}
\caption{(color online) Complete setup for generation of a single-photon state.} \label{fig_total}
\end{figure}
\section{Preparation of single-photon state}
The complete scheme for preparation of single-photon state from $N$ copies of state $|\psi_N\rangle$
is shown in Fig.~\ref{fig_total}. By repeated application of the approximate photon subtraction we can transform any state $|\psi_N\rangle$ to a state
\begin{equation}
|\psi_1\rangle = a_0|0\rangle + a_1|1\rangle,
\end{equation}
with $|a_1|>0$. The parameters $a_0$ and $a_1$ can be made real by a suitable phase shift. This state is then combined with vacuum on a beam splitter with transmittance $t$, \petr{after which a homodyne detection of the amplitude quadrature $x$ of one output mode is performed, projecting the state onto}
\begin{equation}\label{laststep}
|\psi_{\mathrm{out}}\rangle \propto ( a_0 + \sqrt{2} x r a_1) |0\rangle + ta_1|1\rangle .
\end{equation}
If we postselect the events when the measurement outcome is $x = -a_0/(\sqrt{2}r a_1)$, we
remove the vacuum term by destructive quantum interference and obtain the desired single-photon state.
\begin{figure}
\centerline{\psfig{figure=fig3.eps,width=0.6\linewidth}}
\caption{(color online) Complete setup for generation of a single photon state from a pair of two photon states. $t_a$, $t_b$ and $t_c$ denote transmittances of the respective beam splitters, while $BS$ stands for a balanced beam splitter. Numbers 1 to 5 are used to label the modes involved.} \label{fig_twophoton}
\end{figure}
As a demonstration, let us now explicitly show the procedure to create a single-photon state from a pair of two-photon states $|2\rangle$. The full scheme is presented in Fig.~\ref{fig_twophoton}.
It can be easily shown that to generate the single-photon state the feed-forward displacement $\alpha$ should read
\begin{equation}\label{alpha}
\alpha = \frac{r_a}{t_a}(x_3 \sqrt{2}-1),
\end{equation}
where $x_3$ represents a value obtained by the homodyne measurement of the amplitude quadrature $x_3$ of mode 3. The other three homodyne detectors measure amplitude quadratures $x_2$ and $x_5$ of modes $2$ and $5$, respectively, and phase quadrature $p_4$ of mode $4$. Successful preparation of state $|1\rangle$ is heralded by the measurement outcomes
\begin{equation}
x_2 = 0, \quad p_4 = 0, \quad
x_5 = - r_b \frac{t_a^2 + 2(x_3\sqrt{2} -1)r_a^2}{t_a r_a t_b r_c 2\sqrt{2}}.
\label{x5formula}
\end{equation}
\begin{figure}[!t!]
\centerline{\psfig{figure=fig4.eps,width=0.8\linewidth}}
\caption{(color online) Fidelity (left, blue solid line) and probability of success (right, green dashed line) of the preparation of the single photon state from a pair of two photon states with respect to the post-selection threshold $X$. } \label{fig_twophotonreal}
\end{figure}
In real experimental practice we cannot condition on the projection on a single quadrature eigenstate $|x = \xi\rangle$, as this corresponds to an event with zero probability of success. Instead, we have to accept all events when the measured value falls within a narrow interval centered at $\xi$, thus realizing a POVM element
\begin{equation}\label{homodyne}
\Pi_{k,x=\xi} = \int_{-X}^{X}|x = \xi+q\rangle_k\langle x = \xi + q| dq,
\end{equation}
where the parameter $X$ determines the half-width of the post-selection interval and $k$ labels the mode that is measured. This of course effects the output state. The global input state encompassing five modes, as can be seen in Fig.~\ref{fig_twophoton}, can be expressed as
\begin{equation}
|\psi_{\mathrm{in}}\rangle = |2\rangle_1|0\rangle_2|0\rangle_3|2\rangle_4|0\rangle_5.
\end{equation}
After after interactions on all beam splitters and the feed-forward loop the output state for a single particular measured value $x_3$ reads
\begin{equation}
|\psi_{\mathrm{out}}(x_3)\rangle = U_{c,15}U_{\mathrm{BS},24}U_{b,12}D_4(\alpha)\langle x_3|_3U_{a,34}|\psi_{\mathrm{in}}\rangle.
\end{equation}
Here, $U_{j,kl}$ represents a unitary transformation of a beam splitter $j = a,b,c,\mathrm{BS}$ coupling a pair of modes $k,l$, $D_4(\alpha)$ represents the displacement performed on mode 4 and $\alpha$ is given by Eq. (\ref{alpha}). The final state is given by
\begin{eqnarray}\label{rhofin}
\rho_{\mathrm{1}}(x_3) &=& \frac{\mathrm{Tr}_{2345}[ \Pi_{2,x=0} \Pi_{4,p=0}\Pi_{5,x=x_5}|\psi_{\mathrm{out}}\rangle\langle\psi_{\mathrm{out}}|]}{P_S(x_3)},\nonumber
\end{eqnarray}
where $x_5$ is given by Eq. (\ref{x5formula}) and we have avoided to explicitly mark the dependence of $|\psi_{\mathrm{out}}\rangle$ on the value $x_3$ for the sake of brevity. $\mathrm{Tr}_{2345}$ stands for the partial trace over all modes other than mode 1 and $P_S(x_3)$ denotes the probability of success
\begin{equation}
P_S(x_3) = \mathrm{Tr}[ \Pi_{2,x=0}\Pi_{4,p=0}\Pi_{5,x=x_5}|\psi_{\mathrm{out}}\rangle\langle\psi_{\mathrm{out}}|].
\end{equation}
This, however, still corresponds only to the scenario when a particular value $x_3$ was detected. To obtain the final result, we need to average
the state (\ref{rhofin}) over all possible experimental outcomes, arriving at
\begin{equation}\label{rhofin2}
\rho_{1} = \frac{1}{P_S}\int_{-\infty}^{\infty} P_S(x_3) \rho_{1}(x_3) d x_3,
\end{equation}
with a probability of success
$ P_S = \int_{-\infty}^{\infty} P_S(x_3) d x_3.$
Figure~\ref{fig_twophotonreal} shows the performance of the procedure with respect to homodyne detection with nonzero threshold $X$. As the measure of quality we employ the fidelity, $F = \langle 1|\rho_{1}|1\rangle$, which in this case reliably quantifies the content of the single-photon state in the overall mixture. The transmittances of the beam splitters were optimized as to maximize the probability of success $P_S$ in the limit of very narrow acceptance windows ($X\rightarrow 0$): $t_a = 0.62$, $t_b = 0.79$ and $t_c = 0.90$. The trade-off between fidelity and the success probability
is clearly visible in Fig.~\ref{fig_twophotonreal}.
\begin{figure}[!t!]
\centerline{\psfig{figure=fig5.eps,width=0.60\linewidth}}
\caption{(color online) Complete setup for implementation of the nonlinear sign gate using only Gaussian operations and two photon states as a resource. $t_a$ and $t_b$ transmittances of the respective beam splitters, while $BS$ stands for a balanced beam splitter. } \label{fig_kerrtwophotonscheme}
\end{figure}
\section{An example: nonlinear SIGN gate}
Finally we are going to present a full implementation of a non-Gaussian operation using only Gaussian operations and measurements and ancillary states $|\psi_N\rangle$. The resource states are again going to be the two-photon states $|2\rangle$ from which the single-photon states are extracted by means of procedure depicted in Fig.~\ref{fig_twophoton}. The non-Gaussian operation under consideration is the nonlinear sign gate \cite{Knill01}
which transforms a generic state $|\psi_{\mathrm{in}}\rangle = c_0 |0\rangle + c_1|1\rangle + c_2|2\rangle$ into $|\psi_{\mathrm{out}}\rangle = c_0|0\rangle + c_1|1\rangle - c_2|2\rangle$. This represents a unitary evolution induced by a Kerr-type Hamiltonian $\hat{H}=\frac{\pi}{2}\hat{n}(\hat{n}-1)$ on a three-dimensional Hilbert space spanned by $|0\rangle,|1\rangle,|2\rangle$.
A celebrated result in linear-optics quantum computing is that this gate
can be implemented with help of only beam splitters, one ancillary single-photon state, and two measurements, one projecting on a single-photon state, the other on the vacuum state \cite{NSG}, see Fig.~\ref{fig_kerrtwophotonscheme}. The single-photon state projection can be performed with help of a scheme in Fig.~1(a) while the projection on the vacuum state is a Gaussian operation. The transmittances of the beam splitter must satisfy $t_a^2 = (3-\sqrt{2})/7 \approx 0.23$ and $t_b = t_a/(1-2t_a^2) \approx 0.87$ \cite{NSG}.
The performance of the gate can be evaluated by using the quantum process fidelity. Consider a maximally entangled state on the Hilbert space $\mathcal{H}_{N=2}^{\otimes 2}$, $|\Phi_{012}\rangle = (|00\rangle + |11\rangle + |22\rangle)/\sqrt{3}$. Applying the nonlinear sign gate on one of the modes transforms the state into
$ |\Phi'_{012}\rangle = (|00\rangle + |11\rangle - |22\rangle)/\sqrt{3}.$
With the help of this state the gate could be applied by means of teleportation to an arbitrary unknown state \cite{Gottesman99}. In this sense, the measure of quality of the state $|\Phi'_{012}\rangle$ can serve as a tool to evaluate the quality of operation.
Using similar calculations as before, we can determine the mixed two-mode state $\rho_{012}$ produced by the scheme and the success probability of the scheme for finite acceptance windows on homodyne detections. The fidelity of the operation can now be expressed as
$F = \langle \Phi'_{012}|\rho_{012}|\Phi'_{012}\rangle.$
Figure~\ref{fig_kerrtwophoton} shows the resulting relations between the fidelity, the post-selection threshold $X$, and the probability of success.
\begin{figure}[!t!]
\centerline{\psfig{figure=fig6.eps,width=0.8\linewidth}}
\caption{(color online) Fidelity (left, blue solid line) and probability of success (right, green dashed line) for implementation of the nonlinear sign gate with respect to the post-selection threshold $X$. } \label{fig_kerrtwophoton}
\end{figure}
\section{Conclusions}
In summary, we have demonstrated that a steady supply of pure non-Gaussian states possessing finite expansion in the Fock-state basis, together with the experimentally readily accessible Gaussian operations and Gaussian measurements, is sufficient for universal quantum state manipulation and engineering. \petr{The required ancilla non-Gaussian states could be generated e.g. using squeezing operations, coherent displacements and conditional single-photon subtraction \cite{Fiurasek05}. The conditional photon subtraction can be performed reliably with avalanche photo-diode detectors even though their overall detection efficiency is of the order of 50\% or even lower. The low efficincy only reduces the success probability of the state-preparation scheme but not the fidelity of the prepared state \cite{Fiurasek05}. In contrast, such detectors are unsuitable for direct implementation of measurement induced non-Gaussian operations using the schemes proposed in Refs. \cite{CKW,Fiurasek03} becuase the low efficiency would imply reduced fidelity of the gate. In our approach we thus replace direct single-photon detection by an indirect detection relying on off-line produced non-Gaussian states and homodyne detection. In this way it is possible to achieve high fidelity at the expense of probabilistic nature of the scheme.}
Our generic scheme involves several optimization possibilities and its efficiency can be improved by tuning the transmittances of beam splitters and the widths of the acceptance windows of homodyne measurements. Moreover, it is likely that for each particular task the efficiency can be improved significantly by using specific dedicated scheme tailored to a given resource state $|\psi_N\rangle.$
\petr{Besides states with finite Fock-state expansion also other classes of states could be sufficient for universal CV quantum gate engineering. However, dealing with completely generic states in infinite-dimensional Hilbert space of the quantized electromagnetic field is extremly difficult due to the \emph{a-priori} infinite number of parameters. It is unlikely that the question of sufficiency of a given state for universal CV quantum gate engineering could be decided in a completely general way. Instead, partial ad-hoc solutions could be provided for certain finite-parametric classes of states (e.g. the cubic phase state proposed in Ref. \cite{Gottesman01}). Identifying such potentially useful classes of states is an interesting open problem which, however, is beyond the scope of our present work.}
Our findings shall find applications in advanced optical quantum information processing and quantum state engineering. On more fundamental side, our results shed more light on the quantum information processing power of non-Gaussian states and they help to bridge the gap between single-photon and continuous-variable approaches.
\begin{acknowledgments}
This work was supported by MSMT under projects LC06007, MSM6198959213, and 7E08028,
and also by the EU under the FET-Open project COMPAS (212008).
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,128
|
Q: Annotations Not Appearing With Core Data I'm creating an app where annotations are created by a long gesture tap on a map and saved in Core Data. I've checked out this resource, but I'm still having trouble with the annotations appearing on the map. Things were fine in Swift 2 for me, not sure what has changed and what I need to do to fix since it looks right to me. Thanks! Here is my code:
import UIKit
import MapKit
import CoreData
class MapViewController: UIViewController, MKMapViewDelegate {
//Variables
var locationManager = CLLocationManager()
var currentPins = [Pin]()
var gestureBegin: Bool = false
var sharedContext: NSManagedObjectContext {
return CoreDataStack.sharedInstance().managedObjectContext
}
//Fetch All Pins
func fetchAllPins() -> [Pin] {
let fetchRequest = NSFetchRequest<NSFetchRequestResult>(entityName: "Pin")
do {
return try sharedContext.fetch(fetchRequest) as! [Pin]
} catch {
print("Error In Fetch!")
return [Pin]()
}
}
//Outlets
@IBOutlet weak var mapView: MKMapView!
//Core Dat
override func viewDidLoad() {
super.viewDidLoad()
let longPressRecognizer = UILongPressGestureRecognizer(target: self, action: #selector(MapViewController.longPressGesture(longPress:)))
longPressRecognizer.minimumPressDuration = 1.5
mapView.addGestureRecognizer(longPressRecognizer)
self.mapView.delegate = self
addSavedPinsToMap()
}
//Add Saved Pin To Map
func addSavedPinsToMap() {
currentPins = fetchAllPins()
print("Pins Count in Core Data Is \(currentPins.count)")
for currentPin in currentPins {
let annotation = MKPointAnnotation()
annotation.coordinate = currentPin.coordinate
mapView.addAnnotation(annotation)
}
}
//Long Press Gesture Recognizer
func longPressGesture(longPress: UIGestureRecognizer) {
//Take Point
let touchPoint = longPress.location(in: self.mapView)
//Convert Point To Coordinate From View
let touchMapCoordinate = mapView.convert(touchPoint, toCoordinateFrom: mapView)
//Init Annotation
let annotation = MKPointAnnotation()
annotation.coordinate = touchMapCoordinate
//Init New Pin
let newPin = Pin(latitude: annotation.coordinate.latitude, longitude: annotation.coordinate.longitude, context: sharedContext)
//Save To Core Data
CoreDataStack.sharedInstance().saveContext()
//Adding New Pin To Pins Array
currentPins.append(newPin)
//Add New Pin To Map
mapView.addAnnotation(annotation)
}
} // End Class
Here is my Core Data file:
import Foundation
import CoreData
import MapKit
//Pin Class
public class Pin: NSManagedObject {
var coordinate: CLLocationCoordinate2D {
return CLLocationCoordinate2D(latitude: latitude, longitude: longitutde)
}
convenience init(latitude: Double, longitude: Double, context: NSManagedObjectContext) {
if let ent = NSEntityDescription.entity(forEntityName: "Pin", in: context) {
self.init(entity: ent, insertInto: context)
self.latitude = latitude
self.longitutde = longitude
} else {
fatalError("Unable To Find Entity Name!")
}
}
}
And the other Core Data file:
import Foundation
import CoreData
extension Pin {
@nonobjc public class func fetchRequest() -> NSFetchRequest<Pin> {
return NSFetchRequest<Pin>(entityName: "Pin")
}
@NSManaged public var latitude: Double
@NSManaged public var longitutde: Double
}
A: Figured out that I didn't have my tap gesture setup properly within my VC. Once I made sure the outlet was connected to the VC, all worked fine.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,540
|
module ApplicationHelper
def current_user
@current_user ||= User.find_by(id: session[:user_id])
end
def logged_in?
return true if session[:user_id] != nil
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,348
|
Business opens temporary pop-up shop after storefront damage from truck crash
Wednesday, November 13, 2019 6:05 PM EST
A Larchmont business has opened a temporary pop-up shop as it waits for construction to be completed to its damaged storefront.
Write On! Westchester says it will move to Boston Post Road while its storefront undergoes renovations.
Nearly six months ago, a truck slammed into the Palmer Avenue storefront, affecting multiple businesses. Video from the incident in June shows the front of the red Ford F-150 inside the stores after it jumped the curb.
Police say no drugs or alcohol were involved, and did not give any new information regarding the driver or the cause of the crash.
One of the storefronts affected by the crash - Elli Travel Group -permanently moved to 235 Larchmont Ave.
Write On! Westchester says it is not letting ongoing construction put a damper on sales. That's why the stationary store is relocating temporarily at Bread and Cocoa on Boston Post Road with a pop-up shop.
"If the landlords can't supply some of the local people who don't have their storefronts with a pop-up for a month or two then let me. Just let somebody come in — and she's not taking up any space you know, just adding to the atmosphere," says Laurie Vanderwoude, Bread and Cocoa owner.
Vanderwoude's help couldn't have come at a better time.
"People are busy doing holiday cards. I mean they like to order now before Thanksgiving so they can send it out so they can arrive by Christmas," says Write On! Westchester owner Susie Yamaguchi.
Building management for the Palmer Avenue stores says construction is expected to be completed just in time for warmer weather in early spring.
Breaking news and headlines in your inbox. Sign up for News 12 email alerts!
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,863
|
<input type="text" placeholder="email" name="username" ng-model="form.username"><br>
<input type="text" placeholder="password" name="password" ng-model="form.password"><br>
<button name="login" id="login" ng-click="login()">login</button>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,831
|
\section{Introduction}
\paragraph{\textbf{Motivations}}
Derived algebraic geometry is a far-reaching enhancement of classical algebraic geometry.
We refer to Toën-Vezzosi \cite{HAG-I,HAG-II} and Lurie \cite{Lurie_Thesis,DAG-V} for foundational works.
The prototypical idea of derived algebraic geometry originated from intersection theory:
Let $X$ be a smooth complex projective variety.
Let $Y, Z$ be two smooth closed subvarieties of complementary dimension.
We want to compute their intersection number.
When $Y$ and $Z$ intersect transversally, it suffices to count the number of points in the set-theoretic intersection $Y\cap Z$.
When $Y$ and $Z$ intersect non-transversally, we have two solutions:
The first solution is to perturb $Y$ and $Z$ into transverse intersection;
the second solution is to compute the Euler characteristic of the derived tensor product $\mathcal O_Y\otimes^\mathrm L_{\mathcal O_X}\mathcal O_Z$ of the structure sheaves.
The second solution can be seen as doing perturbation in a more conceptual and algebraic way.
It suggests us to consider spaces with a structure sheaf of derived rings instead of ordinary rings.
This is one main idea of derived algebraic geometry.
Besides intersection theory, motivations for derived algebraic geometry also come from deformation theory, cotangent complexes, moduli problems, virtual fundamental classes, homotopy theory, etc.\ (see Toën \cite{Toen_Derived_2014} for an excellent introduction).
All these motivations apply not only to algebraic geometry, but also to analytic geometry.
Therefore, a theory of derived analytic geometry is as desirable as derived algebraic geometry.
The \emph{purpose} of this paper is to define a notion of derived space in non-archimedean analytic geometry and then study their basic properties.
A non-archimedean field is a field with a complete nontrivial non-archimedean absolute value.
By non-archimedean analytic geometry, we mean the theory of analytic geometry over a non-archimedean field $k$, initiated by Tate \cite{Tate_Rigid_1971}, then systematically developed by Raynaud \cite{Raynaud_Geometrie_analytique_rigide_1974}, Berkovich \cite{Berkovich_Spectral_1990,Berkovich_Etale_1993}, Huber \cite{Huber_Generalization_1994,Huber_Etale_1996} and other mathematicians with different levels of generalizations.
The survey \cite{Conrad_Several_approaches_2008} by Conrad gives a friendly overview of the subject.
We will restrict to the category of quasi-paracompact\footnote{A rigid $k$-analytic\xspace space is called quasi-paracompact if it has an admissible affinoid covering of finite type.} quasi-separated rigid $k$-analytic spaces, which is the common intersection of the various approaches to non-archimedean analytic geometry mentioned above.
For readers more familiar with Berkovich spaces, we remark that this category is equivalent to the category of paracompact strictly $k$-analytic spaces in the sense of Berkovich (cf.\ \cite[\S 1.6]{Berkovich_Etale_1993}).
A more direct motivation of our study comes from mirror symmetry.
Mirror symmetry is a conjectural duality between Calabi-Yau manifolds (cf.\ \cite{Yau_Essays_1992,Voisin_Symetrie_1996,Cox_Mirror_symmetry_1999,Hori_Mirror_symmetry_2003}).
More precisely, mirror symmetry concerns degenerating families of Calabi-Yau manifolds instead of individual manifolds.
An algebraic family of Calabi-Yau manifolds over a punctured disc gives rise naturally to a non-archimedean analytic space over the field $\mathbb C(\!( t)\!)$ of formal Laurent series.
In \cite[\S 3.3]{Kontsevich_Homological_2001}, Kontsevich and Soibelman suggested that the theory of Berkovich spaces may shed new light on the study of mirror symmetry.
Progresses along this direction are made by Kontsevich-Soibelman \cite{Kontsevich_Affine_2006} and by Tony Yue Yu \cite{Yu_Balancing_2013,Yu_Gromov_2014,Yu_Tropicalization_2014,Yu_Enumeration_cylinders_2015,Yu_Enumeration_cylinders_II_2016}.
The works by Gross, Hacking, Keel, Siebert \cite{Gross_Real_Affine_2011,Gross_Mirror_Log_published,Gross_Tropical_2011} are in the same spirit.
More specifically, in \cite{Yu_Enumeration_cylinders_2015}, a new geometric invariant is constructed for log Calabi-Yau surfaces, via the enumeration of holomorphic cylinders in non-archimedean geometry.
These invariants are essential to the reconstruction problem in mirror symmetry.
In order to go beyond the case of log Calabi-Yau surfaces, a general theory of virtual fundamental classes in non-archimedean geometry must be developed.
The situation here resembles very much the intersection theory discussed above, because moduli spaces in enumerative geometry can often be represented locally as intersections of smooth subvarieties in smooth ambient spaces.
The virtual fundamental class is then supposed to be the set-theoretic intersection after perturbation into transverse situations.
However, perturbations do not necessarily exist in analytic geometry.
Consequently, we need a more general and more robust way of constructing the virtual fundamental class, whence the need for derived non-archimedean geometry.
\bigskip
\paragraph{\textbf{Basic ideas and main results}}
Our previous discussion on intersection numbers suggests the following definition of a derived scheme:
\begin{defin}[cf.\ \cite{Toen_Derived_2014}]\label{def:derived_scheme}
A \emph{derived scheme} is a pair $(X,\mathcal O_X)$ consisting of a topological space $X$ and a sheaf $\mathcal O_X$ of commutative simplicial rings on $X$, satisfying the following conditions:
\begin{enumerate}[(i)]
\item The ringed space $(X,\pi_0(\mathcal O_X))$ is a scheme.
\item For each $j\ge 0$, the sheaf $\pi_j(\mathcal O_X)$ is a quasi-coherent sheaf of $\pi_0(\mathcal O_X)$-modules.
\end{enumerate}
\end{defin}
In order to adapt \cref{def:derived_scheme} to analytic geometry, we need to impose certain analytic structures on the sheaf $\mathcal O_X$.
For example, we would like to have a notion of norm on the sections of $\mathcal O_X$;
moreover, we would like to be able to compose the sections of $\mathcal O_X$ with convergent power series.
A practical way to organize such analytic structures is to use the notions of pregeometry and structured topos introduced by Lurie \cite{DAG-V}.
We will review these notions in \cref{sec:definitions} (see also the introduction of \cite{Porta_DCAGI} for an expository account of these ideas).
We will define a pregeometry $\cT_{\mathrm{an}}(k)$ which will help us encode the theory of non-archimedean geometry responsible for our purposes.
After that, we are able to introduce our main object of study: derived $k$-analytic\xspace spaces.
It is a pair $(\mathcal X,\mathcal O_\mathcal X)$ consisting of an $\infty$-topos\xspace $\mathcal X$ and a $\cT_{\mathrm{an}}(k)$-structure $\mathcal O_\mathcal X$, satisfying analogs of \cref{def:derived_scheme} Conditions (i)-(ii).
We will explain more intuitions in \cref{rem:definition_intuition}.
The goal of this paper is to study the basic properties of derived $k$-analytic\xspace spaces and to compare them with ordinary $k$-analytic\xspace spaces.
Here are our main results:
\begin{thm}[cf.\ \cref{thm:fully_faithfulness}]
The category of quasi-paracompact quasi-separated rigid $k$-analytic\xspace spaces embeds fully faithfully into the $\infty$-category\xspace of derived $k$-analytic\xspace spaces.
\end{thm}
\begin{thm}[cf.\ \cref{thm:fiber_products}]
The $\infty$-category\xspace of derived $k$-analytic\xspace spaces admits fiber products.
\end{thm}
Let $(\mathrm{An}_k,\tau_\mathrm{\acute{e}t})$ denote the étale site of rigid $k$-analytic\xspace spaces (cf.\ \cite[\S 8.2]{Fresnel_Rigid_2004}) and let $\mathbf P_\mathrm{\acute{e}t}$ denote the class of étale morphisms.
The triple $(\mathrm{An}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})$ constitutes a geometric context in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
The associated geometric stacks are called \emph{higher $k$-analytic\xspace Deligne-Mumford stacks}.
\begin{thm}[cf.\ \cref{cor:underived_higher_kanal_stacks}]
The $\infty$-category\xspace of higher $k$-analytic\xspace Deligne-Mumford stacks embeds fully faithfully into the $\infty$-category\xspace of derived $k$-analytic\xspace spaces.
The essential image of this embedding is spanned by $n$-localic discrete derived $k$-analytic\xspace spaces.
\end{thm}
\bigskip
\paragraph{\textbf{Outline of the paper}}
In \cref{sec:definitions}, we introduce the pregeometry $\cT_{\mathrm{an}}(k)$ and the notion of derived $k$-analytic\xspace space.
In \cref{sec:pregeometry}, we study the properties of the pregeometry $\cT_{\mathrm{an}}(k)$.
We prove the unramifiedness conditions as well as the compatibility with truncations.
In \cref{sec:fullyfaithfulness}, we construct a functor $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ from the category of $k$-analytic\xspace spaces to the $\infty$-category\xspace of derived $k$-analytic\xspace spaces.
We prove that $\Phi$ is a fully faithful embedding.
In \cref{sec:closed_etale}, we study closed immersions and étale morphisms under the embedding $\Phi$.
In \cref{sec:fiber_products}, we prove the existence of fiber products between derived $k$-analytic\xspace spaces.
In \cref{sec:essential_image}, we characterize the essential image of the embedding $\Phi$.
Moreover, we compare derived $k$-analytic\xspace spaces with higher $k$-analytic\xspace stacks in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
\ifpersonal
\bigskip
\paragraph{\textbf{Personal note of outline: }}
In \cref{sec:pregeometry}, we prove the unramifiedness of $\cT_{\mathrm{an}}(k)$ (\cref{cor:Tkan_unramified}) and the unramifiedness of the morphism $\cT_{\mathrm{disc}}(k)\to\cT_{\mathrm{an}}(k)$ (\cref{prop:unramified_transformation}) following \cite[\S 4]{DAG-IX}.
For the first, we first show that a closed immersion of $k$-analytic\xspace spaces induces a closed immersion of $\infty$-topoi\xspace.
\cref{lem:descent_for_closed_subtopoi} is a gluing lemma which allows us to reason only for affinoid spaces.
\cref{lem:alg_conservative} and \cref{prop:alg_effective_epi} are two auxiliary results concerning the morphism $\cT_{\mathrm{disc}}(k)\to\cT_{\mathrm{an}}(k)$.
\cref{prop:closed_fiber_products_Top} is a corollary of unramifiedness, which shows the interest of the definition of unramifiedness. That is the property of unramifiedness which will be used later.
Unramifiedness of transformation of pregeometries implies that pullback of structured topoi along such morphism preserves closed immersions and pullbacks along closed immersions.
The main result of \cref{sec:fullyfaithfulness} is \cref{thm:fully_faithfulness}.
We define the functor $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ as follows.
First we define the functor $\Phi$ on objects by \cref{lem:inclusion}.
In order to define $\Phi$ on morphisms, we need to show that the mapping spaces are discrete (\cref{prop:discrete_mapping_spaces_I}), so we do not need to worry about higher homotopies.
In order to prove \cref{prop:discrete_mapping_spaces_I},
we construct an auxiliary functor $\Upsilon\colon\mathrm{An}_k\to\mathrm{LRT}$ and prove that it is fully faithful (\cref{lem:first_fully_faithful}).
Then \cref{prop:discrete_mapping_spaces_I} follows from \cref{lem:alg_faithful} and \cref{lem:alg_homotopy_monomorphism}.
Now the proof of \cref{thm:fully_faithfulness} is done as follows.
The faithfullness of $\Phi$ is easy, which is the second paragraph of the proof of \cref{thm:fully_faithfulness}.
The fullness is proved in the following way.
By construction, $\Upsilon$ is $\Phi$ composed with truncation and algebraization.
Given a morphism between $\varphi\colon\Phi(X)\to\Phi(Y)$, first we apply algebraization.
Then we use \cref{lem:first_fully_faithful} to obtain a morphism $f\colon X\to Y$, which induces another morphism $\Phi(f)\colon\Phi(X)\to\Phi(Y)$, whose algebraization equals that of $\varphi$.
\cref{lem:alg_faithful} says that for 0-truncated topoi, algebraization is faithful.
Therefore, $f$ is what we want.
\cref{sec:fiber_products} shows the existence of fiber products.
\cref{prop:closed_fiber_products_dAn} shows the existence of fiber products along a closed immersion.
It is analog of \cite[Proposition 12.10]{DAG-IX}.
Lurie deduces (v) from (iv).
We will first prove (v) and then deduce (iv).
In order to deduce (iv) from (v), we need \cref{lem:sheaves_coherent_modules}, which is analog of \cite[12.11]{DAG-IX}.
\cref{lem:closed_devissage} shows that a derived $k$-analytic\xspace space can locally be embedded into non-derived smooth $k$-analytic\xspace spaces. It is analog of \cite[12.13]{DAG-IX}.
\cref{lem:products_dAn} shows the existence of products over a point.
It is analog of \cite[12.14]{DAG-IX}.
Finally, we are able to deduce \cref{thm:fiber_products} as in \cite[12.12]{DAG-IX}.
Lurie's proof in the complex analytic case is a bit easier because the underlying topological space of the fiber product of complex analytic spaces is just the fiber product of topological spaces.
\cref{sec:essential_image}.
\cite[12.8]{DAG-IX} cannot literally hold in $k$-analytic\xspace case.
Because the category of $k$-analytic\xspace spaces is not closed under étale equivalence relations (cf.\ \cite{Conrad_Non-archimedean_analytification_2009}).
Moreover, $k$-analytic\xspace spaces gives rise to $1$-localic topoi and not to $0$-localic ones.
So we present a different statement and a different proof here.
In fact we didn't understand Lurie's proof of \cite{Conrad_Non-archimedean_analytification_2009}, which involves loop spaces.
\fi
\bigskip
\paragraph{\bfseries Notations and terminology}
We refer to Bosch-Güntzer-Remmert \cite{Bosch_Non-archimedean_1984} and Fresnel-van der Put \cite{Fresnel_Rigid_2004} for the classical theory of non-archimedean analytic geometry, to Lurie \cite{HTT,Lurie_Higher_algebra} for the theory of $\infty$-categories\xspace, and to Lurie \cite{DAG-V} for the theory of structured spaces.
Throughout the paper, by $k$-analytic\xspace spaces, we mean quasi-paracompact quasi-separated rigid $k$-analytic\xspace spaces.
We denote by $\mathrm{Set}$ the category of sets and by $\mathcal S$ the $\infty$-category\xspace of spaces.
For any small $\infty$-category\xspace $\mathcal C$ equipped with a Grothendieck topology $\tau$ and any presentable $\infty$-category\xspace $\mathcal D$, we denote by $\mathrm{PSh}_\mathcal D(\mathcal C)$ the $\infty$-category\xspace of $\mathcal D$-valued presheaves on $\mathcal C$ and by $\mathrm{Sh}_\mathcal D(\mathcal C,\tau)$ the $\infty$-category\xspace of $\mathcal D$-valued sheaves on the $\infty$-site\xspace $(\mathcal C,\tau)$.
We will refer to $\mathcal S$-valued presheaves (resp.\ sheaves) simply as presheaves (resp.\ sheaves), and denote $\mathrm{PSh}(\mathcal C) \coloneqq \mathrm{PSh}_\mathcal S(\mathcal C)$, $\mathrm{Sh}(\mathcal C,\tau) \coloneqq \mathrm{Sh}_\mathcal S(\mathcal C,\tau)$.
We denote the Yoneda embedding by
\[ h\colon\mathcal C\to\mathrm{PSh}(\mathcal C),\qquad X\mapsto h_X.\]
\bigskip
\paragraph{\textbf{Related works and further developments}}
Our approach is very much based on the foundational works of Lurie \cite{DAG-V,DAG-VII,DAG-VIII,DAG-IX} on derived algebraic geometry and derived complex analytic geometry.
In \cite{Porta_DCAGI,Porta_DCAGII}, Mauro Porta studied the theories of analytification and deformation in derived complex analytic geometry, more specifically, the analytification functor, relative flatness, derived GAGA theorems, square-zero extensions, analytic modules and cotangent complexes.
The papers by Ben-Bassat and Kremnitzer \cite{Ben-Bassat_Non-archimedean_2013}, by Bambozzi and Ben-Bassat \cite{Bambozzi_Dagger_2015}, and by Paugam \cite{Paugam_Overconvergent_2014} suggest other approaches to derived analytic geometry.
In order to apply derived non-archimedean analytic geometry to enumerative geometry, mirror symmetry as well as other domains of mathematics, we must show that derived non-archimedean analytic spaces arise naturally in these contexts.
The key to the construction of derived structures is to prove a representability theorem in derived non-archimedean geometry.
This will be the main goal of our subsequent work \cite{Porta_Yu_Representability}.
\bigskip
\paragraph{\textbf{Acknowledgements}}
We are grateful to Vladimir Berkovich, Antoine Chambert-Loir, Brian Conrad, Antoine Ducros, Bruno Klingler, Maxim Kontsevich, Jacob Lurie, Marco Robalo, Matthieu Romagny, Pierre Schapira, Carlos Simpson, Michael Temkin, Bertrand Toën and Gabriele Vezzosi for valuable discussions.
The authors would also like to thank each other for the joint effort.
This research was partially conducted during the period Mauro Porta was supported by Simons Foundation grant number 347070 and the group GNSAGA, and Tony Yue Yu served as a Clay Research Fellow.
\section{Basic definitions} \label{sec:definitions}
Intuitively, a derived non-archimedean analytic\xspace space is a ``topological space'' $\mathcal X$ equipped with a structure sheaf $\mathcal O_\mathcal X$ of ``derived non-archimedean analytic rings''.
In order to give the precise definition, we introduce the notions of pregeometry and structured topos following \cite{DAG-V}.
\begin{defin}[{\cite[3.1.1]{DAG-V}}]
A \emph{pregeometry} is an $\infty$-category\xspace $\mathcal T$ equipped with a class of \emph{admissible} morphisms and a Grothendieck topology generated by admissible morphisms, satisfying the following conditions:
\begin{enumerate}[(i)]
\item The $\infty$-category\xspace $\mathcal T$ admits finite products.
\item The pullback of an admissible morphism along any morphism exists, and is again admissible.
\item For morphisms $f,g$, if $g$ and $g\circ f$ are admissible, then $f$ is admissible.
\item Every retract of an admissible morphism is admissible.
\end{enumerate}
\end{defin}
We now define two pregeometries responsible for derived non-archimedean geometry.
\begin{construction}
We define a pregeometry $\cT_{\mathrm{an}}(k)$ as follows:
\begin{enumerate}[(i)]
\item the underlying category of $\cT_{\mathrm{an}}(k)$ is the category of smooth $k$-analytic spaces;
\item a morphism in $\cT_{\mathrm{an}}(k)$ is admissible if and only if it is étale;
\item the topology on $\cT_{\mathrm{an}}(k)$ is the étale topology (cf.\ \cite[\S 8.2]{Fresnel_Rigid_2004}).
\end{enumerate}
\end{construction}
\begin{construction}
We define a pregeometry $\cT_{\mathrm{disc}}(k)$ as follows:
\begin{enumerate}[(i)]
\item the underlying category of $\cT_{\mathrm{disc}}(k)$ is the full subcategory of the category of $k$-schemes spanned by affine spaces $\Spec(k[x_1, \ldots, x_n])$;
\item a morphism in $\cT_{\mathrm{disc}}(k)$ is admissible if and only if it is an isomorphism;
\item the topology on $\cT_{\mathrm{disc}}(k)$ is the trivial topology, i.e.\ a collection of admissible morphisms is a covering if and only if it is nonempty.
\end{enumerate}
\end{construction}
\begin{defin}[{\cite[3.1.4]{DAG-V}}] \label{def:structure}
Let $\mathcal T$ be a pregeometry, and let $\mathcal X$ be an $\infty$-topos\xspace.
A \emph{$\mathcal T$-structure} on $\mathcal X$ is a functor $\mathcal O\colon\mathcal T\to\mathcal X$ with the following properties:
\begin{enumerate}[(i)]
\item The functor $\mathcal O$ preserves finite products.
\item Suppose given a pullback diagram
\[
\begin{tikzcd}
U' \arrow{r} \arrow{d} & U \arrow{d}{f} \\
X' \arrow{r} & X
\end{tikzcd}
\]
in $\mathcal T$, where $f$ is admissible.
Then the induced diagram
\[
\begin{tikzcd}
\mathcal O(U') \arrow{r} \arrow{d} & \mathcal O(U) \arrow{d} \\
\mathcal O(X') \arrow{r} & \mathcal O(X)
\end{tikzcd}
\]
is a pullback square in $\mathcal X$.
\item Let $\{U_\alpha\to X\}$ be a covering in $\mathcal T$ consisting of admissible morphisms.
Then the induced map
\[\coprod_\alpha\mathcal O(U_\alpha)\to\mathcal O(X)\]
is an effective epimorphism in $\mathcal X$.
\end{enumerate}
A morphism of $\mathcal T$-structures $\mathcal O\to\mathcal O'$ on $\mathcal X$ is \emph{local} if for every admissible morphism $U\to X$ in $\mathcal T$, the resulting diagram
\[ \begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \mathcal O'(U) \arrow{d} \\
\mathcal O(X) \arrow{r} & \mathcal O'(X)
\end{tikzcd} \]
is a pullback square in $\mathcal X$.
We denote by $\mathrm{Str}^\mathrm{loc}_\mathcal T(\mathcal X)$ the $\infty$-category\xspace of $\mathcal T$-structures on $\mathcal X$ with local morphisms.
A \emph{$\mathcal T$-structured $\infty$-topos\xspace} is a pair $(\mathcal X,\mathcal O_\mathcal X)$ consisting of an $\infty$-topos\xspace $\mathcal X$ and a $\mathcal T$-structure $\mathcal O_\mathcal X$ on $\mathcal X$.
We denote by $\tensor*[^\rR]{\Top}{}(\mathcal T)$ the $\infty$-category\xspace of $\mathcal T$-structured $\infty$-topoi\xspace (cf.\ \cite[Definition 1.4.8]{DAG-V}).
Note that a 1-morphism $f\colon (\mathcal X, \mathcal O_\mathcal X) \to (\mathcal Y, \mathcal O_\mathcal Y)$ in $\tensor*[^\rR]{\Top}{}(\mathcal T)$ consists of a geometric morphism of $\infty$-topoi\xspace $f_*\colon\mathcal X\rightleftarrows\mathcal Y\colon f^{-1}$ and a local morphism of $\mathcal T$-structures $f^\sharp \colon f^{-1} \mathcal O_\mathcal Y \to \mathcal O_\mathcal X$.
\end{defin}
We have a natural functor $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ induced by analytification.
Composing with this functor, we obtain an ``algebraization'' functor
\[
(-)^\mathrm{alg} \colon \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X).
\]
In virtue of \cite[Example 3.1.6, Remark 4.1.2]{DAG-V}, we have an equivalence induced by the evaluation on the affine line
\[\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X) \xrightarrow{\ \sim\ } \mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal X), \]
where $\mathrm{CAlg}_k$ denotes the $\infty$-category\xspace of simplicial commutative algebras over $k$.
We are now ready to introduce our main object of study: derived $k$-analytic\xspace spaces.
\begin{defin}\label{def:derived_space}
A $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $(\mathcal X,\mathcal O_\mathcal X)$ is called a \emph{derived $k$-analytic\xspace space} if $\mathcal X$ is hypercomplete and there exists an effective epimorphism from $\coprod_i U_i$ to the final object of $\mathcal X$ satisfying the following conditions, for every index $i$:
\begin{enumerate}[(i)]
\item The pair $(\mathcal X_{/U_i}, \pi_0(\mathcal O^\mathrm{alg}_\mathcal X | U_i))$ is equivalent to the ringed $\infty$-topos\xspace associated to the étale site on a $k$-analytic\xspace space $X_i$.
\item For each $j\ge 0$, $\pi_j(\mathcal O^\mathrm{alg}_\mathcal X | U_i)$ is a coherent sheaf of $\pi_0(\mathcal O^\mathrm{alg}_\mathcal X | U_i)$-modules on $X_i$.
\end{enumerate}
We denote by $\mathrm{dAn}_k$ the full subcategory of $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ spanned by derived $k$-analytic\xspace spaces.
\end{defin}
\begin{rem}\label{rem:definition_intuition}
Let us explain the heuristic relation between \cref{def:derived_space} and \cref{def:derived_scheme} in the introduction.
Let $(\mathcal X,\mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space as in \cref{def:derived_space}.
Let $\mathbf A^1_k$ be the $k$-analytic\xspace affine line and let $O\coloneqq \mathcal O_\mathcal X (\mathbf A^1_k)\in\mathcal X$.
We have the sum operation $+\colon \mathbf A^1_k \times \mathbf A^1_k \to \mathbf A^1_k$ and the multiplication operation $\bullet\colon \mathbf A^1_k\times \mathbf A^1_k \to \mathbf A^1_k$.
By \cref{def:structure}(i), they induce respectively a sum operation $+\colon O\times O\to O$ and a multiplication operation $\bullet\colon O\times O\to O$ on $O$.
Therefore, intuitively, we can think of $O$ as a sheaf of commutative simplicial rings as in \cref{def:derived_scheme}.
Moreover, the sheaf $O$ is also equipped with analytic structures.
For example, let $\mathbf D^1_k\subset \mathbf A^1_k$ denote the closed unit disc.
By \cref{def:structure}(ii), we obtain a monomorphism $\mathcal O_\mathcal X(\mathbf D^1_k)\hookrightarrow O$.
We can think of $\mathcal O_\mathcal X(\mathbf D^1_k)$ as the subsheaf of $O$ consisting of functions of norm less than or equal to one.
Furthermore, any holomorphic function $f$ on $\mathbf D^1_k$ induces a morphism $f_O\colon \mathcal O_\mathcal X(\mathbf D^1_k)\to O$, which we think of as the composition with $f$.
(See also the discussion after \cref{def:derived_scheme}.)
\end{rem}
\begin{rem}
The hypercompleteness assumption in \cref{def:derived_space} will ensure that the underlying $\infty$-topos\xspace of a derived $k$-analytic\xspace space has enough points (cf.\ \cref{rem:points_of_cX_X}).
\end{rem}
The goal of this paper is to study the basic properties of derived $k$-analytic\xspace spaces and to compare them with ordinary $k$-analytic\xspace spaces as well as with the higher $k$-analytic\xspace stacks introduced in \cite{Porta_Yu_Higher_analytic_stacks_2014}.
Before moving on, we stress that the underlying $\infty$-topos\xspace of a derived $k$-analytic\xspace space is, by definition, hypercomplete.
Therefore, using the notations of \cite[\S 2.2]{DAG-V}, for $X \in \cT_{\mathrm{an}}(k)$, the $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $\Spec^{\cT_{\mathrm{an}}(k)}(X)$ is \emph{not} a derived $k$-analytic\xspace space.
We remedy this problem by introducing the hypercomplete spectrum as follows:
Let $\tensor*[^\rR]{\Top}{}$ (resp.\ $\tensor*[^\rL]{\Top}{}$) denote the $\infty$-category\xspace of $\infty$-topoi\xspace where morphisms are right (resp.\ left) adjoint geometric morphisms.
Denote by $\tensor*[^\rR]{\mathcal{H}\Top}{}$ the full subcategory of $\tensor*[^\rR]{\Top}{}$ spanned by hypercomplete $\infty$-topoi\xspace.
Denote by $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$ the full subcategory of $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ spanned by $\cT_{\mathrm{an}}(k)$-structured $\infty$-topoi\xspace $(\mathcal X, \mathcal O_\mathcal X)$ such that $\mathcal X$ is a hypercomplete. It follows from \cite[6.5.2.13]{HTT} that the inclusion $\tensor*[^\rR]{\mathcal{H}\Top}{} \to \tensor*[^\rR]{\Top}{}$ admits a right adjoint, given by hypercompletion.
This induces a right adjoint to the inclusion $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k)) \hookrightarrow \tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$, as the next lemma shows:
\begin{lem} \label{lem:hyp_right_adjoint}
The inclusion $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k)) \hookrightarrow \tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ admits a right adjoint, which we denote by $\mathrm{Hyp} \colon \tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k)) \to \tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$.
\end{lem}
\begin{proof}
Fix $X\coloneqq(\mathcal X, \mathcal O_\mathcal X) \in \tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$.
Since the hypercompletion $L \colon \mathcal X \to \mathcal X^\wedge$ is left exact, we obtain a well defined functor $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X^\wedge)$ induced by composition with $L$.
Let $X^\wedge \coloneqq (\mathcal X^\wedge, L(\mathcal O_\mathcal X))$ be the resulting hypercomplete $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace.
In $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ there is a natural morphism $\varphi \colon X^\wedge \to X$.
Using the dual of \cite[5.2.7.8]{HTT} it suffices to show that $\varphi$ exhibits $X^\wedge$ as a colocalization of $X$ relative to $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$. In order to prove this, let $Y \coloneqq (\mathcal Y, \mathcal O_\mathcal Y)$ be any hypercomplete $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace.
Using \cite[6.5.2.13]{HTT} we obtain an equivalence
\[ \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal Y, \mathcal X^\wedge) \to \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal Y, \mathcal X). \]
Fix a geometric morphism $g_*\colon\mathcal Y\rightleftarrows\mathcal X^\wedge\colon g^{-1}$ and let $(f^{-1}, f_*)$ denote the induced geometric morphism $\mathcal Y \rightleftarrows \mathcal X$.
We remark that $f^{-1} \simeq g^{-1} \circ L$.
Using \cite[2.4.4.2]{HTT} and \cite[Remark 1.4.10]{DAG-V} we obtain a morphism of fiber sequences:
\[ \begin{tikzcd}[column sep=small]
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}(g^{-1} L(\mathcal O_X), \mathcal O_\mathcal Y) \arrow{r} \arrow{d} & \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(Y, X^\wedge) \arrow{r} \arrow{d} & \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal Y, \mathcal X^\wedge) \arrow{d} \\
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}(f^{-1} \mathcal O_X, \mathcal O_\mathcal Y) \arrow{r} & \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(Y, X) \arrow{r} & \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal Y, \mathcal X).
\end{tikzcd} \]
Since $f^{-1} \simeq g^{-1} \circ L$, we see that the left vertical morphism is an equivalence. Since this holds for all base points in $\Map_{\tensor*[^\rR]{\Top}{}}(\mathcal Y, \mathcal X)$, we conclude that the middle vertical morphism is an equivalence as well, completing the proof.
\end{proof}
\begin{defin}
Given $X \in \cT_{\mathrm{an}}(k)$, we define its \emph{hypercomplete (absolute) spectrum} $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(X)$ to be $\mathrm{Hyp}(\Spec^{\cT_{\mathrm{an}}(k)}(X))$.
\end{defin}
\begin{lem} \label{lem:universal_property_HSpec}
Let $Y \coloneqq (\mathcal Y, \mathcal O_\mathcal Y)$ be a derived $k$-analytic\xspace space and let $X \in \cT_{\mathrm{an}}(k)$.
The natural morphism $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(X) \to \Spec^{\cT_{\mathrm{an}}(k)}(X)$ induces an equivalence
\[ \Map_{\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))}(Y, \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(X)) \xrightarrow{\ \sim\ } \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(Y, \Spec^{\cT_{\mathrm{an}}(k)}(X)). \]
\end{lem}
\begin{proof}
Since $Y$ belongs to $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$, the statement is an immediate consequence of \cref{lem:hyp_right_adjoint}.
\end{proof}
\section{Properties of the pregeometry} \label{sec:pregeometry}
In this section, we study the properties of the pregeometry $\cT_{\mathrm{an}}(k)$ introduced in \cref{sec:definitions}.
More specifically, we will prove the unramifiedness of $\cT_{\mathrm{an}}(k)$, the unramifiedness of the algebraization and the compatibility of $\cT_{\mathrm{an}}(k)$ with $n$-truncations.
\subsection{Unramifiedness}
In order that the collection of closed immersions behaves well with respect to fiber products, our pregeometry $\cT_{\mathrm{an}}(k)$ has to verify a condition of unramifiedness.
\begin{defin}[{\cite[1.3]{DAG-IX}}]\label{def:unramified_pregeometry}
A pregeometry $\mathcal T$ is said to be \emph{unramified} if for every morphism $f\colon X\to Y$ in $\mathcal T$ and every object $Z\in\mathcal T$, the diagram
\[ \begin{tikzcd}
X\times Z \arrow{r} \arrow{d} & X\times Y\times Z \arrow{d} \\
X \arrow{r} & X\times Y
\end{tikzcd} \]
induces a pullback square
\[ \begin{tikzcd}
\mathcal X_{X\times Z} \arrow{r} \arrow{d} & \mathcal X_{X\times Y\times Z} \arrow{d} \\
\mathcal X_X \arrow{r} & \mathcal X_{X\times Y}
\end{tikzcd} \]
in $\tensor*[^\rR]{\Top}{}$, where the symbol $\mathcal X_{(-)}$ denotes the associated $\infty$-topos\xspace.
\end{defin}
Our first goal is to prove that the pregeometry $\cT_{\mathrm{an}}(k)$ is unramified (cf.\ \cref{cor:Tkan_unramified}). In order to do this, we need to describe explicitly the $\infty$-topos\xspace $\mathcal X_X$ associated to a $k$-analytic\xspace space $X$ and prove that the assignment $X \mapsto \mathcal X_X$ is well behaved with respect to closed immersions (cf.\ \cref{prop:preserve_closed_immersion}).
Let $\mathrm{An}_k$ denote the category of $k$-analytic\xspace spaces and let $\mathrm{Afd}_k$ denote the category of $k$-affinoid spaces.
For $X\in\mathrm{An}_k$, let $(\mathrm{An}_X)_\mathrm{\acute{e}t}$ (resp.\ $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}$) denote the category of étale morphisms from $k$-analytic\xspace spaces (resp.\ $k$-affinoid spaces) to $X$.
We equip the categories $(\mathrm{An}_X)_\mathrm{\acute{e}t}$ and $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}$ with the étale topology.
By \cite[Proposition 2.24]{Porta_Yu_Higher_analytic_stacks_2014}, the inclusion $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}\hookrightarrow(\mathrm{An}_X)_\mathrm{\acute{e}t}$ induces an equivalence of $\infty$-topoi\xspace
\begin{equation}\label{eq:afd_in_an}
\mathrm{Sh}((\mathrm{Afd}_X)_\mathrm{\acute{e}t})\xrightarrow{\ \sim\ }\mathrm{Sh}((\mathrm{An}_X)_\mathrm{\acute{e}t}).
\end{equation}
We call the two equivalent $\infty$-topoi\xspace above the \emph{étale $\infty$-topos\xspace associated to $X$}, and denote it by $\mathcal X_X$.
We will denote the site $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}$ by $X_\mathrm{\acute{e}t}$ for simplicity.
\begin{rem}
The $\infty$-topos\xspace $\mathcal X_X$ is not hypercomplete in general.
In the subsequent sections we will also consider its hypercompletion $\mathcal X_X^\wedge$.
\end{rem}
\begin{rem}\label{rem:points_of_cX_X}
Since the site $X_\mathrm{\acute{e}t}$ is a 1-category, the $\infty$-topos\xspace $\mathcal X_X$ is $1$-localic.
It follows that for any $\infty$-topos\xspace $\mathcal Y$ one has an equivalence of $\infty$-categories \[
\Fun_*(\mathcal Y, \mathcal X_X) \simeq \Fun_*(\tau_{\le 0} \mathcal Y, \tau_{\le 0} \mathcal X_X),
\]
where $\Fun_*$ denotes the $\infty$-category of geometric morphisms (taken in $\tensor*[^\rR]{\Top}{}$).
Put $\mathcal Y = \mathcal S$ and observe that $\tau_{\le 0} \mathcal X_X = \mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t})$ and $\tau_{\le 0}(\mathcal S) \simeq \mathrm{Set}$.
We conclude that the points of $\mathcal X_X$ correspond bijectively to the points of the classical 1-topos associated to the site $X_\mathrm{\acute{e}t}$.
The latter is classified by the geometric points of the adic space associated to $X$ in the sense of Huber (cf.\ \cite[Proposition 2.5.17]{Huber_Etale_1996}).
Since the site $X_\mathrm{\acute{e}t}$ is finitary, it follows from \cite[Corollary 3.22]{DAG-VII} that the hypercompletion $\mathcal X_X^\wedge$ is locally coherent.
Therefore, by Theorem 4.1 in loc.\ cit., the $\infty$-topos\xspace $\mathcal X_X^\wedge$ has enough points.
\end{rem}
\begin{rem}\label{rem:spectrum}
As we already discussed in \cref{sec:definitions}, \cite[\S 2.2]{DAG-V} assigns to every $X \in \cT_{\mathrm{an}}(k)$ a $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $\Spec^{\cT_{\mathrm{an}}(k)}(X)$, called the \emph{spectrum} of $X$.
It is characterized by the following universal property: for any $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $(\mathcal Y, \mathcal O_\mathcal Y)$ there is a natural equivalence
\[ \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}( (\mathcal Y, \mathcal O_\mathcal Y), \Spec^{\cT_{\mathrm{an}}(k)}(X) ) \simeq \Map_{\mathrm{Ind}(\mathcal G_\mathrm{an}(k)^{\mathrm{op}})}( X, \Gamma_{\mathcal G}(\mathcal Y, \mathcal O_\mathcal Y) ), \]
where $\mathcal G_{\mathrm{an}}(k)$ denotes a geometric envelope of $\cT_{\mathrm{an}}(k)$ (cf.\ \cite[Theorem 2.2.12]{DAG-V}).
We note that the underlying $\infty$-topos\xspace of $\Spec^{\cT_{\mathrm{an}}(k)}(X)$ can be identified with $\mathcal X_X$.
\end{rem}
We refer to \cite[7.3.2]{HTT} for the notion of closed immersion of $\infty$-topoi\xspace.
\begin{prop}\label{prop:preserve_closed_immersion}
The functor
\begin{align*}
\mathrm{An}_k&\longrightarrow \mathrm{h}(\tensor*[^\rR]{\Top}{}) \\
X&\longmapsto\mathcal X_X
\end{align*}
preserves closed immersions, where $\mathrm h(\tensor*[^\rR]{\Top}{})$ denotes the homotopy category of $\tensor*[^\rR]{\Top}{}$.
\end{prop}
\begin{rem}
It will follow from the results of \cref{sec:fullyfaithfulness} (see in particular \cref{lem:rigidity} and the construction of $\Phi$) that the functor above can be promoted to an $\infty$-functor $\mathrm{An}_k \to \tensor*[^\rR]{\Top}{}$.
\end{rem}
\begin{lem} \label{lem:descent_for_closed_subtopoi}
Let $\mathcal X, \mathcal Y$ be $\infty$-topoi\xspace and let $U \in \mathcal X$.
Let $f^{-1} \colon \mathcal X /U \rightleftarrows \mathcal Y \colon f_*$ be a geometric morphism.
Then $(f^{-1}, f_*)$ is an equivalence if and only if there exists an effective epimorphism $V \to 1_\mathcal X$ such that $\mathcal X_{/V} / (U \times V) \rightleftarrows \mathcal Y_{/ f^{-1}(V)}$ is an equivalence.
\end{lem}
\begin{proof}
To see that the condition is necessary it is enough to take $V \to 1_\mathcal X$ to be the identity of $1_\mathcal X$.
We now prove the sufficiency.
Let us denote by $j^{-1} \colon \mathcal X \leftrightarrows \mathcal X / U \colon j_*$ (resp.\ $i^{-1} \colon \mathcal X \leftrightarrows \mathcal X_{/V} \colon i_*$) the given closed (resp.\ \'etale) morphism of $\infty$-topoi\xspace.
We claim that
\[\mathcal X_{/V} / (U \times V) \simeq (\mathcal X/U)_{/j^{-1}(V)}.\]
Indeed, the left hand side can be identified with the pullback $\mathcal X_{/V} \times_{\mathcal X} \mathcal X/U$ in virtue of \cite[6.3.5.8]{HTT}.
The right hand side can be identified with the same pullback in virtue of \cite[7.3.2.13]{HTT}.
At this point, we obtain a commutative square of geometric morphisms in $\tensor*[^\rR]{\Top}{}$
\[
\begin{tikzcd}
\mathcal Y_{/f^{-1}(V)} \arrow{r} \arrow{d} & (\mathcal X/U)_{/j^{-1}(V)} \arrow{d} \\
\mathcal Y \arrow{r}{f_*} & \mathcal X / U.
\end{tikzcd}
\]
So the lemma follows from the descent property of $\infty$-topoi\xspace \cite[6.1.3.9(3)]{HTT}.
\end{proof}
\begin{lem} \label{lem:qet_structure}
Let $A \to B$ be a surjective morphism of $k$-affinoid algebras.
Let $B \to B'$ be an \'etale morphism of $k$-affinoid $A$-algebras.
Then there exists an \'etale $A$-algebra $A'$ and a pushout square:
\[
\begin{tikzcd}
A \arrow{r} \arrow{d} & B \arrow{d} \\
A' \arrow{r} & B'.
\end{tikzcd}
\]
\end{lem}
\begin{proof}
Since $B \to B'$ is \'etale, by \cite[Proposition 1.7.1]{Huber_Etale_1996}, we can write
\[
B' = B \langle y_1, \ldots, y_m \rangle / (f_1, \ldots, f_m),
\]
such that the Jacobian $J\coloneqq\mathrm{Jac}(f_1, \ldots, f_m)$ is invertible in $B'$.
So
\[\rho\coloneqq\min_{x\in\Sp B'}\abs{J(x)}\]
is positive.
Since $A \to B$ is surjective, the induced morphism
\[
A \langle y_1, \ldots, y_m \rangle \to B \langle y_1, \ldots, y_m \rangle
\]
is surjective as well.
Therefore we can find elements $\overline{f_1}, \ldots, \overline{f_m} \in A \langle y_1, \ldots, y_m \rangle$ lifting $f_1, \ldots, f_m$.
Set
\[
A_0 \coloneqq A \langle y_1, \ldots, y_m \rangle / (\overline{f_1}, \ldots, \overline{f_m}).
\]
Let $\widebar J\coloneqq\mathrm{Jac}(\overline{f_1}, \ldots, \overline{f_m})$.
Let $n$ be a positive integer such that $\rho^n\in\abs{k}$ and let $a$ be an element in $k$ such that $\abs{a}=\rho^n$.
Set $A' \coloneqq A_0 \langle w \rangle / (w \widebar J^n - a)$.
We see that the natural morphism $A_0\to B'$ factors as
\[
A_0 \to A' \to B'.
\]
It follows from the construction that $A \to A'$ is \'etale, and moreover $B' \simeq A' \cotimes_A B$, completing the proof.
\end{proof}
\begin{proof}[Proof of \cref{prop:preserve_closed_immersion}]
Let $f \colon Y \to X$ be a closed immersion in $\mathrm{An}_k$.
Let $U \colon X_\mathrm{\acute{e}t} \to \mathcal S$ be the functor defined by the formula
\[
U(Z) = \begin{cases}
\{*\} & \text{if } Z\times_X Y =\emptyset, \\
\emptyset & \text{otherwise.}
\end{cases}
\]
This is a sheaf and therefore determines a closed subtopos $\mathcal X_X / U$.
The morphism $f$ induces a geometric morphism
\[f^{-1} \colon \mathcal X_X \rightleftarrows \mathcal X_Y \colon f_* .\]
We claim that $f_*$ factors through the closed subtopos $\mathcal X_X / U$.
Indeed, it suffices to check that for every sheaf $G \in \mathcal X_Y$ and every representable sheaf $h_{Z}$ in $\mathcal X_X$ such that $\Map_{\mathcal X_X}(h_{Z}, U) \ne \emptyset$, the space $\Map_{\mathcal X_X}(h_{Z}, f_*(G))$ is contractible.
This is true, because we have
\[
\Map_{\mathcal X_X}(h_{Z}, f_*(G)) \simeq G(Z\times_X Y) = G(\emptyset) \simeq \{*\}.
\]
We denote by $(f^{-1},f_*)$ again the induced adjunction
\begin{equation} \label{eq:induced_adjunction}
f^{-1} \colon \mathcal X_X / U \rightleftarrows \mathcal X_Y \colon f_* .
\end{equation}
We conclude our proof by the following lemma.
\end{proof}
\begin{lem} \label{lem:closed_immersion_closed_subtopos}
The adjunction in \cref{eq:induced_adjunction} is an equivalence.
\end{lem}
\begin{proof}
By \cref{lem:descent_for_closed_subtopoi}, we can assume that both $X$ and $Y$ are affinoid.
Note that $\mathcal X_X / U$ and $\mathcal X_Y$ are $1$-localic $\infty$-topoi\xspace in virtue of \cite[7.5.4.2]{HTT} and \cite[Lemma 1.2.6]{DAG-VIII}.
Therefore it suffices to show that the adjunction $(f^{-1}, f_*)$ induces an equivalence when restricted to $1$-truncated objects of $\mathcal X_X / U$ and $\mathcal X_Y$.
Let us prove that the functor $f_*$ is conservative.
Let $\alpha \colon F \to F'$ be a morphism in $\mathcal X_Y$ and suppose that $f_*(\alpha)$ is an equivalence.
By the equivalence \eqref{eq:afd_in_an}, it is enough to show that $\alpha$ induces equivalences $F(Y') \to F'(Y')$ for every étale morphism $Y' \to Y$.
Using \cref{lem:qet_structure}, we can form a pullback diagram
\[
\begin{tikzcd}
Y' \arrow{r} \arrow{d} & X' \arrow{d} \\
Y \arrow{r} & X ,
\end{tikzcd}
\]
where $X' \to X$ is étale.
It follows that
\[
F(Y') = (f_* F)(X') \to (f_* F')(X') = F(Y')
\]
is an equivalence.
We are left to check that the unit of the adjunction $(f^{-1}, f_*)$ is an equivalence over $1$-truncated objects.
For this, it suffices to check that for every $1$-truncated sheaf $F \in \mathcal X_X$,
the unit $u \colon F \to f_* f^{-1} F$ induces an equivalence on sheaves of homotopy groups.
Since both $F$ and $f_* f^{-1} F$ are $1$-truncated, they are hypercomplete objects.
Therefore, it suffices to check that $\eta^{-1}(u)$ is an equivalence for every geometric morphism $\eta^{-1} \colon \mathcal X_X \to \mathcal S\colon\eta_*$.
Such a geometric morphism corresponds to a geometric point $x$ of the adic space associated to $X$ (cf.\ \cref{rem:points_of_cX_X}).
Let $\{V_\alpha\}$ be a system of étale neighborhoods of $x$.
We have $\eta^{-1}(G)=\colim G(V_\alpha)$.
If $x$ does not meet $Y$, we see that $\eta^{-1}(G)$ is contractible whenever $G \in \mathcal X_X / U$.
In particular $\eta^{-1}(u)$ is an equivalence for every $1$-truncated $F \in \mathcal X_X / U$.
Otherwise, $x$ lifts to a geometric morphism $\eta_1^{-1} \colon \mathcal X_Y \to \mathcal S$, satisfying $\eta^{-1} = \eta_1^{-1} \circ f^{-1}$.
So we have
\begin{align*}
\eta^{-1}(f_* f^{-1} F) & \simeq \colim (f_* f^{-1} F)(V_\alpha) \\
& \simeq \colim (f^{-1} F)(V_\alpha \times_{X} Y) \\
& \simeq \eta_1^{-1} f^{-1} F \simeq \eta^{-1} F ,
\end{align*}
completing the proof.
\end{proof}
\begin{prop} \label{prop:closed_immersion_pullback_of_topoi}
Let
\[ \begin{tikzcd}
W \arrow{r} \arrow{d} & Y \arrow{d}{g} \\
X \arrow{r}{f} & Z
\end{tikzcd} \]
be a pullback square in $\mathrm{An}_k$ and assume that $f$ is a closed immersion. The induced square of $\infty$-topoi\xspace
\[ \begin{tikzcd}
\mathcal X_W \arrow{r} \arrow{d} & \mathcal X_Y \arrow{d}{g_*} \\
\mathcal X_X \arrow{r}{f_*} & \mathcal X_Z
\end{tikzcd} \]
is a pullback diagram in $\tensor*[^\rR]{\Top}{}$.
\end{prop}
\begin{proof}
Let $U_X$ be the sheaf on the étale site $Z_\mathrm{\acute{e}t}$ of $Z$ defined by
\[ U_X(T) \coloneqq \begin{cases} \{*\} & \text{if } T \times_Z X = \emptyset \\ \emptyset & \text{otherwise.} \end{cases} \]
Define $U_W$ to be the sheaf on the étale site $Y_\mathrm{\acute{e}t}$ of $Y$ in a similar way.
Using \cref{lem:closed_immersion_closed_subtopos} twice, we can rewrite the induced square of $\infty$-topoi\xspace as
\[ \begin{tikzcd}
\mathcal X_Y / U_W \arrow{r} \arrow{d} & \mathcal X_Y \arrow{d}{g_*} \\
\mathcal X_Z / U_X \arrow{r} & \mathcal X_Z .
\end{tikzcd} \]
In virtue of \cite[7.3.2.13]{HTT}, we only need to show that $g^{-1} U_X \simeq U_W$.
First of all, let us observe that there exists a map $U_X \to g_* U_W$: indeed, if $T \to X$ is étale with $T \to Z$ a smooth morphism such that $T \times_Z X = \emptyset$, then we also have $(T \times_Z Y) \times_Y W \simeq (T\times_Z W) \times_Z Y = \emptyset$, and therefore $g_*(U_W)(T) = U_W(T \times_Z Y) = \Delta^0$.
This allows to define the desired map, which induces by adjunction a morphism $g^{-1} U_X \to U_W$.
By construction, $U_W$ is $(-1)$-truncated and \cite[5.5.6.16]{HTT} shows that $g^{-1} U_X$ is $(-1)$-truncated too.
Therefore they are both hypercomplete.
So it suffices to check that $g^{-1} U_X \to U_W$ is an isomorphism on the stalks of $\mathcal X_Y$. This is true because a geometric point $\eta_* \colon \mathcal S \to \mathcal X_Y$ factors through $\mathcal X_W$ if and only if $g_* \circ \eta_*$ factors through $\mathcal X_X$ (cf.\ \cref{rem:points_of_cX_X}).
\end{proof}
\begin{cor} \label{cor:Tkan_unramified}
The pregeometry $\cT_{\mathrm{an}}(k)$ is unramified.
\end{cor}
\begin{proof}
We check that \cref{def:unramified_pregeometry} is satisfied.
Let $X, Y, Z \in \cT_{\mathrm{an}}(k)$ and let $f \colon Y \to X$ be any morphism.
The diagram
\[ \begin{tikzcd}
X \arrow{r} \arrow{d}{\mathrm{id}_X \times f} & Y \arrow{d}{\Delta} \\
X \times Y \arrow{r} & Y \times Y
\end{tikzcd} \]
is a pullback diagram. Since $Y$ is separated, $Y$ is a closed immersion, and therefore the same goes for $X \to X \times Y$.
We can therefore use \cref{prop:closed_immersion_pullback_of_topoi} to conclude that the induced square
\[ \begin{tikzcd}
\mathcal X_{X \times Z} \arrow{r} \arrow{d} & \mathcal X_X \arrow{d} \\
\mathcal X_{X \times Y \times Z} \arrow{r} & \mathcal X_{X \times Y}.
\end{tikzcd} \]
is a pullback diagram in $\tensor*[^\rR]{\Top}{}$.
\end{proof}
\subsection{Algebraization}
The functor $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ induced by analytification is a transformation of pregeometries in the following sense:
\begin{defin}[{\cite[3.2.1]{DAG-IX}}]
A \emph{transformation of pregeometries} from $\mathcal T$ to $\mathcal T'$ is a functor $\theta\colon\mathcal T\to\mathcal T'$ such that
\begin{enumerate}[(i)]
\item it preserves finite products;
\item it sends admissible morphisms in $\mathcal T$ to admissible morphisms in $\mathcal T'$;
\item it sends coverings in $\mathcal T$ to coverings in $\mathcal T'$;
\item it sends any pullback in $\mathcal T$ along an admissible morphism to a pullback in $\mathcal T'$.
\end{enumerate}
\end{defin}
In the following, we study some properties of the transformation of pregeometries $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$.
\begin{lem} \label{lem:alg_conservative}
Let $\mathcal X$ be an $\infty$-topos\xspace.
The algebraization functor
\[
(-)^\mathrm{alg} \colon \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X)
\]
induced by composition with the transformation $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ is conservative.
\end{lem}
\begin{proof}
Let $f \colon \mathcal O \to \mathcal O'$ be a local morphism of $\cT_{\mathrm{an}}(k)$-structures on $\mathcal X$ such that $f^\mathrm{alg} \colon \mathcal O^\mathrm{alg} \to \mathcal O^{\prime \mathrm{alg}}$ is an equivalence.
We will show that for every $X\in\cT_{\mathrm{an}}(k)$,
the induced morphism $\mathcal O(X) \to \mathcal O'(X)$ is an equivalence.
Since $X$ is smooth, there exists an affinoid G-covering $\{\Sp B_i\to X\}$ such that every $\Sp B_i$ admits an étale morphism to a $k$-analytic\xspace affine space.
So we obtain a commutative square
\[
\begin{tikzcd}
\coprod \mathcal O(\Sp B_i) \arrow{d} \arrow{r} & \coprod \mathcal O'(\Sp B_i) \arrow{d} \\
\mathcal O(X) \arrow{r} & \mathcal O'(X) ,
\end{tikzcd}
\]
where the vertical morphisms are effective epimorphisms.
Moreover, since admissible open immersions are étale and $f$ is a local morphism, we see that the above square is a pullback.
We are therefore reduced to show that $\mathcal O(\Sp B) \to \mathcal O'(\Sp B)$ is an equivalence whenever $\Sp B$ admits an étale morphism to a $k$-analytic\xspace affine space $\mathbf A^n_k$.
Since $f$ is a local morphism, we have in this case a pullback square
\[
\begin{tikzcd}
\mathcal O(\Sp B) \arrow{r} \arrow{d} & \mathcal O(\mathbf A^n_k) \arrow{d} \\
\mathcal O'(\Sp B) \arrow{r} & \mathcal O'(\mathbf A^n_k).
\end{tikzcd}
\]
Let $\mathbb A^n_k$ denote the $n$-dimensional algebraic affine space over $k$.
Since $\mathcal O(\mathbf A^n_k) = \mathcal O^\mathrm{alg}(\mathbb A^n_k) \to \mathcal O^{\prime \mathrm{alg}}(\mathbb A^n_k) = \mathcal O'(\mathbf A^n_k)$ is an equivalence by our assumption, we deduce that $\mathcal O(\Sp B) \to \mathcal O'(\Sp B)$ is an equivalence as well, completing the proof.
\end{proof}
\begin{prop} \label{prop:alg_effective_epi}
Let $\mathcal X$ be an $\infty$-topos\xspace and let $f \colon \mathcal O \to \mathcal O'$ be a morphism in $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)$.
The following conditions are equivalent:
\begin{enumerate}[(i)]
\item The morphism $f$ is an effective epimorphism, i.e.\ for every $U \in \cT_{\mathrm{an}}(k)$ the morphism $\mathcal O(U) \to \mathcal O'(U)$ is an effective epimorphism in $\mathcal X$.
\item The morphism $f^\mathrm{alg} \colon \mathcal O^\mathrm{alg} \to \mathcal O^{\prime \mathrm{alg}}$ is an effective epimorphism.
\item The morphism $\mathcal O(\mathbf A^1_k) \to \mathcal O'(\mathbf A^1_k)$ is an effective epimorphism.
\end{enumerate}
\end{prop}
\begin{proof}
It follows directly from the definition of effective epimorphism of $\cT_{\mathrm{an}}(k)$-structures that (i) implies (ii) and (ii) implies (iii).
Let us show that (iii) implies (i).
Let $X \in \cT_{\mathrm{an}}(k)$. Choose an étale covering $\{U_i \to X \}$ such that each $U_i$ admits an étale morphism to $\mathbf A^n_k$.
Since $f$ is a local morphism, we have the following pullback square:
\[ \begin{tikzcd}
\coprod \mathcal O(U_i) \arrow{r} \arrow{d} & \coprod \mathcal O'(U_i) \arrow{d} \\
\mathcal O(X) \arrow{r} & \mathcal O'(X) .
\end{tikzcd} \]
The vertical arrows are effective epimorphisms, and therefore it suffices to check that the upper horizontal map is an effective epimorphism.
Since $f$ is a local morphism, we see that the diagram
\[ \begin{tikzcd}
\mathcal O(U_i) \arrow{r} \arrow{d} & \mathcal O'(U_i) \arrow{d} \\
\mathcal O(\mathbf A^n_k) \arrow{r} & \mathcal O'(\mathbf A^n_k)
\end{tikzcd} \]
is a pullback diagram.
So it suffices to show that $\mathcal O(\mathbf A^n_k) \to \mathcal O'(\mathbf A^n_k)$ is an effective epimorphism. This follows from the hypothesis and the fact that both $\mathcal O$ and $\mathcal O'$ commute with products.
\end{proof}
\begin{defin}[{\cite[10.1]{DAG-IX}}]
Let $\theta\colon\mathcal T'\to\mathcal T$ be a transformation of pregeometries, and $\Theta\colon\mathcal T\mathrm{op}(\mathcal T)\to\mathcal T\mathrm{op}(\mathcal T')$ the induced functor given by composition with $\theta$.
We say that $\theta$ is \emph{unramified} if the following conditions hold:
\begin{enumerate}[(i)]
\item The pregeometries $\mathcal T$ and $\mathcal T'$ are unramified.
\item For every morphism $f\colon X\to Y$ in $\mathcal T$ and every object $Z\in\mathcal T$, the diagram
\[ \begin{tikzcd}
\Theta\Spec^\mathcal T(X\times Z) \arrow{r} \arrow{d} & \Theta\Spec^\mathcal T(X) \arrow{d} \\
\Theta\Spec^\mathcal T(X\times Y\times Z) \arrow{r} & \Theta\Spec^\mathcal T(X\times Y)
\end{tikzcd} \]
is a pullback square in $\mathcal T\mathrm{op}(\mathcal T')$.
\end{enumerate}
\end{defin}
\begin{prop} \label{prop:unramified_transformation}
The transformation of pregeometries $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ is unramified.
\end{prop}
\begin{proof}
For $X\in\cT_{\mathrm{an}}(k)$, we denote the spectrum $\Spec^{\cT_{\mathrm{an}}(k)}(X)$ by $(\mathcal X_X, \mathcal O_X)$.
For a morphism $X \to Y$ in $\cT_{\mathrm{an}}(k)$,
we denote by $\mathcal O_Y^\mathrm{alg} | X$ the image of $\mathcal O_Y^\mathrm{alg}$ under the pullback functor $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X_Y) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X_X)$.
We have to show that for every morphism $f \colon X \to Y$ in $\cT_{\mathrm{an}}(k)$ and every $Z\in\cT_{\mathrm{an}}(k)$,
the commutative square
\begin{equation} \label{eq:desired_pushout}
\begin{tikzcd}
\mathcal O_{X \times Y}^\mathrm{alg} | (X \times Z) \arrow{r} \arrow{d} & \mathcal O_X^\mathrm{alg} | (X \times Z) \arrow{d} \\
\mathcal O_{X \times Y \times Z}^\mathrm{alg} | (X \times Z) \arrow{r} & \mathcal O_{X \times Z}^\mathrm{alg}
\end{tikzcd}
\end{equation}
is a pushout in $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X_{X \times Z}) \simeq \mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal X_{X \times Z})$.
Form the pushout
\[ \begin{tikzcd}
\mathcal O_{X \times Y}^\mathrm{alg} | (X \times Z) \arrow{r} \arrow{d} & \mathcal O_X^\mathrm{alg} | (X \times Z) \arrow{d} \\
\mathcal O_{X \times Y \times Z}^\mathrm{alg} | (X \times Z) \arrow{r} & \mathcal A
\end{tikzcd} \]
in $\mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal X_{X \times Z})$.
Let $\mathcal A^\wedge$ be the hypercompletion of $\mathcal A$.
We will prove below that $\mathcal A^\wedge$ is equivalent to $\mathcal O_{X \times Z}^\mathrm{alg}$.
Assuming this, we see that $\mathcal A^\wedge$ is discrete.
It follows that $\mathcal A$ is discrete as well, and therefore it is hypercomplete.
We thus conclude that the square \eqref{eq:desired_pushout} is a pushout.
So we are reduced to show that the map $\mathcal A^\wedge \to \mathcal O_{X \times Z}^\mathrm{alg}$ is an equivalence.
Both sheaves are hypercomplete and \cref{rem:points_of_cX_X} shows that $\mathcal X_{X \times Z}^\wedge$ has enough points.
Thus, it suffices to show that for every geometric point $(x,z)$ of the adic space associated to $X \times Z$ in the sense of Huber, the diagram
\begin{equation}\label{eq:pushout_stalks}
\begin{tikzcd}
\mathcal O_{(x,y)}^\mathrm{alg} \arrow{r} \arrow{d} & \mathcal O_x^\mathrm{alg} \arrow{d} \\
\mathcal O_{(x,y,z)}^\mathrm{alg} \arrow{r} & \mathcal O_{(x,z)}^\mathrm{alg}
\end{tikzcd}
\end{equation}
is a pushout square, where we set $y \coloneqq f(x)$.
Choose a fundamental system of étale affinoid neighborhoods $\{V_\alpha\}$ of $(x,y)$ in $X \times Y$.
Set $U_\alpha \coloneqq V_\alpha \times_{X \times Y} X$ and observe that $\{U_\alpha\}$ forms a fundamental system of étale affinoid neighborhoods of $x$ in $X$.
Choose moreover a fundamental system $\{W_\beta\}$ of étale affinoid neighborhoods of $z$ in $Z$.
We have pullback squares
\begin{equation}\label{eq:neighborhood_systems}
\begin{tikzcd}
U_\alpha \times W_\beta \arrow{r} \arrow{d} & U_\alpha \arrow{d} \\
V_\alpha \times W_\beta \arrow{r} & V_\alpha.
\end{tikzcd}
\end{equation}
Assume $U_\alpha = \Sp A_\alpha$, $V_\alpha = \Sp B_\alpha$ and $W_\beta = \Sp C_\beta$.
Since $U_\alpha \to V_\alpha$ is a closed immersion, the pullback above corresponds to a pushout in the category of $k$-algebras\begin{equation}\label{eq:algebraic_neighborhood_systems}
\begin{tikzcd}
B_\alpha \arrow{r} \arrow{d} & B_\alpha \cotimes_k C_\beta \arrow{d} \\
A_\alpha \arrow{r} & A_\alpha \cotimes_k C_\beta
\end{tikzcd}
\end{equation}
Taking limit in Diagram \ref{eq:neighborhood_systems} (or equivalently, taking colimit in Diagram \ref{eq:algebraic_neighborhood_systems}),
we observe that Diagram \ref{eq:pushout_stalks} is a pushout diagram in the category of $k$-algebras.
Since the projections $V_\alpha \times W_\beta \to V_\alpha$ are flat, we see that every morphism $B_\alpha \to B_\alpha \cotimes_k C_\beta$ is flat.
As a consequence, $\mathcal O_{(x,y)}^\mathrm{alg} \to \mathcal O_{(x,y,z)}^\mathrm{alg}$ is flat.
The pushout \eqref{eq:pushout_stalks} is therefore a derived pushout square, completing the proof.
\end{proof}
Intuitively, the pregeometry $\cT_{\mathrm{an}}(k)$ enables us to consider structure sheaves with ``non-archimedean analytic structures'' in addition to the usual algebraic structures.
The unramifiedness of the transformation $\cT_{\mathrm{disc}}(k)\to\cT_{\mathrm{an}}(k)$ in \cref{prop:unramified_transformation} will imply that for certain purposes, this additional analytic structure can be ignored.
Here is a simple example illustrating this phenomenon:
Consider the completed tensor product $A \cotimes_B C$ of three $k$-affinoid algebras.
When $C$ is finitely presented as a $B$-module, we have an isomorphism $A \cotimes_B C\simeq A\otimes_B C$.
That is, in this case, for the purpose of tensor product, the analytic structure on affinoid algebras can be ignored.
The proposition below elaborates on this idea:
\begin{prop} \label{prop:closed_fiber_products_Top}
Let $f \colon (\mathcal Y, \mathcal O_{\mathcal Y}) \to (\mathcal X, \mathcal O_{\mathcal X})$ and $g \colon (\mathcal X', \mathcal O_{\mathcal X'}) \to (\mathcal X, \mathcal O_{\mathcal X})$ be morphisms in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$.
Assume that the induced map $\theta \colon f^{-1} \mathcal O_{\mathcal X}^\mathrm{alg} \to \mathcal O_{\mathcal Y}^\mathrm{alg}$ is an effective epimorphism.
Then:
\begin{enumerate}[(i)]
\item \label{item:pullback_structured_Top} There exists a pullback diagram
\[ \begin{tikzcd}
(\mathcal Y', \mathcal O_{\mathcal Y'}) \arrow{r}{f'} \arrow{d}{g'} & (\mathcal X', \mathcal O_{\mathcal X'}) \arrow{d}{g} \\
(\mathcal Y, \mathcal O_{\mathcal Y}) \arrow{r}{f} & (\mathcal X, \mathcal O_{\mathcal X})
\end{tikzcd} \]
in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$. If moreover $(\mathcal X, \mathcal O_\mathcal X), (\mathcal X', \mathcal O_{\mathcal X'}), (\mathcal Y, \mathcal O_\mathcal Y) \in \tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$, then $\mathrm{Hyp}(\mathcal Y', \mathcal O_{\mathcal Y'})$ is equivalent to the pullback computed in $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$.
\item \label{item:pullback_Top} The underlying diagram of $\infty$-topoi\xspace
\[ \begin{tikzcd}
\mathcal Y' \arrow{r} \arrow{d} & \mathcal X' \arrow{d} \\
\mathcal Y \arrow{r} & \mathcal X
\end{tikzcd} \]
is a pullback square in $\tensor*[^\rR]{\Top}{}$.
If moreover
\[(\mathcal X, \mathcal O_\mathcal X), (\mathcal X', \mathcal O_{\mathcal X'}), (\mathcal Y, \mathcal O_\mathcal Y) \in \tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k)),\] then $(\mathcal Y')^\wedge$ is equivalent to the pullback computed in $\tensor*[^\rR]{\mathcal{H}\Top}{}$.
\item \label{item:pushout_analytic_algebras} The diagram
\[ \begin{tikzcd}
f^{\prime -1} g^{-1} \mathcal O_{\mathcal X}^\mathrm{alg} \arrow{r} \arrow{d} & f^{\prime -1} \mathcal O_\mathcal Y^\mathrm{alg} \arrow{d} \\
g^{\prime -1} \mathcal O_{\mathcal Y}^\mathrm{alg} \arrow{r} & \mathcal O_{\mathcal Y'}^\mathrm{alg}
\end{tikzcd} \]
is a pushout square in $\mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal Y')$. If moreover $(\mathcal X, \mathcal O_\mathcal X)$, $(\mathcal X', \mathcal O_{\mathcal X'})$, $(\mathcal Y, \mathcal O_\mathcal Y) \in \tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$, the same holds after applying the hypercompletion functor $L \colon \mathcal Y' \to (\mathcal Y')^\wedge$.
\item \label{item:effective_epi} The map $\theta' \colon f^{\prime -1} \mathcal O_{\mathcal X'} \to \mathcal O_{\mathcal Y'}$ is an effective epimorphism. If moreover $(\mathcal X, \mathcal O_\mathcal X)$, $(\mathcal X', \mathcal O_{\mathcal X'})$, $(\mathcal Y, \mathcal O_\mathcal Y) \in \tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$, the same holds after applying the hypercompletion functor $L \colon \mathcal Y' \to (\mathcal Y')^\wedge$
\end{enumerate}
\end{prop}
\begin{proof}
We first deal with the non-hypercomplete case.
\cref{prop:alg_effective_epi} shows that the morphism $f^{-1} \mathcal O_{\mathcal X} \to \mathcal O_{\mathcal Y}$ is an effective epimorphism.
Moreover, $\cT_{\mathrm{an}}(k)$ is unramified in virtue of \cref{cor:Tkan_unramified}.
Therefore \cite[Theorem 1.6]{DAG-IX} implies the first two statements.
Combining \cref{prop:unramified_transformation}, \cref{prop:alg_effective_epi} and \cite[Proposition 10.3]{DAG-IX}, we deduce the other two statements.
We now assume that $(\mathcal X, \mathcal O_\mathcal X), (\mathcal X', \mathcal O_{\mathcal X'}), (\mathcal Y, \mathcal O_\mathcal Y) \in \tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$.
Then (\ref{item:pullback_structured_Top}) and (\ref{item:pullback_Top}) follow from what we already proved and the fact that $\mathrm{Hyp}$ commutes with limits, being a right adjoint by \cref{lem:hyp_right_adjoint}.
On the other side, (\ref{item:pushout_analytic_algebras}) and (\ref{item:effective_epi}) follow from the fact that the hypercompletion functor $L \colon \mathcal Y' \to (\mathcal Y')^\wedge$ commutes with colimits and finite limits.
\end{proof}
\subsection{Truncations}
Now we discuss the compatibility of the pregeometry $\cT_{\mathrm{an}}(k)$ with $n$-truncations.
\begin{defin}[{\cite[3.3.2]{DAG-V}}]
Let $\mathcal T$ be a pregeometry and let $n \ge -1$ be an integer.
The pregeometry $\mathcal T$ is said to be \emph{compatible with $n$-truncations} if for every $\infty$-topos\xspace $\mathcal X$, every $\mathcal T$-structure $\mathcal O \colon \mathcal T \to \mathcal X$ and every admissible morphism $U \to V$ in $\mathcal T$, the induced square
\[ \begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le n}(\mathcal O(U)) \arrow{d} \\
\mathcal O(V) \arrow{r} & \tau_{\le n}(\mathcal O(V))
\end{tikzcd} \]
is a pullback in $\mathcal X$.
\end{defin}
This definition is equivalent to say that for every $\mathcal T$-structure $\mathcal O \colon \mathcal T \to \mathcal X$ the composition $\tau_{\le n} \circ \mathcal O$ is again a $\mathcal T$-structure and the canonical morphism $\mathcal O \to \tau_{\le n} \circ \mathcal O$ is a local morphism of $\mathcal T$-structures, where $\tau_{\le n} \colon \mathcal X \to \mathcal X$ denotes the truncation functor of the $\infty$-topos\xspace $\mathcal X$.
In order to prove that $\cT_{\mathrm{an}}(k)$ is compatible with $n$-truncations for every $n \ge 0$, it will be convenient to introduce a pregeometry slightly different from $\cT_{\mathrm{an}}(k)$.
\begin{construction}
We define a pregeometry $\cT_{\mathrm{an}}^G(k)$ as follows:
\begin{enumerate}
\item the underlying category of $\cT_{\mathrm{an}}^G(k)$ is the category of smooth $k$-analytic spaces;
\item a morphism in $\cT_{\mathrm{an}}^G(k)$ is admissible if and only if it is an admissible open embedding;
\item the topology on $\cT_{\mathrm{an}}^G(k)$ is the G-topology.
\end{enumerate}
\end{construction}
\begin{lem} \label{lem:G_pregeometry_0_truncations}
The pregeometry $\cT_{\mathrm{an}}^G(k)$ is compatible with $n$-truncations for every $n \ge 0$.
\end{lem}
\begin{proof}
Since admissible open immersions are monomorphisms, the lemma is a direct consequence of \cite[3.3.5]{DAG-V}.
\end{proof}
\begin{lem} \label{lem:etale_analytic_domain}
Let $U \to V$ be an étale morphism in $\cT_{\mathrm{an}}(k)$.
There exists a G-covering $\{V_i\to V\}_{i\in I}$, G-coverings $\{U_{ij}\to U\times_V V_i\}_{j\in J_i}$ for every $i\in I$, smooth algebraic $k$-varieties $Y_i$ and $X_{ij}$, étale morphisms $X_{ij}\to Y_i$, admissible open immersions $V_i\hookrightarrow Y_i^\mathrm{an}$ and $U_{ij}\hookrightarrow X_{ij}^\mathrm{an}$ such that the morphism $U_{ij}\to V_i$ equals the restriction of the morphism $X_{ij}^\mathrm{an}\to Y_i^\mathrm{an}$ to $V_i$ for every $i\in I$ and $j\in J_i$.
\end{lem}
\begin{proof}
Since $V$ is smooth, there exists an affinoid G-covering $\{V_i\to V\}_{i\in I}$ such that every $V_i$ admits an étale morphism to a polydisc $\mathbf D^{n_i}$.
By \cite[Proposition 1.7.1]{Huber_Etale_1996}, the affinoid algebra associated to $V_i$ has a presentation of the form
\[k\langle T_1,\dots,T_{n_i},T'_1,\dots,T'_{m_i}\rangle/(f_1,\dots,f_{m_i})\]
such that the determinant $\det\big(\frac{\partial f_\alpha}{\partial T'_\beta}\big)_{\alpha,\beta=1,\dots,m_i}$ is invertible in $k\langle T_1,\dots,T_{n_i}\rangle$.
By \cite[Chap.\ III Theorem 7 and Remark 2]{Elkik_Solutions_1973}, there exists a smooth affine scheme $Y_i$ and an étale morphism $Y_i\to\mathbb A^{n_i}_k$ such that $V_i$ is isomorphic to the fiber product $Y_i^\mathrm{an}\times_{(\mathbb A^{n_i}_k)^\mathrm{an}} \mathbf D^{n_i}$.
We now fix $i\in I$.
Since the morphism $U\to V$ is étale, by base change, the morphism $U\times_V V_i\to V_i$ is étale.
So the composition
\[ U\times_V V_i\to V_i\to\mathbf D^{n_i}\]
is étale.
Let $\{U_{ij}\to U\times_V V_i\}_{j\in J_i}$ be an affinoid G-covering.
For every $j\in J_i$, by \cite[Proposition 1.7.1]{Huber_Etale_1996}, the affinoid algebra associated to $U_{ij}$ has a presentation of the form
\[k\langle T_1,\dots,T_{n_i},T'_1,\dots,T'_{m_{ij}}\rangle/(f_1,\dots,f_{m_{ij}})\]
such that the determinant $\det\big(\frac{\partial f_\alpha}{\partial T'_\beta}\big)_{\alpha,\beta=1,\dots,m_{ij}}$ is invertible in $k\langle T_1,\dots,T_{n_i}\rangle$.
By \cite[Chap.\ III Theorem 7]{Elkik_Solutions_1973} again, there exists a smooth affine scheme $Z_{ij}$ and an étale morphism $Z_{ij}\to\mathbb A^{n_i}_k$ such that $U_{ij}$ is isomorphic to the fiber product $Z_i^\mathrm{an}\times_{(\mathbb A^{n_i}_k)^\mathrm{an}} \mathbf D^{n_i}$.
Let $X_{ij}\coloneqq Y_i\times_{\mathbb A^{n_i}_k} Z_{ij}$.
By the universal property of the fiber product, there exists a unique map $r\colon U_{ij}\to X^\mathrm{an}_{ij}$ making the following diagram commutative:
\[\begin{tikzcd}
U_{ij} \arrow{d} \arrow[swap]{rd}{r} \arrow{rrd}{t} & &\\
U\times_V V_i \arrow{d} & X^\mathrm{an}_{ij}\arrow{d}\arrow{r}{s} & Z^\mathrm{an}_{ij}\arrow{d}\\
V_i \arrow[hookrightarrow]{r} & Y^\mathrm{an}_i \arrow{r} & (\mathbb A^{n_i}_k)^\mathrm{an}
\end{tikzcd}\]
The map $t$ is an admissible open immersion, so it is in particular étale.
The map $s$ is étale by base change, so it is étale.
Since $t=s\circ r$, we deduce that the map $r$ is étale.
Moreover, the map $t$ is a monomorphism, so the map $r$ is also a monomorphism.
Since the map $r$ is étale, we deduce that it is an admissible open immersion.
\end{proof}
\begin{lem} \label{lem:pullbacks_are_local}
Let $\mathcal X$ be an $\infty$-topos\xspace and let
\[ \begin{tikzcd}
U \arrow{r} \arrow{d} & W \arrow{r} \arrow{d} & Z \arrow{d} \\
V \arrow{r}{p} & Y \arrow{r} & X
\end{tikzcd} \]
be a diagram in $\mathcal X$.
Assume that the left and the outer squares are pullbacks and that $p$ is an effective epimorphism.
Then the right square is a pullback as well.
\end{lem}
\begin{proof}
Let $W' \coloneqq Y \times_X Z$.
We obtain a commutative diagram
\[ \begin{tikzcd}
U \arrow{r} \arrow{d} & W' \arrow{r} \arrow{d} & Z \arrow{d} \\
V \arrow{r}{p} & Y \arrow{r} & X .
\end{tikzcd} \]
Since the outer square is a pullback by our assumption, the left square is a pullback as well.
The universal property of pullbacks induces a morphism $\alpha \colon W \to W'$.
By hypothesis, the induced map $\alpha \times_Y V \colon W \times_Y V \to W' \times_Y V$ is an equivalence.
Since $p$ is an effective epimorphism, the pullback functor $p^{-1} \colon \mathcal X_{/Y} \to \mathcal X_{/V}$ is conservative (cf.\ \cite[6.2.3.16]{HTT}).
We conclude that $\alpha$ is an equivalence, completing the proof.
\end{proof}
\begin{thm} \label{thm:compatibility_truncations}
The pregeometry $\cT_{\mathrm{an}}(k)$ is compatible with $n$-truncations for every $n \ge 0$.
\end{thm}
\begin{proof}
When $n \ge 1$, the statement is a direct consequence of \cite[3.3.5]{DAG-V}.
We now prove the case $n = 0$.
Let $\mathcal X$ be an $\infty$-topos\xspace and let $\mathcal O \in \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)$.
For the purpose of this proof, we will say that a morphism $f \colon U \to V$ in $\cT_{\mathrm{an}}(k)$ is \emph{compatible} if the induced diagram
\begin{equation} \label{eq:compatibility_zero_truncations}
\begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le 0} \mathcal O(U) \arrow{d} \\
\mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(V)
\end{tikzcd}
\end{equation}
is a pullback square.
We need to show that every étale morphism is compatible.
Let us start by observing the following properties of compatible morphisms:
\begin{enumerate}
\item \label{item:compatible_analytic_domain} Admissible open immersions are compatible.
This follows from \cref{lem:G_pregeometry_0_truncations}.
\item \label{item:compatible_analytification_etale} If $f \colon X \to Y$ is an \'etale morphism of smooth $k$-varieties, then the analytification $f^\mathrm{an} \colon X^\mathrm{an} \to Y^\mathrm{an}$ is compatible.
Indeed, let $\cT_{\mathrm{\acute{e}t}}(k)$ be the pregeometry of \cite[Definition 4.3.1]{DAG-V}.
The analytification functor induces a morphism of pregeometries $\varphi \colon \cT_{\mathrm{\acute{e}t}}(k) \to \cT_{\mathrm{an}}(k)$.
We have $\mathcal O(X^\mathrm{an}) = (\mathcal O \circ \varphi)(X)$ and $\mathcal O(Y^\mathrm{an}) = (\mathcal O \circ \varphi)(Y)$.
Since $\mathcal O \circ \varphi$ is a $\cT_{\mathrm{\acute{e}t}}(k)$-structure on $\mathcal X$, the statement follows from the fact that $\cT_{\mathrm{\acute{e}t}}(k)$ is compatible with $0$-truncations (cf.\ \cite[Proposition 4.3.28]{DAG-V}).
\item \label{item:compatible_composition} Compatible morphisms are stable under composition. This follows from the composition property of pullback squares (cf.\ \cite[4.4.2.1]{HTT}).
\item \label{item:compatible_pullback} Suppose given a pullback square
\[ \begin{tikzcd}
U \arrow{r}{g} \arrow{d}{f'} & V \arrow{d}{j} \\
X \arrow{r}{f} & Y
\end{tikzcd} \]
where $f$ is compatible and $j$ is an admissible open immersion.
Then $f'$ is compatible.
To see this, consider the commutative diagram
\[ \begin{tikzcd}
\mathcal O(U) \arrow{d} \arrow{r} & \mathcal O(X) \arrow{d} \arrow{r} & \tau_{\le 0} \mathcal O(X) \arrow{d} \\
\mathcal O(V) \arrow{r} & \mathcal O(Y) \arrow{r} & \tau_{\le 0} \mathcal O(Y).
\end{tikzcd} \]
Since admissible open immersions are in particular étale morphisms and since $\mathcal O$ is a $\cT_{\mathrm{an}}(k)$-structure, we see that the left square is a pullback diagram.
On the other side, the right square is a pullback because $f$ is compatible by assumption.
We conclude that the outer square in the commutative diagram
\[ \begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le 0} \mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le 0} \mathcal O(X) \arrow{d} \\
\mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(Y)
\end{tikzcd} \]
is a pullback square.
We remark that $\tau_{\le 0} \circ \mathcal O$ is a $\cT_{\mathrm{an}}^G(k)$-structure in virtue of \cref{lem:G_pregeometry_0_truncations}.
So by \cite[Proposition 3.3.3]{DAG-V}, the right square is a pullback as well.
It follows that the left square is also a pullback, completing the proof of the claim.
\item \label{item:compatible_G_local_source} Being compatible is G-local on the source.
Indeed, let $f \colon X \to Y$ be a morphism in $\cT_{\mathrm{an}}(k)$ and assume there exists a G-covering $\{X_i\}$ of $X$ such that each composite $f_i \colon X_i \to X \to Y$ is compatible.
We want to prove that $f$ is compatible as well.
Consider the commutative diagram
\[ \begin{tikzcd}
\coprod \mathcal O(X_i) \arrow{r} \arrow{d} & \mathcal O(X) \arrow{r} \arrow{d} & \mathcal O(Y) \arrow{d} \\
\coprod \tau_{\le 0} \mathcal O(X_i) \arrow{r} & \tau_{\le 0} \mathcal O(X) \arrow{r} & \tau_{\le 0} \mathcal O(Y).
\end{tikzcd} \]
Since G-coverings are étale coverings, it follows from the properties of $\cT_{\mathrm{an}}(k)$-structures that the total morphism $\coprod \mathcal O(U_i) \to \mathcal O(U)$ is an effective epimorphism. Since $\tau_{\le 0}$ commutes with coproducts (being a left adjoint) and with effective epimorphisms (cf.\ \cite[7.2.1.14]{HTT}), we conclude that the total morphism $\coprod \tau_{\le 0} \mathcal O(U_i) \to \tau_{\le 0} \mathcal O(U)$ is an effective epimorphism as well.
Since each $X_i \to X$ is an admissible open immersion, Property (\ref{item:compatible_analytic_domain}) implies that the left square is a pullback.
Moreover, the outer square is a pullback by hypothesis.
Thus, \cref{lem:pullbacks_are_local} shows that the right square is a pullback as well, completing the proof of this property.
\end{enumerate}
Let now $f \colon U \to V$ be an étale morphism in $\cT_{\mathrm{an}}(k)$.
We will prove that $f$ is compatible.
Using \cref{lem:etale_analytic_domain} we obtain a G-covering $\{V_i\to V\}_{i\in I}$, G-coverings $\{U_{ij}\to U\times_V V_i\}_{j\in J_i}$ for every $i\in I$, smooth algebraic $k$-varieties $Y_i$ and $X_{ij}$, étale morphisms $X_{ij}\to Y_i$, admissible open immersions $V_i\hookrightarrow Y_i^\mathrm{an}$ and $U_{ij}\hookrightarrow X_{ij}^\mathrm{an}$ such that the morphism $U_{ij}\to V_i$ equals to restriction of the morphism $X_{ij}^\mathrm{an}\to Y_i^\mathrm{an}$.
In particular we can factor $U_{ij} \to V_i$ as the composition
\[ \begin{tikzcd}
U_{ij} \arrow{r} & X_{ij}^\mathrm{an} \times_{Y_i^\mathrm{an}} V_i \arrow{r} & V_i
\end{tikzcd} \]
where the first morphism is an admissible open immersion and the second is compatible by Property (\ref{item:compatible_pullback}) of compatible morphisms.
Therefore, Property (\ref{item:compatible_composition}) implies that $U_{ij} \to V_i$ is compatible.
Finally, using Property (\ref{item:compatible_G_local_source}) we conclude that the morphisms $U \times_V V_i \to V_i$ are compatible.
We are therefore reduced to prove the following statement: given a morphism $f \colon U \to V$, suppose that there exists a G-covering $\{v_i \colon V_i \to V\}$ such that each base change $f_i \colon U_i \coloneqq U \times_V V_i \to V_i$ is compatible, then $f$ is compatible.
We consider the commutative diagram
\[ \begin{tikzcd}
\coprod \mathcal O(U_i) \arrow{r} \arrow{d} & \mathcal O(U) \arrow{d} \arrow{r} & \tau_{\le 0} \mathcal O(U) \arrow{d} \\
\coprod \mathcal O(V_i) \arrow{r} & \mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(V).
\end{tikzcd} \]
Since $\mathcal O$ is a $\cT_{\mathrm{an}}(k)$-structure, the total morphism $\coprod \mathcal O(U_i) \to \mathcal O(U)$ is an effective epimorphism.
Moreover, since each $V_i \to V$ is an admissible open immersion, so in particular étale, we see that the left square is a pullback.
By hypothesis, the outer square is a pullback as well, so we conclude the proof using \cref{lem:pullbacks_are_local}.
\end{proof}
\begin{cor} \label{cor:truncation_derived_kanal_spaces}
Let $(\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space.
Then $(\mathcal X, \pi_0( \mathcal O_\mathcal X ))$ is also a derived $k$-analytic\xspace space.
Moreover, we have $(\pi_0(\mathcal O_\mathcal X))^\mathrm{alg} \simeq \pi_0(\mathcal O_\mathcal X^\mathrm{alg})$.
\end{cor}
\begin{proof}
It follows from \cref{thm:compatibility_truncations} that $\pi_0(\mathcal O_\mathcal X)$ is a $\cT_{\mathrm{an}}(k)$-structure on $\mathcal X$.
Let $\varphi \colon \cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ be the transformation of pregeometries induced by the analytification functor.
Then we have by definition
\[ (\pi_0(\mathcal O_\mathcal X))^\mathrm{alg} = (\pi_0^{\mathcal X} \circ \mathcal O_\mathcal X ) \circ \varphi \simeq \pi_0^\mathcal X \circ ( \mathcal O_\mathcal X \circ \varphi) = \pi_0( \mathcal O_\mathcal X^\mathrm{alg}) , \]
where $\pi_0^\mathcal X$ denotes the truncation functor of the $\infty$-topos\xspace $\mathcal X$.
In particular, we see that $(\mathcal X, \pi_0(\mathcal O_\mathcal X)^\mathrm{alg})$ is a derived $k$-analytic\xspace space.
\end{proof}
\begin{defin}
A $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos $(\mathcal X, \mathcal O_\mathcal X)$ is said to be \emph{discrete} if $\mathcal O_\mathcal X$ is a discrete object in $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)$.
We denote by $\tensor*[^\rR]{\Top}{}^0(\cT_{\mathrm{an}}(k))$ the full subcategory of $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ spanned by discrete $\cT_{\mathrm{an}}(k)$-structured $\infty$-topoi\xspace.
We say that a derived $k$-analytic\xspace space $(\mathcal X, \mathcal O_\mathcal X)$ is \emph{discrete} if it is discrete as a $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace.
We denote by $\mathrm{dAn}_k^0$ the full subcategory of $\mathrm{dAn}_k$ spanned by discrete derived $k$-analytic\xspace spaces.
\end{defin}
Choose a geometric envelope $\mathcal G_{\mathrm{an}}(k)$ for $\cT_{\mathrm{an}}(k)$ and let $\mathcal G_{\mathrm{an}}(k) \to \mathcal G_{\mathrm{an}}^{\le 0}(k)$ be a $0$-stub for $\mathcal G_{\mathrm{an}}(k)$ (cf.\ \cite[Definition 1.5.10]{DAG-V}).
It follows from \cite[Proposition 1.5.14]{DAG-V} that
\[ \tensor*[^\rR]{\Top}{}(\mathcal G_{\mathrm{an}}^{\le 0}(k)) \simeq \tensor*[^\rR]{\Top}{}^0(\cT_{\mathrm{an}}(k)) . \]
The relative spectrum (cf.\ \cite[§ 2.1]{DAG-V}) is a functor
\[ \Spec^{\cG_{\mathrm{an}}^{\le 0}(k)}_{\cG_{\mathrm{an}(k)}} \colon \tensor*[^\rR]{\Top}{}(\mathcal G_{\mathrm{an}}(k)) \to \tensor*[^\rR]{\Top}{}(\mathcal G_{\mathrm{an}}^{\le 0}(k)) \simeq \tensor*[^\rR]{\Top}{}^0(\cT_{\mathrm{an}}(k)) , \]
which we refer to as the \emph{truncation functor}.
Using \cref{thm:compatibility_truncations}, we can identify the action of this functor on objects with the assignment
\[ (\mathcal X, \mathcal O_\mathcal X) \mapsto (\mathcal X, \pi_0(\mathcal O_\mathcal X)) . \]
The following proposition is an analogue of \cite[Proposition 3.13]{Porta_DCAGI} and of \cite[Proposition 2.2.4.4]{HAG-II}:
\begin{prop} \label{prop:truncation_and_finite_limits}
Let $i \colon \mathrm{dAn}_k^0 \to \mathrm{dAn}_k$ denote the natural inclusion functor. Then:
\begin{enumerate}[(i)]
\item \label{item:truncation_derived_analytic_spaces} The functor $\Spec^{\cG_{\mathrm{an}}^{\le 0}(k)}_{\cG_{\mathrm{an}(k)}} \colon \mathcal T\mathrm{op}(\cT_{\mathrm{an}}(k)) \to \mathcal T\mathrm{op}^0(\cT_{\mathrm{an}}(k))$ restricts to a functor $\mathrm{t}_0 \colon \mathrm{dAn}_k \to \mathrm{dAn}_k^0$.
\item \label{item:truncation_right_adjoint} The functor $i$ is left adjoint to the functor $\mathrm{t}_0$.
\item \label{item:truncated_spaces_embeds_fully_faithfully} The functor $i$ is fully faithful.
\end{enumerate}
\end{prop}
\begin{proof}
The statement (\ref{item:truncated_spaces_embeds_fully_faithfully}) holds by definition of the functor $i$.
It follows from \cref{cor:truncation_derived_kanal_spaces} that the functor $\Spec^{\cG_{\mathrm{an}}^{\le 0}(k)}_{\cG_{\mathrm{an}(k)}}$ respects the $\infty$-category\xspace of derived $k$-analytic\xspace spaces.
Therefore the statements (\ref{item:truncation_derived_analytic_spaces}) and (\ref{item:truncation_right_adjoint}) follow immediately.
\end{proof}
\section{Fully faithful embedding of $k$-analytic\xspace spaces} \label{sec:fullyfaithfulness}
In this section, we construct a functor $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ from the category of $k$-analytic\xspace spaces to the category of derived $k$-analytic\xspace spaces.
We will prove that $\Phi$ is a fully faithful embedding.
First we will define the functor $\Phi$ on objects, then we will define it on morphisms.
Let $\mathrm{dAn}_k^{1, 0}$ be the full subcategory of $\mathrm{dAn}_k$ spanned by derived $k$-analytic\xspace spaces $(\mathcal X, \mathcal O_{\mathcal X})$ such that $\mathcal X$ is $1$-localic and $\mathcal O_{\mathcal X}$ is $0$-truncated.
\begin{defin}
Let $X$ be a $k$-analytic\xspace space and let $\mathcal X_X$ be the étale $\infty$-topos\xspace associated to $X$.
We define a functor $\mathcal O_X\colon\cT_{\mathrm{an}}(k)\to\mathcal X_X$ by the formula
\[ \mathcal O_X (M) (U) = \Hom_{\mathrm{An}_k} ( U, M ).\]
\end{defin}
\begin{lem} \label{lem:inclusion}
Let $X$ be a $k$-analytic\xspace space.
Then $\mathcal O_X$ is a 0-truncated $\cT_{\mathrm{an}}(k)$-structure on the $\infty$-topos\xspace $\mathcal X_X$.
Let $\mathcal X_X^\wedge$ denote the hypercompletion of $\mathcal X_X$ and let $\Phi(X)$ denote the pair $(\mathcal X_X^\wedge,\mathcal O_X)$.
Then $\Phi(X)$ is a derived $k$-analytic\xspace space.
\end{lem}
\begin{proof}
In order to prove that $\mathcal O_X$ is a $\cT_{\mathrm{an}}(k)$-structure on $\mathcal X_X$, it suffices to verify that if $\{ M_i \to M\}$ is an étale covering of $M\in\cT_{\mathrm{an}}(k)$, then the induced map $\coprod_i \mathcal O_X (M_i) \to \mathcal O_X (M)$ is an effective epimorphism in $\mathcal X_X$.
Observe that for any $U$ in the étale site on $X$ and any morphism $U \to M$, there exists an étale covering $\{U_j\to U\}$ such that the composite morphisms $U_j\to M$ factor though $\coprod M_i \to M$.
So we conclude using \cite[Corollary 2.9]{Porta_Yu_Higher_analytic_stacks_2014}.
Since $\mathcal O_X$ is $0$-truncated by construction, it is hypercomplete.
Therefore the second statement follows from the first.
\end{proof}
In order to define the functor $\Phi$ on morphisms, our strategy is to prove that the mapping spaces $\Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y))$ are discrete for all $X, Y \in \mathrm{An}_k$ (cf.\ \cref{prop:discrete_mapping_spaces_I}).
In this way we can promote $\Phi$ to an $\infty$-functor without the need to specify higher homotopies.
We begin by introducing an auxiliary functor $\Upsilon$.
Let $\mathrm{LRT}$ denote the $2$-category of locally ringed 1-topoi and let $\Upsilon\colon\mathrm{An}_k\to\mathrm{LRT}$ be the functor sending every $k$-analytic\xspace space to the associated locally ringed étale 1-topos.
For $X\in\mathrm{An}_k$, we denote by $\mathcal O_X^\mathrm{alg}$ the structure sheaf of $k$-algebras of $\Upsilon(X)$.
\begin{lem}\label{lem:affine_embedding}
Let $X$ be a $k$-affinoid space, $Y$ a $k$-analytic space and $\alpha\colon X\to Y$ a morphism.
Then there exists a positive integer $N$ and a monomorphism $\beta\colon X\hookrightarrow\mathbf D^N_Y$ over $Y$, where $\mathbf D^N_Y$ denotes the unit polydisc over $Y$.
\end{lem}
\begin{proof}
Let $A\coloneqq\Gamma(\mathcal O_X)$.
Write $A=k\langle x_1,\dots,x_n\rangle /I$ as a quotient of a Tate algebra.
Denote by $a_1,\dots,a_n$ the images of $x_1,\dots,x_n$ in $A$.
We cover $X$ by finitely many rational domains $U_i$ such that $\alpha(U_i)$ is contained in an affinoid domain $V_i\subset Y$.
Write \[\Gamma(\mathcal O_{U_i})=A\Big\langle\frac{b_{i1}}{b_{i0}},\dots,\frac{b_{in_i}}{b_{i0}}\Big\rangle,\]
where $b_{i0},\dots,b_{in_i}$ is a collection of elements in $A$ with no common zero.
Let $c_{i0},\dots,c_{in_i}$ be elements in $k$ such that $\abs{c_{ij}}\ge\rho(b_{ij})$ for $j=0,\dots,n_i$, where $\rho(\cdot)$ denotes the spectral radius.
Consider the morphism
\[\Gamma(\mathcal O_{V_i})\langle y_1,\dots,y_n, y_{i0},\dots,y_{in_i}\rangle\to\Gamma(\mathcal O_{U_i})\]
that sends $y_j$ to $a_j$ and $y_{ij}$ to $b_{ij}/c_{ij}$.
It induces a monomorphism $U_i\hookrightarrow\mathbf D^{n+n_i+1}_{V_i}$.
Let $N\coloneqq n+\sum_{i=1}^m(n_i+1)$.
Consider the unit polydisc $\mathbf D^N_Y$ over $Y$.
We denote by $y_i,y_{ij}$ for $i=1,\dots,m$, $j=0,\dots,n_i$ the coordinate functions on $\mathbf D^N_Y$.
Let $\beta\colon X\to\mathbf D^N_Y$ be the morphism that sends $y_i$ to $a_i$ and $y_{ij}$ to $b_{ij}/c_{ij}$ for all $i=1,\dots,m$, $j=0,\dots,n_i$.
Let $Z_i$ be the admissible open subset in $\mathbf D^N_Y$ given by the inequalities $\abs{c_{i0}\cdot y_{ij}}\le\abs{c_{i0}\cdot y_{i0}}$ for $j=1,\dots,n_i$.
Let $Z'_i\coloneqq Z_i\times_Y V_i$.
We see that $\beta^{-1}(Z'_i)$ is $U_i$.
By construction, $\beta|_{U_i}\colon U_i\to Z'_i$ is a monomorphism.
We conclude that $\beta\colon X\to\mathbf D^N_Y$ is a monomorphism.
\end{proof}
\begin{lem}\label{lem:rigidity}
Let $f\colon X\to Y$ be a morphism of $k$-analytic\xspace spaces.
Let
\[(f,f^\#)\colon\big(\mathrm{Sh}_\mathrm{Set} (X_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_X\big) \to \big(\mathrm{Sh}_\mathrm{Set} (Y_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_Y\big)\]
denote the induced morphism of locally ringed 1-topoi.
Let $t$ be a 2-morphism from $(f,f^\#)$ to itself.
Then $t$ equals the identity.
\end{lem}
\begin{proof}
Using \cref{lem:affine_embedding}, the same proof of \cite[Tag 04IJ]{stacks-project} applies.
\end{proof}
\begin{lem}\label{lem:first_fully_faithful}
The functor
\begin{align*}
\Upsilon \colon \mathrm{An}_k &\longrightarrow \mathrm{LRT}\\
X &\longmapsto (\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}), \mathcal O_X^\mathrm{alg})
\end{align*}
is fully faithful.
\end{lem}
\begin{proof}
Let $X,Y$ be two $k$-analytic\xspace spaces.
Let \[(g,g^\#)\colon\big(\mathrm{Sh}_\mathrm{Set} (X_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_X\big) \to \big(\mathrm{Sh}_\mathrm{Set} (Y_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_Y\big)\]
be a morphism of locally ringed 1-topoi.
We would like to show that there exists a unique morphism of $k$-analytic\xspace spaces $f\colon X\to Y$ which induces $(g,g^\#)$.
We proceed along the same lines as \cite[Tag 04JH]{stacks-project}.
Let $g^{-1}\colon\mathrm{Sh}_\mathrm{Set} (Y_\mathrm{\acute{e}t}) \leftrightarrows \mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}) \colon g_*$ denote the morphism of 1-topoi.
First, we assume that $X=\Sp A$, $Y=\Sp B$ for some $k$-affinoid algebras $A$ and $B$.
Since $B=\Gamma(Y_\mathrm{\acute{e}t},\mathcal O^\mathrm{alg}_Y)$ and $A=\Gamma(X_\mathrm{\acute{e}t},\mathcal O^\mathrm{alg}_X)$, we see that $g^\#$ induces a map of affinoid algebras $\varphi\colon B\to A$.
Let $f=\Sp\varphi\colon X\to Y$.
Let us show that $f$ induces $(g,g^\#)$.
Let $V\to Y$ be an affinoid space étale over $Y$.
Assume $V=\Sp C$.
By \cite[Proposition 1.7.1]{Huber_Etale_1996}, we can write
\[ C=B\langle x_1,\dots,x_n\rangle / (r_1,\dots,r_n), \]
where $r_1,\dots,r_n\in B\langle x_1,\dots,x_n\rangle$ and the determinant $\mathrm{Jac}(r_1,\dots,r_n)$ is invertible in $C$.
Now the sheaf $h_V$ on $Y_\mathrm{\acute{e}t}$ is the equalizer of the two maps
\[
\xymatrix{
\prod_{i=1}^n \mathcal O^\mathrm{alg}_Y \ar@<0.6ex>[r]^{a} \ar@<-0.6ex>[r]_{b} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_Y
}
\]
where $b=0$ and $a(h_1,\dots,h_n)=\big(r_1(h_1,\dots,h_n),\dots,r_n(h_1,\dots,h_n)\big)$.
We have the following commutative diagram
\begin{equation}\label{eq:equalizers}
\xymatrix{
g^{-1} h_V \ar@{.>}[d]^{\alpha} \ar[r] & \prod_{i=1}^n g^{-1} \mathcal O^\mathrm{alg}_Y \ar[d]^{\prod g^\#} \ar@<0.6ex>[r]^{g^{-1} a} \ar@<-0.6ex>[r]_{g^{-1} b} & \prod_{j=1}^n g^{-1}\mathcal O^\mathrm{alg}_Y \ar[d]^{\prod g^\#}\\
h_{X\times_Y V}\ar[r] & \prod_{i=1}^n \mathcal O^\mathrm{alg}_X \ar@<0.6ex>[r]^{a'} \ar@<-0.6ex>[r]_{b'} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_X,
}
\end{equation}
where $b'=0$, $a'(h_1,\dots,h_n)=\big(\varphi(r_1)(h_1,\dots,h_n),\dots,\varphi(r_n)(h_1,\dots,h_n)\big)$, the two horizontal lines are equalizer diagrams and the dotted arrow $\alpha$ is obtained by the universal property of equalizers.
We claim that the map $\alpha\colon g^{-1} h_V\to h_{X\times_Y V}$ is an isomorphism.
Let us check this on the stalks.
Let $\bar x$ be a geometric point of the adic space $X^\mathrm{ad}$ associated to $X$ in the sense of Huber.
Denote by $p$ the associated point of the 1-topos $\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t})$ (cf.\ \cref{rem:points_of_cX_X}).
Applying localization at $p$ to Diagram \eqref{eq:equalizers}, we would like to show that $\alpha_p\colon (g^{-1} h_V)_p\to (h_{X\times_Y V})_p$ is an isomorphism.
Set $q\coloneqq g\circ p$.
This is a point of the 1-topos $\mathrm{Sh}_\mathrm{Set}(Y_\mathrm{\acute{e}t})$.
We denote by $\bar y$ the corresponding geometric point of the adic space $Y^\mathrm{ad}$ associated to $Y$.
Then the localization of the map $g^\#$ at $p$ has the following description
\[(g^\#)_p\colon\mathcal O^\mathrm{alg}_{Y,\bar y} = \mathcal O^\mathrm{alg}_{Y,q} = (g^{-1} \mathcal O^\mathrm{alg}_Y)_p \longrightarrow \mathcal O^\mathrm{alg}_{X,p} = \mathcal O^\mathrm{alg}_{X,\bar x}.\]
It suffices to treat the two cases: either $V\to Y$ is finite étale, or $V\to Y$ is an affinoid domain embedding.
In the first case, there exists an étale neighborhood $U$ of $\bar y$ in $Y^\mathrm{ad}$ such that the pullback morphism $V\times_Y U\to U$ splits.
Then the equalizer of
\begin{equation} \label{eq:O_Y_V}
\xymatrix{
\prod_{i=1}^n \mathcal O^\mathrm{alg}_Y(U) \ar@<0.6ex>[r]^{a} \ar@<-0.6ex>[r]_{b} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_Y(U)
}
\end{equation}
is isomorphic to the equalizer of
\begin{equation} \label{eq:ky}
\xymatrix{
\prod_{i=1}^n k(\bar y) \ar@<0.6ex>[r]^{a} \ar@<-0.6ex>[r]_{b} & \prod_{j=1}^n k(\bar y),
}
\end{equation}
where $k(\bar y)$ denotes the residue field of $\bar y$.
Similarly, there exists an étale neighborhood $U'$ of $\bar x$ in $X^\mathrm{ad}$ such that the pullback morphism $X\times_Y V\times_X U'\simeq V\times_Y U'\to U'$ splits.
Then the equalizer of
\begin{equation} \label{eq:O_X_V}
\xymatrix{
\prod_{i=1}^n \mathcal O^\mathrm{alg}_X(U') \ar@<0.6ex>[r]^{a'} \ar@<-0.6ex>[r]_{b'} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_X(U')
}
\end{equation}
is isomorphic to the equalizer of
\begin{equation} \label{eq:kx}
\xymatrix{
\prod_{i=1}^n k(\bar x) \ar@<0.6ex>[r]^{a'} \ar@<-0.6ex>[r]_{b'} & \prod_{j=1}^n k(\bar x).
}
\end{equation}
Since the equalizer of \cref{eq:ky} and the equalizer of \cref{eq:kx} are isomorphic by construction,
we deduce that the equalizer of \cref{eq:O_Y_V} and the equalizer of \cref{eq:O_X_V} are isomorphic.
Taking colimits over all such étale neighborhoods, we conclude that $\alpha_p\colon (g^{-1} h_V)_p\to (h_{X\times_Y V})_p$ is an isomorphism.
Then let us consider the second case where $V\to Y$ is an affinoid domain embedding.
If the geometric point $\bar y$ can be lifted to a geometric point in $V$, then for any étale neighborhood $U$ of $\bar y$ in $Y^\mathrm{ad}$ refining $V$, the equalizer of \cref{eq:O_Y_V} consists of a single element.
The same goes for the equalizer of \cref{eq:O_X_V}.
If the geometric point cannot be lifted to a geometric point in $V$, then the equalizer of \cref{eq:O_Y_V} is empty, so is the equalizer of \cref{eq:O_X_V}.
We conclude that $\alpha_p\colon (g^{-1} h_V)_p\to (h_{X\times_Y V})_p$ is an isomorphism.
Now the same argument in \cite[Tag 04I6]{stacks-project} shows that the isomorphisms $g^{-1} h_V\to h_{X\times_Y V}$ are functorial with respect to $V$ and that the map $f\colon X\to Y$ indeed induces the morphism of locally ringed 1-topoi $(g,g^\#)$ we started with.
Finally, the argument in \cite[Tag 04I7]{stacks-project} allows us to deduce the general case from the affinoid case considered above.
\end{proof}
\begin{lem}\label{lem:alg_faithful}
Let $\mathcal X$ be an $\infty$-topos\xspace.
The induced functor
\[
\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\tau_{\le 0} \mathcal X)
\]
is faithful.
\end{lem}
\begin{proof}
We can factor the functor $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\tau_{\le 0} \mathcal X)$ as
\[ \begin{tikzcd}
\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \arrow{r} & \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\tau_{\le 0} \mathcal X) \arrow{r}{U} & \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\tau_{\le 0} \mathcal X) ,
\end{tikzcd} \]
where $\cT_{\mathrm{\acute{e}t}}(k)$ is the pregeometry introduced in \cite[Definition 4.3.1]{DAG-V}.
Combining \cite[Propositions 4.3.16, 2.6.16 and Remark 2.5.13]{DAG-V} we see that the functor $U$ is faithful.
So we are reduced to prove the same statement for
\[ \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\tau_{\le 0} \mathcal X) . \]
The mapping spaces of $\tau_{\le 0} \mathcal X$ are discrete by definition.
It follows from \cite[2.3.4.18]{HTT} that we can find a minimal $1$-category $\mathcal D$ and a categorical equivalence $\tau_{\le 0} \mathcal X \simeq \mathcal D$.
Let $F, G \in \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal D)$. We want to show that the natural morphism
\[
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal D)}(F,G) \to \Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\mathcal D)}(F^\mathrm{alg}, G^\mathrm{alg})
\]
is a homotopy monomorphism.
Since $F$ and $G$ take values in the $1$-category $\mathcal D$, both mapping spaces above are sets.
We want to prove that the given map is a monomorphism.
Since $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\mathcal D)$ is a $1$-category, two natural transformations $\varphi$ and $\psi$ represent the same object in $\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\mathcal D)}(F^\mathrm{alg}, G^\mathrm{alg})$ if and only if they are equal, in the sense that
\[
\begin{tikzcd}
F^\mathrm{alg}(X) \arrow{r}{\varphi^\mathrm{alg}_{X}} \arrow{d}[swap]{\mathrm{id}_{F^\mathrm{alg}(X)}} & G^\mathrm{alg}(X) \arrow{d}{\mathrm{id}_{G^\mathrm{alg}(X)}} \\
F^\mathrm{alg}(X) \arrow{r}{\psi^\mathrm{alg}_{X}} & G^\mathrm{alg}(X)
\end{tikzcd}
\]
commutes for every $X \in \cT_{\mathrm{\acute{e}t}}(k)$.
Fix $U \in \cT_{\mathrm{an}}(k)$.
We first assume that $U$ is isomorphic to an affinoid domain in $X^\mathrm{an}$ for a smooth $k$-variety $X$.
Since $U \to X^\mathrm{an}$ is a monomorphism, we have a pullback square
\[\begin{tikzcd}
U \arrow[-, double equal sign distance]{r} \arrow[-, double equal sign distance]{d} & U \arrow[hook]{d} \\
U \arrow[hook]{r} & X^\mathrm{an}.
\end{tikzcd}\]
Since $U \to X$ is an affinoid embedding, it is étale, so it is an admissible morphism in $\cT_{\mathrm{an}}(k)$.
Applying the functor $F$, we obtain another pullback square
\[\begin{tikzcd}
F(U) \arrow[-, double equal sign distance]{r} \arrow[-, double equal sign distance]{d} & F(U) \arrow{d} \\
F(U) \arrow{r} & F(X^\mathrm{an}).
\end{tikzcd}\]
So $F(U) \to F(X^\mathrm{an})$ is a monomorphism in the category $\mathcal D$.
We have a commutative cube
\[
\begin{tikzcd}[column sep=small, row sep=small]
{} & F(U) \arrow{dl} \arrow{rr} \arrow[dotted]{dd} & & F(X^\mathrm{an}) \arrow{dl} \\
G(U) \arrow[crossing over]{rr} & & G(X^\mathrm{an}) \\
{} & F(U) \arrow{rr} \arrow{dl} & & F(X^\mathrm{an}) \arrow[-, double equal sign distance]{uu} \arrow{dl} \\
G(U) \arrow{rr} \arrow[-, double equal sign distance]{uu} & & G(X^\mathrm{an}) \arrow[-, double equal sign distance, crossing over]{uu}
\end{tikzcd}
\]
where the dotted arrow exists by the universal property of the pullbacks.
Since $F(U) \to F(X^\mathrm{an})$ is a monomorphism, the dotted arrow is in fact the identity of $F(U)$.
Let us now consider a general $U \in \cT_{\mathrm{an}}(k)$.
Choose a G-covering of $U$ by affinoid domains $\{U_i \to U\}$ such that each $U_i$ is isomorphic to an affinoid domain in $X_i^\mathrm{an}$ for some smooth $k$-variety $X_i$.
Set $U^0 \coloneqq \coprod U_i$ and consider the \v{C}ech nerve $U^\bullet \to U$. Observe that both $F(U^\bullet)$ and $G(U^\bullet)$ are groupoid objects in the 1-topos $\mathcal D$ and that their realizations are respectively $F(U)$ and $G(U)$. Since we have a commutative square of groupoids
\[
\begin{tikzcd}
F(U^\bullet) \arrow{r}{\varphi_{U^\bullet}} \arrow[-, double equal sign distance]{d} & G(U^\bullet) \arrow[-, double equal sign distance]{d} \\
F(U^\bullet) \arrow{r}{\psi_{U^\bullet}} & G(U^\bullet),
\end{tikzcd}
\]
the square
\[
\begin{tikzcd}
F(U) \arrow{r}{\varphi_U} \arrow[-, double equal sign distance]{d} & G(U) \arrow[-, double equal sign distance]{d} \\
F(U) \arrow{r}{\psi_U} & G(U)
\end{tikzcd}
\]
commutes as well.
Since the identity is functorial, the proof is now complete.
\end{proof}
\begin{lem}\label{lem:1_truncated_mapping_space}
Let $\mathcal T$ be a pregeometry and let $(\mathcal X, \mathcal O_\mathcal X)$, $(\mathcal Y, \mathcal O_\mathcal Y)$ be $\mathcal T$-structured $\infty$-topoi\xspace such that $\mathcal X$ and $\mathcal Y$ are $1$-localic and the structure sheaves $\mathcal O_\mathcal X$, $\mathcal O_\mathcal Y$ are discrete.
Then $\Map_{\tensor*[^\rR]{\Top}{}(\mathcal T)}((\mathcal X, \mathcal O_{\mathcal X}), (\mathcal Y, \mathcal O_{\mathcal Y}))$ is $1$-truncated.
Moreover, the canonical morphism
\[ \Map_{\tensor*[^\rR]{\Top}{}(\mathcal T)}((\mathcal X, \mathcal O_{\mathcal X}), (\mathcal Y, \mathcal O_{\mathcal Y})) \to \Map_{\tensor*[^\rR]{\Top}{}_1(\mathcal T)}((\tau_{\le 0} \mathcal X, \mathcal O_{\mathcal X}), (\tau_{\le 0} \mathcal Y, \mathcal O_{\mathcal Y})) \]
is an equivalence, where $\tensor*[^\rR]{\Top}{}_1$ denotes the $\infty$-category of 1-topoi with morphisms being right adjoint geometric morphisms.
\end{lem}
\begin{proof}
Consider the coCartesian fibration $\tensor*[^\rL]{\Top}{}(\mathcal T) \to \tensor*[^\rL]{\Top}{}$.
We know from \cite[Remark 1.4.10]{DAG-V} that the fiber over an $\infty$-topos\xspace $\mathcal X$ is equivalent to $\mathrm{Str}^\mathrm{loc}_{\mathcal T}(\mathcal X)$.
Let $f^{-1} \colon \mathcal X \rightleftarrows \mathcal Y \colon f_*$
be a geometric morphism between $\mathcal X$ and $\mathcal Y$.
Using \cite[2.4.4.2]{HTT} and \cite[Remark 1.4.10]{DAG-V} we obtain a fiber sequence
\[ \begin{tikzcd}
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)}(f^{-1} \mathcal O_\mathcal Y, \mathcal O_\mathcal X) \arrow{r} & \Map_{\tensor*[^\rR]{\Top}{}(\mathcal T)}((\mathcal X, \mathcal O_{\mathcal X}), (\mathcal Y, \mathcal O_{\mathcal Y})) \arrow{r} & \Fun_*(\mathcal X, \mathcal Y) ,
\end{tikzcd} \]
where the fiber is taken at the geometric morphism $(f^{-1}, f_*)$.
Since both $\mathcal X$ and $\mathcal Y$ are $1$-localic, there is an equivalence
\[ \Fun_*(\mathcal X, \mathcal Y) \simeq \Fun_*(\tau_{\le 0} \mathcal X, \tau_{\le 0} \mathcal Y). \]
Therefore $\Fun_*(\mathcal X, \mathcal Y)$ is $1$-truncated.
On the other side, $\mathcal O_\mathcal X$ is $0$-truncated, so $\Map_{\mathrm{Str}^\mathrm{loc}_{\mathcal T}(\mathcal X)}(f^{-1} \mathcal O_\mathcal Y, \mathcal O_\mathcal X)$ is discrete.
The second statement follows as well.
\end{proof}
\begin{lem} \label{lem:alg_homotopy_monomorphism}
Let $X = (\mathcal X, \mathcal O_{\mathcal X})$ and $Y = (\mathcal Y, \mathcal O_{\mathcal Y})$ be two $\cT_{\mathrm{an}}(k)$-structured $\infty$-topoi\xspace.
Let $X^\mathrm{alg} \coloneqq (\mathcal X, \mathcal O_\mathcal X^\mathrm{alg})$ and $Y^\mathrm{alg} \coloneqq (\mathcal Y, \mathcal O_\mathcal Y^\mathrm{alg})$ be the underlying $\cT_{\mathrm{disc}}(k)$-structured $\infty$-topoi\xspace.
Assume that $\mathcal X$ and $\mathcal Y$ are $1$-localic and that $\mathcal O_{\mathcal X}$ and $\mathcal O_{\mathcal Y}$ are $0$-truncated.
Then the canonical map
\[ \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(X, Y) \to \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) \]
induces monomorphisms on $\pi_0$ and on $\pi_1$ (for every choice of base point).
\end{lem}
\begin{proof}
Let $f_* \colon \mathcal X \rightleftarrows \mathcal Y \colon f^{-1}$ be a geometric morphism in $\tensor*[^\rR]{\Top}{}$.
We have a commutative diagram in $\mathcal S$:
\[ \begin{tikzcd}[column sep=small]
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)}(f^{-1} \mathcal O_\mathcal Y, \mathcal O_{\mathcal X}) \arrow{r} \arrow{d} & \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))} (X, Y) \arrow{r} \arrow{d} & \Fun_*(\mathcal X, \mathcal Y) \arrow[-, double equal sign distance]{d} \\
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X)}(f^{-1} \mathcal O_{\mathcal Y}^\mathrm{alg}, \mathcal O_{\mathcal X}^\mathrm{alg}) \arrow{r} & \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) \arrow{r} & \Fun_*(\mathcal X, \mathcal Y).
\end{tikzcd} \]
Using \cite[2.4.4.2]{HTT} and \cite[Remark 1.4.10]{DAG-V} we see that the two horizontal lines are fiber sequences.
Moreover, since $\mathcal O_\mathcal X$ and $\mathcal O_\mathcal Y$ are $0$-truncated, we can use \cref{lem:alg_faithful} to deduce that the first vertical map is a homotopy monomorphism.
Passing to the long exact sequences of homotopy groups and applying the five lemma, we obtain monomorphisms
\begin{gather*}
\pi_0 \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(X, Y) \to \pi_0 \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) , \\
\pi_1 \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(X, Y) \to \pi_1 \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) ,
\end{gather*}
completing the proof.
\end{proof}
\begin{lem} \label{lem:mapping_space_hypercompletion_localic}
Let $\mathcal Y$ be an $n$-localic $\infty$-topos\xspace and let $\mathcal X$ be any $\infty$-topos\xspace.
Then there is a canonical equivalence in the homotopy category of spaces $\mathcal H$:
\[ \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal X^\wedge, \mathcal Y^\wedge) \simeq \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal X, \mathcal Y) . \]
\end{lem}
\begin{proof}
Using \cite[6.5.2.13]{HTT} we see that the canonical morphism
\[ \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal X^\wedge, \mathcal Y^\wedge) \to \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal X^\wedge, \mathcal Y) \]
is an equivalence.
Since $\mathcal Y$ is $n$-localic, the restriction
\[ \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal X^\wedge, \mathcal Y) \to \Map_{\tensor*[^\rR]{\Top}{}_n}(\tau_{\le n - 1} (\mathcal X^\wedge), \tau_{\le n - 1} \mathcal Y) \]
is an equivalence as well.
On the other side, the restriction
\[ \Map_{\tensor*[^\rR]{\Top}{}}(\mathcal X, \mathcal Y) \to \Map_{\tensor*[^\rR]{\Top}{}_n}(\tau_{\le n - 1}\mathcal X, \tau_{\le n - 1} \mathcal Y) \]
is also an equivalence.
We now conclude by observing that $\tau_{\le n - 1} \mathcal X \simeq \tau_{\le n - 1}(\mathcal X^\wedge)$.
\end{proof}
\begin{prop} \label{prop:discrete_mapping_spaces_I}
Let $X, Y \in \mathrm{An}_k$. Then $\Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}(\Phi(X), \Phi(Y))$ is discrete.
\end{prop}
\begin{proof}
It follows from \cref{lem:mapping_space_hypercompletion_localic} that
\begin{equation} \label{eq:from_hypercomplete_to_non-hypercomplete}
\Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y)) \simeq \Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}((\mathcal X_X, \mathcal O_X), (\mathcal X_Y, \mathcal O_Y) ) .
\end{equation}
On the other side, \cref{lem:1_truncated_mapping_space} shows that the right hand side is $1$-truncated and
\begin{equation} \label{eq:from_sheaves_of_spaces_to_sheaves_of_sets}
\Map_{\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))}((\mathcal X_X, \mathcal O_X), (\mathcal X_Y, \mathcal O_Y) ) \simeq \Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{an}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y))
\end{equation}
We can now apply \cref{lem:alg_homotopy_monomorphism} to conclude that the canonical map
\begin{multline*}
\Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{an}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y)) \to\\
\Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{disc}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X^\mathrm{alg}), (\tau_{\le 0}\mathcal X_Y, \mathcal O_Y^\mathrm{alg}))
\end{multline*}
induces monomorphisms on $\pi_0$ and on $\pi_1$.
It follows from \cref{lem:first_fully_faithful} that the canonical map
\[ \Hom_{\mathrm{An}_k}(X,Y) \to \pi_0 \Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{disc}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X^\mathrm{alg}), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y^\mathrm{alg})) \]
is an isomorphism.
At this point, we can invoke \cref{lem:rigidity} to deduce that, for every choice of base point, we have
\[ \pi_1 \Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{disc}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X^\mathrm{alg}), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y^\mathrm{alg})) = 0 . \]
Thus, we conclude that
\[ \pi_1 \Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{an}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y)) = 0 \]
for every choice of base point.
It follows from the equivalences \eqref{eq:from_hypercomplete_to_non-hypercomplete} and \eqref{eq:from_sheaves_of_spaces_to_sheaves_of_sets} that $\Map_{\mathrm{dAn}_k}(\Phi(X),\allowbreak \Phi(Y))$ is discrete, completing the proof.
\end{proof}
We can now promote $X \mapsto \Phi(X)$ to an $\infty$-functor.
\todo{Be careful with $X_\mathrm{\acute{e}t}$ below.}
Let $\mathcal C$ temporarily denote the full subcategory of $\mathrm{dAn}_k$ spanned by the objects which are equivalent to $\Phi(X)$ for some $X \in \mathrm{An}_k$.
\cref{prop:discrete_mapping_spaces_I} shows that mapping spaces in $\mathcal C$ are discrete, hence $\mathcal C$ is equivalent to a $1$-category.
Fix a morphism $f \colon X \to Y$ in $\mathrm{An}_k$.
It induces a morphism of sites
\[ \varphi \colon Y_\mathrm{\acute{e}t} \to X_\mathrm{\acute{e}t} \]
given by base change along $f$.
Since all the morphisms in $X_\mathrm{\acute{e}t}$ and $Y_\mathrm{\acute{e}t}$ are étale, it follows that $X_\mathrm{\acute{e}t}$ and $Y_\mathrm{\acute{e}t}$ have fiber products.
Moreover, $\varphi$ is left exact.
Therefore, it follows from \cite[Lemma 2.16]{Porta_Yu_Higher_analytic_stacks_2014} that the induced adjunction
\[ \varphi^s \colon \mathcal X_Y \rightleftarrows \colon \mathcal X_X \colon \varphi_s \]
is a geometric morphism of $\infty$-topoi\xspace.
In particular, we obtain an induced geometric morphism $\mathcal X_Y^\wedge \rightleftarrows \mathcal X_X^\wedge$, which we denote by
\[ f^{-1} \colon \mathcal X^\wedge \rightleftarrows \mathcal X_X^\wedge \colon f_* . \]
We obtain in this way a well defined morphism $(\mathcal X_X^\wedge, \mathcal O_X) \to (\mathcal X_Y^\wedge, \mathcal O_Y)$.
Since mapping spaces in $\mathcal C$ are discrete, we see that this assignment is functorial.
We denote the resulting $\infty$-functor by
\[ \Phi \colon \mathrm{An}_k \to \mathrm{dAn}_k . \]
\begin{thm} \label{thm:fully_faithfulness}
The functor $\Phi\colon \mathrm{An}_k \rightarrow \mathrm{dAn}_k$ is fully faithful.
\end{thm}
\begin{proof}
Let $X, Y \in \mathrm{An}_k$. We want to show that
\[
\Hom_{\mathrm{An}_k}(X,Y) \to \Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y))
\]
is an equivalence.
\cref{lem:1_truncated_mapping_space} allows us to identify $\Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y))$ with
\[
\Map_{\tensor*[^\rR]{\Top}{}_1(\cT_{\mathrm{an}}(k))}\big((\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}), \mathcal O_X), (\mathrm{Sh}_\mathrm{Set}(Y_\mathrm{\acute{e}t}), \mathcal O_Y)\big) .
\]
Let us first prove the faithfulness.
Let $f, g \colon X \to Y$ be two morphisms and assume that $\Phi(f) = \Phi(g)$.
Since the question of $f$ being equal to $g$ is local on both $X$ and $Y$, we can assume that both $X$ and $Y$ are affinoid.
In this case, $f$ (resp.\ $g$) can be recovered as global section of the natural transformation $\Phi(f)(\mathbf A^1_k)$ (resp.\ $\Phi(g)(\mathbf A^1_k)$), where $\mathbf A^1_k$ denote the $k$-analytic\xspace affine line.
Therefore we have $f = g$.
Let us now turn to the fullness.
Let
\[ (f, f^\sharp) \colon (\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}), \mathcal O_X) \to (\mathrm{Sh}_\mathrm{Set}(Y_\mathrm{\acute{e}t}), \mathcal O_Y) \]
be a morphism in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$.
After forgetting along the morphism $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$, we get a morphism of locally ringed 1-topoi.
\cref{lem:first_fully_faithful} implies that this morphism comes from a map $\varphi \colon X \to Y$.
This means that $\Phi(\varphi)^\mathrm{alg}$ and $(f,f^\sharp)^\mathrm{alg}$ coincide.
\cref{lem:alg_faithful} implies that $\Phi(\varphi)$ and $(f,f^\sharp)$ coincide as well, completing the proof.
\end{proof}
\section{Closed immersions and étale morphisms} \label{sec:closed_etale}
In this section, we study closed immersions and étale morphisms under the fully faithful embedding $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$.
\begin{defin}[{\cite[1.1]{DAG-IX}},{\cite[2.3.1]{DAG-V}}]\label{def:closed_immersion_and_etale}
Let $\mathcal T$ be a pregeometry and $\mathcal X$, $\mathcal Y$ two $\infty$-topoi\xspace.
A morphism $\mathcal O\to\mathcal O'$ in $\mathrm{Str}^\mathrm{loc}_\mathcal T(\mathcal X)$ is said to be an \emph{effective epimorphism} if for every object $X\in\mathcal T$, the induced map $\mathcal O(X)\to\mathcal O'(X)$ is an effective epimorphism in $\mathcal X$.
A morphism $f\colon(\mathcal X, \mathcal O_\mathcal X)\to(\mathcal Y, \mathcal O_\mathcal Y)$ in $\tensor*[^\rR]{\Top}{}(\mathcal T)$ is called a \emph{closed immersion} (resp.\ an \emph{étale morphism}) if the following conditions are satisfied:
\begin{enumerate}[(i)]
\item the underlying geometric morphism $f_*\colon\mathcal X\to\mathcal Y$ is a closed immersion (resp.\ an étale morphism) of $\infty$-topoi\xspace;
\item the morphism of structure sheaves $f^{-1} \mathcal O_\mathcal Y \to \mathcal O_\mathcal X$ is an effective epimorphism (resp.\ an equivalence) in $\mathrm{Str}^\mathrm{loc}_\mathcal T(\mathcal Y)$.
\end{enumerate}
\end{defin}
\begin{lem} \label{lem:hypercompletion_closed_immersions}
The hypercompletion functor $\tensor*[^\rR]{\Top}{} \to \tensor*[^\rR]{\mathcal{H}\Top}{}$ preserves closed immersions.
\end{lem}
\begin{proof}
Let $f_* \colon \mathcal X \rightleftarrows \mathcal Y \colon f^{-1}$ be a closed immersion of $\infty$-topoi\xspace.
By definition we can find a $(-1)$-truncated object $U \in \mathcal Y$ such that the geometric morphism $f_*$ is equivalent to the induced geometric morphism $j_* \colon \mathcal Y / U \rightleftarrows \mathcal Y \colon j^{-1}$.
Since $U$ is $(-1)$-truncated, it belongs to $\mathcal Y^\wedge$.
It is therefore enough to prove that $(\mathcal Y / U)^\wedge \simeq \mathcal Y^\wedge / U$.
The geometric morphism $\mathcal Y / U \to \mathcal Y$ induces by passing to hypercompletions a morphism $(\mathcal Y / U)^\wedge \to \mathcal Y^\wedge$ which by construction fits in the commutative diagram
\[ \begin{tikzcd}
\mathcal Y / U \arrow{r}{j_*} & \mathcal Y \\
(\mathcal Y / U)^\wedge \arrow{u}{i_{U*}} \arrow{r}{j_*^\wedge} & \mathcal Y^\wedge \arrow{u}{i_*} .
\end{tikzcd} \]
Since $j_*$, $i_*$ and $i_{U*}$ are fully faithful, the same goes for $j_*^\wedge$.
Observe that by \cite[7.3.2.5]{HTT}, an object $V \in \mathcal Y^\wedge$ belongs to $\mathcal Y^\wedge / U$ if and only if $V \times U \simeq U$.
Since both $i_*$ and $j_*$ commute with products, we conclude that $j_*^\wedge$ factors through $\mathcal Y^\wedge / U$.
This provides us a fully faithful functor $(\mathcal Y / U)^\wedge \to \mathcal Y^\wedge / U$.
In order to complete the proof, it is enough to prove that it is essentially surjective.
The canonical map $\mathcal Y^\wedge / U \to \mathcal Y^\wedge \to \mathcal Y$ factors through $\mathcal Y / U$.
Now it suffices to prove that this functor can be further factored through $(\mathcal Y / U)^\wedge$.
This follows from the fact that $j_*$ respects the collection of $\infty$-connected morphisms.
To see this, let $V \in \mathcal Y / U$. Since $U$ is $(-1)$-truncated, we see that for every $n \ge 0$ one has:
\[ \tau_{\le n}(V) \times U \simeq \tau_{\le n}(V) \times \tau_{\le n}(U) \simeq \tau_{\le n}(V \times U) \simeq \tau_{\le n}(U) \simeq U . \]
In particular, $\tau_{\le n}(V)$ belongs to $\mathcal Y / U$ as well.
It follows that $j_*$ commutes with truncations, and therefore with $\infty$-connected morphisms.
\end{proof}
\begin{lem} \label{lem:different_closed_immersion}
Let $f^{-1} \colon \mathcal X \rightleftarrows \mathcal Y \colon f_*$ be a closed immersion of $\infty$-topoi\xspace.
Let $F \in \mathcal X$, $G \in \mathcal Y$ and let $f^{-1} F \to G$ be a morphism in $\mathcal Y$.
If the morphism $F \to f_* G$ is an effective epimorphism, then so is the morphism $f^{-1} F \to G$.
\end{lem}
\begin{proof}
Since $f^{-1}$ is left exact, it commutes with effective epimorphisms.
Therefore, $f^{-1} F \to f^{-1} f_* G$ is an effective epimorphism.
Since $f_*$ is fully faithful, we see that $f^{-1} f_* G \simeq G$, hence completing the proof.
\end{proof}
\begin{thm} \label{thm:Phi_classes_of_morphisms}
Let $f \colon X \to Y$ be a morphism in $\mathrm{An}_k$.
Then:
\begin{enumerate}[(i)]
\item The morphism $f$ is an étale morphism if and only if $\Phi(f)$ is an \'etale morphism.
\item The morphism $f$ is a closed immersion if and only if $\Phi(f)$ is a closed immersion.
\end{enumerate}
\end{thm}
\begin{proof}
We start by dealing with étale morphisms.
Assume first that $f$ is an étale morphism.
If $X$ is affinoid, it determines an object in the site $Y_\mathrm{\acute{e}t}$.
Let us denote by $U$ this object. It follows from \cite[5.1.6.12]{HTT} that the adjunction $f_* \colon \mathcal X_X \rightleftarrows \mathcal X_Y \colon f^{-1}$ induced by $f$ can be identified with the \'etale morphism $(\mathcal X_Y)_{/U} \rightleftarrows \mathcal X_Y$.
Since $X$ is an ordinary $k$-analytic\xspace space, $U$ is $0$-truncated and therefore it is hypercomplete.
It follows that we can identify the adjunction
\[ f_* \colon \mathcal X_X^\wedge \rightleftarrows \mathcal X_Y^\wedge \colon f^{-1} \]
with the \'etale morphism $j_* \colon (\mathcal X_Y^\wedge)_{/U} \rightleftarrows \mathcal X_Y^\wedge \colon j^{-1}$.
Moreover, since $f$ is étale, we see that $(f^{-1} \mathcal O_Y)(V) = \mathcal O_Y(V)$.
In particular, we deduce that $f^{-1} \mathcal O_Y = \mathcal O_X$.
In other words, $\Phi(f)$ is \'etale.
If now $X$ is arbitrary, we choose an étale covering $\{X_i \to X\}$ such that every $X_i$ is affinoid.
The above argument shows that the induced morphisms $\mathcal X_{X_i} \rightleftarrows \mathcal X_X$ and $\mathcal X_{X_i} \rightleftarrows \mathcal X_Y$ are \'etale.
It follows that $f_* \colon \mathcal X_X \rightleftarrows \mathcal X_Y \colon f^{-1}$ is \'etale as well.
Let us now assume that $\Phi(f)$ is \'etale.
We will prove that $f$ is étale.
The question being local on $X$ and $Y$, we can assume that they are affinoid, say $X = \Sp B$, $Y = \Sp A$.
By hypothesis, $f^{-1} \mathcal O_Y \to \mathcal O_X$ is an equivalence.
Since the morphism of $\infty$-topoi\xspace $f_* \colon \mathcal X_X^\wedge \rightleftarrows \mathcal X_Y^\wedge\colon f^{-1}$ is \'etale, we see that, for every $U \to X$ étale, one has
\[ f^{-1}(\mathcal O_Y)(U) = \mathcal O_Y(U) . \]
Consider the sheaf $\mathbb L_{\mathcal O_X / f^{-1} \mathcal O_Y}$ on $\mathcal X_X^\wedge$ defined by
\[ C \mapsto \mathbb L^\mathrm{an}_{\mathcal O_X(C) / f^{-1} \mathcal O_Y(C)} = \mathbb L^\mathrm{an}_{C / f^{-1} \mathcal O_Y(C)} , \]
where the symbol $\mathbb L^\mathrm{an}$ denotes the analytic cotangent complex (cf.\ \cite[\S 7.2]{Gabber_Almost_2003}).
Since $f^{-1} \mathcal O_Y \simeq \mathcal O_X$, this sheaf is identically zero.
On the other side, if $\eta^{-1} \colon \mathcal X_X^\wedge \to \mathcal S$ is a geometric point, then
\[ \eta^{-1}(\mathbb L^\mathrm{an}_{\mathcal O_A / f^{-1} \mathcal O_B}) \simeq \mathbb L^\mathrm{an}_{\eta^{-1} \mathcal O_A / \eta^{-1} f^{-1} \mathcal O_B}. \]
We can identify $\eta^{-1} f^{-1} \mathcal O_B$ with a strictly henselian $B$-algebra $B'$.
Since the map $B \to B'$ is formally \'etale, we conclude that
\[ \mathbb L^\mathrm{an}_{\eta^{-1} \mathcal O_A / \eta^{-1} f^{-1} \mathcal O_B} \simeq \mathbb L^\mathrm{an}_{\eta^{-1} \mathcal O_A / B}. \]
This is also the stalk of the sheaf on $\mathcal X_X^\wedge$ defined by
\[ C \mapsto \mathbb L^\mathrm{an}_{C / B}. \]
Therefore, this sheaf vanishes as well.
In particular, $\mathbb L^\mathrm{an}_{A / B} \simeq 0$, completing the proof.
\todo{A proof without cotangent complex maybe better.}
We now turn to closed immersions.
Assume first that $f$ is a closed immersion in $\mathrm{An}_k$.
\cref{prop:preserve_closed_immersion} and \cref{lem:hypercompletion_closed_immersions} show that the induced geometric morphism $f_* \colon \mathcal X_Y^\wedge \rightleftarrows \mathcal X_X^\wedge \colon f^{-1}$ is a closed immersion of $\infty$-topoi\xspace.
We are left to show that the morphism $f^{-1} \mathcal O_X \to \mathcal O_Y$ is an effective epimorphism.
In virtue of \cref{prop:alg_effective_epi}, it suffices to show that $(f^{-1} (\mathcal O_X))(\mathbf A^1_k) \to \mathcal O_Y(\mathbf A^1_k)$ is an effective epimorphism, where $\mathbf A^1_k$ denote the $k$-analytic\xspace affine line.
Observe that $(f^{-1} (\mathcal O_X))(\mathbf A^1_k) \simeq f^{-1} (\mathcal O_X(\mathbf A^1_k))$.
Since $(f^{-1}, f_*)$ is a closed immersion of $\infty$-topoi\xspace, \cref{lem:different_closed_immersion} shows that it is sufficient to check that
\begin{equation} \label{eq:closed_immersion_effective_epi}
\mathcal O_X(\mathbf A^1_k) \to f_* ( \mathcal O_Y(\mathbf A^1_k))
\end{equation}
is an effective epimorphism in $\mathcal X_X^\wedge$.
This question is local on $\mathcal X_X^\wedge$, so we can assume that $X$ is an affinoid space.
Observe now that $\mathcal O_X(\mathbf A^1_k)$ is the underlying sheaf of (discrete) spaces associated to the structure sheaf of $X$. In the same way, $f_*(\mathcal O_Y(\mathbf A^1_k))$ is the underlying sheaf of spaces associated to the pushforward of the structure sheaf of $Y$.
Both are coherent on $X$, and $f_*(\mathcal O_Y(\mathbf A^1_k))$ is the quotient of $\mathcal O_X(\mathbf A^1_k)$ by some coherent sheaf of ideals.
In particular, the map \eqref{eq:closed_immersion_effective_epi} is an effective epimorphism.
Assume now that $\Phi(f)$ is a closed immersion.
We want to prove that $f$ is a closed immersion as well.
The question is local both on the source and on the target, so we can assume that $X$ and $Y$ are affinoid, say $X = \Sp A$ and $Y = \Sp B$.
In this case, it follows from the proof of \cref{thm:fully_faithfulness} that $f$ corresponds to the morphism
\[ A = \mathcal O_X(\mathbf A^1_k)(X) \to B = \mathcal O_Y(\mathbf A^1_k)(Y) . \]
Therefore, we only have to show that this morphism is surjective.
Let $U = \Sp C \to X$ be an étale morphism.
Then it follows again from the proof of \cref{thm:fully_faithfulness} that
\begin{align*}
f_* \mathcal O_Y(\mathbf A^1_k)(U) & = \mathcal O_Y(\mathbf A^1_k)(Y \times_X U) = B {\cotimes}_A C \\
& = f_* \mathcal O_Y(\mathbf A^1_k)(X) {\cotimes}_{\mathcal O_X(\mathbf A^1_k)(X)} \mathcal O_X(\mathbf A^1_k)(U) .
\end{align*}
In particular, $f_* \mathcal O_Y(\mathbf A^1_k)$ is a coherent sheaf of $\mathcal O_X(\mathbf A^1_k)$-modules.
We can thus apply Tate's acyclicity theorem to conclude that $A \to B$ is surjective, completing the proof.
\end{proof}
\section{Existence of fiber products} \label{sec:fiber_products}
The goal of this section is to prove the existence of fiber products of derived $k$-analytic\xspace spaces.
First we will prove the existence of fiber products along a closed immersion (\cref{prop:closed_fiber_products_dAn}).
Then we will prove the existence of products over a point (\cref{lem:products_dAn}).
We will deduce the existence of fiber products in the general case from the two special cases above, plus \cref{lem:closed_devissage}, which shows that any derived $k$-analytic\xspace space can locally be embedded into a non-derived smooth $k$-analytic\xspace space.
\begin{lem} \label{lem:sheaves_coherent_modules}
Let $f \colon (\mathcal X, \mathcal O_{\mathcal X}) \to (\mathcal Y, \mathcal O_{\mathcal Y})$ be a map of derived $k$-analytic\xspace spaces such that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ and $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) \simeq \Phi(Y)$ for two $k$-analytic\xspace spaces $X, Y \in \mathrm{An}_k$.
Assume that $\mathcal F$ is a connective sheaf of $\mathcal O_{\mathcal Y}^\mathrm{alg}$-modules on $\mathcal Y$ and that each $\pi_n \mathcal F$ is a coherent sheaf of $\pi_0 \mathcal O_{\mathcal Y}^\mathrm{alg}$-modules. Then the tensor product $\mathcal F' \coloneqq f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_{\mathcal Y}^\mathrm{alg}} \mathcal O_{\mathcal X}^\mathrm{alg}$ is connective, and each $\pi_n \mathcal F'$ is a coherent sheaf of $\pi_0(\mathcal O_{\mathcal X}^\mathrm{alg})$-modules.
\end{lem}
\begin{proof}
The connectivity of $\mathcal F' \coloneqq f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y} \mathcal O_\mathcal X$ follows from the compatibility of the tensor product with the $t$-structure (cf.\ \cite[Proposition 2.1.3(6)]{DAG-VIII}.)
In order to prove that the homotopy groups $\pi_k \mathcal F'$ are coherent $\pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}$-modules, we first remark that the question is local both on $\mathcal X$ and on $\mathcal Y$.
so we can assume that $X$ and $Y$ are affinoid, say $X = \Sp A$ and $Y = \Sp B$.
We follow closely the proof of \cite[Lemma 12.11]{DAG-IX}.
Thus, we start by proving that for every integer $m \ge - 1$ there exists a sequence of morphisms
\[ 0 = \mathcal F(-1) \to \mathcal F(0) \to \mathcal F(1) \to \cdots \to \mathcal F(m) \to \mathcal F \]
of $\mathcal O_\mathcal Y^\mathrm{alg}$-modules with the following properties:
\begin{enumerate}[(i)]
\item For $0 \le i \le m$, the fiber of $\mathcal F(i-1) \to \mathcal F(i)$ is equivalent to a direct sum of finitely many copies of $\mathcal O_\mathcal Y^\mathrm{alg}[i]$.
\item For $0 \le i \le m$, the fiber of $\mathcal F(i) \to \mathcal F$ is $i$-connective.
\item For $-1 \le i \le m$, the homotopy groups $\pi_j \mathcal F(i)$ are coherent $\pi_0(\mathcal O_\mathcal Y^\mathrm{alg})$-modules, which vanish for $j < 0$.
\end{enumerate}
We proceed by induction on $m$.
If $m = -1$, we simply take $\mathcal F(-1) = 0$. The fiber of $\mathcal F(-1) \to \mathcal F$ is then $\mathcal F[1]$, which is $(-1)$-connective because $\mathcal F$ is connective.
Assume now that we are given a sequence
\[ 0 = \mathcal F(-1) \to \mathcal F(0) \to \cdots \to \mathcal F(m) \to \mathcal F \]
satisfying the conditions above.
Let $\mathcal G$ be the fiber of the map $\mathcal F(m) \to \mathcal F$, so $\mathcal G$ is $m$-connective.
We have an exact sequence
\[ \pi_{m+1} \mathcal F(m) \to \pi_{m+1} \mathcal F \to \pi_m \mathcal F' \to \pi_m \mathcal F(m) \to \pi_m \mathcal F , \]
from which we deduce that $\pi_m \mathcal F'$ is a coherent sheaf of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
In particular, there exists a positive integer $l$ and a surjection $B^l \to \mathcal G(Y)$.
This induces an epimorphism $(\pi_0 \mathcal O_\mathcal Y^\mathrm{alg})^l \to \mathcal G[-m]$.
Composing with the canonical map $(\mathcal O_\mathcal Y^\mathrm{alg})^l \to (\pi_0 \mathcal O_\mathcal Y^\mathrm{alg})^l$, we obtain a map
\[ (\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal G . \]
Let $\mathcal F(m+1)$ be the cofiber of the composite map $(\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal G \to \mathcal F(m)$.
Then the property (i) is satisfied by construction and the property (iii) follows from the long exact sequence associated to the cofiber sequence $(\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal F(m) \to \mathcal F(m+1)$.
Let $\mathcal G'$ denote the fiber of the map $\mathcal F(m+1) \to \mathcal F$, so we have a fiber sequence
\[ (\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal G \to \mathcal G' . \]
Passing to the long exact sequence, we deduce that $\mathcal G'$ is $(m+1)$-connective, proving the property (ii).
Let us now prove that the homotopy groups of $\mathcal F' \coloneqq f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg}$ are coherent sheaves of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
Fix an integer $n \ge 0$.
Choose a sequence
\[ 0 \to \mathcal F(-1) \to \mathcal F(0) \to \cdots \to \mathcal F(n+1) \to \mathcal F \]
satisfying the properties (i), (ii) and (iii) above.
In particular, the fiber of $\mathcal F(n+1) \to \mathcal F$ is $(n+1)$-connective and therefore the same goes for the map
\[ f^{-1} \mathcal F(n+1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \to f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} . \]
So we obtain an isomorphism
\[ \pi_n \left( f^{-1} \mathcal F(n+1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right) \to \pi_n \left( f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right) . \]
We can therefore replace $\mathcal F$ by $\mathcal F(n+1)$.
We will now prove that for $-1 \le i \le n+1$, $\pi_n \left( f^{-1} \mathcal F(i) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right)$ is a coherent sheaf of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
We proceed by induction on $i$.
The case $i = -1$ is trivial.
To deal with the inductive step, we note that the property (i) implies the existence of a fiber sequence
\[ (\mathcal O_\mathcal Y^\mathrm{alg})^l[i] \to \mathcal F(i) \to \mathcal F(i+1) . \]
We therefore obtain a long exact sequence
\begin{multline*}
\cdots \to ( \pi_{n-i} \mathcal O_\mathcal X^\mathrm{alg} )^l \to \pi_n ( f^{-1} \mathcal F(i) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} ) \to\\
\pi_n( f^{-1} \mathcal F(i+1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} ) \to ( \pi_{n - i - 1} \mathcal O_\mathcal X^\mathrm{alg} )^l \to \cdots
\end{multline*}
We conclude that $\pi_n \left( f^{-1} \mathcal F(i + 1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right)$ is a coherent sheaf of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
\end{proof}
\begin{prop} \label{prop:closed_fiber_products_dAn}
Assume we are given maps of derived $k$-analytic\xspace spaces $f \colon (\mathcal Y, \mathcal O_{\mathcal Y}) \allowbreak{}\to (\mathcal X, \mathcal O_{\mathcal X})$ and $(\mathcal X', \mathcal O_{\mathcal X'}) \to (\mathcal Y, \mathcal O_{\mathcal Y})$.
Assume moreover that $f$ is a closed immersion.
Then we have the following statements:
\begin{enumerate}[(i)]
\item \label{item:dAn_fiber_products_Top} There exists a pullback diagram $\sigma$:
\[ \begin{tikzcd}
(\mathcal Y', \mathcal O_{\mathcal Y'}) \arrow{r}{f'} \arrow{d} & (\mathcal X', \mathcal O_{\mathcal X'}) \arrow{d} \\
(\mathcal Y, \mathcal O_{\mathcal Y}) \arrow{r}{f} & (\mathcal X, \mathcal O_{\mathcal X})
\end{tikzcd} \]
in the $\infty$-category $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$.
\item \label{item:dAn_fiber_products_topoi} The image of $\sigma$ in $\tensor*[^\rR]{\mathcal{H}\Top}{}$ is a pullback diagram of hypercomplete $\infty$-topoi\xspace.
\item \label{item:dAn_closed_immersion} The map $f'$ is a closed immersion.
\item \label{item:dAn_fiber_product_dAn} The structured $\infty$-topos\xspace $(\mathcal Y', \mathcal O_{\mathcal Y'})$ is a derived $k$-analytic\xspace space.
\item \label{item:truncation_pullback} Assume that $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) = \Phi(Y)$, $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) = \Phi(X)$ and $(\mathcal X', \pi_0 \mathcal O_{\mathcal X'}) = \Phi(X')$. Then $(\mathcal Y', \pi_0 \mathcal O_{\mathcal Y'})$ can be identified with $\Phi(Y \times_X X')$.
\end{enumerate}
\end{prop}
\begin{proof} The statements (\ref{item:dAn_fiber_products_Top}), (\ref{item:dAn_fiber_products_topoi}) and (\ref{item:dAn_closed_immersion}) follow from \cref{prop:closed_fiber_products_Top}.
We now prove (\ref{item:truncation_pullback}).
Observe that the map $f$ induces a closed immersion $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) \to (\mathcal X, \pi_0 \mathcal O_{\mathcal X})$.
So by \cref{thm:Phi_classes_of_morphisms}, it corresponds to a closed immersion $\varphi \colon Y \to X$ of $k$-analytic\xspace spaces.
On the other side, the map $\Phi(X') \to \Phi(X)$ corresponds to a map $X' \to X$ by \cref{thm:fully_faithfulness}.
Let $Y' \coloneqq Y \times_X X'$ be the fiber product computed in $\mathrm{An}_k$.
Then \cref{prop:closed_immersion_pullback_of_topoi} allows us to identify $\mathcal X_{Y'}\coloneqq\mathrm{Sh}(Y'_\mathrm{\acute{e}t})^\wedge$ with $\mathcal Y'$.
It follows from the universal property of the fiber product that there exists a map in $\tensor*[^\rR]{\mathcal{H}\Top}{}(\cT_{\mathrm{an}}(k))$
\[ (\mathcal Y', \mathcal O_{Y'}) \to (\mathcal Y', \mathcal O_{\mathcal Y'}) \]
Moreover, it follows from \cref{prop:closed_fiber_products_Top}(iii) that we have an identification
\[ \mathcal O_{\mathcal Y'}^\mathrm{alg} \simeq f^{\prime -1} \mathcal O_{X'}^\mathrm{alg} \otimes_{f^{\prime -1} g^{-1} \mathcal O_{X}} g^{\prime -1} \mathcal O_{Y} . \]
Using \cite[7.2.1.22]{Lurie_Higher_algebra}, we obtain an equivalence
\[ \pi_0(\mathcal O_{\mathcal Y'}^\mathrm{alg}) \simeq \mathrm{Tor}_0^{ f^{\prime -1} g^{-1}( \pi_0 \mathcal O_X^\mathrm{alg} )}(f^{\prime -1} \pi_0( \mathcal O_{X'}^\mathrm{alg} ), g^{\prime -1}( \pi_0 \mathcal O_Y^\mathrm{alg} )) . \]
As $\pi_0( \mathcal O_\mathcal X) \to f_* \pi_0( \mathcal O_\mathcal Y)$ is surjective, we see that the same formula can be used to describe $\mathcal O_{Y'}$.
Hence $\pi_0(\mathcal O_{\mathcal Y'}) \simeq \mathcal O_{Y'}$.
This proves (\ref{item:truncation_pullback}).
We are left to prove the statement (\ref{item:dAn_fiber_product_dAn}).
The assertion is local on $\mathcal Y'$, so we can assume that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) = \Phi(X)$, $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) = \Phi(Y)$ and $(\mathcal X', \pi_0 \mathcal O_{\mathcal Y}) = \Phi(X')$ for $k$-analytic\xspace spaces $X, X'$ and $Y$.
It follows from (\ref{item:truncation_pullback}) that $(\mathcal Y', \pi_0 \mathcal O_{\mathcal Y'}^\mathrm{alg})$ is a $k$-analytic\xspace space.
Moreover, since $f$ is a closed immersion, we see that for each $n \ge 0$ the pushforward $f_* \pi_n \mathcal O_{Y}^\mathrm{alg}$ is a coherent sheaf of $\pi_0 \mathcal O_X^\mathrm{alg}$-modules on $X$.
Using \cref{lem:sheaves_coherent_modules} and \cref{prop:closed_fiber_products_Top}, we conclude that for each $n \ge 0$, the pushforward $f'_* \pi_n \mathcal O_{\mathcal Y'}^\mathrm{alg}$ is a coherent sheaf of $\pi_0 \mathcal O_{\mathcal X'}^\mathrm{alg}$-modules.
Then each $\pi_n \mathcal O_{\mathcal Y'}^\mathrm{alg}$ is a coherent sheaf of $\pi_0 \mathcal O_{\mathcal Y'}^\mathrm{alg}$-modules.
This completes the proof.
\end{proof}
\begin{lem} \label{lem:closed_devissage}
Let $(\mathcal X, \mathcal O_{\mathcal X})$ be a derived $k$-analytic\xspace space and let $\mathbf 1_\mathcal X$ be the final object of $\mathcal X$.
Then there exists an effective epimorphism $\coprod U_i \to \mathbf 1_\mathcal X$ and a collection of closed immersions $(\mathcal X_{/U_i}, \mathcal O_{\mathcal X}|_{U_i}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i)$, where $V_i$ is a smooth $k$-analytic\xspace space.
\end{lem}
\begin{proof}
We can assume without loss of generality that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ for a $k$-affinoid space $X$.
So we have a closed immersion into a $k$-analytic\xspace polydisc $X \hookrightarrow \mathbf D^n_k$. Composing with the affinoid domain embedding $\mathbf D^n_k \hookrightarrow \mathbf A^n_k$, we obtain an embedding $X \hookrightarrow \mathbf A^n_k$.
This embedding is given by $n$ global sections $f_1, \ldots, f_n \in \pi_0(\mathcal O_{\mathcal X}^\mathrm{alg})(X)$.
Let $\{u_i \colon U_i \to X\}_{i \in I}$ be an étale covering such that each restriction $f_j \circ u_i$ is represented by some $\widetilde{f}_{ij} \in \mathcal O_{\mathcal X}(\mathbf A^1_k)(U_i)$.
Combining \cref{lem:universal_property_HSpec} and \cite[Theorem 2.2.12]{DAG-V}, we deduce that these global sections determine a morphism of derived $k$-analytic\xspace spaces
\[ \varphi_i \colon (\mathcal X_{/U_i}, \mathcal O_{\mathcal X}|_{U_i}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(\mathbf A^n_k). \]
Choose a factorization of $U_i \to X \to \mathbf D^n_k$ as $U_i \xrightarrow{p} V_i \xrightarrow{g} \mathbf D^n_k$, where $p$ is a closed immersion and $g$ is étale.
The composite map $V_i \to \mathbf D^n_k \to \mathbf A^n_k$ is étale and therefore by \cref{thm:Phi_classes_of_morphisms}(i) the induced morphism of derived $k$-analytic\xspace spaces $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(\mathbf A^n_k)$ is \'etale.
Then \cite[Remark 2.3.4]{DAG-V} shows that the map $\varphi_i$ factors through $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i)$ if and only if the underlying morphism of $\infty$-topoi\xspace factors through $\mathcal X_{V_i}$.
The latter holds by construction.
Moreover, the truncation of $\psi_i \colon (\mathcal X_{/U_i}, \mathcal O_{\mathcal X}|_{U_i}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i)$ corresponds to the map $U_i \to V_i$, which is a closed immersion. It follows that $\psi_i$ is a closed immersion as well, completing the proof.
\end{proof}
\begin{lem} \label{lem:products_dAn}
Let $(\mathcal X, \mathcal O_{\mathcal X})$ and $(\mathcal Y, \mathcal O_{\mathcal Y})$ be derived $k$-analytic\xspace spaces.
We have the following statements:
\begin{enumerate}[(i)]
\item There exists a product $(\mathcal Z, \mathcal O_{\mathcal Z}) \simeq (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal Y, \mathcal O_{\mathcal Y})$ in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$.
\item The structured $\infty$-topos\xspace $(\mathcal Z, \mathcal O_{\mathcal Z})$ is a derived $k$-analytic\xspace space.
\item Assume that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ and $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) \simeq \Phi(Y)$. Then $(\mathcal Z, \pi_0\mathcal O_{\mathcal Z})$ is equivalent to $\Phi(X \times Y)$.
\item Assume that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ where $X$ is a separated $k$-analytic\xspace space.
Then the diagonal map $\delta \colon (\mathcal X, \mathcal O_{\mathcal X}) \to (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal X, \mathcal O_{\mathcal X})$ is a closed immersion.
\end{enumerate}
\end{lem}
\begin{proof}
The statements (i) and (ii) are local on $(\mathcal X, \mathcal O_{\mathcal X})$ and $(\mathcal Y, \mathcal O_{\mathcal Y})$, so we can assume in virtue of \cref{lem:closed_devissage} that there exists closed immersions $(\mathcal X, \mathcal O_{\mathcal X}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V)$ and $(\mathcal Y, \mathcal O_{\mathcal Y}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(W)$, where $V$ and $W$ are smooth $k$-analytic\xspace spaces.
\cref{prop:closed_fiber_products_dAn} allows us to reduce to the case $(\mathcal X, \mathcal O_{\mathcal X}) \simeq \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V)$ and $(\mathcal Y, \mathcal O_{\mathcal Y}) \simeq \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(W)$.
In this case, we have
\[(\mathcal Z, \mathcal O_{\mathcal Z}) \simeq \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V \times W).\]
The statement (iii) follows from the construction of $(\mathcal Z, \mathcal O_{\mathcal Z})$ we described and \cref{prop:closed_fiber_products_dAn}(\ref{item:truncation_pullback}).
We are left to prove the statement (iv).
The statement (ii) shows that the induced map
\[ \pi_0(\delta) \colon (\mathcal X, \pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}) \to (\mathcal X, \pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}) \times (\mathcal X, \pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}) \]
corresponds to $\Phi(\Delta) \colon \Phi(X) \to \Phi(X \times X)$.
Since $X$ is separated, $\Delta \colon X \to X \times X$ is a closed immersion and therefore \cref{thm:Phi_classes_of_morphisms} implies that $\Phi(\Delta)$ is a closed immersion.
Now, the assertion follows from \cref{prop:alg_effective_epi}.
\end{proof}
Now we can deduce the main result of this section:
\begin{thm} \label{thm:fiber_products}
The $\infty$-category\xspace $\mathrm{dAn}_k$ admits fiber products.
\end{thm}
\begin{proof}
Let $(\mathcal Y,\mathcal O_\mathcal Y)\to(\mathcal X,\mathcal O_\mathcal X)\leftarrow(\mathcal X',\mathcal O_{\mathcal X'})$ be maps of derived $k$-analytic\xspace spaces.
We would like to construct the fiber product.
Working locally on $\mathcal X$, we can assume that $(\mathcal X,\pi_0\mathcal O_\mathcal X^\mathrm{alg})\simeq\upsilon(X)$ for a separated $k$-analytic\xspace space $X$.
Using \cref{lem:products_dAn}(i), we deduce the existence of two products $(\mathcal Z,\mathcal O_\mathcal Z)\coloneqq(\mathcal X',\mathcal O_\mathcal X')\times(\mathcal Y,\mathcal O_\mathcal Y)$ and $(\mathcal X,\mathcal O_\mathcal X)\times(\mathcal X,\mathcal O_\mathcal X)$ in $\mathcal T\mathrm{op}(\cT_{\mathrm{an}}(k))$.
By \cref{lem:products_dAn}(iv), the diagonal map $\delta \colon (\mathcal X, \mathcal O_{\mathcal X}) \to (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal X, \mathcal O_{\mathcal X})$ is a closed immersion.
We now apply \cref{prop:closed_fiber_products_dAn} to produce a fiber product
\[ \begin{tikzcd}
(\mathcal Y',\mathcal O_{\mathcal Y'}) \arrow{r} \arrow{d} & (\mathcal Z,\mathcal O_\mathcal Z) \arrow{d} \\
(\mathcal X,\mathcal O_\mathcal X) \arrow{r} & (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal X, \mathcal O_{\mathcal X}).
\end{tikzcd} \]
Note that $(\mathcal Y',\mathcal O_{\mathcal Y'})$ is the fiber product of $(\mathcal Y,\mathcal O_\mathcal Y)\to(\mathcal X,\mathcal O_\mathcal X)\leftarrow(\mathcal X',\mathcal O_{\mathcal X'})$ , completing the proof.
\end{proof}
\begin{comment}
We record the following fact for later use: \todo{We never use it. It was needed in the old essential image. Shall we remove it?}
\begin{prop} \label{prop:Phi_etale_pullback}
Assume that the square $\sigma$
\[ \begin{tikzcd}
X' \arrow{r} \arrow{d} & Y' \arrow{d} \\
X \arrow{r}{j} & Y
\end{tikzcd} \]
is a pullback in $\mathrm{An}_k$ and that $j$ is an étale morphism.
Then $\Phi$ preserves this pullback.
\end{prop}
\begin{proof}
Using \cite[6.3.5.8]{HTT} we deduce that $\Phi(\sigma)$ is a pullback square in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$.
In particular, it is a pullback also in $\mathrm{dAn}_k$.
\end{proof}
\end{comment}
\section{Comparison between derived spaces and non-derived stacks} \label{sec:essential_image}
In this section, we will characterize the essential image of the embedding $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ constructed in \cref{sec:fullyfaithfulness}.
Moreover, we will compare derived $k$-analytic\xspace spaces with higher $k$-analytic\xspace stacks in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
\subsection{Construction of the comparison functor}
On the $\infty$-category\xspace $\mathrm{dAn}_k$ of derived $k$-analytic\xspace spaces, we define the étale topology $\tau_\mathrm{\acute{e}t}$ to be the Grothendieck topology generated by collections of étale morphisms $\{U_i\to U\}$ such that $\coprod U_i\to U$ is an effective epimorphism (cf.\ \cref{def:closed_immersion_and_etale}).
\begin{rem}
The restriction of $\tau_\mathrm{\acute{e}t}$ to the full subcategory $\mathrm{An}_k$ of $\mathrm{dAn}_k$ coincides with the étale topology $\tau_\mathrm{\acute{e}t}$.
\end{rem}
\begin{lem}
Every representable presheaf on $\mathrm{dAn}_k$ is a hypercomplete sheaf for the topology $\tau_\mathrm{\acute{e}t}$.
\end{lem}
\begin{proof}
Let $X \coloneqq (\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space.
The universal property of \'etale morphisms (cf.\ \cite[Remark 2.3.4]{DAG-V}) shows that a $\tau_\mathrm{\acute{e}t}$-hypercovering of $X$ can be identified with a hypercovering $U^\bullet$ of $\mathbf 1_\mathcal X$ in the $\infty$-topos\xspace $\mathcal X$.
Given such a hypercovering, the associated $\tau_\mathrm{\acute{e}t}$-hypercovering $X^\bullet$ of $X$ is described by $X^n \coloneqq (\mathcal X_{/U^n}, \mathcal O_\mathcal X |_{U^n})$.
Therefore, we have to prove that
\[ \colim_\Delta ( \mathcal X_{/U^\bullet}, \mathcal O_\mathcal X |_{U^\bullet} ) \simeq (\mathcal X, \mathcal O_\mathcal X) \]
in the $\infty$-category $\mathrm{dAn}_k$.
Using the statement (3') in the proof of \cite[Proposition 2.3.5]{DAG-V}, we see that it is enough to prove that $\mathcal X \simeq \colim \mathcal X_{/U^\bullet}$ in $\tensor*[^\rR]{\Top}{}$.
Since $\mathcal X$ is hypercomplete, this follows from the descent theory of $\infty$-topoi\xspace (cf.\ \cite[6.1.3.9]{HTT}) and from the fact that $|U^\bullet| \simeq \mathbf 1_\mathcal X$ (cf.\ \cite[6.5.3.12]{HTT}).
\end{proof}
\begin{defin} \label{def:derived_affinoid}
A \emph{derived $k$-affinoid space} is a derived $k$-analytic\xspace space $(\mathcal X, \mathcal O_\mathcal X)$ such that $(\mathcal X, \pi_0(\mathcal O_\mathcal X)) \simeq \Phi(X)$ for some $k$-affinoid space $X$.
We denote by $\mathrm{dAfd}_k$ the full subcategory of $\mathrm{dAn}_k$ spanned by derived $k$-affinoid spaces.
\end{defin}
The Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ induces by restriction a Grothendieck topology on $\mathrm{dAfd}_k$ which we denote again by $\tau_\mathrm{\acute{e}t}$.
We define the functor $\widetilde{\phi}$ as the composition
\[ \begin{tikzcd}
\mathrm{dAn}_k \arrow{r} & \Fun(\mathrm{dAn}_k^{\mathrm{op}}, \mathcal S) \arrow{r} & \Fun( ( \mathrm{dAfd}_k )^{\mathrm{op}}, \mathcal S ) ,
\end{tikzcd} \]
where the first functor is the Yoneda embedding and the second one is the restriction along $\mathrm{dAfd}_k \subset \mathrm{dAn}_k$.
Since the Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ is subcanonical, the functor $\widetilde{\phi}$ factors through $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
We denote by
\[ \phi \colon \mathrm{dAn}_k \to \mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) \]
the induced functor.
Our first goal is to show that $\phi$ is fully faithful.
\begin{lem} \label{lem:affine_site_big_site}
Let $X = (\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space and let $p \colon U \to \mathbf 1_\mathcal X$ be an effective epimorphism.
Let $U^\bullet$ be the \v{C}ech nerve of $p$ and put $X^n \coloneqq (\mathcal X_{/U^n}, \mathcal O_\mathcal X|_{U^n})$.
Then in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ we have
\[ \phi(X) \simeq \colim_{\Delta} \phi(X^\bullet) . \]
\end{lem}
\begin{proof}
Let $j\colon\mathrm{dAfd}_k\hookrightarrow\mathrm{dAn}_k$ denote the inclusion functor.
It is continuous and cocontinuous in the sense of \cite[\S 2.4]{Porta_Yu_Higher_analytic_stacks_2014}.
It induces a pair of adjoint functors
\[ j_s\colon \mathrm{Sh}(\mathrm{dAn}_k,\tau_\mathrm{\acute{e}t})\leftrightarrows\mathrm{Sh}(\mathrm{dAfd}_k,\tau_\mathrm{\acute{e}t})\colon j^s.\]
Since the Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ is subcanonical, we can factor $\phi$ as
\[\mathrm{dAn}_k \xrightarrow{\ \psi\ } \mathrm{Sh}( \mathrm{dAn}_k, \tau_\mathrm{\acute{e}t} ) \xrightarrow{\ j_s\ } \mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} ).\]
Moreover, we have
\[ \psi(X) \simeq \colim_{\Delta} \psi(X^\bullet) . \]
Since the functor $j_s$ is a left adjoint, it commutes with colimits, completing the proof.
\end{proof}
\begin{lem} \label{lem:affine_hypercover}
Let $X = (\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space.
Then there exists a hypercovering $X^\bullet$ of $X$ in $\mathrm{dAn}_k$ such that each $X^n$ is a disjoint union of derived $k$-affinoid spaces.
\end{lem}
\begin{proof}
It follows directly from Definitions \ref{def:derived_space} and \ref{def:derived_affinoid}.
\end{proof}
\begin{prop} \label{prop:phi_fully_faithful}
The functor $\phi \colon \mathrm{dAn}_k \to \mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ is fully faithful.
\end{prop}
\begin{proof}
Let $X, Y \in \mathrm{dAn}_k$ and consider the natural map
\[ \psi_{X,Y} \colon \Map_{\mathrm{dAn}_k}(X,Y) \to \Map_{\mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} )}(\phi(X), \phi(Y)) . \]
Keeping $Y$ fixed, let $\mathcal C$ be the full subcategory of $\mathrm{dAn}_k$ spanned by those $X$ such that $\psi_{X,Y}$ is an equivalence.
Since $\mathcal C$ is stable under colimits, combining Lemmas \ref{lem:affine_site_big_site} and \ref{lem:affine_hypercover}, we are reduced to the case where $X$ belongs to $\mathrm{dAfd}_k$.
In this case, the statement follows entirely from the Yoneda lemma.
\end{proof}
Our second goal is to identify the essential image of the functor $\phi$.
For this, we need to introduce some notations.
\begin{defin}
Let $\mathbf P_\mathrm{\acute{e}t}$ denote the class of étale morphisms in $\mathrm{dAn}_k$.
The triple $(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$ constitutes a geometric context in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
We call the associated geometric stacks \emph{derived $k$-analytic\xspace Deligne-Mumford stacks}.
We denote by $\mathrm{DM}$ the $\infty$-category\xspace of derived $k$-analytic\xspace Deligne-Mumford stacks.
\end{defin}
\begin{defin}
Let $F \in \mathrm{DM}$.
We say that $F$ is \emph{$n$-truncated} if $F(X)$ is $n$-truncated for every $X = (\mathcal X, \mathcal O_\mathcal X) \in \mathrm{dAfd}_k$ such that $\mathcal O_\mathcal X$ is discrete.
We denote by $\mathrm{DM}_n$ the full subcategory of $\mathrm{DM}$ spanned by $n$-truncated $k$-analytic\xspace Deligne-Mumford stacks.
\end{defin}
We denote by $\mathrm{dAn}_k^{\le n}$ the full subcategory of $\mathrm{dAn}_k$ spanned by those derived $k$-analytic\xspace spaces $(\mathcal X, \mathcal O_\mathcal X)$ such that $\mathcal X$ is $n$-localic (cf.\ \cite[6.4.5.8]{HTT}).
With these notations we can now state our main comparison theorem, which is an analogue of \cite[Theorem 3.7]{Porta_DCAGI} and \cite[Theorem 1.7]{Porta_Comparison_2015}.
\begin{thm} \label{thm:functor_of_points_vs_dAnk}
For every integer $n \ge 1$, the functor $\phi \colon \mathrm{dAn}_k \to \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ restricts to an equivalence of $\infty$-categories\xspace $\mathrm{dAn}_k^{\le n} \simeq \mathrm{DM}_n$.
\end{thm}
The proof will occupy the rest of this section.
Before plunging ourselves into the details, let us deduce from this theorem an important application.
Let $(\mathrm{An}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$ be the geometric context consisting of the category of $k$-analytic spaces, the étale topology and the class of étale morphisms.
The associated geometric stacks are called \emph{higher $k$-analytic\xspace Deligne-Mumford stacks}.
They are in particular higher $k$-analytic\xspace stacks considered in \cite{Porta_Yu_Higher_analytic_stacks_2014}.
So all the results in loc.\ cit.\ apply.
\begin{cor} \label{cor:underived_higher_kanal_stacks}
Let $\mathrm{Geom}(\mathrm{An}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})$ denote the $\infty$-category\xspace of higher $k$-analytic\xspace Deligne-Mumford stacks.
There is a fully faithful embedding $\mathrm{Geom}(\mathrm{An}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t}) \to \mathrm{dAn}_k$ whose essential image is spanned by those derived $k$-analytic\xspace spaces $(\mathcal X, \mathcal O_\mathcal X)$ such that $\mathcal X$ is $n$-localic for some $n$ and $\mathcal O_\mathcal X$ is discrete.
\end{cor}
\begin{proof}
Let $(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$ be the geometric context consisting of the category of $k$-affinoid spaces, the étale topology and the class of étale morphisms.
Let $\mathrm{Geom}(\mathrm{Afd}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})$ denote the $\infty$-category\xspace of geometric stacks associated to this geometric context.
By \cite[\S 2.5]{Porta_Yu_Higher_analytic_stacks_2014}, we have an equivalence
\[\mathrm{Geom}(\mathrm{An}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})\simeq\mathrm{Geom}(\mathrm{Afd}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t}).\]
It follows from \cref{thm:fully_faithfulness} that the natural inclusion $j \colon \mathrm{Afd}_k \to \mathrm{dAfd}_k$ is fully faithful.
So the induced functor
\[ j_s \colon \mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t}) \to \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) \]
is fully faithful as well.
We know moreover that $j_s$ preserves geometric stacks.
Therefore $j_s$ factors through the full subcategory $\mathrm{DM} = \bigcup \mathrm{DM}_n$.
Applying \cref{thm:functor_of_points_vs_dAnk}, we obtain the desired fully faithful functor $\mathrm{Geom}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t}) \to \mathrm{dAn}_k$.
Now it suffices to observe that if a geometric stack $X \in \mathrm{dAn}_k^{\le n}$ is discrete, then $\phi(X)$ lies in the essential image of $j_s$.
Indeed, if $X$ is discrete, then
\[ \Map_{\mathrm{dAn}_k}( Y, X ) = \Map_{\mathrm{dAn}_k}(t_0(Y), X). \]
Therefore $\phi(X)$ coincides with the left Kan extension of its restriction along $j$, completing the proof.
\end{proof}
\subsection{The case of algebraic spaces}
Given a derived $k$-analytic\xspace space $X$, we denote by $\mathrm{dAfd}_X$ the overcategory $(\mathrm{dAfd}_k)_{/X}$.
The Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ induces a Grothendieck topology on $\mathrm{dAfd}_X$, which we still denote by $\tau_\mathrm{\acute{e}t}$.
Let $X_\mathrm{big,\acute{e}t}$ denote the Grothendieck site $(\mathrm{dAfd}_X, \tau_\mathrm{\acute{e}t})$.
Let $(\mathrm{dAfd}_X)_\mathrm{\acute{e}t}$ be the full subcategory of the overcategory $\mathrm{dAfd}_X$ spanned by \'etale morphisms $Y \to X$.
The \'etale topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAfd}_X$ restricts to a Grothendieck topology on $(\mathrm{dAfd}_X)_\mathrm{\acute{e}t}$, which we still denote by $\tau_\mathrm{\acute{e}t}$.
Let $X_\mathrm{\acute{e}t}$ denote the Grothendieck site $((\mathrm{dAfd}_X)_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
\begin{rem}
Let $X$ be an ordinary $k$-analytic\xspace space. Let $f \colon (\mathcal Y, \mathcal O_\mathcal Y) \to \Phi(X)$ be an \'etale morphism in $\mathrm{dAn}_k$. Since the morphism $f^{-1} \mathcal O_X \to \mathcal O_\mathcal Y$ is an equivalence, we see that $\mathcal O_\mathcal Y$ is discrete. In particular, if $(\mathcal Y, \mathcal O_\mathcal Y)$ is a derived $k$-affinoid space, then it belongs to the essential image of $\Phi$.
This shows that there is a canonical equivalence $X_\mathrm{\acute{e}t} \simeq \Phi(X)_\mathrm{\acute{e}t}$.
\end{rem}
We have continuous functors between the sites
\[ \begin{tikzcd}
(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t}) \arrow{r}{u} & (X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \arrow{r}{v} & (\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) .
\end{tikzcd} \]
By \cite[\S 2.4]{Porta_Yu_Higher_analytic_stacks_2014}, they induce adjunctions on the $\infty$-categories\xspace of sheaves
\begin{gather*}
u_s \colon \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t}) \rightleftarrows \mathrm{Sh}(X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \colon u^s, \\
v_s \colon \mathrm{Sh}(X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \rightleftarrows \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) \colon v^s .
\end{gather*}
Moreover, since $u$ is left exact, $(u_s, u^s)$ is a geometric morphism of $\infty$-topoi\xspace. In particular, $u_s$ takes $n$-truncated objects to $n$-truncated objects.
On the other side, we can identify the adjunction $(v_s, v^s)$ with the canonical adjunction
\[ \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)} \rightleftarrows \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}), \]
where the right arrow is the forgetful functor.
\begin{defin}
Let $X \in \mathrm{dAfd}_k$, $Y \in \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ and $\alpha \colon Y \to \phi(X)$ a natural transformation.
We say that \emph{$\alpha$ exhibits $Y$ as an \'etale derived algebraic space over $X$} if there exists a $0$-truncated sheaf $F \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ and an equivalence $Y \simeq v_s(u_s(F))$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
\end{defin}
\begin{prop} \label{prop:etale_algebraic_spaces}
Let $X \in \mathrm{dAfd}_k$, $Y \in \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ and $\alpha \colon Y \to \phi(X)$ a natural transformation.
The following statements are equivalent:
\begin{enumerate}[(i)]
\item The natural transformation $\alpha$ exhibits $Y$ as an \'etale derived algebraic space over $X$.
\item There exists a discrete object $U \in \mathcal X$ such that $\phi(j)$ is equivalent to $\alpha$, where $j \colon (\mathcal X_{/U}, \mathcal O_\mathcal X |_U) \to (\mathcal X, \mathcal O_\mathcal X)$ is the induced \'etale morphism.
\item The natural transformation $\alpha$ is $0$-truncated and $0$-representable by étale morphisms.
\end{enumerate}
\end{prop}
\begin{proof}
We first prove the equivalence between (i) and (ii).
If $\alpha$ exhibits $Y$ as an \'etale derived algebraic space over $X$, we can find a $0$-truncated sheaf $U \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ and an equivalence $Y \simeq v_s(u_s(U))$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
Consider $X_U \coloneqq (\mathcal X_{/U}, \mathcal O_\mathcal X |_U)$ and let $j \colon X_U \to X$ be the induced \'etale map.
We want to prove that $\phi(j)$ is equivalent to $\alpha$.
For any $Z = (\mathcal Z, \mathcal O_\mathcal Z) \in \mathrm{dAfd}_k$ and any map $f \colon \phi(Z) \to \phi(X)$, we have a fiber sequence
\[ \begin{tikzcd}
\Map_{\mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} )_{/\phi(X)}}(\phi(Z)_f, u_s(U)) \arrow{r} \arrow{d} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), v_s(u_s(U))) \arrow{d} \\
\{f\} \arrow{r} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X)),
\end{tikzcd} \]
where $\phi(Z)_f$ denotes the object $f \colon \phi(Z) \to \phi(X)$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
Since $\phi$ is fully faithful by \cref{prop:phi_fully_faithful}, we can view $\phi(Z)_f$ as a representable object in $\mathrm{Sh}( X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \simeq \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
Therefore, the Yoneda lemma combined with \cite[4.3.2.15]{HTT} implies that
\[ \Map_{\mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} )_{/\phi(X)}}(\phi(Z)_f, u_s(U)) \simeq \Gamma(\mathcal Z, f^{-1}(U)) . \]
In particular, taking $Z$ to be an atlas for $X_U$ and choosing $f$ to be $j$, we obtain a canonical map $\phi(X_U) \to v_s(u_s(U))$.
For any $Z \in \mathrm{dAfd}_k$, we obtain in this way a commutative square
\[ \begin{tikzcd}
\Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X_U)) \arrow{r} \arrow{d} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X)) \arrow[-, double equal sign distance]{d} \\
\Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), v_s(u_s(U))) \arrow{r} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X)) .
\end{tikzcd} \]
For any morphism $f \colon \phi(Z) \to \phi(X)$, we can combine the fully faithfulness of $\phi$ and \cite[Remark 2.3.4]{DAG-V} to identify the fiber of the top horizontal morphism with $\Gamma(\mathcal Z, f^{-1}(U))$.
The same holds for the lower horizontal morphism in virtue of the above discussion.
Therefore, there is a canonical identification of $\phi(X_U)$ with $Y = v_s(u_s(U))$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$, and a canonical identification of $\phi(j)$ with $\alpha$.
On the other side, if (ii) is satisfied, then $U$ defines an \'etale derived algebraic space $v_s(u_s(U))$ over $X$, which can be identified with $Y$ using the same argument as above.
Let us now prove the equivalence between (i) and (iii).
First, assume that (iii) is satisfied.
In this case, we can define a sheaf $U \colon X_\mathrm{\acute{e}t} \to \mathcal S$ by sending an \'etale map $f \colon Z \to X$ to the fiber product
\[ \begin{tikzcd}
U(Z) \arrow{r} \arrow{d} & Y(Z) \arrow{d}{\alpha_Z} \\
\{*\} \arrow{r}{f} & \phi(X)(Z).
\end{tikzcd} \]
Since $\alpha$ is $0$-truncated, we see that $U$ takes values in $\mathrm{Set}$.
Since both $\phi(X)$ and $Y$ are sheaves, the same goes for $U$.
It follows that $U$ defines a $0$-truncated object in $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
Since $\alpha$ is $0$-representable by \'etale maps we obtain a canonical map $Y \to v_s(u_s(Y))$, and \cite[Remark 2.3.4]{DAG-V} shows that this map is an equivalence.
Finally, let us prove that (i) implies (iii).
Choose a $0$-truncated sheaf $U \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ such that $Y \simeq v_s(u_s(U))$.
We already remarked that in this case $\alpha$ is $0$-truncated.
Choose $V_i \in X_\mathrm{\acute{e}t}$ and sections $\eta_i \in U(V_i)$ which generate $U$, we obtain an effective epimorphism
\[ \coprod \phi(V_i) = \coprod v_s(u_s( V_i )) \to v_s(u_s(U)) \]
in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
Suppose there is a $(-1)$-truncated morphism $v_s(u_s(U)) \to \phi(Z)$ for some $Z \in X_\mathrm{\acute{e}t}$.
In this case, we see that
\[ \phi(V_i) \times_{v_s(u_s(U))} \phi(V_j) \simeq \phi(V_i) \times_{\phi(Z)} \phi(V_j) \]
and therefore the maps $\phi(V_i) \to v_s(u_s(U)) \simeq Y$ is $(-1)$-representable by \'etale maps.
In the general case, the fiber product $Y_{i,j} \coloneqq \phi(V_i) \times_{v_s(u_s(U))} \phi(V_j)$ is again a derived algebraic space \'etale over $X$.
We claim that the canonical map $Y_{i,j} \to \phi(V_i \times_X V_j)$ is $(-1)$-truncated.
Indeed, we have a pullback diagram
\[ \begin{tikzcd}
Y_{i,j} \arrow{r} \arrow{d} & v_s(u_s(U)) \arrow{d} \\
\phi(V_i) \times_{\phi(X)} \phi(V_j) \arrow{r} & v_s(u_s(U)) \times_{\phi(X)} v_s(u_s(U)).
\end{tikzcd} \]
Since the map $\alpha \colon Y \to \phi(X)$ is $0$-truncated, we conclude the proof of the claim by \cite[5.5.6.15]{HTT}.
At this point, we deduce that $Y_{i,j} \to X$ is $(-1)$-representable by \'etale maps, and therefore that each $\phi(V_i) \to Y$ is $0$-representable by \'etale maps.
\end{proof}
\subsection{Proof of \cref{thm:functor_of_points_vs_dAnk}}
We begin with the following analogue of \cite[Lemma 2.7]{Porta_Comparison_2015}
\begin{lem} \label{lem:decreasing_truncated_level}
Let $n \ge 0$ be an integer.
Fix $X = (\mathcal X, \mathcal O_\mathcal X) \in \mathrm{dAn}_k^{\le n+1}$ and let $V \in \mathcal X$ be an object such that $X_V \coloneqq (\mathcal X_{/V}, \mathcal O_\mathcal X |_V)$ is a derived $k$-affinoid space.
Then $V$ is $n$-truncated.
\end{lem}
\begin{proof}
We have to prove that for every object $U \in \mathcal X$, the space
\[ \Map_\mathcal X(U, V) \simeq \Map_{\mathcal X_{/U}}(U, U \times V) \]
is $n$-truncated.
This property is local on $U$, so we can restrict ourselves to the situation where $X_U \coloneqq (\mathcal X_{/U}, \mathcal O_\mathcal X |_U)$ is a derived $k$-affinoid space.
Using \cite[Remark 2.3.4]{DAG-V}, we see that this space fits into a fiber sequence
\[ \Map_{\mathcal X}(U, V) \to \Map_{\mathrm{dAn}_k}( X_U, X_V ) \to \Map_{\mathrm{dAn}_k}( X_U, X ) . \]
Since a derived $k$-analytic\xspace space $(\mathcal Y, \mathcal O_\mathcal Y)$ belongs to $\mathrm{dAfd}_k$ if and only if its truncation $(\mathcal Y, \pi_0(\mathcal O_\mathcal Y))$ does, we can replace $X$ with its truncation.
Let us denote by $F_X \colon \mathrm{Afd}_k \to \mathcal S$ the functor of points associated to $X$ and by $F_V \colon \mathrm{Afd}_k \to \mathcal S$ the functor of points associated to $(\mathcal X_{/V}, \mathcal O_\mathcal X |_V)$.
The arguments above show that it is enough to prove that for every ordinary $k$-affinoid space $Z$, the fibers of $F_X(Z) \to F_V(Z)$ are $n$-truncated.
By hypothesis, $F_V$ is the functor of points associated to some $k$-affinoid space, so it takes values in $\mathrm{Set}$.
Since $F_V(Z)$ is discrete, it suffices to show that $F_X(Z)$ is $(n+1)$-truncated. This follows directly from \cite[Lemma 2.6.19]{DAG-V}.
\end{proof}
\begin{prop} \label{prop:geometricity}
Let $n \ge 1$ and let $X = (\mathcal X, \mathcal O_\mathcal X) \in \mathrm{dAn}_k$ be a derived $k$-analytic\xspace space such that $\mathcal X$ is $n$-localic.
Then $\phi(X)$ belongs to $\mathrm{DM}_n$.
\end{prop}
\begin{proof}
Let $Y = (\mathcal Y, \mathcal O_\mathcal Y) \in \mathrm{dAfd}_k$ be a derived $k$-affinoid space such that $\mathcal O_\mathcal Y$ is discrete.
For every geometric morphism $f^{-1} \colon \mathcal X \leftrightarrows \mathcal Y \colon f_*$ we can use \cite[2.4.4.2]{HTT} to obtain a fiber sequence
\[ \Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}(f^{-1} \mathcal O_\mathcal X, \mathcal O_\mathcal Y ) \to \Map_{\mathrm{dAn}_k}( Y, X ) \to \Map_{\tensor*[^\rR]{\Top}{}}( \mathcal Y, \mathcal X ) , \]
where the fiber is taken at $(f^{-1}, f_*)$.
Since $\mathcal Y$ is $1$-localic and $n \ge 1$, it is also $n$-localic.
Therefore, \cite[Lemma 2.2]{Porta_Comparison_2015} shows that $\Map_{\tensor*[^\rR]{\Top}{}}( \mathcal Y, \mathcal X )$ is $n$-truncated.
Since $\mathcal O_\mathcal Y$ is discrete, we see that $\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}( f^{-1} \mathcal O_\mathcal X, \mathcal O_\mathcal Y )$ is $0$-truncated, hence $n$-truncated.
So
\[\phi(X)(Y) = \Map_{\mathrm{dAn}_k}(Y,X)\]
is $n$-truncated as well.
Let us now prove that $\phi(X)$ is geometric. Combining \cref{cor:representability_diagonal} and \cref{cor:dAfdk_closed_under_tau}, we see that it is enough to prove that $\phi(X)$ admits an atlas.
Choose objects $U_i \in \mathcal X$ such that $(\mathcal X_{/U_i}, \mathcal O_\mathcal X |_{U_i})$ is a derived $k$-affinoid space and that the joint morphism $\coprod U_i \to \mathbf 1_\mathcal X$ is an effective epimorphism.
Put $X_i \coloneqq (\mathcal X_{/U_i}, \mathcal O_\mathcal X |_{U_i})$.
By functoriality we obtain maps $\phi(X_i) \to \phi(X)$.
It follows from \cref{lem:affine_site_big_site} that the total morphism $\coprod \phi(X_i) \to \phi(X)$ is an effective epimorphism.
We are therefore left to prove that $\phi(X_i) \to \phi(X)$ are $(n-1)$-representable by \'etale maps.
First of all, we remark that if $Z \in \mathrm{dAfd}_k$, then for any map $\phi(Z) \to \phi(X)$, using full faithfulness of $\phi$, we obtain
\[ \phi(Z) \times_{\phi(X)} \phi(X_i) \simeq \phi( Z \times_X X_i ) , \]
and $Z \times_X X_i$ is \'etale over $Z$. Therefore we are reduced to prove that the stacks $\phi(X) \times_{\phi(Z)} \phi(X_i)$ are $(n-1)$-geometric.
We prove this by induction on $n$.
If $n = 1$, \cref{lem:decreasing_truncated_level} shows that the objects $U_i$ are discrete.
It follows from \cref{prop:etale_algebraic_spaces} that $\phi(Z) \times_{\phi(X)} \phi(X_i)$ is $0$-geometric.
Now suppose that $\mathcal X$ is $n$-localic and $n > 1$.
\cref{lem:decreasing_truncated_level} shows again that the objects $U_i$ are $(n-1)$-truncated.
Therefore \cite[Lemma 2.3.16]{DAG-V} shows that the underlying $\infty$-topos\xspace of $Z \times_X X_i$ is $(n-1)$-localic.
We conclude by the inductive hypothesis.
\end{proof}
As a consequence of \cref{prop:geometricity}, the functor $\phi \colon \mathrm{dAn}_k \to \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ induces a well defined functor
\[ \phi_n \colon \mathrm{dAn}_k^{\le n} \to \mathrm{DM}_n . \]
In order to achieve the proof of \cref{thm:functor_of_points_vs_dAnk}, we are left to show that $\phi_n$ is essentially surjective.
We will need the following elementary observation:
\begin{lem} \label{lem:equivalence_etale_sites}
Let $X$ be a geometric stack for the geometric context $(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$.
The functor $\mathrm{t}_0 \colon X_\mathrm{\acute{e}t} \to (\mathrm{t}_0(X))_\mathrm{\acute{e}t}$ is an equivalence of sites.
\end{lem}
\begin{proof}
We prove this by induction on the geometric level of $X$.
If $X$ is $(-1)$-geometric we can find $Y = (\mathcal Y, \mathcal O_\mathcal Y) \in \mathrm{dAfd}_k$ such that $X \simeq \phi(Y)$.
Consider the chain of equivalences
\[ (\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))_{/Y})_\mathrm{\acute{e}t} \simeq (\tensor*[^\rR]{\Top}{}_{/Y})_{_\mathrm{\acute{e}t}} \simeq (\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))_{/\mathrm{t}_0(Y)})_\mathrm{\acute{e}t} . \]
We now remark that, if $X \to Y$ is an \'etale map in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$, then $X$ is a derived $k$-analytic\xspace space.
Moreover, a derived $k$-analytic\xspace space belongs to $\mathrm{dAfd}_k$ if and only if its truncation does.
These observations imply that the above equivalence restricts to an equivalence
\[ Y_\mathrm{\acute{e}t} \simeq (\mathrm{t}_0(Y))_\mathrm{\acute{e}t}, \]
thus achieving the proof of the base step of the induction.
Suppose now that $X$ is $n$-geometric and that the statement holds for $(n-1)$-geometric stacks.
Choose an \'etale $n$-groupoid presentation $U^\bullet$ for $X$.
This means that $U^\bullet$ is a groupoid object in the $\infty$-category $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ such that each $U^m$ is $(n-1)$-geometric and that the map $U^0 \to X$ is $(n-1)$-representable by \'etale maps.
Since $\mathrm{t}_0$ commutes with products in virtue of \cref{prop:truncation_and_finite_limits} and it takes effective epimorphisms to effective epimorphisms by \cite[7.2.1.14]{HTT}, we see that $V^\bullet \coloneqq \mathrm{t}_0(U^\bullet)$ is a groupoid presentation for $\mathrm{t}_0(X)$.
Now, let $Y \to \mathrm{t}_0(X)$ be an \'etale map. We see that $Y \times_{\mathrm{t}_0(X)} V^\bullet \to V^\bullet$ is an \'etale map (i.e.\ it is a map of groupoids which is \'etale in each degree). By the inductive hypothesis, we obtain a map of simplicial objects $Z^\bullet \to U^\bullet$, such that
\[ \mathrm{t}_0(Z^\bullet) = Y \times_{\mathrm{t}_0(X)} V^\bullet . \]
Since $Y \times_{\mathrm{t}_0(X)} V^\bullet$ is a groupoid, so is $Z^\bullet$ (here we use again the equivalence guaranteed by the inductive hypothesis).
The geometric realization of $Z^\bullet$ provides us with an \'etale map $Z \to X$. Since $\mathrm{t}_0$ preserves effective epimorphisms, we conclude that $\mathrm{t}_0(Z) = Y$.
This construction is functorial in $Y$, and it provides the inverse to the functor $\mathrm{t}_0$.
\end{proof}
\begin{prop} \label{prop:phi_essentially_surjective}
The functor $\phi_n \colon \mathrm{dAn}_k^{\le n} \to \mathrm{DM}_n$ is essentially surjective.
\end{prop}
\begin{proof}
Let $X\in\mathrm{DM}_n$.
By \cref{lem:equivalence_etale_sites}, $X_\mathrm{\acute{e}t}$ is equivalent to $(\mathrm{t}_0(X))_\mathrm{\acute{e}t}$.
By hypothesis, $\mathrm{t}_0(X)$ is $n$-truncated.
Therefore, \cref{prop:over_n_category} shows that the mapping spaces in $(\mathrm{t}_0(X))_\mathrm{\acute{e}t}$ are $(n-1)$-truncated.
In other words, $(\mathrm{t}_0(X))_\mathrm{\acute{e}t}$ is equivalent to an $n$-category (cf.\ \cite[2.3.4.18]{HTT}).
As a consequence, $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ is $n$-localic.
Put $\mathcal X \coloneqq \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})^\wedge$.
Consider the composition
\[ \cT_{\mathrm{an}}(k) \times X_\mathrm{\acute{e}t}^{\mathrm{op}} \to \mathrm{dAfd}_k \times \mathrm{dAfd}_k^{\mathrm{op}} \xrightarrow{y} \mathcal S , \]
where the last functor classifies the Yoneda embedding (cf.\ \cite[\S 5.2.1]{Lurie_Higher_algebra}).
This induces a well defined functor
\[ \overline{\mathcal O_\mathcal X} \colon \cT_{\mathrm{an}}(k) \to \mathrm{PSh}(X_\mathrm{\acute{e}t}) , \]
which factors through $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
Let $\mathcal O_\mathcal X$ be its hypercompletion.
Since the functor $\cT_{\mathrm{an}}(k) \to \mathrm{dAfd}_k$ preserves products and admissible pullbacks, the same holds for $\mathcal O_\mathcal X$.
Moreover, \cref{lem:affine_site_big_site} implies that $\mathcal O_\mathcal X$ takes $\tau_\mathrm{\acute{e}t}$-coverings to effective epimorphisms.
In other words, $\mathcal O_\mathcal X$ defines a $\cT_{\mathrm{an}}(k)$-structure on $\mathcal X$.
If $\{U_i \to X\}$ is an \'etale $n$-atlas of $X$, each $U_i$ defines an object $V_i$ in $\mathcal X$.
Unraveling the definitions, we see that the $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $(\mathcal X_{/V_i}, \mathcal O_X |_{V_i})$ is canonically isomorphic to $U_i \in \mathrm{dAn}_k$ itself.
Therefore $X' \coloneqq (\mathcal X, \mathcal O_X)$ is a derived $k$-analytic\xspace space.
We are left to prove that $\phi(X') \simeq X$.
We can proceed by induction on the geometric level $n$ of $X$.
If $n = -1$, $\phi(X')$ is the functor represented by $X'$, and the same holds for $X$.
Let now $n \ge 0$.
Choose an \'etale $n$-atlas $\{U_i \to X\}$ for $X$.
Set $U \coloneqq \coprod U_i$ and let $U^\bullet$ denote the \v{C}ech nerve of $U \to X$.
Every map $U^n \to X$ is \'etale.
In particular, the functor $X_\mathrm{\acute{e}t} \to \mathcal S$ sending $Y$ to $\Map_{X_\mathrm{\acute{e}t}}(Y, U^n)$ defines an element $V^n \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
Using \cref{lem:equivalence_etale_sites}, we see that
\[ \Map_{X_\mathrm{\acute{e}t}}(Y, U^n) \simeq \Map_{\mathrm{t}_0(X)_\mathrm{\acute{e}t}}(\mathrm{t}_0(Y), \mathrm{t}_0(U^n)). \]
Since $t_0(U^n)$ is a geometric stack, we conclude that the above space is truncated.
In particular, the object $V^n$ is a truncated object in $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$, so it is hypercomplete.
In other words, $V^n$ belongs to $\mathcal X$.
We can therefore identify $\mathrm{Sh}(U^n_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})^\wedge$ with $\mathcal X_{/V^n}$.
The universal property of \'etale morphisms (cf.\ \cite[Remark 2.3.4]{DAG-V}) shows that we can arrange the $V^n$s into a simplicial object $V^\bullet$ in $\mathcal X$, whose geometrical realization coincides with $\mathbf 1_\mathcal X$.
The inductive hypothesis shows that $\phi( \mathcal X_{/V^\bullet}, \mathcal O_X |_V^\bullet ) \simeq U^\bullet$ as simplicial objects in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
Since $\phi$ commutes with \v{C}ech nerves of \'etale maps and their realizations (in virtue of \cref{lem:affine_site_big_site}), we conclude that $\phi(X')$ is equivalent to $X$ itself.
\end{proof}
The proof of \cref{thm:functor_of_points_vs_dAnk} is now achieved.
\section{Appendices}
\subsection{Complements on overcategories}
The goal of this subsection is to provide a proof of the following basic result, for which we do not know a reference: if $(\mathcal C, \tau)$ is a Grothendieck site and $\mathcal C$ is a $1$-category, then for every $n$-truncated sheaf $X \in \mathrm{PSh}(\mathcal C)$, the overcategory $\mathcal C_{/X}$ is an $(n-1)$-category.
The proof relies on the following lemma:
\begin{lem} \label{lem:fiber_sequence_for_over}
Let $\mathcal C$ be an $\infty$-category.
Let $X \in \mathcal C$ be an object and let $f \colon U \to X$, $g \colon V \to X$ be two $1$-morphisms of $\mathcal C$ viewed as objects of $\mathcal C_{/X}$.
For every morphism $h \colon U \to V$ in $\mathcal C$, choose a $2$-simplex $\sigma \colon \Delta^2 \to \mathcal C$ extending the morphism $\Lambda^2_1 \to \mathcal C$ classified by $h$ and $g$.
Put $f' \coloneqq d_1(\sigma)$.
Then we have a fiber sequence
\[ \mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \to \Map_{\mathcal C_{/X}}(f,g) \to \Map_{\mathcal C}(U,V). \]
\end{lem}
\begin{proof}
It follows from \cite[Proposition 2.1.2.1]{HTT} that the canonical map $p \colon \mathcal C_{/X} \to \mathcal C$ is a right fibration. In particular, it is a Cartesian fibration where every edge of $\mathcal C_{/X}$ is $p$-Cartesian.
The $2$-simplex $\sigma \colon \Delta^2 \to \mathcal C$ can be viewed as an edge of $\mathcal C_{/X}$.
Reviewing the Kan complex $\Map_{\mathcal C}(U,X)$ as an $\infty$-category, we have a canonical equivalence $\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \simeq \Map_{\Map_{\mathcal C}(U,X)}(f,f')$.
The conclusion follows at this point from \cite[Proposition 2.4.4.2]{HTT}.
\end{proof}
\begin{prop} \label{prop:over_n_category}
Let $\mathcal C$ be an $\infty$-category.
Let $X \in \mathcal C$ be an $n$-truncated object.
Let $f \colon U \to X$ and $g \colon V \to X$ be two morphisms viewed as objects in $\mathcal C_{/X}$.
If $V$ is $m$-truncated with $m < n$, then $\Map_{\mathcal C_{/X}}(U, V)$ is $(n-1)$-truncated.
\end{prop}
\begin{proof}
Choosing $f'$ as in \cref{lem:fiber_sequence_for_over}, we obtain a fiber sequence
\[ \mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \to \Map_{\mathcal C_{/X}}(f,g) \to \Map_{\mathcal C}(U,V). \]
Now, $\Map_{\mathcal C}(U,V)$ is $m$-truncated by hypothesis.
On the other hand, we have a pullback diagram
\[ \begin{tikzcd}
\mathrm{Path}_{\Map_{\mathcal C}(U,X)}(f,f') \arrow{r} \arrow{d} & \{*\} \arrow{d}{f'} \\
\{*\} \arrow{r}{f} & \Map_{\mathcal C}(U,X).
\end{tikzcd} \]
Therefore $\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f')$ fits in the pullback diagram
\[ \begin{tikzcd}
\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \arrow{d} \arrow{r} & \Map_{\mathcal C}(U,X) \arrow{d}{\Delta} \\
\{*\} \arrow{r}{(f,f')} & \Map_{\mathcal C}(U,X) \times \Map_{\mathcal C}(U,X) .
\end{tikzcd} \]
Since $X$ is $n$-truncated, it follows that $\Map_{\mathcal C}(U,X)$ is $n$-truncated.
Therefore, \cite[5.5.6.15]{HTT} shows that $\Delta$ is $(n-1)$-truncated.
We deduce that $\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f')$ is $(n-1)$-truncated.
Thus the fiber sequence of \cref{lem:fiber_sequence_for_over} implies that $\Map_{\mathcal C_{/X}}(f,g)$ is $(n-1)$-truncated as well, completing the proof.
\end{proof}
\subsection{Complements on geometric stacks}
\begin{defin}\label{def:C_closed_under_tau}
Let $(\mathcal C,\tau)$ be an $\infty$-site\xspace.
The $\infty$-category\xspace $\mathcal C$ is said to be \emph{closed under $\tau$-descent} if for any morphism from a sheaf $X$ to a representable sheaf $Y$, any $\tau$-covering $\{Y_i\to Y\}$, the representability of $X\times_Y Y_i$ for every $i$ implies the representability of $X$.
\end{defin}
We need the following converse to \cite[Corollary 2.12]{Porta_Yu_Higher_analytic_stacks_2014}:
\begin{lem} \label{lem:factorizing_property_effective_epimorphisms}
Let $(\mathcal C, \tau)$ be a subcanonical $\infty$-site.
Let $F \to G$ be an effective epimorphism in $\mathrm{Sh}(\mathcal C, \tau)$.
For any object $X \in \mathcal C$ and any morphism $h_X \to G$, there exists a $\tau$-covering $\{U_i \to X\}$ such that the composite morphisms $h_{U_i} \to h_X \to G$ factor as
\[ \begin{tikzcd}
{} & {} & F \arrow{d} \\
h_{U_i} \arrow[dashed]{urr} \arrow{r} & h_X \arrow{r} & G .
\end{tikzcd} \]
\end{lem}
\begin{proof}
Using \cite[Proposition 2.11]{Porta_Yu_Higher_analytic_stacks_2014} we see that the morphism $\pi_0(F) \to \pi_0(G)$ is an effective epimorphism of sheaves of sets.
In particular, there exists a covering $\{V_j \to X\}$ such that the composite morphisms $\pi_0( h_{V_j}) \to \pi_0(h_X) \to \pi_0(G)$ factor through $\pi_0(F)$.
Since $\pi_0(F)$ is by definition the sheafification of the presheaf $Y \mapsto \pi_0(F(Y))$ and since
\[ \Map_{\mathrm{Sh}(\mathcal C, \tau)}( \pi_0(h_{V_j}), \pi_0(F) ) \simeq \Map_{\mathrm{Sh}(\mathcal C, \tau)}(h_{V_j}, \pi_0(F)) \simeq \pi_0(F) (V_j) , \]
we can find a $\tau$-covering $\{U_{ij} \to V_j\}$ such that every composite morphism $h_{U_{ij}} \to h_{V_j} \to \pi_0(F)$ factors through $F \to \pi_0(F)$.
Finally, again since $\pi_0(G)$ is the sheafification of the presheaf $Y \mapsto \pi_0(G(Y))$, we can further refine the covering such that the morphisms $h_{U_{ij}} \to F$ are homotopic to the compositions $h_{U_{ij}} \to h_X \to G$. This completes the proof.
\end{proof}
\begin{prop} \label{prop:geometric_stacks_closed_under_tau}
Let $(\mathcal C, \tau, \mathbf P)$ be a geometric context in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
Assume that $\mathcal C$ is closed under $\tau$-descent.
Then the class of $n$-representable morphisms is closed under $\tau$-descent, in the sense that for any morphism $f \colon X \to Y$ with $Y$ a $n$-geometric stack, if there exists an $n$-atlas $\{U_i\}$ of $Y$ such that $X \times_Y U_i$ is $n$-geometric for every $i$, then $F$ is $n$-geometric as well.
\end{prop}
\begin{proof}
The proof goes by induction on the geometric level $n$.
When $n = -1$, this holds because $\mathcal C$ is closed under $\tau$-descent.
Let now $n \ge 0$. Let $\{U_i\}$ be an $n$-atlas of $Y$ such that $X_i \coloneqq X \times_Y U_i$ is $n$-geometric for every $i$.
Choose an $n$-atlas $\{V_{ij}\}$ of $X \times_Y U_i$. The compositions $V_{ij} \to X_i \to X$ provide an $n$-atlas of $X$.
We are therefore left to prove that the diagonal of $X$ is $(n-1)$-representable.
Let $V \coloneqq \coprod V_{ij}$ be the $n$-atlas of $X$ introduced above. By construction, the map $V \to X$ is $(n-1)$-representable.
It follows that the induced map $V \times_X V \to V$ is $(n-1)$-representable as well.
Since $V$ is a disjoint union of $(-1)$-representable stacks, it follows that $V \times_X V$ is $(n-1)$-geometric.
Observe now that $V \times V \to X \times X$ is an effective epimorphism.
Therefore for every morphism $S \to X \times X$ from a $(-1)$-representable stack $S$, by \cref{lem:factorizing_property_effective_epimorphisms}, we can choose a $\tau$-covering $S_i \to S$ such that the composite map $S_i \to S \to X \times X$ factors as
\[ \begin{tikzcd}
{} & {} & V \times V \arrow{d} \\
S_i \arrow[dashed]{urr} \arrow{r} & S \arrow{r} & X \times X .
\end{tikzcd} \]
In order to prove that the diagonal $\Delta_X \colon X \to X \times X$ is $(n-1)$-representable, we have to show that $S \times_{X \times X} X$ is $(n-1)$-geometric.
Using the induction hypothesis, it suffices to show that each stack $S_i \times_{X \times X} X$ is $(n-1)$-geometric.
Note that this stack fits in the following diagram of cartesian squares:
\[ \begin{tikzcd}
S_i \times_{X \times X} X \arrow{r} \arrow{d} & V \times_X V \arrow{r} \arrow{d} & X \arrow{d} \\
S_i \arrow{r} & V \times V \arrow{r} & X \times X .
\end{tikzcd} \]
Since $V \times V$, $V \times_X V$ and $S_i$ are $(n-1)$-geometric, it follows that the same goes for $S_i \times_{X \times X} X$, thus completing the proof.
\end{proof}
\begin{cor} \label{cor:representability_diagonal}
Let $(\mathcal C, \tau, \mathbf P)$ be a geometric context and assume that $\mathcal C$ is closed under $\tau$-descent.
If $X \in \mathrm{Sh}(\mathcal C, \tau)$ admits an $n$-atlas, then it is $n$-geometric.
\end{cor}
\begin{proof}
We have to prove that the diagonal of $X$ is $(n-1)$-representable.
Let $V \to X$ be an $n$-atlas.
Then $V \times V \to X \times X$ is an $n$-atlas for $X\times X$.
By \cref{lem:factorizing_property_effective_epimorphisms}, for any map $S \to X \times X$, with $S$ being representable, we can find a $\tau$-covering $\{ S_i \to S\}$ such that the composite maps $S_i \to S \to X \times X$ factor through $V \times V$.
Using \cref{prop:geometric_stacks_closed_under_tau}, we are reduced to prove that each $S_i \times_{X \times X} X$ is $(n-1)$-geometric.
Consider the diagram
\[ \begin{tikzcd}
S_i \times_{X \times X} X \arrow{r} \arrow{d} & V \times_X V \arrow{r} \arrow{d} & X \arrow{d} \\
S_i \arrow{r} & V \times V \arrow{r} & X \times X .
\end{tikzcd} \]
The right and the outer squares are pullback diagrams by construction.
Therefore, so is the left square.
Now we conclude from the fact that $V \times_X V$ is $(n-1)$-geometric.
\end{proof}
\begin{prop} \label{prop:Afdk_closed_under_tau}
The category $\mathrm{Afd}_k$ of $k$-affinoid spaces is closed under $\tau_\mathrm{\acute{e}t}$-descent.
\end{prop}
\begin{proof}
Let $Y$ be a $k$-affinoid space and let $f \colon F \to h_Y$ be a morphism in $\mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t})$.
Let $\{Y_i \to Y\}_{i \in I}$ be a finite étale covering in the category $\mathrm{Afd}_k$.
Assume that for every index $i$, the fiber product $h_{Y_i} \times_{h_Y} F$ is representable by $X_i \in \mathrm{Afd}_k$.
Put $Y^0 \coloneqq \coprod_{i \in I} Y_i$ and let $Y^\bullet$ be the \v{C}ech nerve of $Y^0 \to Y$.
By assumption, we see that for every integer $n$, $h_{Y^n} \times_{h_Y} F$ is representable.
Choose $X^n \in \mathrm{Afd}_k$ such that $h_{Y^n} \times_{h_Y} F \simeq h_{X^n}$.
Fully faithfulness of the Yoneda embedding implies that we can arrange the objects $X^n$ into a simplicial object $X^\bullet$ in $\mathrm{Afd}_k$.
Let $\bm{\Delta}_s$ be the semisimplicial category.
It follows from \cite[6.5.3.7]{HTT} that the inclusion $\bm{\Delta}_s^\mathrm{op} \subset \bm{\Delta}^\mathrm{op}$ is cofinal.
Let $\bm{\Delta}_{s, \le 2}$ be the full subcategory of $\bm{\Delta}_s$ spanned by the objects $[0]$, $[1]$ and $[2]$.
The inclusion $\bm{\Delta}_{s, \le 2}^\mathrm{op} \subset \bm{\Delta}_s^\mathrm{op}$ is $1$-cofinal, in the sense that for every $[n] \in \bm{\Delta}_s$, the undercategory $(\bm{\Delta}_{s,\le 2}^\mathrm{op})_{[n]/}$ is nonempty and connected.
Let $j \colon \bm{\Delta}_{s, \le 2}^\mathrm{op} \hookrightarrow \bm{\Delta}$ be the composite functor.
Since $\mathrm{Afd}_k$ is a $1$-category, we see that $X^\bullet$ admits a colimit if and only if $X^\bullet_{s, \le 2} \coloneqq X^\bullet \circ j$ does.
The latter statement is true because $\mathrm{Afd}_k$ admits finite colimits.
Let $X$ be the colimit of $X^\bullet$ and let $g \colon X \to Y$ be the canonical map.
We claim that $X^n \simeq Y^n \times_Y X$.
To prove this, it is enough to show that $X^0 \simeq Y^0 \times_Y X$.
We first remark that if the map $X^0 \to Y^0$ is a closed immersion, then the statement follows directly from the fpqc descent of coherent sheaves (cf.\ \cite{Conrad_Descent_for_coherent_2003}).
In the general case, we factor $X^0 \to Y^0$ as $X^0 \hookrightarrow \mathbf D^N_{Y^0} \to Y^0$, where $\mathbf D^N_{Y^0}$ denotes the $N$-dimensional unit polydisc over $Y^0$ and the first arrow is a closed immersion.
Observe that the colimit of $\mathbf D^N_{Y^\bullet}$ is $\mathbf D^N_Y$, and that $\mathbf D^N_{Y^0} \simeq Y^0 \times_Y \mathbf D^N_Y$.
Consider the following diagram:
\[ \begin{tikzcd}
X^0 \arrow{r} \arrow{d} & X \arrow{d} \\
\mathbf D^N_{Y^0} \arrow{r} \arrow{d} & \mathbf D^N_Y \arrow{d} \\
Y^0 \arrow{r} & Y .
\end{tikzcd} \]
Since $X_0 \hookrightarrow \mathbf D^N_{Y^0}$ is a closed immersion, we see that the top square is a pullback.
Moreover, we remarked that the bottom square is also a pullback.
Hence so is the outer square, completing the proof of the claim.
As a consequence, we see that $X^\bullet$ is the \v{C}ech nerve of the étale covering $X^0 \to X$.
In particular, in $\mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t})$ we have
\[ h_X \simeq | h_{X^\bullet} | ,\]
where $\abs{\cdot}$ denotes the geometric realization.
Finally, since $\mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t})$ is an $\infty$-topos\xspace, we obtain:
\[ h_X \simeq | h_{X^\bullet} | \simeq |h_{Y^\bullet} \times_{h_Y} F| \simeq |h_{Y^\bullet}| \times_{h_Y} F \simeq F. \]
This shows that $F$ is representable, thus completing the proof.
\end{proof}
\begin{cor} \label{cor:dAfdk_closed_under_tau}
The category $\mathrm{dAfd}_k$ of derived $k$-affinoid spaces is closed under $\tau_\mathrm{\acute{e}t}$-descent.
\end{cor}
\begin{proof}
Let $Y = (\mathcal Y, \mathcal O_\mathcal Y)$ be a derived $k$-affinoid space.
Let $F \to h_Y$ be a morphism in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
Assume there exists an \'etale covering $Y_i \to Y$ such that each base change $h_{Y_i} \times_{h_Y} F$ is representable by a derived $k$-affinoid space $X_i$.
In particular, $\mathrm{t}_0(h_{Y_i} \times_{h_Y} F) \simeq \mathrm{t}_0(h_{Y_i}) \times_{\mathrm{t}_0(h_Y)} \mathrm{t}_0(F)$ is representable by an ordinary $k$-affinoid space $\mathrm{t}_0(X_i)$.
It follows from \cref{prop:Afdk_closed_under_tau} that $\mathrm{t}_0(F)$ is representable by an ordinary $k$-affinoid space $Z$.
Form the \v{C}ech nerve $G^\bullet$ of $\coprod h_{Y_i} \times_{h_Y} F \to F$.
By hypothesis, each $G^n$ is a disjoint union of derived $k$-affinoid spaces.
Since $\phi$ is fully faithful, we obtain in this way a simplicial object $X^\bullet$ in $\mathrm{dAn}_k$, such that all the face maps are \'etale morphisms.
It follows from \cite[Proposition 2.3.5]{DAG-V} that this simplicial object admits a colimit $Y$ in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ and that the canonical maps $X^n \to X$ are \'etale.
This shows that we can cover $X$ with derived $k$-affinoid spaces.
In particular, $X$ is a derived $k$-analytic\xspace space.
We are left to prove that $X$ is a derived $k$-affinoid space.
Observe that the maps $\mathrm{t}_0(X^n) \to \mathrm{t}_0(X)$ are \'etale.
Since $X$ (resp.\ $X^n$) and $\mathrm{t}_0(X)$ (resp.\ $\mathrm{t}_0(X^n)$) share the same underlying $\infty$-topos\xspace, we can use the statement (3') in the proof of \cite[Proposition 2.3.5]{DAG-V} to conclude that the colimit of $\mathrm{t}_0(X^\bullet)$ in $\tensor*[^\rR]{\Top}{}(\cT_{\mathrm{an}}(k))$ is $\mathrm{t}_0(X)$.
On the other hand, since $\mathrm{t}_0$ commutes with limits, we can further identify $\phi(\mathrm{t}_0(X^\bullet))$ with the \v{C}ech nerve of the map $\coprod \mathrm{t}_0(h_{Y_i} \times_{h_Y} F) \to \mathrm{t}_0(F) \simeq \phi(Z)$.
It follows that $\mathrm{t}_0(X) \simeq Z$ in $\mathrm{dAn}_k$. This shows that $X$ is a derived $k$-affinoid space, and $\phi(X) \simeq F$.
The proof is thus complete.
\end{proof}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,638
|
Q: Extract and explode embedded json fields in apache spark I'm completely new to spark, but don't mind if the answer is in python or Scala. I can't show the actual data for privacy reasons, but basically I am reading json files with a structure like this:
{
"EnqueuedTimeUtc": 'some date time',
"Properties": {},
"SystemProperties": {
"connectionDeviceId": "an id",
"some other fields that we don't care about": "data"
},
"Body": {
"device_id": "an id",
"tabs": [
{
"selected": false,
"title": "some title",
"url": "https:...."
},
{"same again, for multiple tabs"}
]
}
}
Most of the data is of no interest. What I want is a Dataframe consisting of the time, device_id, and url. There can be multiple urls for the same device and time, so I'm looking to explode these into one row per url.
| timestamp | device_id | url |
My immediate problem is that when I read this, although it can work out the structure of SystemProperties, Body is just a string, probably because of variation. Perhaps I need to specify the schema, would that help?
root
|-- Body: string (nullable = true)
|-- EnqueuedTimeUtc: string (nullable = true)
|-- SystemProperties: struct (nullable = true)
| |-- connectionAuthMethod: string (nullable = true)
| |-- connectionDeviceGenerationId: string (nullable = true)
| |-- connectionDeviceId: string (nullable = true)
| |-- contentEncoding: string (nullable = true)
| |-- contentType: string (nullable = true)
| |-- enqueuedTime: string (nullable = true)
Any idea of an efficient (there are lots and lots of these records) way to extract urls and associate with the time and device_id? Thanks in advance.
A: Here's an example for extraction. Basically you can use from_json to convert the Body to something that is more structured, and use explode(transform()) to get the URLs and expand to different rows.
# Sample dataframe
df.show(truncate=False)
+----------------------------------------------------------------------------------------------------------------------------------------------+---------------+----------------+
|Body |EnqueuedTimeUtc|SystemProperties|
+----------------------------------------------------------------------------------------------------------------------------------------------+---------------+----------------+
|{"device_id":"an id","tabs":[{"selected":false,"title":"some title","url":"https:1"},{"selected":false,"title":"some title","url":"https:2"}]}|some date time |[an id] |
+----------------------------------------------------------------------------------------------------------------------------------------------+---------------+----------------+
df.printSchema()
root
|-- Body: string (nullable = true)
|-- EnqueuedTimeUtc: string (nullable = true)
|-- SystemProperties: struct (nullable = true)
| |-- connectionDeviceId: string (nullable = true)
# Extract desired properties
df2 = df.selectExpr(
"EnqueuedTimeUtc as timestamp",
"from_json(Body, 'device_id string, tabs array<map<string,string>>') as Body"
).selectExpr(
"timestamp",
"Body.device_id",
"explode(transform(Body.tabs, x -> x.url)) as url"
)
df2.show()
+--------------+---------+-------+
| timestamp|device_id| url|
+--------------+---------+-------+
|some date time| an id|https:1|
|some date time| an id|https:2|
+--------------+---------+-------+
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 576
|
Uncategorized January 06, 2022 0 Comments
Drops suit against the IGC
Full House Resorts has dropped its legal action contesting the Indiana Gaming Commission (IGC)'s decision to award Churchill Downs a Vigo County casino license.
The move by Full House on Tuesday paves the way for Churchill Downs to build a long-awaited casino in the west central Indiana city of Terry Haute.
Anchor at WTHI-TV Terre Haute, Jon Swaner, shared the news via Twitter:
NEW THIS MORNING: Full House Resorts is DROPPING its lawsuit against the IN Gaming Commission. This means Churchill Downs can begin the process of building a casino in Terre Haute. Where it will be built may still be up for discussion. We'll have much more on this on News 10.
— Jon Swaner (@jonswaner) January 5, 2022
Casino.org cites a letter to IGC general counsel Dennis Mullen from an attorney for the Indianapolis firm representing Full House. The lawyer's letter said the firm neither harbored malicious intent nor desired to delay the Terre Haute project any further.
even if the process were reopened or repeated, the outcome is unlikely to be different"
"Although we disagree with the characterizations that were made regarding the motives and merits of our claims, the comments by the Chairman and the other commissioners made clear that, even if the process were reopened or repeated, the outcome is unlikely to be different," the lawyer wrote.
"Full House's top priority is for the Commission to continue to view the company as having the highest levels of character and integrity," he added.
Full House filed its challenge against the IGC on December 16 in the Marion County Superior Court, Indianapolis. The suit alleged the IGC broke the state's Open Door Law, denied Full House's casino license application, and favored Churchill's proposal with no discussion, comment, explanation, or public debate.
At an IGC meeting less than a week later, executive director Greg Small defended the regulator's process in selecting Churchill and criticized Full House for its actions. IGC chairman Michael McMains went as far as dismissing the lawsuit and appeal as "sour grapes."
Full House operates five casinos in Colorado, Indiana, Mississippi, and Nevada. In December, the Illinois Gaming Board selected the Summerlin, Nevada-based casino developer and operator to develop and run a new American Place casino in Waukegan, Illinois.
Casino.org cites a Tuesday night statement from McMains, saying said he was thankful for Full House's change of heart.
West Central Indiana will soon benefit from this significant economic development project"
"We are pleased that Full House has dismissed these actions and that Vigo County and the greater community of West Central Indiana will soon benefit from this significant economic development project," McMains said.
Coast finally clear
Residents of Terre Haute champing at the bit for a casino can finally start chalking off the days, after a dramatic month of legal challenges.
In late December, an Indiana administrative law judge signed an order dismissing Lucy Luck Gaming's appeal against the IGC's decision not to renew Lucy Luck's casino license in Vigo County. In a settlement reached with the IGC, Lucy Luck got its $5m license fee back.
Churchill's proposal for its Queen of Terre Haute casino stated that the project would bring about 1,000 construction jobs and 500 permanent roles. Along with a 125-room hotel, the Queen of Terre Haute will host 1,000 slot machines, 50 game tables, and a retail sportsbook falling under Churchill's TwinSpires brand.
Speculation has arisen about whether Churchill is mulling relocating its proposed casino site from the western part of Terre Haute to the east. According to a spokesperson for the Kentucky-based firm, Churchill is "still open to exploring options."
The post Churchill Downs Finally Free to Build Terre Haute Casino After Full House Drops Lawsuit appeared first on Latest Online Gambling News.
The post Churchill Downs Finally Free to Build Terre Haute Casino After Full House Drops Lawsuit appeared first on Top Rank Casinos.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,689
|
Q: Enable admin mode view in caliburn.micro I am re-writing a WPF app to use Caliburn.Micro. The app is a menu system that shows Folders and then files within those folders. The problem is that I need to allow the user to switch to an "Admin" mode which will allow additional options. I currently have a FolderView and FileView along with ViewModels for each. I was thinking of having a seperate FolderAdminView and FileAdminView so I can change the UI and enable the additional options. The problem is switching between the two when the user changes modes.
Both a FolderViewModel and FileViewModel can be loaded at the same time so a call to DeactivateItem(ActiveItem, true); will act as a back button and return to the folder view.
I would also like to carry over values from the FolderViewModel to the FolderAdminViewModel since the main difference is UI.
Is there an easy way to swap out items in the WindowManager or an easier way to do this altogether? Could I have one ViewModel but two Views? Is there a way to have both templates in one view and select the correct one there?
A: Can you not just have an IsAdmin property on your ViewModel and bind the visibility of your admin only items to that using a BooleanToVisibilityConverter ?
ViewModel
public bool IsAdmin
{
get
{
//What ever you do to work out if user is admin
//omitted any INotifyPropertyChanged gubbins
}
}
Xaml
<StackPanel Visibility="{Binding IsAdmin,Converter={StaticResource BooleanToVisibiltyConverter}}"></StackPanel>
Converter
public sealed class BooleanToVisibilityConverter : IValueConverter
{
public object Convert(object value, Type targetType, object parameter, CultureInfo culture)
{
var flag = false;
if (value is bool)
{
flag = (bool)value;
}
else if (value is bool?)
{
var nullable = (bool?)value;
flag = nullable.GetValueOrDefault();
}
if (parameter != null)
{
if (bool.Parse((string)parameter))
{
flag = !flag;
}
}
return flag ? Visibility.Visible : Visibility.Collapsed;
}
public object ConvertBack(object value, Type targetType, object parameter, CultureInfo culture)
{
var back = ((value is Visibility) && (((Visibility)value) == Visibility.Visible));
if (parameter != null)
{
if ((bool)parameter)
{
back = !back;
}
}
return back;
}
}
A: You can use the 'context' attached property to specify the context for any views which are loaded e.g.:
<ContentControl x:Name="SomeSubViewModel" cal:View.Context="SomeContext" />
CM uses ToString() on the context object to obtain a value it will use to build the typename during view resolution. This means you can have multiple views for the same viewmodel and therefore add additional functions when the user is in admin mode by binding the View.Context property
You could also create a binding for each item you want to hide on the viewmodel and use a converter to check if the user is logged in etc - obviously it depends on whether you want to duplicate the XAML in two views or have a single view with conditional logic to hide/show areas
Read up on the context property:
http://caliburnmicro.codeplex.com/wikipage?title=All%20About%20Conventions&referringTitle=Documentation
the link above has some examples (in the first few sections)
and here:
http://caliburnmicro.codeplex.com/wikipage?title=Screens%2c%20Conductors%20and%20Composition
in the Multiple Views over the Same ViewModel section
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,081
|
Q: Java Servlet How to filter pages by User Role I'm coding a security web app. And I have a problem. For example , there is a pages called " Manager (JSP) " . I only want admins to see this page.
When user login the website , session create.
if(dao.check(uname, pass)) {
UserAccount user = new UserAccount();
user.setUsername(uname);
HttpSession session = request.getSession();
session.setAttribute("username", uname);
response.sendRedirect("welcome.jsp");
}
Heres the UserAccount Class :
public class UserAccount {
private String username;
private String password;
private List<String> roles;
public String getUsername() {
return username;
}
public void setUsername(String username) {
this.username = username;
}
public String getPassword() {
return password;
}
public void setPassword(String password) {
this.password = password;
}
public List<String> getRoles() {
return roles;
}
public void setRoles(List<String> roles) {
this.roles = roles;
}
}
And i have a Filter class which implements Filter. How program get understand who is admin or not and let to see manager page?
A: You can use a boolean attribute in your UserAccount class , this attribute will hold a value of true if the use is an admin , and false if he is not
public class UserAccount {
private String username;
private String password;
private List<String> roles;
private boolean is_admin;
...}
The redirection to manager page will see first if the current user is admin or not , so he will be denied if he's not an admin
NB : i see that you are using a List of roles , so maybe you're using a Role-Based access control ( like RBAC or RABAC )
In that case maybe you're giving admin permissions through his roles , so here the Filter method will see the user's roles list and check if it contains the role of admin , and if yes it will redirect into the manager page
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,355
|
<?php
// Compatible with sf_escaping_strategy: true
$a_blog_post = isset($a_blog_post) ? $sf_data->getRaw('a_blog_post') : null;
use_helper('a');
?>
<?php if ($a_blog_post->Author): ?>
<?php echo link_to(aHtml::entities($a_blog_post->Author), '@a_blog_admin_addFilter?name=author_id&value='.$a_blog_post->Author->id, 'post=true') ?>
<?php else: ?>
<?php echo link_to("No Author", '@a_blog_admin_addFilter?name=author_id&value=-', 'post=true') ?>
<?php endif ?>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,708
|
Q: (Codility binary gap error) Problem : Cant convert octal code into expected array Its works well when converting to string except binary with prefix 0.
My code:
function dec2bin(dec) {
return dec.toString(2);
}
function getRndInteger(min, max) {
return Math.floor(Math.random() \* (max - min + 1)) + min;
}
const min = 1;
const max = 2147483647;
const nilai = dec2bin(getRndInteger(min, max));
function solution(N) {
**const nilaiArr = Array.from(String(N), Number);**
let temporer = 0;
let save = \[\];
for (let i = 0; i \< nilaiArr.length - 1; i++) {
if (nilaiArr\[i\] == 1 && nilaiArr\[i + 1\] == 0) {
for (let j = i + 1; nilaiArr\[j\] == 0; j++) {
temporer++;
if (nilaiArr\[j + 1\] == 1) {
save.push(temporer);
temporer = 0;
} else if (nilaiArr\[j + 1\] == undefined) {
break;
}
}
} else {
continue;
}
}
if (save.length === 0) {
console.log(0);
} else {
save.sort();
save.reverse();
console.log(save\[0\]);
}
}
solution(nilai);
Problem sample : 010000010001 became 1073745921.
Expected output same as origin binary number. I cant find the answer as long as I scroll in this forum lol. I
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 590
|
'Breaking Bad': Giancarlo Esposito Once Recalled Vince Gilligan's Reaction to Gus Fring's Pre-Death Gesture — 'His Eyes Lit Up'
Amanda Harding
There are plenty of death scenes on Breaking Bad ranging from typical to extraordinary. But none of them are as memorable as the moment Walter White takes down his nemesis, Gus Fring, in an epic suicide bomber explosion that's impossible to forget.
The gore factor of this Breaking Bad scene is one reason fans will always remember. But showrunner Vince Gilligan is equally enamored by a different small detail that actor Giancarlo Esposito came up with on his own.
Walter White and Gus Fring try to kill each other on 'Breaking Bad'
By season 4 of Breaking Bad, it becomes clear that Fring and Heisenberg cannot both exist on the planet at the same time. The two former business associates engage in a dangerous back-and-forth with each trying to kill the other. Since they are both brilliant and strategic, the process is not so simple.
Fring easily circumvents a car bomb Walter White planted. But he's not so perceptive when Walt lures him to the nursing home to visit Hector, who has rigged a bomb to his wheelchair to take out Fring. Walt and Hector both get their revenge on the drug kingpin with one epic blast.
Gus Fring's death is a gory nightmare
RELATED: 'Breaking Bad': Walter White Defeated Criminal Mastermind Gus Fring By Exploiting 1 Major Weakness
The Breaking Bad team borrowed makeup artists from The Walking Dead to create Fring's post-explosion look, which included half his face being blown off. For one terrifying moment, Fring emerges from Hector's room walking upright, which leads fans to wonder if he may have survived the blast. But then the camera pans to the opposite side of his face to reveal it's been blown away.
Several second elapse before Fring collapses and dies, marking the end of the fourth season of Breaking Bad. But his small gesture in this moment is so perfectly Fring that Gilligan became giddy over it.
Gus Fring straightens his tie before collapsing down dead
Giancarlo Esposito as Gustavo "Gus" Fring – Better Call Saul _ Season 3, Gallery | Robert Trachtenberg/AMC
Fring is calm, calculated, methodical, and ruthless. He always dresses impeccably and speaks carefully. One reason he can't stand Walt and Jesse is because they're too sloppy. This does not match his approach to large-scale meth dealing.
And that's precisely why his last little move before death is so appropriate for his character. Esposito explained his vision during a 2018 interview with The Rich Eisen Show.
"So we had discussed a lot of it, and Vince says, 'What might you be doing if this happened?' And I said, 'Well look, Gus is very proper. I close my jacket when I stand up, I open it when I sit down.' I said, 'I would be straightening my tie.' And Vince just—his eyes lit up, and he said, 'Oh my goodness' — and I said, 'Yeah, he's preparing to go.'"
It may seem like a small detail. But that simple tie straightening is the perfect ending for a man who never reacts to extreme situations. Even his own death.
And that, even more than the gore, is why fans will never forget Gus Fring being killed on Breaking Bad.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,337
|
It's no surprise that with the sky rocketing gas prices as of recently many motorists are looking for new and improved ways to save gas/money. It's no surprise that many people have decided to switch over to a moped or scooter. Cheap Mopeds for Sale was created to help people like you find a cheaper vehicle and way of transportation. While at first the idea of owning a moped to some may be a little weird or different, we have had constant feedback from many people who have made the swap from a full sized vehicle and most are happy they made that change. Cheap Mopeds for sale helps find you the best prices on new and used mopeds. We use complex search algorithms to help you search for the brands you want at the best possible prices. Be sure to check back often as our pages are updated several times a day. If you don't find what you are looking for, try to use the search.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,118
|
The environmental performance of partici...
The environmental performance of participatory and collaborative governance: A framework of causal mechanisms
Jens Newig
Edward Challies
Nicolas Wilhelm Jager
Elisa Kochskämper
Ana Adzersen
Professorship for Governance and Sustainability
Center for the Study of Democracy
Many have advocated for collaborative governance and the participation of citizens and stakeholders on the basis that it can improve the environmental outcomes of public decision making, as compared to traditional, top-down decision making. Others, however, point to the potential negative effects of participation and collaboration on environmental outcomes. This article draws on several literatures to identify five clusters of causal mechanisms describing the relationship between participation and environmental outcomes. We distinguish (i) mechanisms that describe how participation impacts on the environmental standard of outputs, from (ii) mechanisms relating to the implementation of outputs. Three mechanism clusters focus on the role of representation of environmental concerns, participants' environmental knowledge, and dialogical interaction in decision making. Two further clusters elaborate on the role of acceptance, conflict resolution, and collaborative networks for the implementation of decisions. In addition to the mechanisms, linking independent with dependent variables, we identify the conditions under which participation may lead to better (or worse) environmental outcomes. This helps to resolve apparent contradictions in the literature. We conclude by outlining avenues for research that builds on this framework for analysis.
Policy Studies Journal
https://doi.org/10.1111/psj.12209
Published - 05.2018
Politics - effectiveness, modes of governance, stakeholder involvement, deliberation, causal hypotheses, collective learning, public policy
Environmental Governance - environmental governance
EDGE - Evaluating the Delivery of participatory environmenal Governance using an Evidence-based research design
Related by journal
How to Explain Major Policy Change Towards Sustainability? Bringing Together the Multiple Streams Framework and the Multilevel Perspective on Socio-Technical Transitions to Explore the German "Energiewende"
Derwort, P., Jager, N. W. & Newig, J., 31.05.2021, In : Policy Studies Journal. p. 1-29 29 p.
From Planning to Implementation: Top-Down and Bottom-Up Approaches for Collaborative Watershed Management
Koontz, T. M. & Newig, J., 08.2014, In : Policy Studies Journal. 42, 3, p. 416-442 27 p.
Other publications by the same author(s)
Case study meta-analysis in the social sciences. Insights on data quality and reliability from a large-N case survey
Jager, N. W., Newig, J., Challies, E., Kochskämper, E. & von Wehrden, H., 01.2022, In : Research Synthesis Methods. 13, 1, p. 12-27 16 p.
Complexity in Water Management and Governance
Kirschke, S. & Newig, J., 2021, Handbook of Water Resources Management: Discourses, Concepts and Examples. Bogardi, J. J., Gupta, J., Nandalal, K. D. W., Salamé, L., van Nooijen, R. R. P., Kumar, N., Tingsanchali, T., Bhaduri, A. & Kolechkina, A. G. (eds.). Cham: Springer Nature AG, p. 801-810 10 p.
Research output: Contributions to collected editions/works › Contributions to collected editions/anthologies › Research
Corrigendum to: Pathways to Implementation: Evidence on How Participation in Environmental Governance Impacts on Environmental Outcomes
Jager, N. W., Newig, J., Challies, E. & Kochskämper, E., 01.07.2021, In : Journal of Public Administration Research and Theory. 31, 3, p. 616-616 1 p.
Research output: Journal contributions › Comments / debate › Research
Demarcating transdisciplinary research in sustainability science—Five clusters of research modes based on evidence from 59 research projects
Jahn, S., Newig, J., Lang, D. J., Kahle, J. & Bergmann, M., 20.12.2021, In : Sustainable Development. 15 p.
Governing Transitions towards Sustainable Agriculture - Taking Stock of an Emerging Field of Research
Melchior, I. C. & Newig, J., 08.01.2021, In : Sustainability. 13, 2, 27 p., 528.
Research output: Journal contributions › Scientific review articles › Research
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,319
|
Q: scrape data from Instagram - errors while running Instascrape I am trying to scrape post, and other information related to that post using instascrape. I am receiving an errors. So kindly help me out in this. If you know any other package that can do the same, kindly let me know.
from selenium.webdriver import Chrome
from instascrape import Profile, scrape_posts
webdriver = Chrome("path/to/chromedriver.exe")
headers = {
"user-agent": "Mozilla/5.0 (Linux; Android 6.0; Nexus 5 Build/MRA58N) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/87.0.4280.88 Mobile Safari/537.36 Edg/87.0.664.57",
"cookie": "sessionid=PASTE_YOUR_SESSIONID_HERE;"
}
joe = Profile("joebiden")
joe.scrape(headers=headers)
output:
JSONDecodeError: Expecting property name enclosed in double quotes: line 1 column 2 (char 1)
A: The Python "JSONDecodeError: Expecting property name enclosed in double quotes: line 1 column 2 (char 1)" occurs when we try to parse an invalid JSON string (e.g. single-quoted keys or values, or a trailing comma). Use the ast.literal_eval() method to solve the error.
Here is a way to go about it
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,319
|
Unimog 437 steht für folgende Fahrzeuge:
Unimog 437.1, 1988 bis 2002
Unimog 437.4, ab 2002
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,432
|
Q: ¿Se podría añadir un favicon con react, de la pagina web, a un enlace proveniente de otra pagina/servidor? No tengo mucha experiencia con react, y necesito saber si es posible agregar el favicon de mi pagina a un enlace proveniente del servidor, en el momento en que se redirige a él y se abre el archivo que se desea visualizar, el servidor que estamos usando es aws .
Y no sé si es posible agregar un favicon a un enlace que provenga de otra pagina. Lo único que puedo encontrar de momento, son las formas de agregar un favicon a tu propia página web y eso ya lo tengo :(
Espero que entiendan mi pregunta, porque es difícil para mí explicarlo exactamente, ¡pero agradecería tu ayuda!
A: Todos tus componentes de React se renderizan en un <div /> que está en el archivo public/index.html dentro de tu proyecto de React.
Así que si necesitas agregar un favicon en todas tus páginas, bastaría con ponerlo solo en ese HTML para que se vea en toda tu App.
También se puede poner el link de un favicon de un sitio externo a tu proyecto de esa manera.
Para hacerlo, se usa la etiqueta de HTML <link rel="icon">.
Por ejemplo:
<link rel="icon" href="LINK DEL FAVICON QUE VIENE DE OTRO SITIO WEB" />
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,152
|
(function() {
var File, Files, Image, ImageThumbnailView, ImageToolsView, ImageView, Images, Page, PageThumbnailView, PageView, Pages, Project, ProjectRouter, Table, Tables, ThumbnailListView, ThumbnailView, allImages;
Project = Backbone.Model.extend({
initialize: function() {
return this.set({
'files': new Files(this.get('files'))
});
}
});
File = Backbone.Model.extend({
initialize: function() {
return this.set({
'pages': new Pages(this.get('pages'))
});
}
});
Files = Backbone.Collection.extend({
model: File
});
Page = Backbone.Model.extend({
initialize: function() {
allImages.add(this.get('images'));
return this.set({
'images': new Images(this.get('images'))
});
}
});
Pages = Backbone.Collection.extend({
model: Page
});
Image = Backbone.Model.extend({
initialize: function() {
return this.set({
'tables': new Tables(this.get('tables'))
});
}
});
Images = Backbone.Collection.extend({
model: Image
});
Table = Backbone.Model.extend({
initialize: function() {
return this.set({
'data': JSON.parse(this.get('data'))
});
}
});
Tables = Backbone.Collection.extend({
model: Table
});
ThumbnailListView = Backbone.View.extend({
tagName: 'ul',
id: 'thumbnails',
render: function() {
var _this = this;
this.options.collection.each(function(li, i) {
var _ref;
return _this.$el.append((new _this.options.itemClass({
item: li,
selected: li.id === ((_ref = _this.options.selected) != null ? _ref.id : void 0),
number: i
})).render());
});
return this.$el;
}
});
ThumbnailView = Backbone.View.extend({
tagName: "li",
className: "thumbnail",
render: function() {
var a;
a = $('<a>', {
href: this.getLink()
});
a.append($('<img>', {
src: Carpenter.options["static"] + this.getImage()
}));
if (this.options.selected) {
this.$el.addClass('selected');
}
return this.$el.append(a);
}
});
PageThumbnailView = ThumbnailView.extend({
getImage: function() {
return this.options.item.get('thumbnail');
},
getLink: function() {
return "#file/" + (this.options.item.get('file')) + "/page/" + this.options.item.id;
}
});
ImageThumbnailView = ThumbnailView.extend({
getImage: function() {
return this.options.item.get('path');
},
getLink: function() {
return "#image/" + this.options.item.id;
}
});
PageView = Backbone.View.extend({
tagName: "div",
className: "page",
render: function() {
this.$el.append($('<img>', {
src: Carpenter.options["static"] + this.options.page.get('image')
}));
this.options.page.get('images').each(function(img) {});
return this.$el;
}
});
ImageView = Backbone.View.extend({
tagName: "div",
className: "image",
render: function() {
this.$el.append($('<img>', {
src: Carpenter.options["static"] + this.options.image.get('path')
}));
return this.$el;
}
});
ImageToolsView = Backbone.View.extend({
tagName: "div",
className: "image-tools",
events: {
'click .find-tables': 'findTables',
'click .show-tables': 'showTables'
},
findTables: function(e) {
e.preventDefault();
return $.post("./image/" + this.options.image.id + "/analyze");
},
showTables: function(e) {
var out;
out = '';
this.options.image.get('tables').each(function(table) {
var cell, row, _i, _j, _len, _len1, _ref;
out += '<table class="table table-bordered">';
_ref = table.get('data');
for (_i = 0, _len = _ref.length; _i < _len; _i++) {
row = _ref[_i];
out += '<tr>';
for (_j = 0, _len1 = row.length; _j < _len1; _j++) {
cell = row[_j];
if (cell) {
cell.colspan = cell.colspan || 1;
cell.rowspan = cell.rowspan || 1;
out += "<td rowspan='" + cell.rowspan + "' colspan='" + cell.colspan + "'>" + cell.text + "</td>";
}
}
out += '<tr>';
}
return out += '</table>';
});
return $('#page-container').html(out);
},
render: function() {
this.$el.append($('<button>', {
'class': 'btn find-tables'
}).text('Analyze'));
this.$el.append($('<button>', {
'class': 'btn show-tables'
}).text('Show Tables'));
return this.$el;
}
});
allImages = new Images;
ProjectRouter = Backbone.Router.extend({
initialize: function(options) {
return this.project = new Project(options.project);
},
routes: {
'': 'index',
'file/:fileid': 'showFileFirstPage',
'file/:fileid/page/:pageid': 'showPage',
'image/:imageid': 'showImage'
},
index: function() {
var img;
img = allImages.first();
if (img) {
return this.navigate('image/' + img.id, {
trigger: true
});
}
},
showFileFirstPage: function(fileid) {
var file, page;
file = this.project.get('files').get(fileid);
page = file.get('pages').first();
return Carpenter.router.navigate("file/" + file.id + "/page/" + page.id, {
trigger: true
});
},
showPage: function(fileid, pageid) {
var file, page, pageView, thumbnails;
file = this.project.get('files').get(fileid);
page = file.get('pages').get(pageid);
thumbnails = new ThumbnailListView({
itemClass: PageThumbnailView,
collection: file.get('pages'),
selected: page
});
$('#thumbnail-container').html(thumbnails.render());
pageView = new PageView({
page: page
});
return $('#page-container').html(pageView.render());
},
showImage: function(imageid) {
var imageTools, imageView, img, thumbnails;
img = allImages.get(imageid);
thumbnails = new ThumbnailListView({
itemClass: ImageThumbnailView,
collection: allImages,
selected: img
});
$('#thumbnail-container').html(thumbnails.render());
imageView = new ImageView({
image: img
});
$('#page-container').html(imageView.render());
imageTools = new ImageToolsView({
image: img
});
return $('#tools-container').html(imageTools.render());
}
});
window.Carpenter = {
setup: function(project, options) {
this.options = options;
this.router = new ProjectRouter({
project: project
});
return Backbone.history.start();
}
};
}).call(this);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 286
|
Went with my two kids for brunch and was surprised at how family friendly it is — they had games and toys all over the place (probably for the young at heart, but we enjoyed it!!). The menu was inventive while still offering the breakfast staples the kids could enjoy. Good portion sizes, too. The table next to us got a Bloody Mary that looked AMAZING!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,574
|
Are you a Funeral Director?
Log in to access Forms, Price Lists and Other Resources
From the original watercolor painting "Philadelphia Skyline" by N. P. Santoleri © 1999 www.Santoleri.com
Philadelphia Crematories, Inc.
History of Cremation
Cremation FAQs
Selecting a Crematory
Frequently Asked Questions About Cremation
Becoming more educated regarding cremation will help you gain confidence in your decision-making process.
Is a casket required for a cremation to take place?
Is embalming required prior to cremation?
Can a cremation be witnessed by the family?
Are there any religions that do not approve of cremation?
Can more than one cremation be performed at once?
What usually happens after the cremation is finished?
What do cremated remains look like?
Are all cremated remains returned to the family?
Are urns required to collect the cremated remains?
What options are available with the cremated remains?
How can you be certain that all remains are kept separate, and you receive the correct remains?
For more information on cremation, please visit The Cremation Association of North America's website at: www.cremationassociation.org
Q: What is Cremation?
A: Cremation is the process of reducing human remains to its basic elements in the form of bone fragments through flame, heat and vaporization (usually 1800 - 2000 degrees Fahrenheit for two hours or more). Cremation occurs at a crematory in a special kind of furnace called a cremation chamber or retort. The resulting bone fragments are further reduced in size through a mechanical process and are referred to as "cremated remains". (It may surprise many to learn that ashes are not the final result since cremated remains have neither the appearance nor the chemical properties of ashes.) After processing, the cremated remains are placed in an urn or other container suitable for memorialization, transport or interment. Depending upon the size of the deceased's skeletal makeup, there are normally four to eight pounds of cremated remains resulting. Top
Q: Is a casket required for a cremation to take place?
A: A casket is not required for a cremation to take place. In most states, all that is required is an alternative container which can be constructed of wood or cardboard, and is cremated along with the deceased. Top
Q: Is embalming required prior to cremation?
A: No. Public health law states that "any human remains held 24 hours beyond death, and not yet cremated or interred at a cemetery, shall be either embalmed, or kept under refrigeration." Top
Q: Can a cremation be witnessed by the family?
A: Yes, in most situations, the cremation provider will permit family members to be in attendance when the body is placed into the cremation chamber. Actually, a few religious groups include this as integral part of their funeral practice. Top
Q: Are there any religions that do not approve of cremation?
A: Most major religions readily accept cremation, with the exception of Islam and Orthodox Judaism. Today, all of the Christian denominations allow cremation and are pleased for their members who choose it. (The Catholic Church approves cremation, but advocates the interment of the cremated remains in a cemetery.) Buddhists favor cremation, and for Hindus, cremation is the orthodox method of disposition. Top
Q: Can more than one cremation be performed at once?
A: No. Not only is it a practical impossibility, but it is illegal to do so. The majority of modern cremation chambers are not of adequate size to house more than one adult. Top
Q: What usually happens after the cremation is finished?
A: All organic bone fragments and all non-consumed metal items are collected into a stainless steel cooling pan located in the lower front of the cremation chamber. All non-consumed items, such as metal from clothing, joint replacements, and dental bridgework, are divided from the cremated remains. This separation is accomplished through visual inspection as well as using a strong magnet for smaller and minute metallic objects. Items such as dental gold and silver are non-recoverable and may be commingled with the cremated remains. Remaining bone fragments are then processed in a machine to a consistent size, sealed in a plastic bag, and then placed into a cremated remains container selected by the family. Top
Q: What do cremated remains look like?
A: Processed cremated remains are a mixture of powdery and granular substances, and are very light gray to white in color. The remains of an average sized adult usually weigh between four to eight pounds. Top
Q: Are all cremated remains returned to the family?
A: With the exclusion of minuscule and microscopic particles, which are impossible to remove from the cremation chamber and processing machine, all of the cremated remains are returned to the family. Top
Q: Are urns required to collect the cremated remains?
A: There is no law requiring an urn. Nevertheless, the cremated remains must be held in some type of container. A more traditional urn may be desired if the cremated remains are to be memorialized at home, at a public memorial service, or the remains are to be interred at a cemetery. A family member may also supply a container or containers suitable for holding the cremated remains. Top
Q: What options are available with the cremated remains?
A: After the cremation, the cremated remains may be returned to a designated family member or friend to be kept at home, scattered or buried on private property, or released to a cemetery for burial or above-ground entombment. (Philadelphia Crematories, Inc. also offers a sea scattering service three miles off the southern New Jersey coast.) Cremated remains are often divided to satisfy various memorialization requests. There are smaller urns and even jewelry which hold "keepsake" portions of cremated remains. Cremation offers infinite possibilities to memorialize a loved one. Top
Q: How can you be certain that all remains are kept separate, and you receive the correct remains?
A: All responsible cremation providers have thorough operating policies and procedures in order to provide the highest level of service and reduce the possibility of human error. If you have questions, ask the cremation provider what procedures they use, and if you are allowed to witness all or any of the procedures relating to the cremation, including retrieval, processing, and packaging of the cremated remains. It is not only your right, but also your responsibility to gain a feeling of confidence in your cremation provider's facility, employees, policies, and procedures. Choosing your cremation provider is one of the most critical decisions you need to make. Top
7350 State Road
Phone: 215-708-7747 (24-hour Availability)
Email: info@philadelphiacrematories.com
Saturdays 8:00am to 2:00pm
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,013
|
Neurocythere is een uitgestorven geslacht van kreeftachtigen uit de klasse van de Ostracoda (mosselkreeftjes).
Soorten
Neurocythere (Neurocythere) bipartita (Nienholz, 1967) Gruendel, 1973 †
Neurocythere (Neurocythere) bradiana (Jones, 1884) Gruendel, 1973 †
Neurocythere bipartita (Wienholz, 1967) Gruendel, 1973 †
Neurocythere bradiana (Jones, 1884) Gruendel, 1973 †
Neurocythere caesa (Triebel, 1951) Gruendel, 1973 †
Neurocythere carinilia (Sylvester-Bradley, 1948) Gruendel, 1973 †
Neurocythere composita (Wienholz, 1967) Gruendel, 1973 †
Neurocythere craticula (Jones & Sherborn, 1888) Gruendel, 1973 †
Neurocythere kirtlingtonensis Ware & Whatley, 1980 †
Neurocythere oertlii (Bizon, 1958) Gruendel, 1973 †
Uitgestorven kreeftachtigen
Progonocytheridae
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,480
|
Posted in General and tagged art, langa on Wednesday, 13 May, 2015 by Paul.
Posted in General and tagged art, langa, sport on Tuesday, 12 May, 2015 by Paul.
Posted in General and tagged langa on Monday, 11 May, 2015 by Paul.
Posted in General and tagged langa on Sunday, 10 May, 2015 by Paul.
Posted in General and tagged graffiti, langa on Saturday, 9 May, 2015 by Paul.
Posted in General and tagged graffiti, langa on Friday, 8 May, 2015 by Paul.
Posted in General and tagged graffiti, langa on Thursday, 7 May, 2015 by Paul.
Posted in General and tagged graffiti, langa on Wednesday, 6 May, 2015 by Paul.
Posted in General and tagged langa on Tuesday, 5 May, 2015 by Paul.
Posted in General and tagged graffiti, langa on Monday, 4 May, 2015 by Paul.
Ikhaya le Langa is a project of the 2014 World Design Capital.
Posted in General and tagged langa on Sunday, 3 May, 2015 by Paul.
Posted in General and tagged langa on Saturday, 2 May, 2015 by Paul.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,075
|
\section{Introduction}
\pagenumbering{arabic}
Since the prediction of gravitational waves (GW) by Albert Einstein in 1916~\cite{Einstein:1916}
on the basis of general relativity, they have been
an object of intensive studies. Gravitational
waves are thought to be fluctuations in the curvature of space-time,
which propagate as waves, traveling outward from the source.
Although gravitational radiation has not yet been directly detected, it has been indirectly shown to exist
because it increases the pulsar orbital frequency \cite{Hulse:1974eb}
in good agreement with theoretical predictions.
Roughly speaking there are two groups of
possible sources of gravitational radiation which may be registered by
gravitational wave detectors either on the Earth or by space missions. The first group includes
energetic phenomena in the contemporary universe, such as emission of GWs by black hole or
compact star binaries, supernova explosions, and possibly some other catastrophic phenomena.
The second group contains gravitational radiation coming from the early Universe, which
creates today an isotropic background usually with rather low frequency.
Such gravitational radiation could be produced at inflation, phase transitions in the primeval plasma,
by the decay or interaction of topological defects, e.g. cosmic
strings, {\it etc}.
The graviton (gravitational wave) production in the Friedmann-Robertson-Walker metric was first considered
by Grishchuk~\cite{Grishchuk:1974ny}, who
noticed that the graviton wave equation is not conformal invariant and thus such quanta can
be produced by conformal flat external gravitational field. Generation of gravitational waves
at the De Sitter (inflationary)
stage was studied by Starobinsky~\cite{Starobinsky:1979ty} (see also ref.~\cite{Rubakov:1982df}).
The stochastic homogeneous background of the low frequency gravitational waves is
now one of the very important predictions of inflationary cosmology, which may present a final proof of
inflation.
In this work we discuss one more source of gravitational wave (GW) radiation in the early
Universe, namely, the interaction between primordial black holes (PBH). We consider
relatively light PBH, such that they evaporated before the big bang
nucleosynthesis (BBN) and so
they are not constrained by the light element abundances.
Cosmological scenario with early formed and evaporated primordial
black holes producing gravitons was considered
in ref.~\cite{Dolgov:2000ht}. Here we will remain in essentially the same frameworks and
study in addition the GW emission in different processes with PBH.
According to ref.~\cite{Hawking:1976de, DonPage:1976} the life-time of evaporating black hole
with initial mass $M$ is equal to:
\begin{eqnarray}
\tau_{BH} = \frac{10240\,\pi}{N_{eff}}\, \frac{M^3}{m_{Pl}^4}\,,
\label{tau-BH}
\end{eqnarray}
where the Planck mass is $m_{Pl}=2.176\cdot 10^{-5}$ g and
$N_{eff}$ is the number of particle species with masses
smaller than the black hole temperature:
\begin{eqnarray}
T_{BH}=\frac{m_{Pl}^2}{8\pi M}\, .
\label{T-BH}
\end{eqnarray}
To avoid a conflict with BBN the black holes should had been evaporated
before cosmological time $t\approx 10^{-2}$ s~\cite{Carr:2009jm} and thus their mass would be bounded from above by
\begin{eqnarray}
M < 1.75\cdot 10^8 \left(\frac{N_{eff}}{100}\right)^{1/3}\,\,{\rm g}.
\label{M-upper-bound}
\end{eqnarray}
The temperature of such PBHs should be higher than $3\cdot 10^4 $ GeV and
correspondingly $N_{eff} \geq 10^2$.
On the other hand, as is discussed in what follows,
the PBH mass is bounded from below e.g. by equation (\ref{M-large-Delta}).
This is the mass range of PBHs considered in this work. Such PBH are not
constrained by any astronomical data, which are applicable to heavier
ones~\cite{Carr:2009jm}--\cite{Josan:2009qn}.
Primordial black holes should interact in the early Universe creating gravitational radiation.
Below we estimate the efficiency of GW emission in
several processes with PBH. In sec.~\ref{s-BH-prod} some
mechanisms of PBH production and PBH evolution in the early
Universe are briefly described. We stress, in particular, a very important role played by
the clumping of PBH due to gravitational instability at the
matter dominated stage.
In section \ref{s-onset} we consider the initial
interaction between the PBHs when they started to "feel" each other and accelerate
with respect to the background cosmological expansion.
In sec. \ref{s-brems} the
quantum bremsstrahlung of gravitons at PBH collisions is discussed, which is quite similar to
the electromagnetic bremsstrahlung at Coulomb scattering of electrically charged particles.
Next, in sec. \ref{s-classical} we consider the classical emission of GW at accelerated motion
of a pair of BHs in their mutual gravitational field. In sec.~\ref{s-energy-loss} we evaluate the
energy loss of PBHs due to their mutual interaction. It may be relevant to the estimation of the
probability of formation of PBH binaries. The gravitational radiation from PBH binaries in high
density clusters is discussed in sec.~\ref{s-binaries}.
In sec. \ref{s-evaporation} we calculate the present day energy density of gravitons produced
at PBH evaporation. In sec. \ref{s-GW-background} we review some mechanisms of the
production of stochastic background of GWs. In sec. \ref{GW-detectors} the status
of existing and planned detectors of GWs is discussed. In sec. \ref{conclusion} we conclude.
\section{Production and evolution of PBH in the early universe.} \label{s-BH-prod}
Formation of primordial black holes from the primordial density perturbations in the early
Universe was first considered by Zeldovich and
Novikov \cite{Novikov}
and later by Hawking and Carr \cite{Hawking:1971ei, Carr:1974nx}.
PBHs would be formed
when the density contrast, $\delta\rho /\rho$, at horizon was of the order of unity or,
in other words, when the Schwarzschild radius of the perturbation was of the order of the
horizon scale. If PBH was formed at the radiation dominated stage, when the cosmological
energy density was $\rho(t) = 3 m_{Pl}^2/(32\pi t^2) $, and the horizon was
$l_h = 2t$, the mass of PBH would be:
\begin{equation}
M(t)={ m_{Pl}^2t}\simeq 4\cdot 10^{38} \,\left(\frac{t}{\rm{sec}}\right)\,{\rm g}
\label{BH-mass}
\end{equation}
where $t$ is the time elapsed since Big Bang.
The fraction of the cosmological energy density of PBH produced by such mechanism depends upon the spectrum
of the primordial density perturbations. We denote this fraction $\Omega_p$ and take it as a free parameter of the
model. The data on the large scale structure of the Universe and on the angular fluctuations of the cosmic
microwave background radiation (CMB) show that the spectrum of the primordial density fluctuations is almost
flat Harrison-Zeldovich one. For such spectrum the probability of PBH production is quite low and
$\Omega_p \ll 1$.
However, the flatness of the spectrum is verified only for astronomically large scales, comparable with the
galactic ones. The form of the spectrum for masses below $10^{10}$ g is not known. Inflation predicts that
the spectrum remains flat for all the scales but there exist scenarios with strong deviation from flatness at small
scales. In particular, in ref.~\cite{Dolgov:1992pu, Dolgov:2008wu} a model of PBH formation has been proposed which leads to log-normal
mass spectrum of the produced PBH:
\begin{eqnarray}
\frac{dN}{dM} = C \exp \left[\frac{(M-M_0)^2}{M_1^2}\right],
\label{dN-dM}
\end{eqnarray}
where $C$, $M_0$, and $M_1$ are some model dependent parameters. Quite
naturally the central value of PBH mass distribution may be in the desired range $M_0 < 10^{9}$ g.
In this model the value of $\Omega_p$ may be much larger than in the conventional model
based on the flat spectrum of the primordial fluctuations. We will not further speculate on the
value of $\Omega_p$ and on the form of the mass spectrum of PBH. In what follows we assume
for an order of magnitude estimate that the spectrum is well localized near some fixed
mass value and that $\Omega_p$ is an arbitrary parameter. Different mechanisms of PBH production are reviewed e.g. in
ref.~\cite{Scenario1, Khlopov:2008qy}.
We assume that PBHs were produced in radiation dominated (RD) Universe, when the cosmological energy density
was equal to
\begin{eqnarray}
\rho_{R} = \frac{3 m_{Pl}^2}{32 \pi t^2}\,.
\label{rho-RD}
\end{eqnarray}
If we neglect the PBH evaporation and possible coalescence, their number density would remain
constant in the co-moving volume, $n_{BH} (t) a^3(t) = const $.
In what follows the instant decay approximation for evaporation is used. The cosmological evolution
of PBHs with more realistic account of their decay was studied in ref.~\cite{bcl-91}.
Since the black holes were non-relativistic at
production, their relative contribution to the cosmological energy density rose as the cosmological scale
factor, $a(t)$ :
\begin{eqnarray}
\Omega_{BH} (t) = \Omega_p \left(\frac{a(t)}{a_p}\right),
\label{Omega-of-t-3}
\end{eqnarray}
where $a_p$ is the value of the scale factor at the PBH production and at RD-stage
$a(t)/a_p = (t/t_p)^{1/2}$. The moment $t_p$ of the black hole production is connected
with the PBH mass through eq. (\ref{BH-mass}). Hence
\begin{eqnarray}
t_p = \frac{M}{ m_{Pl}^2}\,.
\label{t-p}
\end{eqnarray}
Thus if PBHs lived long enough, they would dominate the cosmological energy density
and the Universe would become matter dominated at $t>t_{eq}$, where
\begin{eqnarray}
t_{eq} = \frac{M}{m_{Pl}^2 \Omega_p^2} = \frac{r_g}{2\Omega_p^2} \,,
\label{t-eq}
\end{eqnarray}
and $r_g = 2M/m_{Pl}^2$ is the gravitational (Schwartzschild) radius of a black hole.
In what follows we assume that all PBHs have the same mass $M$, but
the results can be simply generalized by integration over the PBH mass spectrum.
Evidently at RD stage the number density of PBHs drops as:
\begin{eqnarray}
n_{BH}(t) = n_p \left(\frac{a_p}{a(t)}\right)^3 =
n_p \left(\frac{t_p}{t}\right)^{3/2}\,,
\label{n-of-t-RD}
\end{eqnarray}
while at MD stage
\begin{eqnarray}
n_{BH}(t) = n_p\left(\frac{t_p}{t_{eq}}\right)^{3/2}\,\left(\frac{t_{eq}}{t}\right)^2\,.
\label{n-of-t-MD}
\end{eqnarray}
Cosmological mass fraction of BH as a function of time behaves as
\begin{eqnarray}
\Omega_{BH} (t) = \frac{n_{BH}(t) M}{\rho_c} = \frac{16\pi}{3}\, r_g
t^2 n_{BH}(t) \,,
\label{Omega-of-t-2}
\end{eqnarray}
i.e. ${\Omega_{BH} \sim t^{1/2}}$ at RD stage.
After the onset of the PBH dominance,
$\Omega_{BH} $ approached unity and remained constant
till the PBH evaporation when
${\Omega_{BH} }$ quickly dropped down to zero and the universe became
dominated by relativistic particles produced by PBH evaporation. All relics from the earlier RD stage would be diluted
by the redshift factor $(t_{eq}/\tau_{BH})^{2/3}$. In particular the energy density of GWs produced at inflation would be diminished by this factor with respect to the standard predictions. Such dilution may cause problems with baryogenesis. However, these problems may be resolved if baryogenesis
took place at the process of PBH evaporation through the mechanism suggested by Zeldovich~\cite{zeld-BG} and quantitatively studied in ref.~\cite{dad-BG, dolgov1980}. Somewhat similar model of baryogenesis by heavy particle decay
(e.g. by bosons of GUT) created at PBH evaporation was considered in ref.~\cite{barrow-BG, Barrow1981, Barrow1991, Baumann:2007yr}.
To survive till equilibration the PBHs should live long enough
so that their evaporation time $t_{ev}$ would be larger than $t_{eq}$ or
$\tau_{BH}>t_{eq}-t_{p}$ which can be
translated into the bound on the PBH mass:
\begin{eqnarray}
M >\left(\frac{N_{eff}}{3.2\cdot 10^4}\right)^{1/2}m_{Pl}
\,\left(\frac{1}{\Omega_p^2}-1\right)^{1/2}
\simeq 5.6\cdot 10^{-2}\,\left(\frac{N_{eff}}{100}\right)^{1/2}\frac{m_{Pl} }{\Omega_p}
\label{M-for-t-MD}
\end{eqnarray}
where $\Omega_p\ll 1$ and $M$ is mass of PBHs at production
\footnote{In fact in equation \eqref{M-for-t-MD} there must be the PBHs mass
at the equilibrium time, $M(t_{eq})$. Due to evaporation the
PBH mass as a function of time is given by
$M(t)=M(t_p)(1-t/\tau_{BH})^{1/3}$ and it is easy to see that for
$\tau_{BH}>t_{eq}$ it gives $M=M(t_p)\simeq M(t_{eq})$, so hereafter
we refer to $M$ as the mass of PBH at production.}.
Both constraints (\ref{M-upper-bound}) and (\ref{M-for-t-MD}) would be satisfied if
\begin{eqnarray}
\Omega_p > 0.7\cdot 10^{-14} \left(\frac{N_{eff}}{100}\right)^{1/6}.
\label{Omega-p-lower-limit}
\end{eqnarray}
For example, if $\Omega_p = 10^{-10}$, the black holes should be heavier than $1.2\cdot 10^{4}$ g.
When the Universe became dominated by non-relativistic
PBHs, primordial density perturbations,
$\Delta=\delta\rho/\rho$, should rise as the cosmological scale factor.
They could reach unity at cosmological time $t_1$ satisfying the condition:
\begin{eqnarray}
\Delta_{in}\left(\frac{t_1} {t_{eq}}\right)^{2/3} \sim 1\, ,
\label{delta-rho}
\end{eqnarray}
where $\Delta_{in}$ is the initial magnitude of the primordial
density perturbations. To be more accurate, the evolution of density
perturbations depends upon the moment when they cross horizon, see
below, eq. (\ref{Delta-of-t1}). For the moment we neglect this
complication to make some simple estimates.
The initial density contrast is usually assumed to be of the order of
$ { \Delta_{in} \sim 10^{-5}-10^{-4}}$ which is not
necessarily true at small scales and may be much larger, especially in the model
of ref.~\cite{Dolgov:1992pu,Dolgov:2008wu}.
Evidently the BH life-time, ${\tau_{BH}}$, must be long
enough, so that the density fluctuations in BH matter would rise up
to the values of the order of unity. The condition $t_{ev}>t_1$ or equivalently $\tau_{BH} > t_1-t_p$
leads to the following restriction on the PBH mass:
\begin{eqnarray}
M > M_{low}=\left(\frac{N_{eff}}{3.2\cdot 10^4}\right)^{1/2}\frac{m_{Pl}}{\Omega_p \Delta_{in}^{3/4}}
\simeq 1.2 \cdot 10^3 \,{\rm g}\,\left(\frac{10^{-6}}{\Omega_p}\right)
\left(\frac{10^{-4}} {\Delta_{in}}\right)^{3/4} \left(\frac{N_{eff}}{100} \right)^{1/2}\,.
\label{M-large-Delta}
\end{eqnarray}
We can see that eq. \eqref{M-large-Delta} puts a stronger lower
limit on PBHs mass than eq. \eqref{M-for-t-MD}. The limits are
comparable only if $\Delta_{in} \approx 1$.
Using eqs. \eqref{M-large-Delta} and \eqref{M-upper-bound} we get
a stronger than (\ref{Omega-p-lower-limit}) restriction on $\Omega_p$:
\begin{equation}
\Omega_p > 0.7\cdot 10^{-11}\left(\frac{10^{-4}}{\Delta_{in}}\right)^{3/4} \left(\frac{N_{eff}}{100}\right)^{1/6}.
\label{Omega-p-strongerlower-limit}
\end{equation}
After ${\Delta}$ reached unity, the rapid structure formation would take
place and high density clusters of PBHs would be formed.
As we see in what follows, generation
of gravitational waves would be especially efficient from such high
density clusters of primordial black holes.
Let us assume that the spectrum of perturbations is the flat
Harrison-Zeldovich one and that a perturbation with some wave length
$\lambda$ crossed horizon at moment $t_{in}$. The mass inside horizon
at this moment was:
\begin{eqnarray}
M_b (t_{in} ) = m_{Pl}^2 t_{in}.
\label{M-b}
\end{eqnarray}
It is the mass of the would-be high density cluster of PBHs.
This initial time is supposed to be larger than $t_{eq}$ (\ref{t-eq}),
i.e. the horizon crossing took place already at MD-stage. For flat
spectrum of perturbations density contrast,
$\Delta = \delta\rho/\rho$, at horizon crossing
is the same for all wave lengths. After horizon crossing the
perturbations would continue to grow up as the scale factor,
$\Delta (t) = \Delta _{in} (t/t_{in})^{2/3}$. Such rise would
continue till moment $t_1(t_{in})$ such that:
\begin{eqnarray}
\Delta [t_1(t_{in})]= \Delta_{in} [t_1(t_{in}) /t_{in}]^{2/3} =1\,\,\,
\rm{or}\,\,\, t_1 (t_{in} )=t_{in} \Delta_{in}^{-3/2}\,.
\label{Delta-of-t1}
\end{eqnarray}
The radius of the PBH cluster rose almost as the cosmological scale factor
till $t=t_1(t_{in})$. After the density contrast has reached unity the cluster would
decouple from the common cosmological expansion. In other words,
the cluster stopped to expand
together with the universe and, on the opposite, it would begin to shrink
when gravity takes over the free streaming of PBHs. So the cluster size would drop down
and both $n_{BH}$ and $\rho_b$ would rise. The
density contrast would quickly rise from unity to $\Delta_b=\rho_b/\rho_c\gg 1$,
where $\rho_c$ and $\rho_b$ are respectively the average cosmological
energy density and the density of PBHs in the cluster (bunch). It looks
reasonable that the density contrast of the evolved cluster could rise up
to ${\Delta = 10^{5}-10^{6}}$, as in the contemporary galaxies.
After the size of the cluster stabilized,
the number density of PBH, $n_{BH}$, as well as
their mass density, $\rho_{BH}$,
would be constant too. But the density contrast, $\Delta_b$
would continue to rise as $(t/t_1)^2$ because $\rho_c$ drops down as $1/t^2$.
From time $t=t_1$ to $t=\tau_{BH}$ the density contrast would
additionally rise by the factor:
\begin{eqnarray}
\Delta (\tau_{BH})= \Delta (t_1) \left(\frac{\tau_{BH}}{t_1}\right)^2 =
\Delta(t_i)\left(\frac{M}{M_{low}}\right)^4,
\label{Delta-of-tau}
\end{eqnarray}
where $t_1$ and $M_{low}$ are given by eqs. (\ref{delta-rho} ) and
(\ref{M-large-Delta}) respectively.
The size of the high density clusters of PBH would be
\begin{eqnarray}
R_b = \Delta_b^{-1/3} t_1^{2/3} t_{in}^{1/3}\
\label{R-b}
\end{eqnarray}
and the average distance between the PBHs in the bunch can be estimated as:
\begin{eqnarray}
d_b = \left( M /M_b \right)^{1/3} R_b = \Delta_b^{-1/3} t_1^{2/3} r_g^{1/3}=
2^{-2/3} \Delta_b^{-1/3} \Delta_{in}^{-1} \Omega_p^{-4/3} r_g\,.
\label{d-b}
\end{eqnarray}
It does not depend upon $t_{in}$. Here eqs. (\ref{delta-rho}) and (\ref{t-eq}) have been used.
The virial velocity inside the cluster would be
\begin{eqnarray}
v = \sqrt{\frac{2M_b}{m_{Pl}^2 R_b}} =2^{1/2} \Delta_b^{1/6}
\Delta_{in}^{1/2} \approx 0.14
\left(\frac{\Delta_b}{10^6}\right)^{1/6}
\left(\frac{\Delta_{in}}{10^{-4}}\right)^{1/2}\,.
\label{v-b}
\end{eqnarray}
So PBHs in the cluster can be moderately relativistic.
Later, when $t=\tau_{BH}$, black holes would decay producing relativistic matter and the Universe
would return to the normal RD regime. However, the previous history of the earlier RD stage
would be forgotten.
For the future discussion it is convenient to introduce the average
distance between the PBHs at arbitrary time,
$d = n_{BH}^{-1/3}$, where $n_{BH} = \rho_{BH}/M$ is the number density of PBHs. Since
\begin{eqnarray}
\Omega_p = \frac{\rho_p}{\rho_c} = \frac{32\pi t_p^2 M n_p}{3
m_{Pl}^2} = \frac{32\pi}{3}\,\left(\frac{t_p}{d_p}\right)^3 \,,
\label{Omega-p-of-d}
\end{eqnarray}
the average distance between PBHs at the production moment is equal to
\begin{eqnarray}
d_p = (4\pi/3)^{1/3} r_g \Omega_p^{-1/3}\,.
\label{d-p}
\end{eqnarray}
When the mutual gravitational attraction of PBH may be neglected, $d$ rises as cosmological scale
factor, $a(t)$.
Gravitational waves produced in the early universe will be hopefully registered in the present epoch. The sensitivity of GW detectors strongly depends upon the frequency of the signal. The frequency $f_*$ of GW produced at time $t_*$ during PBH evaporation, is redshifted down to the present day value, $f$, according to:
\begin{eqnarray}
f = f_*\left[\frac{a(t_*)}{a_0}\right]=0.34\, f_*\, \frac{T_0}{T_{*}} \left[\frac{100}{g_S (T_*)}\right]^{1/3}\,,
\label{Omega-brems-today}
\end{eqnarray}
where $T_0= 2.725$ K \cite{Fixsen:2009ug} is the
temperature of the cosmic microwave background
radiation at the present time, $T_*\equiv T(t_{*})$ is the plasma
temperature at the moment of radiation of the gravitational waves, and
$g_S(T_*)$ is the number of species contributing to the entropy of the primeval plasma at temperature $T_*$.
It is convenient to express $T_0$ in frequency units, $T_0 = 2.7 \,{\rm K} = 5.4\cdot 10^{10}$ Hz.
The temperature of the primeval plasma after the PBH evaporation
can be approximately found from:
\begin{eqnarray}
\rho = \frac{m_{Pl}^2}{6\pi t^2} = \frac{\pi^2 g_*(T_*) T_*^4}{30}\,,
\label{rho-evap}
\end{eqnarray}
where ${g_* (T_*)\approx 10^2}$ is the contribution of
different particle species to the energy
density at temperature ${T_*}$ and $t_1<t<t_{ev}$.
For relativistic plasma $g_* (T) = g_S(T)$.
Since $t_{ev}=\tau_{BH}+t_p\simeq \tau_{BH}$, we obtain from
equation \eqref{rho-evap} at time $t_*=\tau_{BH}$:
\begin{eqnarray}
T_*(\tau_{BH}) = \left[\frac{30}{6\pi^3 g_S (T_*)}\right]^{{1}/{4}}
\left(\frac{N_{eff}}{3.2\cdot 10^4}\right)^{{1}/{2}}\,
\frac{m_{Pl}^{{5}/{2}}}{M^{{3}/{2}}}\,.
\label{T-tau-BH}
\end{eqnarray}
Substituting the numbers we find:
\begin{eqnarray}
T_*(\tau_{BH})
\approx 0.011 m_{Pl} \left[\frac{100}{g_S(T_*)}\right]^{1/4}
\left(\frac{N_{eff}}{100}\right)^{1/2}
\left(\frac{m_{Pl}}{M}\right)^{{3}/{2}}\,.
\label{tau-BH-num}
\end{eqnarray}
For comparison at the PBH production moment the temperature of the primeval plasma was:
\begin{eqnarray}
T_p \approx 0.2 m_{Pl} \left(\frac{m_{Pl}}{M} \right)^{1/2} \,.
\label{T-p}
\end{eqnarray}
Using eqs. (\ref{Omega-brems-today}) and (\ref{tau-BH-num}), we find
that the present day frequency of the GWs, emitted at $T_*$ (\ref{T-tau-BH}) with frequency
$f_*$, would be equal to:
\begin{eqnarray}
f = 1.7\cdot 10^{12}\textrm{Hz} \left[\frac{100} {g_S(T_*)}\right]^{1/12}
\left( \frac{100}{N_{eff}}\right)^{1/2}
\left(\frac{f_*}{m_{Pl}}\right)
\left(\frac{M}{m_{Pl}}\right)^{3/2}.
\label{omega-0}
\end{eqnarray}
If we take the maximum frequency of the emitted gravitons ${f_{max*}\approx r_g^{-1}= m_{Pl}^2/2M}$, the GW maximum frequency today would be:
\begin{eqnarray}
f_{max} \approx 8.6\cdot 10^{11}\textrm{Hz} \left(\frac{M}{m_{Pl}}\right)^{1/2}
= 5.8 \cdot 10^{16}\,{\rm Hz}\left(\frac{M}{10^5 {\rm g}}\right)^{1/2}\,.
\label{omega-today}
\end{eqnarray}
\section{Onset of GW radiation \label{s-onset} }
Once PBHs enter inside each other cosmological horizon\footnote{The cosmological horizon is the distance which PBHs started interacting with each other exchanging gravitons and should not be confused with the black hole event horizon.} they start to interact and
thus to radiate gravitational waves due
to their mutual acceleration. The corresponding time moment $t_h$ is
determined by the condition $2t_h=d(t_h)$ and remembering that
it happened still at RD stage, we find
\begin{equation}
t_h = \frac{1}{2}\left(\frac{4\pi}{3}\right)^{2/3} r_g \Omega_p^{-2/3}.
\end{equation}
For ${t>t_h}$, the curvature effects can be neglected and the PBH
motion is completely determined by the Newtonian gravity:
\begin{equation}
\ddot{\mathbf{r}} = - \frac{M_{BH}}{m_{Pl}^2 r^2} \,\frac{\mathbf{r}}{r}
\label{ddot-r}
\end{equation}
with the initial conditions $r_i\equiv |\mathbf{r}_i|= d(t_i)$ and
$|\dot{\mathbf{r}_i}|=H(t_i)|\mathbf{r}_i|$, where $\mathbf{r}$ is the
position vector of PBHs. For ${t_{i} = t_h}$ their relative initial
velocity $\dot{ |\mathbf{r}_i}| = v_{i}=1$ and non-relativistic
approximation is invalid. To avoid that we should choose ${t_{i} >
t_h}$ such that ${v_{i} \ll 1}$. The solution of the equation of
motion demonstrates that the effects of mutual attraction at this
stage and production of GW are weak.
After PBHs enter inside each other horizon and Newtonian gravity can
be applied, their acceleration toward each other becomes essential
when their Hubble velocity drops below the capture velocity.
The corresponding time moment, $t_c$, when it happened, is
determined from the condition:
\begin{equation}
\frac{1}{2}v^2(t_c) \equiv \frac{1}{2} [H(t_c) d(t_c)]^2 \lesssim \frac{M_{BH}}{m_{pl}^2d(t_c)}.
\label{velocity}
\end{equation}
If it took place at the RD regime,
the corresponding time moment would be equal to:
\begin{equation}
t_c = \frac{8\pi^2}{9}\,\frac{r_g}{\Omega_p^2},
\end{equation}
and the density parameter of PBHs at $t=t_c$ would be
\begin{equation}
\Omega_{BH}(t_c) = \Omega_p \left(\frac{t_c}{t_p}\right)^{1/2} =
\frac{4\pi}{3} > 1\,.
\end{equation}
Thus at ${t=t_c}$ the universe is already matter dominated and we have
to use the non-relativistic expansion law, ${a \sim t^{2/3}}$, starting
from the moment $t=t_{eq}$ (\ref{t-eq}). Accordingly the average distance
between BHs, when ${t>t_{eq}}$, grows as:
\begin{eqnarray}
d(t) = d_p
\left(\frac{t_{eq}}{t_p}\right)^{1/2}\,\left(\frac{t}{t_{eq}}\right)^{2/3}\,.
\label{d-of-t-MD}
\end{eqnarray}
Now we find that the condition that the Hubble
velocity, $v_H =(2/3t_c) d_c$
is smaller than the virial one, for average values, reads:
\begin{eqnarray}
\frac{4d_p^3}{9r_g t_p^{3/2} t_{eq}^{1/2}} <1\,.
\label{t2-nonrel}
\end{eqnarray}
One can see that this condition is never fulfilled.
However, this negative result does not mean that the acceleration of
BHs and GW emission are suppressed, because of the mentioned above
effect of rising density perturbations.
\section{Bremsstrahlung of gravitons. \label{s-brems} }
PBH scattering in the early Universe should be accompanied by the graviton emission almost exactly as the scattering of charged
particles is accompanied by the emission of photons.
The cross-section of the graviton bremsstrahlung in particle collisions was calculated in
ref.~\cite{Barker:1969jk} for the case of two spineless particles
(here black holes) with masses $m$ and $M$ under assumption that $m\ll M$.
In non-relativistic approximation, $\mathbf{p}^2\ll m^2$, the differential cross section reads:
\begin{equation}
\mathrm{d}\sigma=\frac{64M^2m^2}{15m_{pl}^6}\frac{\mathrm{d}\xi}{\xi}
\left[ 5\sqrt{1-\xi} +\frac{3}{2} (2-\xi) \ln
\frac{1+\sqrt{1-\xi}}{1-\sqrt{1-\xi}}\, \right]\,,
\label{sigma-brems}
\end{equation}
where $\xi$ is the ratio of the emitted graviton frequency,
$\omega=2\pi f$, to the kinetic energy of the incident black hole,
i.e. $\xi = 2m \omega / {\bf p}^2$.
We will use expression (\ref{sigma-brems}) for an order of magnitude estimate assuming that it
is approximately valid
for arbitrary $m$ and $M$, in particular, for $m\sim M$.
The energy density of gravitational waves emitted at the time interval $t$ and $t+\mathrm{d}t$
in the frequency range $\omega$ and $\omega+\mathrm{d}\omega$ is given by
\begin{eqnarray}
\frac{\mathrm{d}\rho_{GW}}{\mathrm{d}\omega} =
v_{rel} n_{BH}^2 \omega\left(\frac{\mathrm{d}\sigma}{\mathrm{d}\omega}\right)\mathrm{d}t\, ,
\label{d-dot-rho-domega}
\end{eqnarray}
where $n_{BH}$ is the number density of PBH and $v_{rel}$ is their
relative velocity.
The energy emitted in the frequency interval $\omega\in[0, \omega_{max}]$
per unit time is proportional to the integral
\begin{equation}
I(\omega_{max}) = \frac{{\bf p}^2}{2m}\varint_{0}^{\xi{max}} d\xi
\left[ 5\sqrt{1-\xi} +\frac{3}{2} (2-\xi) \ln
\frac{1+\sqrt{1-\xi}}{1-\sqrt{1-\xi}}\, \right]\,.
\label{integral}
\end{equation}
The maximum value of the frequency of the emitted gravitons should be smaller than
either the kinetic energy of the colliding BHs, $E_{kin} = p^2/(2M)$ or the BH
inverse gravitational radius, $ 1/r_g =m_{Pl}^2/2 M$, depending on
which of the two is smaller. Their ratio is $E_{kin} r_g = M^2 v^2 / m_{Pl}^2 $, so
for $M < m_{Pl} v^{-1} $ the maximum frequency would be the PBH kinetic energy
and in this case $\xi_{max} =1$. It corresponds to the
situation when PBH is nearly captured. It looses practically all its kinetic energy,
which goes to the graviton.
For PBHs in the high density clusters, when $v\sim 0.1$, the maximum frequency
would be $\omega_{max} \sim 1/r_g$ for all PBHs heavier than $10 m_{Pl}$. In this case
$\xi_{max}= (m_{Pl}/Mv)^2$.
The first, rather exotic case, when $M<m_{Pl}/v$
can be realized only if $\Omega_p \geq 0.01$,
see eq. (\ref{M-for-t-MD}). If $\xi_{max} =1$, then $\omega_{max} \sim {\bf p}^2/2m$
and the integral can be taken analytically:
\begin{equation}
I (\omega_{max}=p^2/2M) =\frac{25}{3}\frac{{\bf p}^2}{2m} = \frac{25}{3}\omega_{max}.
\label{I-of-omega-max}
\end{equation}
In this case the energy taken by GWs is of the order of the kinetic energy of PBH and correspondingly
$\Omega_{GW} \sim M n_{bh} v^2 /\rho_{BH} = v^2$.
Below we will consider more natural situation when $M > m_{Pl}\, v^{-1}$. Integral (\ref{integral})
in the limit of small $\xi_{max} $ is
\begin{eqnarray}
I(\omega_{max}=1/r_g ) = \frac{p^2}{2M} \,\xi_{max}\,\left[8 + 3 \ln (4/\xi_{max})\right]
\label{I-of-omega-max-2}
\end{eqnarray}
This expression is accurate within 30\% up to $\xi_{max} =1$. So in what follows we will use
this result as $I(\omega_{max}) \approx 25 \omega_{max}/3$, keeping in mind that normally
$\omega_{max} = 1/r_g \ll p^2/2M$.
The fraction of the cosmological
energy density of the emitted gravitational waves which has been produced during
time interval $t$ and $t+\mathrm{d}t$, which is smaller than or comparable to the cosmological
time $t_1\lesssim t\lesssim\ t_{ev}\simeq\tau_{BH}$, can be obtained by the integration of
equation \eqref{d-dot-rho-domega}
over $\omega$ from $0$ to $\omega_{max}$ taking into account that the energy density of GWs
goes with the redshift as $(1+z)^{-4}$, and the integration over cosmological
time, $t$, which is connected with the redshift by the relation\footnote{In this paper we consider flat space with
curvature $k=0$ and neglect cosmological constant, $\Lambda=0$.}
\begin{equation}
\mathrm{d}t=-\frac{\mathrm{d}z}{H_*\,(1+z)\left[\Omega_{BH*}(1+z)^3+\Omega_{r*}(1+z)^4\right]^{1/2}}\,,
\label{redshift}
\end{equation}
where $H_*$, $\Omega_{BH*}$, and $\Omega_{r*}$ are respectively the Hubble parameter, the matter density
parameter, and the radiation density parameter evaluated at cosmological time $t_*=\tau_{BH}$, just
before the PBH decay. Recall that we use the instant decay approximation, so
the Universe at $t = \tau_{BH}$ was still at MD stage. In this case all quantities such
as $H_*$ and $\rho_c$ are taken at this stage:
$H_*=2/3t_*$, $\rho_c=m_{Pl}^2/6\pi \tau_{BH}^2$, $\Omega_{BH*}=1$, and $\Omega_{r*}=0$.
We need to calculate the energy density of GWs at the moment of the PBH evaporation. The rate of GW
production is given by eq. (\ref{d-dot-rho-domega}). To take
into account the redshift of the energy density of the gravitational waves
we have to divide $d\rho_{GW}/d\omega$ by $(1+z)^4$, to substitute
$\omega = (1+z)\omega_*$, where $\omega_*$ is the GW frequency at $t=\tau_{BH}$, and to express
time through the redshift
as $\mathrm d t = (3/2)\tau_{BH}(1+z)^{-5/2} \mathrm d z$. As a result we obtain at $t_*=\tau_{BH}$:
\begin{eqnarray}
\mathrm d\rho_{GW} (\tau_{BH}) = \frac{32 M^2 v_{rel}}{5 m_{Pl}^6} [\rho_{BH}^{(cluster)}]^2
\tau_{BH} (1+z)^{-13/2} f[\omega_* (z+1)] \mathrm d[(1+z)\omega_*] \mathrm dz\,.
\label{d-rho-GW-brems}
\end{eqnarray}
Here $\rho_{BH}^{(cluster)}$ is the energy density of the PBHs in the cluster (which is
denoted above as $\rho_b$). Note that $\rho_{BH}^{(cluster)} = const$ before the PBH decay. We parametrize
this quantity as $\rho_{BH}^{(cluster)} = \rho_{BH}^{(c)} (\tau_{BH})\Delta(\tau_{BH})$, where
$\rho_{BH}^{(c)} (\tau_{BH})= m_{Pl}^2/(6\pi \tau^2_{BH})$ is the average cosmological energy density of PBH and
$\Delta(\tau_{BH})$
is given by eq. (\ref{Delta-of-tau}), see also the discussion above this equation.
Function $f (\omega)$ is the function of $\xi = 2m\omega/p^2$ in the square brackets of eq. (\ref{sigma-brems}).
To find the cosmological energy fraction of GWs at $t=\tau_{BH}$
we need to integrate the expression above
over frequency, using eq.~(\ref{I-of-omega-max}),
and over redshift and to divide it by the total average cosmological energy
density $\rho_{BH}^{(c)} (\tau_{BH})= m_{Pl}^2/(6\pi \tau^2_{BH})$. Since we have to average over the whole
cosmological volume, one factor $\Delta$ disappears and we remain with the first power of $\Delta$.
So the cosmological energy fraction of GWs would be:
\begin{eqnarray}
{\Omega_{GW} (\omega_{max}, \tau_{BH})}
\,\approx 16 Q\, \left(\frac{v_{rel}}{0.1}\right)\,
\left(\frac{\Delta}{10^5}\right)\,
\left(\frac{N_{eff}}{100}\right)\, \left(\frac{\omega_{max}}{M}\right)\,.
\label{rho-brems}
\end{eqnarray}
Here coefficient $Q$ reflects the uncertainty in the
cross-section due to the unaccounted for Sommerfeld enhancement~\cite{sommer, sakharov}.
Note that $\Delta$ may be considerably larger than $10^5$.
With ${v_{rel} = 0.1}$, ${\Delta = 10^5}$, $Q=100$, and $f_{max} = r_g^{-1}$
the fraction of the cosmological energy density of the GWs emitted by the
bremsstrahlung of gravitons
from the PBHs collisions, when the Universe age was equal to the life-time of the
PBH, could reach:
\begin{eqnarray}
\Omega_{GW}(\tau_{BH}) \sim 3.8\cdot 10^{-17} \left(\frac{10^5\,\textrm{g}}{M}\right)^2\,.
\label{Omega-max}
\end{eqnarray}
It looks that for very light PBH, $M < 50 m_{Pl}$ , the fraction of
GW might exceed unity, which is evidently a senseless result. However,
one should remember the lower bound on the PBH mass (\ref{M-large-Delta}) and that
${ m_{Pl}/M < \Omega_p/20}$ and $m_{Pl}/ M < 10^{-7} (\Omega_p/ 10^{-6} ) $.
It may be interesting to calculate the contribution to $\Omega_{GW} (\tau_{BH}) $ from the earlier
period before the cluster formation. The mass density of PBHs at that stage was equal to the cosmological
energy density but since it was quite high and the effect is proportional to the density squared, the
contribution from this period might be non-negligible. The result can be obtained from eq.~(\ref{d-rho-GW-brems}),
where $\rho_{BH}$ is taken equal to the average cosmological energy density. Since $\rho_{c}$ evolves with time
we need to insert into the integral over $\mathrm dz$ the factor $(1+z)^6$ where the redshift is taken from some initial
time, presumably $t_i=t_{eq}$, down to the moment of the cluster formation, $t_1$. So the energy density
of gravitational waves produced by bremsstrahlung from $t=t_{eq}$ (\ref{t-eq}) till $t=t_{1}$ (\ref{delta-rho})
would be:
\begin{eqnarray}
\mathrm d\rho_{GW}^{(1)} =
\frac{32 M^2 v_{rel}}{5 m_{Pl}^6} [\rho_{BH}^{(c)} (t_1)]^2
t_{1} (1+z)^{-1/2} f[\omega_* (z+1)] \mathrm d[(1+z)\omega_*] \mathrm dz\,,
\label{d-rho-GW1-brems}
\end{eqnarray}
where $\rho_{BH}^{(c)} = m_{Pl}^2/(6\pi t_1^2)$ and $(1+z)$ runs from 1 up to $ (t_1/t_{eq})^{2/3}$. We have
introduced an upper index $(1)$ to indicate that this is the energy density of GWs generated before the cluster
formation time $t=t_1$. The integration over $z$ gives the enhancement factor
$(1+z_{max})^{1/2} = (t_1/t_{eq})^{1/3} $. According to
eqs. (\ref{t-eq}) and (\ref{delta-rho}), this ratio is $\Delta_{in}^{-1/2} \sim 10^2$. Another enhancement
factor comes from a larger cosmological energy density $\rho^{(c)} (t_1) =\rho^{(c)} (\tau_{BH})(\tau_{BH}/t_1)^2$.
The other factor $\rho_{BH}^{(c)} (t_1)$ disappears in the ratio $\Omega_{GW} = \rho_{GW}/\rho^{(c)}$.
On the other hand, $\Omega_{GW}$ is redshifted by $(\tau_{BH}/t_1)^{2/3}$.
Correspondingly
\begin{eqnarray}
\frac{\Omega_{GW}^{(1)}(\tau_{BH})}{\Omega_{GW}(\tau_{BH})} =
\frac{11 \Delta_{in}^{-1/2}} {\Delta (\tau_{BH})} \,\frac{v_{rel}^{(1)}}{v_{rel}} {\left(\frac{\tau_{BH}}{t_1}\right)^{1/3}}\,,
\label{Om1-over-Om}
\end{eqnarray}
where the coefficient 11 came from the ratio of the integrals over $z$ of eqs. (\ref{d-rho-GW-brems}) and
(\ref{d-rho-GW1-brems}) and
\begin{eqnarray}
\left(\frac{\tau_{BH}}{t_1}\right)^{1/3} = \left(\frac{32170}{N_{eff} }\right)^{1/3} \Omega_p^{2/3}
\left(\frac{M}{m_{Pl}} \right)^{2/3}\,.
\label{tau-over-t1}
\end{eqnarray}
The ratio of relative velocities of PBHs before and after the cluster formation, ${v_{rel}^{(1)}}/{v_{rel}}$,
is tiny, according to the estimates of sec.~\ref{s-onset}, and this introduces another strong suppression
factor to the production of GWs at an earlier stage.
In accordance with eq. (\ref{Delta-of-tau}) the density contrast rises as
$\Delta = \Delta(t_1) (\tau_{BH}/ t_1)^2$, where $\Delta(t_1)$ is supposed to be large, say, $10^4-10^5$
due to the fast rise of density perturbations at MD stage after they reached unity. Thus
the generation of GWs in high density PBH clusters is much more efficient than at the earlier stage.
The density parameter of the gravitational waves at the present time
is related to cosmological time $t_*$ as:
\begin{equation}
\Omega_{GW}(t_0)=\Omega_{GW}(t_{*})\left(\frac{a(t_{*})}{a(t_0)}\right)^4\left(\frac{H_{*}}{H_0}\right)^2\,,
\label{Omega-GW-of-H}
\end{equation}
where $H_0 =100 h_0 $~km/s/Mpc is the Hubble parameter and $h_0=0.74\pm 0.04$ \cite{Komatsu:2010fb, Nakamura:2010zzi}.
Using expression for redshift \eqref{Omega-brems-today} and taking the emission time $t_*=\tau_{BH}$ we obtain:
\begin{equation}
\Omega_{GW}(t_0)=1.67\times 10^{-5} h_0^{-2}\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{1/3}\Omega_{GW}(\tau_{BH})\,.
\label{generalexpression}
\end{equation}
Now using both equations $(\ref{Omega-max})$ and $(\ref{generalexpression})$ we find that
the total density parameter of gravitational waves
integrated up to the maximum frequency is:
\begin{eqnarray}
h_0^{2}\Omega_{GW} (t_0) \approx 0.6\cdot 10^{-21}\,K
\left(\frac{10^5\,\textrm{g}}{M}\right)^{2}\,,
\label{Omega-brems-0}
\end{eqnarray}
where $K$ is the numerical coefficient:
\begin{eqnarray}
K = \left(\frac{v_{rel}}{0.1}\right)\,
\left(\frac{\Delta}{10^5}\right)\,
\left(\frac{N_{eff}}{100}\right)\,
\left(\frac{Q} {100}\right)\,
\left(\frac{100}{g_S (T(\tau_{BH}))}\right)^{1/3}\,.
\label{kappa}
\end{eqnarray}
Presumably $K$ is of order unity but since
$\Delta$ may be much larger than one, see eq. (\ref{Delta-of-tau}), $K$ may also be large.
\section{GW from PBH scattering. Classical treatment. \label{s-classical}}
Classical radiation of gravitational waves by non-relativistic masses
is well described in quadrupole approximation, see e.g.
books~\cite{Landau:1987gn, Misner:1974qy, Schutz:1985jx}. However,
as we have seen, in high density clusters of PBH, their relative velocity
could be high, see eq.~(\ref{v-b}), and relativistic corrections may be
non negligible. This problem was studied by Peters \cite{Peters:1970mx},
who considered emission of the GWs by two
bodies with masses $M$ and $m$, where the former is supposed to be
heavy and at rest and the latter, lighter one, moves with velocity $v$.
For non-relativistic motion, when $v \ll 1$, and the minimal
distance between the bodies is larger than their gravitational radii,
the energy of gravitational waves emitted in a single scattering process is equal to:
\begin{equation}
\delta E_{GW}=\frac{37\pi}{15}\frac{M^2m^2v}{b^3m_{Pl}^6},\qquad v\ll 1\,,
\label{delta-E-nonrel}
\end{equation}
where $b$ is the impact parameter.
For the relativistic motion, $1-v^2 < 1$, the emitted energy is:
\begin{equation}
\delta E_{GW}=\frac{M^2m^2}{b^3m_{Pl}^6(1-v^2)^{3/2}}\,.
\label{delta-E-rel}
\end{equation}
The frequency of the emitted gravitational waves in this process is peaked
near $\omega \sim 2\pi/\delta t$,
where $\delta t$ is the transition time which,
for non-relativistic motion is $\delta t = b/v$ according to
ref.~\cite{Peters:1970mx}, while for
the relativistic one it is equal to $\delta t\sim b(1-v^2)^{1/2}$.
For an order of magnitude estimate let us take $M\sim m$, then the
radiated energy, as a function of frequency, would be:
\begin{equation}
\delta E_{GW}(\omega)\approx\frac{M^4}{m_{Pl}^6} \,\omega^3\,.
\label{delta-E-of-omega}
\end{equation}
This and the previous equations
are true for sufficiently large impact parameter, $b\gg r_g$ for which
the space-time between the scattered PBHs may be considered as flat and
their gravitational mass defect can be neglected. The energy loss in
a single scattering event cannot be larger than
\begin{eqnarray}
\delta E_{max} = \frac{p\,q}{M}\,,
\label{delta-E-max}
\end{eqnarray}
where $p = Mv_{rel}$ is the relative momentum of two scattered PBHs and $q$ is
the momentum transfer which by an order of magnitude is $q = 1/b$.
Here and in what follows we use non-relativistic approximation.
So equations \eqref{delta-E-nonrel} and \eqref{delta-E-rel} can be true only
for
\begin{eqnarray}
b > b_{min}=\sqrt{\frac{37\pi}{15}} \,\frac{M^2}{m_{Pl}^3}\,.
\label{b-min}
\end{eqnarray}
For smaller impact parameters the radiation of gravitational waves would
be considerably stronger but the approximation used becomes invalid. For
the (near) ``head-on'' collision of black holes a bound state of two BH (a binary)
or a larger black hole could be formed and the energy loss might be comparable
to the BH mass due to gravitational mass defect. However, we are
interested in gravitational waves at the low frequency part of
the spectrum, such that they could be registered by existing or
not-so-distant-future GW detectors. For such low frequency
gravitational waves the approximation used here is an adequate one.
The differential cross-section of the gravitational scattering of two PBHs in non-relativistic regime,
$q^2\ll 2M^2$, can be taken as:
\begin{eqnarray}
\mathrm d\sigma = \frac{M^2}{m_{Pl}^2}\, \frac{\mathrm dq^2}{q^4}
=\frac{2M^2}{m_{Pl}^2} b \mathrm db\, .
\label{d-sigma-dq2}
\end{eqnarray}
The differential energy density of GWs emitted at time and frequency intervals
$[t,\,t+\mathrm{d}t]$ and $[\omega,\,\, \omega+\mathrm d\omega]$ respectively
can be calculated as follows. The rate of the energy emission by GWs is
\begin{eqnarray}
\mathrm d\dot\rho_{GW} = \mathrm d\sigma n^2_{BH} v_{rel} \delta E_{GW}\,,
\label{rho-dot-brems}
\end{eqnarray}
where we take for $\delta E$ non-relativistic expression (\ref{delta-E-nonrel}). We assume that the
impact parameter is related to the radiated frequency as $\omega = 2\pi v_{rel}/b$, as is discussed
below eq. (\ref{delta-E-rel}). So $b\mathrm db = b^3\mathrm d\omega /(2\pi v_{rel})$. So we find:
\begin{eqnarray}
\mathrm{d}\rho_{GW} = \frac{74 \pi v_{rel}} {15} \,\rho_{BH}^2\,
\frac{M^4}{m_{Pl}^8} \,\frac{\mathrm d\omega}{2\pi}\, \mathrm dt\,.
\label{d-dot-rho-GW}
\end{eqnarray}
The energy density parameter of GW at the moment of BH evaporation can be obtained integrating
this expression over time and frequency. Thus we obtain:
\begin{eqnarray}
\Omega_{GW} (\tau_{BH})= 2\cdot 10^{-10}\left(\frac{v_{rel}}{0.1}\right)^2
\left(\frac{\Delta_b}{10^5}\right)\,
\left(\frac{N_{eff}}{100}\right)\,
\left(\frac{10^5\,\textrm{g}}{M}\right)\,.
\label{Omega-GW-of-t}
\end{eqnarray}
If we do not confine ourselves to the impact parameter bounded by
condition (\ref{b-min}) and allow for $b \sim r_g$, the energy
density of GWs at the moment of PBHs evaporation might be comparable to unity.
\begin{figure}[!htb]
\centering
\includegraphics[scale=.8]{classicalscattering}
\caption{Log-log plot of density parameter today, $h_0^{2}\Omega_{GW}$, as
a function of expected frequency today in classical approximation for $N_{eff}\sim 100$,\, $g_S\sim 100$,
$\Delta_b\sim 10^5$, and $v_{rel}\sim 0.1$ for different values of PBH mass $M\sim 1$ g (solid line) and $M\sim 10^5$ g (dashed line).}
\label{fig:classical}
\end{figure}
Let us now take into account the redshift of GWs emitted at different moments during the
the life-time of the high density clusters. The energy density of GWs emitted at some time
$t$ is redshifted to the moment of BH decay as $1/(z+1)^4$. The
frequency of GW is redshifted as $\omega = (z+1)\omega_*$, where $\omega_*$ is the
frequency of GWs at $t=\tau_{BH}$. Integration over time or redshift is trivial and we find
from equation \eqref{d-dot-rho-GW} that the energy
density parameter of gravitational waves per logarithmic interval of frequency or the spectral density parameter, which is defined
according to ref.~\cite{Thorne:1987} as:
\begin{equation}
\Omega_{GW}(f; t)\equiv \frac{1}{\rho_c}\,\frac{\mathrm{d}\rho_{GW}}{\mathrm{d}\ln f},
\label{spectraldensityparameter}
\end{equation}
at time $t=\tau_{BH}$ is equal to:
\begin{eqnarray}
\Omega_{GW} (f_*; \tau_{BH}) \approx 8.5 \,\left(\frac{v_{rel}}{0.1}\right)
\left(\frac{\Delta_b}{10^5}\right)\,
\left(\frac{N_{eff}}{100}\right)\,
\left(\frac{M}{m_{Pl}^2}\right)\,f_*\,.
\label{Omega-GW-of-t-and-f}
\end{eqnarray}
Now using equations \eqref{omega-0} and \eqref{generalexpression}
we can calculate the relative energy density of GWs per logarithmic frequency at the present time:
\begin{equation}
h_0^{2}\Omega_{GW} (f; t_0)\approx 1.23\cdot 10^{-12}\alpha '\, \left(\frac{f}{\textrm{GHz}}\right)\,\left(\frac{10^5\,\textrm{g}}{M}\right)^{1/2}\,,
\label{Omega-GW-of-f-t_0}
\end{equation}
where $\alpha '$ is the coefficient at least of order of unity:
\begin{equation}
\alpha '= \left(\frac{v_{rel}}{0.1}\right)
\left(\frac{\Delta_b}{10^5}\right)
\left(\frac{N_{eff}}{100}\right)^{3/2}
\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{1/4}\,.
\end{equation}
It may be much larger if $\Delta_b\gg 10^5$.
As we mentioned above, the classical approximation is valid if the impact parameter is bounded
from below by equation \eqref{b-min}. Since the frequency of the radiated GWs is of the order of $v/b$, the
maximum present day frequency of GWs,
produced at cosmological time $t=\tau_{BH}$,
for which the classical non-relativistic approximation is still valid, would be:
\begin{equation}
f_{max}\sim 9\cdot 10^5 \textrm{Hz}\left(\frac{v_{rel}}{0.1}\right)\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{1/12}
\left(\frac{100}{N_{eff}}\right)^{1/2}\left(\frac{10^5\,\textrm{g}}{M}\right)^{1/2}\,.
\end{equation}
For $M=10^5$ g the minimum impact parameter is $b_{min} \approx 10^{-13}$ cm. The frequency of the order
of 1 Hz today corresponds to the impact parameter 6 orders of magnitude larger. If we demand that the impact parameter
should be smaller than the average distance between PBHs in the clusters, then using equations (\ref{d-b}) and \eqref{b-min} we find that it can be true if the following condition is fulfilled:
\begin{eqnarray}
\Omega_p < 1.8 \cdot 10^{-6}\left(\frac{10^5 {\rm g}}{M}\right)^{3/4}\left(\frac{10^5}{\Delta_b}\right)^{1/4}\left(\frac{10^{-4}}{\Delta_{in}}\right)^{3/4}\,.
\label{Omega-p-M}
\end{eqnarray}
\section{Energy loss of PBHs \label{s-energy-loss}}
We calculate here the total energy loss of PBHs
in the high density clusters, in order to understand how probable could be
the formation of the PBH binaries.
First, let us estimate the total energy loss of PBHs due to the
graviton bremsstrahlung. The loss of the kinetic energy per unit time
due to the graviton emission is:
\begin{eqnarray}
-\left(\frac{\mathrm{d}E_{kin}}{\mathrm{d}t}\right)_{brem} = n_{BH} v_{rel} \varint_0^{\omega_{max}} \mathrm{d}\omega\,\omega
\left(\frac{\mathrm{d}\sigma}{\mathrm{d}\omega}\right)_{brem} \,,
\label{dot-E-kin}
\end{eqnarray}
where $\omega_{max}$ is defined in sec. \ref{s-brems}.
The total loss of kinetic energy of a single PBH
during the time interval equal to the PBH life-time,
$\delta E_{kin} =-\dot E_{kin} \tau_{BH}$, normalized to the original kinetic energy
of the PBH can be estimated as
\begin{eqnarray}
\frac{\delta E_{kin}}{E_{kin}} = 6\cdot 10^4 \kappa_2 \left(\frac{m_{Pl}}{M}\right)^2\,,
\label{delta-E-over-E}
\end{eqnarray}
where
\begin{eqnarray}
\kappa_2 = \left(\frac{0.1}{v_{rel}}\right)\,
\left(\frac{\Delta_b}{10^5}\right)\,
\left(\frac{N_{eff}}{100}\right)\,
\left(\frac{Q} {10}\right)\, .
\label{kappa-2}
\end{eqnarray}
Clearly the energy loss is essential for very light PBHs which
could form dense clusters only if $\Omega_p$ is sufficiently high,
see eq.~(\ref{M-large-Delta}).
The energy loss due to classical GW emission might be somewhat more efficient.
According to the previous section the energy loss by a single PBH per
unit time is:
\begin{eqnarray}
\Delta \dot E_{class} =
n_{BH} v\varint_{b_{min}}^\infty \,db \left(\frac{\mathrm{d}\sigma}{\mathrm{d}b}\right)_{class}\delta E (b)\,,
\label{Delta-E}
\end{eqnarray}
where $\delta E (b)$ and $b_{min}$ are given respectively by eqs. (\ref{delta-E-nonrel})
and (\ref{b-min}).
Taking the integral over $b$ and time we find for the fractional energy loss of PBH due
to classical emission of the gravitational waves:
\begin{eqnarray}
\frac{\Delta E_{class}}{E_{kin} } = 0.9\cdot 10^3\, \frac{\Delta_b}{10^5}\,
\frac{N_{eff}} {100}\, \frac{m_{Pl}} {M}\,.
\label{Delta-E-class}
\end{eqnarray}
One should remember however that this energy loss comes from the PBHs scattering
with rather large impact parameter $b>b_{min}$. For smaller $b$, when the simple approximation
used in this work is inapplicable, the energy loss might be much larger.
Moreover, according to eqs. (\ref{t-eq}), (\ref{delta-rho}), and (\ref{Delta-of-tau}) the density
amplification factor $\Delta_b$ may be much larger than $10^{5}$:
\begin{eqnarray}
\Delta_b (\tau_{BH}) = 10^4\,\Delta (t_1)\, \Delta_{in}^3 \Omega_p^4 \left(\frac{100}{N_{eff}} \right)^2
\left(\frac{M}{m_{Pl}}\right)^4\,,
\label{Detla-amplf}
\end{eqnarray}
where we may expect e.g. that $\Delta (t_1) \sim10^5$, $\Delta_{in} \sim 10^{-4}$, and
$\Omega_{p} \sim 10^{-6}$.
PBHs in the high density clouds could also loose their
energy by dynamical friction, see e.g. book~\cite{Binney:2008}. A
particle moving in the cloud of other particles would
transfer its energy to these particles due to their gravitational interaction.
However, one should keep in mind that the case of
dynamical friction is essentially different from the energy loss due to
gravitational radiation. In the latter case the energy leaks out of the
system cooling it down, while dynamical friction does not change the
total energy of the cluster. Nevertheless a particular pair of black holes moving
toward each other with acceleration may transmit their energy to the
rest of the system and became gravitationally captured forming a binary.
For an order of magnitude estimate we will use the
Chandrasekhar's formula which is valid for a heavy particle moving in
the gas of lighter particles having the Maxwellian velocity
distribution with dispersion $\sigma$. The deceleration of
a BH moving at velocity $v_{BH}$ with
respect to the rest frame of the gas is given by
\begin{equation}
\label{eq:dynfric}
\frac{d}{dt}\vec{v}_{BH} = - 4 \pi \, G_N^2 \, M_{BH} \,
\rho_b \, \ln \Lambda \,
\frac{\vec{v}_{BH}}{v_{BH}^3} \, \left[ {\rm erf}(X) -
\frac{2 X \exp(-X^2)}{\sqrt{\pi}} \right]\,,
\end{equation}
where $X \equiv v_{BH}/(\sqrt{2}\sigma)$, erf is the error function,
$\rho_b$ is the density of the background particles, and
$\ln\Lambda \approx \ln(M_*/M_{BH})$ is the Coulomb logarithm, which is
defined as \cite{Binney:2008}:
\begin{displaymath}
\ln\Lambda = \ln \frac{b_{max} m_{Pl}^2\, \sigma^2}{M_{BH} + m} \, .
\end{displaymath}
Here $b_{max}$ is the maximum impact parameter, $\sigma^2$ is the
mean square velocity of the gas and m is the mass of particles in the gas.
Numerical simulations show that $b_{max}$ can be assumed to be of the order
the radius of the cloud, $R_b$, which is given by equation \eqref{R-b}. Since
$\sigma^2 \sim M_b/(m_{Pl}^2 R_b)$, a reasonable estimate of $\Lambda$ is
$M_b/M_{BH}$.
Equation \eqref{eq:dynfric} was solved in ref.~\cite{Bambi:2008uc} in two
limits $v>\sigma$ and $v<\sigma$. In both cases the characteristic
dynamical friction time was of the order of:
\begin{eqnarray}
\tau_{DF} = \frac{\sigma^3 m_{Pl}^4}{4\pi\, M_{BH} \rho_b \ln \Lambda }
\approx \left(\frac{\sigma}{0.1}\right)^3 \left[\frac{25}{\ln (10^{-6}/\Omega_p)}\right]
\left(\frac{100}{N_{eff}}\right) \left(\frac{M}{1\,{\rm g}}\right)
\left(\frac{10^6}{\Delta}\right)\tau_{BH}\,.
\label{tau-DF}
\end{eqnarray}
For PBH masses below a few grams dynamical friction would be an efficient mechanism
of PBH cooling leading to frequent binary formation. Moreover,
dynamical friction could result in the collapse of small PBHs into much
larger BH with the mass of the order of $M_b$ (\ref{M-b}). This
process would be accompanied by a burst of GW emission.
\section{Gravitational waves from PBH binaries \label{s-binaries}}
Binary systems of PBH could be formed with
non-negligible probability in the high density clusters.
As we have seen in the previous section, PBHs
could loose their energy due to emission of gravitational waves and
due to dynamical friction~\cite{Binney:2008}. As a result
they would be mutually captured. Determination of the capture probability is a complicated
task, which could probably be solved by numerical simulation. Since it is outside
of the scope of the present work, we simply assume that the mass or number fraction of
PBH binaries in the high density bunches of PBH is equal to $\epsilon$,
where $\epsilon$ is a dimensionless parameter which is hopefully
not too small in comparison with unity.
Gravitationally bound systems of two massive bodies in circular orbit are known to emit gravitational
waves with stationary rate and fixed frequency which is twice the
rotation frequency of the orbit. In this approximation
orbital frequency, $\omega_{orb}$, and orbit radius, $R$, are fixed. Luminosity of GW radiation from a single
binary in the stationary approximation is well known, see e.g. book~\cite{Landau:1987gn}:
\begin{eqnarray}
L_s \equiv \dot E = \frac{32 M_1^2 M_2^2 (M_1+M_2)}{5 R^5 m_{Pl}^8}=
\frac{32}{5}\,m_{Pl}^2\left(\frac{M_c\,\omega_{orb}}{m_{Pl}^2}\right)^{10/3}\,,
\label{L-binary}
\end{eqnarray}
where $M_1$, $M_2$ are the masses of two bodies in the binary system and $M_c$ is the chirp mass which is defined as
\begin{equation}
M_c=\frac{(M_1\,M_2)^{3/5}}{(M_1+M_2)^{1/5}}
\end{equation}
and
\begin{eqnarray}
\omega^2_{orb} = \frac{M_1+M_2}{m_{Pl}^2 R^3}\,.
\label{omega-rot2}
\end{eqnarray}
In the case of elliptic orbit with large semi-axis $a$ and eccentricity $e$ the luminosity is
somewhat larger (if $R=a$):
\begin{eqnarray}
L_e = \frac{32 M_1^2 M_2^2 (M_1+M_2)}{5 a^5 m_{Pl}^8\, \left(1-e^2\right)^{7/2}}\,
\left(1+\frac{73 e^2}{24} +\frac{37e^4}{96} \right)\,.
\label{L-e--binary}
\end{eqnarray}
The emission of GWs costs energy which is provided by the sum of the
kinetic and potential energy of the system. To compensate the energy loss the radius of the binary
system decreases and the frequency rises making the stationary approximation invalid.
As a result the system goes into the so called inspiral regime.
Ultimately the two rotating bodies coalesce and produce a burst of gravitational waves. To reach this stage
the characteristic time of the coalescence should be shorter than the life-time of the system. In our case it is
the life-time of PBH with respect to the evaporation.
In the inspiral regime the initially circular orbit may remain approximately circular
if radial velocity of the orbit, $\dot R$, is much smaller than the tangential velocity, $\omega_{orb}R$.
This regime is called quasi-circular motion and is valid as long as (see e.g. book~\cite{Maggiore:1900zz}):
\begin{equation}
\dot\omega_{orb}\ll \omega_{orb}^2.
\label{quasi-circularcondition}
\end{equation}
Equation \eqref{quasi-circularcondition} can be translated into the lower bound on the radius of the orbit:
\begin{eqnarray}
R\gg r_g^{(eff)} = \frac{M_1+M_2}{m_{Pl}^2}\,,
\label{circular-2}
\end{eqnarray}
which is the condition of the validity of the Newtonian approximation.
It was shown by Peters \cite{Peters:1963ux} that the orbits with initial $e_0=0$ would remain quasi-circular
as far as condition \eqref{quasi-circularcondition} is fulfilled,
while for the orbits with $e_0\neq 0$ the eccentricity rapidly approaches zero due to back reaction of the
gravitational radiation.
Most probably binaries are formed in elliptic orbits with high eccentricity. However in the calculation
of the GW emission by binaries we assume for simplicity that all orbits are circular.
The result would be a lower bound on GW emission, hopefully not too far from the real case.
In what follows we will consider both stationary and inspiral regimes
since they both might be realized for different values of the
parameters. We will use the instant decay approximation, when the PBH mass
is supposed to be constant till $t=\tau_{BH}$ and then BH would
instantly disappear. The case of the realistic decrease of PBH mass will
be considered elsewhere.
The stationary orbit approximation would be valid if time of coalescence, $\tau_{co}$, would be much
larger than the BH life-time, $\tau_{co}>\tau_{BH}$. The former can be found as follows (see e.g.
book~\cite{Landau:1987gn}). According to the virial theorem the total (kinetic plus potential) energy
of the system is ${\cal E} = -M_1 M_2 /(2R m_{Pl}^2)$. Since luminosity (\ref{L-binary})
is $L_s = -d{\cal E}/dt$, the radius varies with time according to
\begin{eqnarray}
\dot R = - \frac{64 M_1 M_2 (M_1+M_2) } { 5 R^3 m_{Pl}^6}\,.
\label{dot-R}
\end{eqnarray}
Correspondingly
\begin{eqnarray}
R(t) = R_0 \left( \frac{t_0 + \tau_{co} -t}{\tau_{co}} \right)^{1/4}\,,
\label{R-of-t}
\end{eqnarray}
where $R_0$ is the initial value of the radius, $t_0$ is the initial time, and the coalescence
time is given by:
\begin{eqnarray}
\tau_{co} = \frac{5 R_0^4\,m_{Pl}^6}{256 M_1 M_2 (M_1+M_2)}\,.
\label{t-coaelescence}
\end{eqnarray}
The condition $\tau_{co}>\tau_{BH}$ can be translated into the lower bound on $R$
(for $M_1=M_2$):
\begin{eqnarray}
R> R_{min} = 4.6\cdot 10^5\,\left(\frac{100}{N_{eff}}\right)^{1/4}
\left(\frac{M}{10^5\,\textrm{g}}\right)^{1/2}\,r_g\,.
\label{R-min}
\end{eqnarray}
Keeping in mind that the frequency of GWs emitted at circular motion of the binary is twice
the orbital frequency, $f_{s} = \omega_{orb}/\pi$ we find from
equation \eqref{omega-rot2} that lower bound (\ref{R-min})
leads to the following upper bound on the GW frequency:
\begin{eqnarray}
f_s < \omega_{max}/\pi \approx 2\cdot 10^{24}\textrm{Hz}\,\left(\frac{N_{eff}}{100}\right)^{3/8}\left(\frac{10^5\,\textrm{g}}{M}\right)^{7/4}\,.
\label{omega-max}
\end{eqnarray}
On the other hand, the radius of the binary orbit should be smaller
than the average distance between PBHs in the cluster (\ref{d-b}) and probably quite close to it.
Using eqs. (\ref{d-b}) and (\ref{R-min}) we find:
\begin{eqnarray}
\frac{R_{min}} {d_b} = 1.3\cdot 10^{-5} \left(\frac{\Delta_b}{10^5}\right)^{1/3}\,
\left(\frac{\Delta_{in}}{10^{-4}}\right)\, \left(\frac{\Omega_p}{10^{-6}}\right)^{4/3}\,
\left(\frac{M}{10^5 g}\right)^{1/2}\,.
\label{Rmin-over-db}
\end{eqnarray}
So it seems natural that $R_{min} \ll d_b$ and the PBH binaries should be mostly in
the quasi-stationary regime. $R_{min}$ would be equal to $d_b$ roughly speaking for quite
large mass fraction of the produced PBHs, $\Omega_p > 10^{-3}$.
The condition $R_{min} = d_b$ gives a lower bound on orbital frequency, $\omega_{orb}$:
\begin{equation}
\omega_{orb}> \omega_{min}\approx 9.4\cdot 10^{17}\, \textrm{sec}^{-1}\,\left(\frac{\Delta_b}{10^5}\right)^{1/2}\left(\frac{\Delta_{in}}{10^{-4}}\right)^{3/2}
\left(\frac{\Omega_p}{10^{-6}}\right)^2\,\left(\frac{10^5\,\textrm{g}}{M}\right)\,.
\label{omega-min}
\end{equation}
During the inspiral phase, for which $\tau_{co}<\tau_{BH}$, we expect that binaries emit GWs in the frequency range:
\begin{equation}
2\cdot 10^{24}\textrm{Hz}\,\left(\frac{N_{eff}}{100}\right)^{3/8}\left(\frac{10^5\,\textrm{g}}{M}\right)^{7/4}<
f < 0.6\cdot 10^{33}\textrm{Hz}\,\left(\frac{10^5\,\textrm{g}}{M}\right)\,.
\label{f-range-insp}
\end{equation}
The upper bound corresponds to $\omega \sim 1/r_g$.
The frequency spectrum of the gravitational waves in inspiral but quasi-circular motion can be found
in the adiabatic approximation as follows. Since the gravitational
waves are emitted in a narrow band near twice the orbital frequency,
the spectrum of the luminosity (\ref{L-binary}) can be approximated as:
\begin{eqnarray}
\mathrm d \dot E = \frac{32 M_1^2 M_2^2 (M_1+M_2)} {5 R^5 (t) m_{Pl}^8} \,
\delta \left( \omega - 2\omega_{orb} (R) \right) \mathrm d\omega
\label{d-dot-E}
\end{eqnarray}
To find the energy spectrum we have to integrate this expression over
time from initial time, $t_{min}=t_0$, to maximum time
$t_{max} = min [\tau_{BH}+ t_p, \tau_{co}+t_0]$, where $t_0$ and $t_p$
are respectively the time of the binary formation (it may be different
for different binaries but here we neglect this possible spread) and the time of PBH formation (it is different
for PBH with different masses). Note that the coalescence time, $\tau_{co}$
is also different for binaries with different initial radius $R_0$.
Using eqs. (\ref{omega-rot2}) and (\ref{dot-R}) and the expression
$\mathrm dt = (\mathrm dR/\mathrm dt)^{-1} (\mathrm dR/\mathrm d\omega_{orb}) \mathrm d\omega_{orb}$, we find:
\begin{eqnarray}
\frac{\mathrm d E}{\mathrm d\ln \omega} = \frac{2^{1/3}\omega^{2/3}}{3}\frac{M_1M_2} {m_{Pl}^{4/3}(M_1+M_2)^{1/3}}
\label{dE-dlog-bin}
\end{eqnarray}
in agreement with refs.~\cite{Maggiore:1900zz}, \cite{Phinney:2001di}. This expression is
valid for the frequencies in the interval determined by eq. (\ref{omega-rot2}) with $R_{max}=R_0$
and $R_{min}=R(t_{max})$.
In expression (\ref{dE-dlog-bin}) we have not taken in account the redshift which is different
for different frequencies and thus this leads to spectrum distortion. According to
eqs. (\ref{omega-rot2}) and (\ref{R-of-t}) frequency $\omega$ is emitted at the
time moment:
\begin{eqnarray}
t(\omega) = t_0 +\tau_{co} \left[ 1 - \left( \frac{ \omega_{min}} {\omega} \right)^{8/3} \right]\,,
\label{t-of-omega}
\end{eqnarray}
where
\begin{eqnarray}
\omega_{min} = 2 \left(\frac{M_1+M_2}{m_{Pl}^2} \right)^{1/2} \, R_0^{-3/2}
\label{omega-min-2}
\end{eqnarray}
is the minimal frequency emitted at initial moment $t=t_0$.
To the moment of the PBH evaporation the frequency of the GWs emitted at $t=t(\omega)$
is redshifted by the frequency dependent factor:
\begin{eqnarray}
\omega_* = \frac{\omega}{ 1+ z(\omega) } = \left[\frac{t (\omega)} {t_p+\tau_{BH} }\right]^{2/3} \omega,
\label{omega-star}
\end{eqnarray}
where $\omega_*$ is the frequency of GWs at $t= t_p + \tau_{BH}$. This equation
implicitly determines $\omega$ as a function of $\omega_*$.
The spectrum of the gravitational waves at PBH evaporation can be obtained from eq. (\ref{dE-dlog-bin})
dividing it by $(1+z)$ (the redshift of the graviton energy, $E$)
and with substitution $\omega = (z+1) \omega_*$. Correspondingly
\begin{eqnarray}
\mathrm d\omega = \frac{z+1}{1-\omega_* (\mathrm dz/\mathrm d\omega) }\,\mathrm d\omega_*
\label{d-omega}
\end{eqnarray}
As a result we find:
\begin{eqnarray}
\frac{\mathrm d E_*}{\mathrm d\ln \omega_*} = \frac{2^{1/3}\omega_*^{2/3}}{3}\frac{M_1M_2} {m_{Pl}^{4/3}(M_1+M_2)^{1/3}}\,
\frac{\left[1-\omega_* (\mathrm dz/\mathrm d\omega)\right]^{-1}}{(1 +z)^{1/3}}\,.
\label{dE*-dlog}
\end{eqnarray}
Here $z(\omega)$ should be taken as a function of $\omega_*$ according to eq. (\ref{omega-star}) and
$\omega_*$ varies between $\omega_{min} $ and $\omega_{max}$ divided by the corresponding red-shift
factor. In particular, $\omega_{* (min)} = \omega_{min} [t_0/(t_p+\tau_{BH})]^{2/3}$.
Note that $R_0$ enters explicitly into eq. (\ref{dE*-dlog}), while in eq. (\ref{dE-dlog-bin}) it enters only through
the limits in which $\omega$ varies. Because of that the frequency spectrum depends upon the distribution
of binaries over their initial radius, $R_0$. As is shown below, it is especially profound in the case of long
coalescence time when the frequency spectrum of a single binary with fixed $R$ is close to delta-function.
In the stationary approximation, when
the change of the orbit radius can be neglected, we expect that a single binary emits
GWs in a narrow band of frequencies close to twice the orbital frequency. However the
distribution of binaries over their initial
radius, $ \mathrm dn_{BIN} = F(R_0) \mathrm dR_0 $ spreads up the spectrum. Here
$\mathrm dn_{BIN}$ is the number density of binaries with the radius in the interval $[R_0, R_0+\mathrm dR_0]$.
Since in this approximation the radius is approximately constant we do not distinguish between $R$ and $R_0$.
The cosmological energy density of the gravitational waves
emitted per unit time is equal to:
\begin{eqnarray}
\mathrm d \dot\rho_{GW}^{(stat)} =\frac{2F(R)\,R}{3} \frac{n_{BH}^c}{n_{BH}^b}\,\frac{\mathrm d\omega}{\omega}\, L_s\,,
\label{d-dot-rho1}
\end{eqnarray}
where $n_{BH}^b$ is the number density of PBH in the high density bunch (cluster),
$n_{BH}^c$ is the average cosmological number density of PBH,
$R = R(\omega_{orb})$ according to eq. (\ref{omega-rot2}), and we used the relation
$\mathrm dR = -2(R/3)\,(\mathrm d\omega/\omega)$. Distribution, $F(R)$, is normalized as:
\begin{eqnarray}
\int \mathrm dR F(R) = n_{BIN} = \epsilon n_{BH}^b\,.
\label{int-F-of-R}
\end{eqnarray}
We assume for simplicity that $F(R) $ does not depend upon $R$ in some
interval $[R_1,\,R_2]$ and vanishes outside it.
So $F(R) = \epsilon n_{BH}^b/(R_1-R_2)$.
A more realistic fit to the PBH distribution over radius could be a Gaussian one:
\begin{eqnarray}
F(R) = \frac{ 1 } {\sqrt{2\pi} \,\sigma} \epsilon n_{BH}
\exp\left[ - { \left( R- \langle R \rangle \right)^2 }/ {2 \sigma^2}\right]\,,
\label{F-of-R-gauss}
\end{eqnarray}
where $\sigma$ is the mean-square deviation of $R$ from the average value $\langle R \rangle$.
The small factor $n_{BH}^c/n_{BH}^b $ enters eq.~(\ref{d-dot-rho1})
because we are interested in
the cosmological energy density of GWs averaged over the whole universe volume.
The cosmological number density of PBH is expressed through their energy density
as $n_{BH} = \rho_{BH}/M = \rho_c (t)/M$.
The number density of binaries in the cluster is parametrized according to:
\begin{eqnarray}
n_{BIN}(t) = \epsilon (t)\, n^b_{BH}(t) =\epsilon (t)\, \rho_c(t)
\Delta (t) /M \,,
\label{n-BIN}
\end{eqnarray}
where, we remind, $\rho_c (t)$ is the total cosmological energy density and
$\Delta (t) = \rho_b /\rho_c \gg 1$ is the density
contrast of the cluster.
The time dependence of $n_{BH}^b$
disappears when the cluster reaches the stationary state,
see discussion in Sec.~\ref{s-BH-prod}, and $\Delta (t)$ evolves according to
equation~(\ref{Delta-of-tau}). When the stationary orbit approximation is valid,
$\epsilon $ remains constant.
Collecting all the factors and integrating eq. (\ref{d-dot-rho1}) over time with an account of the
frequency redshift, $\omega = \omega_* (1+z)$ and the total redshift of the energy
density of GWs, $\rho_{GW}(t_*)=\rho_{GW}(t)/(1+z)^4$, we find:
\begin{eqnarray}
\mathrm d \rho_{GW}^{(stat)}(\omega_*; \tau_{BH})=
\frac{2^{7/3}}{5}\left[\frac{n^c_{BH}(\tau_{BH})}{n^b_{BH}}\right]\frac{(M_1^2 M_2^2)(\tau_{BH}+t_p)}{(M_1+M_2)^{1/3} m_{Pl}^{16/3}}\,F(R) \,\omega_*^{5/3} \mathrm d\omega_*\,
\varint_{x_{min}}^{1} x^{11/6}\,\mathrm dx\,,
\label{d-rho-stat}
\end{eqnarray}
where $x=a(t)/a(t_*)=1/(1+z)$, $x_{min}=a(t_0)/a(t_*)$, $t_0$ is the time moment of binary formation and we make use of equation \eqref{redshift}.
Dividing this result by the critical energy density just before PBHs complete evaporation,
$n_{BH}(\tau_{BH})\approx \rho_c(\tau_{BH})/M$, we find the cosmological fraction of the energy density of GWs at $t=\tau_{BH}$ per logarithmic interval of frequency $f = \omega/(2\pi)$ (below we assume that all BHs have equal masses, $M$):
\begin{eqnarray}
\Omega_{GW}^{(stat) }(f_*; \tau_{BH}) = \frac{3\cdot 2^{17/3}}{85}\,
\frac{\epsilon\cdot(t_p+\tau_{BH})}{R_1-R_2}\,
\left(\frac{\pi f_* M} {m_{Pl}^2} \right)^{8/3}\,[1-x_{min}^{17/6}]
\label{dOmega-dlogf*}
\end{eqnarray}
where for the sake of a simple estimate we assumed that $F(R) = const$. We assume also that all the binaries are formed at the same time, $t_0\ll \tau_{BH}$ and so $x_{min}\ll 1$.
Note that the frequency of GWs coming from the binaries with radii between $R_1$ and $R_2$ is confined according
to eq. (\ref{omega-rot2}).
To make an order of magnitude estimate of the fraction of the energy density of GWs at the moment of PBH evaporation we take $(R_1 -R_2) \sim R_1 \sim R(\omega)$, where $R(\omega)$ is determined
by equation (\ref{omega-rot2}) and take into account that the stationary approximation is valid if the radii of the binaries are bounded from below by eq.~(\ref{R-min}). Hence, if the stationary regime is realized, the spectral density parameter today would be:
\begin{eqnarray}
h_0^{2}\Omega_{GW}^{(stat)} (f; t_0) \approx 10^{-8} \epsilon
\left[\frac{N_{eff}}{100}\right]^{2/3}\left[\frac{100}{g_S(T(\tau_{BH}))}\right]^{1/18} \left[\frac{M}{10^5\,{\rm g}}\right]^{1/3}\,\left[\frac{f}{{\rm {GHz}}}\right]^{10/3}
\label{Omega-upper-limit}
\end{eqnarray}
\begin{figure}[!htb]
\centering
\includegraphics[scale=.8]{binaries.eps}
\caption{Log-log plot of density parameter today $h_0^{2}\Omega_{GW}$ as a function of expected frequency today for
PBHs binaries in the stationary approximation for $\beta\sim 1$, $\epsilon\sim 10^{-5}$, $N_{eff}\sim 100$, $g_S\sim 100$, PBH mass $M\sim 10^7$ g (solid line) and $M\sim 1$ g (dashed line).}
\label{fig:binaries}
\end{figure}
The expected range of the present day frequencies of the GWs from the binaries in the stationary approximation
is given by eqs. \eqref{omega-min} and \eqref{omega-max}.
The emitted frequency is determined by the binary radius, so a single binary emits GWs with a very narrow spectrum.
However, the distribution of binaries over their radius could lead to a significant spread of the spectrum.
In principle the frequencies emitted may have any value in the specified above range.
The minimal present day frequency of such GWs today can be found by plugging eq. \eqref{omega-min}
into eq. \eqref{omega-0}:
\begin{equation}
f_{}\geq 4.3\, \textrm{Hz}\,\beta\,\left(\frac{10^5\,\textrm{g}}{M}\right)^{1/2}\,,
\end{equation}
where $\beta$ is given by
\begin{equation}
\beta=\left(\frac{\Delta_b}{10^5}\right)^{1/2}\left(\frac{\Delta_{in}}{10^{-4}}\right)^{3/2}
\left(\frac{\Omega_p}{10^{-6}}\right)^2\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{1/12}\left(\frac{100}{N_{eff}}\right)^{1/2}\,.
\end{equation}
For binaries formed with $R>R_{min}$, see equations \eqref{omega-0}, \eqref{R-min} and \eqref{omega-max},
the frequency of emitted GWs today is bounded from above by:
\begin{equation}
f_{}\leq 5.7 \cdot 10^7\,\textrm{Hz}\,\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{1/12}\left(\frac{100}{N_{eff}}\right)^{1/8}\left(\frac{10^5\,\textrm{g}}{M}\right)^{1/4}\,.
\end{equation}
Let us estimate now the energy density of GWs in the inspiral case, when $\tau_{co}< \tau_{BH}$ and
the GW emission from a single binary proceeds in a wide range of frequencies due to shrinking of the binary radius.
The radiation frequency spans from $f_{s, min}$, which is the GW frequency at the initial PBH separation,
to $f_{s, max}$ which corresponds to GWs emitted at $R\sim r_g$.
The energy spectrum of GWs is given by eq. (\ref{dE-dlog-bin}) where, in what follows, we change
to cyclic frequency, $f =\omega/2\pi$.
After the cluster evolution was over, the number density of PBHs in high density clusters remained approximately
constant till the PBH evaporation, but in the inspiral phase the fraction of binaries, $\epsilon (t)$, decreased due to
their coalescence. So the tail of the distribution function at small initial $R_0$ is eaten up, and the average value
of $R$ drops down. In distribution function, $F(R_0)$, we have to substitute instead $R_0 $ its expression through
$R$ and time according to
\begin{eqnarray}
R_0 \rightarrow \left[ R^4 + \left(\frac{256 M_1M_2 (M_1+M_2)} {5 m_{Pl}^6}\right)\,(t-t_0)\right]^{1/4}
\label{R0-of-R}
\end{eqnarray}
with the corresponding change of $R^3_0 \mathrm dR_0 \rightarrow R^3 \mathrm dR$.
To calculate the cosmological energy fraction of GWs at the PBH evaporation moment we can proceed
along the same lines as we have done deriving eq. (\ref{dE*-dlog}) introducing additional factor
$F(R_0) \mathrm dR_0$ which depends upon time according to eq. (\ref{R0-of-R}). However, at the level
of calculations in the present model with many unknown parameters it can be sufficient to neglect
such subtleties and to use a simplified estimate:
\begin{equation}
\frac{\mathrm{d}\rho_{GW}}{\mathrm{d}(\log f_s)}=\epsilon_{co} n_{BH}^c(t)\frac{\mathrm{d}E_{GW}}{\mathrm{d}(\log f_s)}\,,
\end{equation}
where $\epsilon_{co}$ is the fraction of binaries with
coalescence time shorter or equal to PBH life-time. For an estimate by an order of magnitude we assume
also that the number of binaries is independent on the redshift. To some extend the decrease of the binary
number may be compensated by their continuous formation. We neglect possible difference of binary
masses and take $M_1 = M_2$. We approximately take the redshift into account from the moment
of the coalescence to the PBH decay, $(z_{co}+1) \approx (\tau_{BH}/\tau_{co})^{2/3}$. This corresponds
to the assumption that
the binaries radiated all GWs only at the moment of $\tau_{co}$. So the $f_* = f (1+z_{co})$.
Thus we obtain as an order of magnitude estimate:
\begin{eqnarray}
\Omega_{GW}(f_*,\tau_{BH}) = \frac{\epsilon_{co} }{3}
\left( \frac{\pi f_* M} {m_{Pl}^2 } \right)^{2/3}\, (z_{co}+1)^{-1/3}\,.
\label{Omega-GW-insp}
\end{eqnarray}
Using equations \eqref{omega-0} and \eqref{generalexpression},
we find that the energy density parameter of gravitational waves
today is equal to:
\begin{equation}
h_0^{2}\Omega_{GW}(f)\approx 5 \cdot 10^{-9}\epsilon_{co}
\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{5/18}\left(\frac{N_{eff}}{100}\right)^{1/3}
\left(\frac{f}{10^{12}\textrm{Hz}}\right)^{2/3}\left(\frac{10^5\,\textrm{g}}{M}\right)^{1/3}\,,
\end{equation}
where we neglected possibly weak redshift dilution of GWs by the factor $(\tau_{co}/\tau_{BH})^{2/9}$.
If the system goes to the inspiral phase, then according to equation (\ref{f-range-insp}) we would expect today
a continuous spectrum in the range from $f_{min}\sim 0.9\cdot 10^7 \textrm{Hz} $ to $f_{max}\sim 3\cdot 10^{14}\,\textrm{Hz}$.
However if we take into account the redshift of the early formed binaries from the moment of their formation to the PBH decay, the lower value of the frequency may move to about 1 Hz.
\section{Gravitons from PBH evaporation \label{s-evaporation}}
In the previous sections we have considered only gravitational waves emitted through mutual acceleration of PBHs in the high density clusters.
On the other hand PBHs could directly produce gravitons by evaporation. This process in connection with creation of cosmological background of relic GWs
was considered in ref.~\cite{Dolgov:2000ht} and later in ref.~\cite{Anantua:2008am}. In the last reference a possible clumping
of PBHs at the matter dominated stage was also considered. Though such clumping does not influence the probability of the GW emission by PBHs, it may change the mass spectrum of PBHs due to their merging.
The PBHs reduce their mass according to the equation:
\begin{equation}
M(t)=M_0\left(1-\frac{t-t_p}{\tau_{BH}}\right)^{1/3}\,,
\label{reducingmass}
\end{equation}
where $M_0$ is the initial mass of an evaporating BH and $t_p$ is the time of BH production after Big Bang.
Equation \eqref{reducingmass} shows that the BH mass can be approximately considered as constant till
the moment of the evaporation and may be
approximated as $\theta (t-t_p-\tau_{BH})$. Due to evaporation a BH emits all kind of particles with masses $m<T_{BH}$
and, in particular, gravitons.
The total energy emitted by BH per unit time and frequency $\omega$ (energy) of the emitted particles,
is approximately given by the equation (see, e. g. book \cite{Frolov:1998wf}):
\begin{equation}
\left(\frac{\mathrm{d}E}{\mathrm{d}t\mathrm{d}\omega}\right)=
\frac{2N_{eff}}{\pi}\,\frac{M^2 }{m_{Pl}^4}\frac{\omega^3}{e^{\omega/T_{BH}}-1}\,,
\end{equation}
where $T$ is the BH temperature \eqref{T-BH}.
Due to the impact of the gravitational field of BH on the propagation of the evaporated particles, their spectrum
is distorted \cite{DonPage:1976} by the so called grey factor $g(\omega)$, but we disregard it in what follows.
Let us now estimate the amount of the gravitational radiation from the graviton evaporation.
After their production PBHs started to emit thermal gravitons independently on the PBH clustering.
Hence the thermal graviton emission depends only on PBH number density, $n_{BH}$.
The energy density of gravitons in logarithmic frequency band emitted in the time
interval $t$ and $t+ \mathrm d t$ is
\begin{equation}
\frac{\mathrm d \rho_{GW}(\omega; t)}{\mathrm d \omega}=
10^{-2}n_{BH}(t)\,\left(\frac{\mathrm{d}E}{\mathrm{d}t\mathrm\,{d}\omega}\right)\mathrm d t\,,
\label{d-rho-evap}
\end{equation}
where factor $10^{-2}$ takes into account that about one percent of the emitted energy goes into gravitons.
The density parameter of GWs per logarithmic frequency interval at cosmological time $t_* =\tau_{BH}$
can be obtained by integrating expression (\ref{d-rho-evap}) over redshift with an account of the drop-off
of the graviton energy density by $(1+z)^{-4}$ and the redshift of the emitted frequency
so that at $t_*=\tau_{BH}$: $\omega= \omega_* (1+z)$. Note that in the instant decay approximation
the BH temperature remains constant. One has also to take into account that the number density of PBH
behaves as $n_{BH}(t) = n_{p}(t_p)(1+z)^3$, so if we normalize our result to $n_{BH} (\tau_{BH})$, the integrand
should be multiplied by $(1+z)^3$. Finally we obtain:
\begin{eqnarray}
\frac{\mathrm d\rho_{GW}(\omega_*, \tau_{BH})} {\mathrm d\ln\omega_*} = \frac{0.03 N_{eff}M \omega^4_*}
{ \pi m_{Pl}^4}\,
(3\tau_{BH})\,{\rho_{BH}(\tau_{BH})}\,I\left(\frac{\omega_*}{T_{BH}}\right)\,,
\label{rho-evap-tau}
\end{eqnarray}
where
\begin{equation}
I\left(\frac{\omega_*}{T_{BH}}\right)\equiv\varint_{0}^{z_{max}} \frac{ \mathrm dz \left(1+z \right)^{1/2}}
{\exp\left[(z+1) \omega_*/T_{BH} \right] -1}\,,
\end{equation}
and
\begin{equation}
1+z_{max} = \left(\frac{\tau_{BH}}{t_{eq}}\right)^{2/3}\left(\frac{t_{eq}}{t_p}\right)^{1/2}=\left(\frac{32170}{N_{eff}}\right)^{2/3}\left(\frac{M}{m_{Pl}}\right)^{4/3}\Omega_p^{1/3}\,,
\end{equation}
where the effective time of integration is equal to $3\tau_{BH}$ because of the instant decay approximation.
One can check that in this case the total evaporated energy would be equal to the PBH mass.
The spectral density parameter of GWs at $t=\tau_{BH}$ is equal to:
\begin{eqnarray}
\Omega_{GW}(\omega_*; \tau_{BH})\approx
\frac{ 2.9\cdot 10^3 M^4 \omega_*^4}{\pi\, m_{Pl}^8}\,I\left(\frac{\omega_*}{T_{BH}}\right)\,.
\label{evaporation-BH}
\end{eqnarray}
The spectrum is not a thermal one, though rather similar to it. It has more power at small frequencies due to
redshift of higher frequencies into lower band and less power at high $\omega_*$.
The spectral density parameter reaches maximum at $\omega_*^{peak}/T_{BH} = 2.8$. Accordingly the maximum value of the spectral density parameter
when PBHs completely evaporated is equal to:
\begin{equation}
\Omega_{GW}^{peak}(\omega_*^{peak}; \tau_{BH}) \approx 3.8\cdot 10^{-3}\,.
\label{evaporation-BH1}
\end{equation}
\begin{figure}[!htb]
\centering
\includegraphics[scale=.8]{blackbody4.eps}
\caption{Log-log plot of the density parameter per logarithmic frequency, $h_0^2\Omega_{GW}(f; t_0)$, as a function of frequency today,
$f$, for the case $g_S\sim 100$, $N_{eff}\sim 100$, black hole mass $M=1$ g (solid line) and black hole mass $M=10^{5}$ g (dashed line).
We can see that the spectrum has a maximum which is sharp and of order $h_0^2\Omega_{GW}(f_{peak})\sim 10^{-7}$.}
\label{fig:evaporation}
\end{figure}
Integrating equation (\ref{evaporation-BH}) first over $\omega_*$ and then over redshift, we find that the total fraction of
energy of GWs is $0.006$ which is reasonably (in view of the used approximations) close to the expected $0.01$.
At BBN the energy fraction of such GWs would be about 0.005. So the total number of
additional effective neutrino species would be close to 0.045,
where 0.03 comes from neutrino heating by $e^+e^-$ annihilation and 0.01 comes from the
plasma corrections (see e.g. review~\cite{nu-rev}). Of course the GWs produced by the considered
mechanism are safely below the BBN bound~\cite{Abbott:2009ws}.
Using equation \eqref{generalexpression} and taking into account the redshift
from $t=\tau_{BH}$ to the present time, we find that the total density parameter of GWs today due to PBH evaporation would be about $10^{-7}$.
The total energy density of GWs from the PBH evaporation is quite large but it is concentrated at high frequencies. According to eq. (\ref{omega-0}) the redshifted peak frequency emitted at time $t_*=\tau_{BH}$ becomes today:
\begin{eqnarray}
f^{(peak)} = 2\cdot 10^{15}\,\textrm{Hz}\,\left(\frac{g_S(T(\tau_{BH}))}{100}\right)^{1/12}\,
\left(\frac{100}{N_{eff}}\right)^{1/2}\left(\frac{M}{10^5\,\textrm{g}}\right)^{1/2}\,.
\label{T_0}
\end{eqnarray}
The energy density of GWs at small $f$ drops down in accordance with equation (\ref{evaporation-BH1}). The spectral density today
can be calculated from equation (\ref{evaporation-BH}) with an account of the redshift to the present day:
\begin{equation}
h_0^2\Omega_{GW}(f; t_0)=1.36\cdot 10^{-27}\left(\frac{N_{eff}}{100}\right)^2\left(\frac{10^5\,\textrm{g}}{M}\right)^2\left(\frac{f}{10^{10}\,\textrm{Hz}}\right)^4\cdot I\left(\frac{2\pi\cdot f}{T_0}\right)\,,
\end{equation}
where we used $\omega=2\pi f$ and $T_0$ is the BH temperature redshifted to the present time:
\begin{equation}
T_0=\left[\frac{a(\tau_{BH})}{a(t_0)}\right]\,T_{BH}=
4.53\cdot 10^{15}\,\textrm{Hz}\left(\frac{100}{g_S(T(\tau_{BH}))}\right)^{1/12}
\left(\frac{100}{N_{eff}}\right)^{1/2}\left(\frac{M}{10^5\,\textrm{g}}\right)^{1/2}\,.
\end{equation}
\section{ Stochastic background of gravitational waves. An overview \label{s-GW-background}}
Stochastic background of relic gravitational waves can be produced by several mechanisms.
The theoretical predictions are model depended due to the uncertainties in the cosmological framework and on
the values of the redshift from the production epoch. Below we briefly describe some of the production scenarios. For a more detailed review on stochastic background of GWs production mechanisms and their spectra the reader can consult more specific ref. \cite{Allen:1996vm, Maggiore:1999vm, Grishchuk:2007uz}.
\emph{Inflationary models.} It was established long ago that gravitational waves could be produced in cosmology
due to an amplification of vacuum fluctuations by external gravitational field (quantum particle production).
It was first studied by Grishchuck~\cite{Grishchuk:1974ny} and first applied to an inflationary model by
Starobinsky\cite{Starobinsky:1979ty}. The gravitational waves could be quite efficiently produced at inflation. Their
spectrum at large wavelengths is independent on the details of inflationary models.
The frequency band of these gravitons today is quite wide and the associated density parameter is very low.
The predicted density parameter of gravitational waves in the frequency range from $3\times 10^{-18}$ Hz$<f<$ $10^{-16}$ Hz is
\begin{equation}
h_0^2\Omega_{GW}(f)\simeq 6.71\cdot 10^{-10}\left(\frac{10^{-18}\textrm{Hz}}{f}\right)^2\left(\frac{H}{10^{15}\textrm{Gev}}\right)^2\,,
\label{inflation1}
\end{equation}
while in the frequency range $2\times 10^{-15}$ Hz$<f<f_{max}\simeq $ $10^{9}$ Hz the spectrum is flat and the density parameter is
\begin{equation}
h_0^2\Omega_{GW}(f)\simeq 6.71\cdot 10^{-14}\left(\frac{H}{10^{15}\textrm{Gev}}\right)^2\,,
\label{inflation2}
\end{equation}
where $H$ is the Hubble parameter at inflation.
A near scale-invariant spectrum over a wide range of frequencies is a key prediction of the standard inflationary model \cite{Fabbri:1983us, Abbott:1984fp}. The relative amplitude of GWs spectrum to density perturbations spectrum is usually expressed in terms of the ratio, $r$, of tensor to scalar perturbations. From observations of WMAP, the current limit on B-mode of the CMB polarization demands $r\lesssim 0.22$ which rule out some models of inflation \cite{Komatsu:2008hk, Peiris:2006sj}. The spectrum of GWs can be expressed in terms of the tensorial spectral index, $n_t$, and is almost flat
in the frequency range $2\times 10^{-15}$ Hz $<f<f_{max}\simeq$ $10^{10}$ Hz. The density parameter is proportional to a power
of the frequency:
\begin{equation}
h_0^2\Omega_{GW}(f)\propto f^{n_t}.
\end{equation}
Since the tensorial spectral index is negative, $n_t<0$, the spectrum is decreasing rather than flat. Depending on inflationary model the value of the tensorial spectral index changes and there are some models which predict $r\sim 10^{-3}$.
\emph{Pre-heating phase} at the end of inflation. At this stage the energy of scalar field $\phi$ is spent to
generate new particles and heat the Universe. The first estimate of the density parameter
of GWs during the
pre-heating phase was done by Klebnihkov and Tkachev \cite{Khlebnikov:1997di} who found the density
parameter of the order $h_0^2\Omega_{GW}\sim 10^{-11}$ for the gravitational waves with the present day
frequency $f\sim 10^{6}$ Hz, in the models with quartic potential, $\lambda\phi^4$. Later, this mechanism was
reconsidered by Easther and Lim \cite{Easther:2006gt, Easther:2007vj} who studied the models with the potentials
of the form $\lambda\phi^4$ and $m^2\phi^2$. The authors have found numerically
that $h_0^2\Omega_{GW}\sim 10^{-10}$ in the frequency range $f\sim 10^8-10^9$ Hz.
\emph{First order phase transitions}. At the end of inflation, first-order phase transitions could have generated a large amount of gravitational waves.
At such transitions the bubble nucleation of true vacuum states and percolation can occur accompanied by the bubble collisions.
In a series of papers~ \cite{Turner:1990rc,Kosowsky:1991ua, Kosowsky:1992vn, Kamionkowski:1993fg}
the energy of gravitational waves generated from bubble collisions at strongly first-order phase transitions
was estimated and the results were later extended
to the electroweak first-order phase transitions. The amount of GWs from strongly first-order phase transition
at its end is of the order $1.3\cdot 10^{-3}(\tau/H)$, where $\tau$ is the duration of the phase transition, $H$ is the Hubble
constant, and the peak frequency is $\omega_*^{peak}=3.8/\tau$.
The present day density parameter of GWs produced at the electroweak first-order phase transition
was found to be of the order $\Omega_{GW}\sim 10^{-22}$ with characteristic
frequency $f\sim 4\cdot 10^{-3}$. Since later it has been found out, that there is no first order electroweak phase transition
in the standard model \cite{Kajantie:1995kf}, the mechanism was reconsidered
by Grojean and Servant \cite{Grojean:2006bp}. The authors estimated the GW production in the temperature
range 100 GeV-$10^7$ GeV.
The spectrum of the GWs today in this temperature range extends from $10^{-3}$ Hz to $10^{2}$ Hz. The associated density parameter was found to be quite large, $h_0^2\Omega_{GW}(f_{peak})\sim 10^{-9}$ depending on the parameters of the model.
\emph{Topological defects and cosmic strings.} Practically in all inflationary models the gravitational wave spectrum
is almost flat in the frequency range from $10^{-15}$ Hz$<f<f_{max}\simeq $ $10^{10}$ Hz with some variations
coming from pre-heating and reheating phases for which the frequency is peaked near GHz region. There are other mechanisms
of GWs production e.g. by cosmic strings which predict almost flat spectrum in a wide range of frequencies. Many of the
proposed observational tests for the existence of cosmic strings are based on their gravitational
interactions \cite{Vilenkin:1984ib, Vilenkin:1986hg}. Particularly interesting are GWs produced by closed string
loops which oscillate in relativistic regime.
The spectrum of the gravitational waves produced by such relativistic oscillations
is almost flat in the region $10^{-8}$ Hz$<f<f_{max}\simeq $ $10^{10}$ Hz with a peak at low frequency near
$f\sim 10^{-12}$ Hz. The density parameter in the frequency range $f\gg 10^{-4}$ Hz, according to ref. \cite{Hogan:2006we},
is equal to:
\begin{equation}
h_0^2\Omega_{GW}(f)\simeq 10^{-8}\left(\frac{G\mu}{10^{-8}}\right)^{1/2}\left(\frac{\gamma}{50}\right)^{1/2}\left(\frac{\alpha}{0.1}\right)^{1/2},
\end{equation}
where $G\mu$, $\alpha$ and $\gamma$ are respectively the string tension, the initial loop size as a fraction of the
Hubble radius and the radiation efficiency. From the pulsar timing data the authors of ref. \cite{DePies:2007bm} constrained
the density parameter of GWs from the cosmic strings in the frequency range $f\gg 10^{-6}$ Hz and put the limit
\begin{equation}
h_0^2\Omega_{GW}(f)\lesssim 10^{-8}.
\end{equation}
It is generally assumed that at the end of inflation the inflaton oscillates and eventually decays. If non-topological
solitons, the so called Q-balls, are produced at the inflaton decay, such Q-balls could be a source of GWs. According
to the calculations of ref. \cite{Mazumdar:2010pn} the density parameter of such GWs would be of the order of
$h_0^2\Omega_{GW}\sim 10^{-9}$ with a peak frequency $f\sim 10^{10}$ Hz.
\section{Gravitational waves detectors. Present status\label{GW-detectors}}
For most of the models mentioned above, the stochastic background of GWs is beyond the sensitivity of the current and planned interferometers. We have seen that inflationary models predict almost flat spectrum of GWs in a wide range of frequencies. There is a narrow band of frequencies of this background that falls into the range of the present detectors such as LIGO and VIRGO. Unfortunately, the density parameter predicted by inflationary models is too low to be detected by the present detectors. Almost all the models mentioned above predict the density parameter of the order $h_0^2\Omega_{GW}\lesssim 10^{-5}$ and actual LIGO and VIRGO are not able to detect such a quantity because of the
frequency dependence of the density parameter. This can be seen from the relation between the expected
amplitude of stochastic gravitational waves $h_c(f)$ with the density parameter as
presented in the ref. \cite{Maggiore:1999vm}
\begin{equation}
h_c(f)=1.3\times 10^{-18}\sqrt{h_0^2\Omega_{GW}(f)}\left(\frac{1 \textrm{Hz}}{f}\right).
\end{equation}
Present detectors such as LIGO and VIRGO with enhanced technologies
operate in the frequency range 1 Hz - $10^4$ Hz and can reach respectively the
strain sensitivity $h_{rms}\sim 10^{-23}$ Hz$^{-1/2}$ and $h_{rms}\sim 10^{-22}$ Hz$^{-1/2}$ in the frequency band $f\sim 10^2 - 10^3$.
\begin{figure}[!htb]
\centering
\includegraphics[scale=.75] {bbn}
\caption{$\log[h_0^2\Omega_{GW}$(f)] vs. $\log$(f [Hz]) for different models of production of stochastic background of GWs as given in ref. \cite{Abbott:2009ws}.}
\label{fig:models}
\end{figure}
The planned detectors such as \href{http://www.advancedligo.mit.edu/summary.html}{Advanced LIGO}, \href{https://wwwcascina.virgo.infn.it/advirgo/}{Advanced VIRGO} and \href{http://lisa.nasa.gov/}{LISA} have better chances to detect this stochastic background. In fact, LISA can reach the density parameter of the order $h_0^2\Omega_{GW}\lesssim 10^{-11}$ at frequency $f\sim 10^{-3}$ Hz and Advanced LIGO can reach a $h_0^2\Omega_{GW}\lesssim 10^{-9}$ at frequency $f\sim 10^{2}$ Hz. These planned detectors can register the stochastic background of GWs coming from cosmic strings and the pre-bing bang stage. The gap between LISA and the ground based detectors will be covered by DECIGO/BBO detectors which will operate in the frequency range from 0.1 Hz to 10 Hz and have $10^3$ better sensitivity than LISA
from 0.1 Hz to 1 Hz \cite{Takahashi:2003wm}. DECIGO will be able to observe the stochastic background of GWs produced at inflation and can reach $h_0^2\Omega_{GW}\sim 10^{-20}$ at $f\sim 1$ Hz after 3 years of observation \cite{Seto:2001qf, Kawamura:2008zz}. All the above mentioned GWs detectors cover a frequency range $10^{-7}$ Hz - $10^{3}$ Hz and the high-frequency range will hopefully explored by future high-frequency GWs detectors. The principle of a high-frequency detector is based on the electromagnetic-gravitational resonance first proposed by Braginsky and Mensky \cite{Braginsky:1971, Braginsky:1972, Braginsky:1972bz}. Actually there is a renewed interest on these new detectors which a prototype has been constructed at Birmingham University \cite{Cruise:2000za, Cruise:2005uq, Cruise:2006zt} and which reaches a strain sensitivity of the order $h_{rms}\sim 10^{-14}$ Hz$^{-1/2}$ at $f\sim 10^8$ Hz. The main goal of this detector is detection of high-frequency stochastic background of GWs from the early Universe and black hole interactions in higher dimensional gravitational theories.
\section{Summary and results\label{conclusion}}
We have analyzed the formation and evolution of light primordial black holes in the early Universe which created a transient matter domination regime in contrast to the present standard cosmology, where the early Universe after inflation was normally radiation dominated. PBHs with masses less than $M\sim 10^8$ g evaporated before primordial nucleosynthesis leaving no trace. Thus the fraction of the energy density of such PBHs, $\Omega_p$, in this case is a free parameter of the model, not constrained by any existing observations.
At MD stage the PBHs could form high density clusters which
would be efficient sources of the primordial GWs. PBHs could have dominated the Universe for a short time of the order of their lifetime,
$\tau_{BH}$, generating relic gravitational waves by various mechanisms of their mutual interactions
as well as due to their evaporation. In the former case we have shown that production of GWs is most efficient after BH density started to dominate over radiation. After that moment, high density clusters of PBHs could have been formed, leading to an efficient production of GWs.
To survive till cluster formation the PBH mass at production must be bounded from below by $M\sim 4\cdot 10^{-5}$ g $\Omega_p^{-1}\Delta_{in}^{-3/4} N_{eff}^{1/2}$ which leads to a lower bound $\Omega_p>10^{-14}\Delta_{in}^{-3/4} N_{eff}^{1/2}$. According to the standard cosmology the amplitude of primordial density perturbations is of order of $\sim 10^{-4}$, which in our case leads to a lower bound on the density parameter of PBHs, $\Omega_p\gtrsim 10^{-11}$.
In this context we have calculated the density parameter of GWs today from scattering of PBHs in both classical and quantum regime, GWs emission
from binaries, and from black hole evaporation. We have shown that a substantial amount of gravitational waves has been emitted by all mechanism considered here. In the case of scattering of PBHs we considered \emph{only} scattering between them neglecting the possibility of PBHs mergers, which results in an underestimate on $h_0^2\Omega_{GW}$. Even in this case the density parameter is substantial at high frequencies reaching values of the order of $h_0^2\Omega_{GW}\sim 10^{-9}$ at $f\sim$ GHz for classical scattering and
the total density parameter $h_0^2\Omega_{GW}\sim 10^{-10}$ for very light primordial black holes. In the low frequency limit the density parameter in the classical case is of the order of $h_0^2\Omega_{GW}\sim 10^{-17}-10^{-20}$ in the frequency range $f\sim 10^{-1}-10^{2}$ Hz which falls into the detection band of DECIGO/BBO.
The number of PBHs that form binaries after cluster formation is subjected to uncertainties and in this paper we parametrized it through factor $\epsilon$. The exact value of this parameter could be calculated elsewhere by numerical calculations. Since the density in such clusters is very high we expect that $\epsilon$ is not very small in comparison with unity. In fig.\ref{fig:binaries} the expected value of the density parameter today is presented. We can see that a large amount of gravitational waves has been emitted in the high frequency regime with $h_0^2\Omega_{GW}\sim 10^{-14}- 10^{-12}$ at frequency $f\sim 10^{10}$ Hz depending on the BH initial mass. In the low frequency part of the spectrum the spectral density parameter is utterly negligible making it impossible to detect GWs produced by this mechanism at present and probably in the near future. In our derivation we have considered both stationary and inspiral phases of binaries leading to a wide range of the frequencies emitted. We have considered only binaries in circular orbits and the problem with elliptical orbits will be treated later.
If elliptical orbits were frequent,
the amount of GWs will be presumably higher over a wide range of frequencies. We assumed that all binaries are formed with initial radius less than the average distance between PBHs and greater than the gravitational radius $r_g$. In this case the frequency spectrum has a cutoff in both low and high frequency bands of the spectrum.
Another mechanism of graviton productions considered here is the PBHs evaporation. This mechanism is independent on the structure
formation during the PBH domination. In fig.\ref{fig:evaporation} we show the density parameter as a function of frequency for BH masses $1$ g and $10^5$ g. Having a near blackbody spectrum, the frequency of the emitted gravitons
can have any value, but unfortunately the GWs spectrum has a peak in the high frequency region which today make a substantial contribution into the cosmological energy density of the order of $h_0^2\Omega_{GW}(f_{peak})\sim 10^{-7}$.
The mechanisms considered in this paper could create a rather high cosmological fraction of the energy density of the relic gravitational waves at very high frequencies and gives an opportunity on investigating the high GWs spectrum by present and future detectors. Unfortunately at the lower part of the spectrum $\Omega_{GW}$ significantly drops down. Still the planned interferometers DECIGO/BBO could be sensitive to the predicted GWs. It is noteworthy that the mechanism of GWs generation suggested here kills or noticeably diminishes GWs from inflation by the redshift of the earlier generated GWs at the PBH (MD) stage.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,208
|
\section{Introduction}
Consider the usual definitions of discrete channels in information theory. It is assumed that,
transmissions of symbols from a discrete alphabet take place and a fraction of the transmissions may result in erroneous reception.
The sender is allowed to ``encode'' information in to an array of symbols, called a codeword. The collection of
all possible codewords is called a ``code'' (or ``codebook''). Without much loss of generality, we can assume that all transmitted codewords
are equally likely, in which case the log-size of a code signify the amount of information that can be transmitted with the code.
In a completely adversarial channel, the adversary is allowed to
see the transmitted set of symbols (codeword) completely and then decides which of the transmitted symbols are to
be corrupted (it is allowed to corrupt a given fraction of all symbols).
Recently, in a series of papers \cite{langberg2009binary,dey2013upper,haviv2011beating}, the study of
online or causal adversarial channels is initiated, in particular, for binary-input channels.
Let us start by giving an informal definition of a causal adversarial channel.
In the causal adversarial model, an adversary is allowed to see
the transmitted codeword only causally (i.e., at any instance it sees only the past transmitted symbols), and decides whether to corrupt the current transmitted symbol.
An upper bound on the capacity (maximum rate of reliable information transfer)
of such channel is presented in \cite{dey2013upper}. One of the most
interesting observation is that, such channels are limited by the ``Plotkin bound,'' of coding theory: whenever
the fraction of error introduced by the adversary surpasses $\frac14$, the capacity is zero (assuming binary input). On the
other hand, by ``random coding'' method, a lower bound is established in \cite{haviv2011beating}.
This lower bound beats the famous Gilbert-Varshamov bound, the best available lower bound for a
completely adversarial channel.
We below describe an adversarial channel model that
is weaker (in terms of adversary limitations) than the above causal channel.
In particular, the adversary is not even allowed to see the past
transmitted symbols, but decides whether to corrupt a symbol based on only
the current transmission.
Our initial aim is to see
whether the channel capacity is still dictated by the Plotkin bound.
\subsection{A memoryless (truly online) adversary}
In this work we consider the code to be {\em deterministic}, in a sense that is described below. Also,
we assume that the input alphabet to be binary ($\{0,1\}$).
A code $\cC$ is simply a subset of $\ff_2^n$.
The size of the code denotes the number of messages encodable with this code; and therefore
the amount of information encodable is $\log |\cC|$. In here and subsequently, all logarithms are base-2, unless otherwise mentioned.
The {\em rate} of the code is $\frac{\log |\cC|}{n}$.
Given the code, the adversarial channel consists of $n$ (possibly random) functions $f_{\cC}^i:\ff_2 \to \ff_2, i=1,\dots,n.$
Suppose a randomly and uniformly chosen codeword $$\bfx \equiv(x_1,x_2,\ldots,x_n) \in \cC$$ is transmitted. At the $i$th time instant, the
adversary will produce $e_i = f_{\cC}^i(x_i)$, taking only the current transmitted symbol $x_i$ as argument (and of course,
taking into account the code $\cC$, which is known to the adversary). For, $i = 1,\dots,n$, $e_i$ is the indicator of an error at the
$i$th position. I.e., the channel produces $y_i =x_i+e_i$, at the $i$th time-instance, where the addition is of course over $\ff_2$.
\begin{definition}
The adversary is called {\em weakly-$p$-limited}, $0\le p\le 1$, if the
expected (with respect to the randomness in $f_{\cC}^i$s and $\bfx$) Hamming weight of the error-vector $\bfe = (e_1,e_2, \dots, e_n) =
( f_{\cC}^1(x_i),\dots, f_{\cC}^n(x_n)) \equiv f_{\cC}(\bfx)$ is
\begin{equation}
{\mathbb E}\wt(\bfe) \le pn.
\end{equation}
A more restrictive adversary ({\em strongly-$p$-limited}) must have,
\begin{equation}\label{eq:pn}
\Pr(\wt(\bfe)/n < p+\epsilon ) = 1- o(1), \forall \epsilon >0.
\end{equation}
\end{definition}
A code is associated with a (possibly randomized) decoder $\phi: \ff_2^n \to \cC$. For a given pair of transmitted codeword and error vector, $ \bfx \in \cC, \bfe\in \ff_2^n$, the decoder makes
an error if, $\phi(\bfx +\bfe) \ne \bfx$.
Given $\cC$ and $p$, define ${\rm Adv}_w(\cC,p)$ to be the collection of all weakly-$p$-limited adversary strategies. That is,
$f_{\cC}\equiv \{f_{\cC}^i:\ff_2 \to \ff_2, i=1,\dots,n\} \in {\rm Adv}_w(\cC,p)$ if and only if, ${\mathbb E}\wt(f_{\cC}(\bfx))\le pn.$
Similarly, we can name the collection of all strongly-$p$-limited adversary strategies as ${\rm Adv}_s(\cC,p)$.
Our results, as in the case of causal adversarial channels of \cite{langberg2009binary}, holds for the case
of {\em average probability of error} \footnote{It is relatively easy to see that the worst-case probability of error
does not lead to anything different than the completely adversarial channel. For the same reason linear
codes do not lead to any improvement for these channels over completely adversarial
channel. We refer to \cite{dey2013upper} for further discussion. In general, the notion of average vs. worst-case
error probability leading to different capacities for {\em arbitrarily varying channels}
is well-known (for example, see \cite{ahlswede1970capacity} or \cite{lapidoth1998reliable}).}.
The average probability of error
is defined to be,
$$
P^w_{\cC}(p) = \max_{f_{\cC}\in {\rm Adv}_w(\cC,p) } \frac{1}{|\cC|} \sum_{\bfx \in \cC} \Pr(\phi(\bfx+f_{\cC}(\bfx)) \ne \bfx),
$$
and,
$$
P^s_{\cC}(p) = \max_{f_{\cC}\in {\rm Adv}_s(\cC,p) } \frac{1}{|\cC|} \sum_{\bfx \in \cC} \Pr(\phi(\bfx+f_{\cC}(\bfx)) \ne \bfx).
$$
The maximum possible size of ``good'' codes are:
\begin{equation}
M^w_\epsilon(n,p) \equiv \max_{\cC\subseteq\ff_2^n: P^w_{\cC}(p)\le \epsilon} |\cC|,
\end{equation}
and,
\begin{equation}
M^s_\epsilon(n,p) \equiv \max_{\cC\subseteq\ff_2^n: P^s_{\cC}(p)\le \epsilon} |\cC|.
\end{equation}
Now, define the {\em capacities} to be,
\begin{equation}
C_w(p) \equiv \inf_{\epsilon>0} \limsup_{n\to \infty} \frac{\log M^w_\epsilon(n,p)}{n},
\end{equation}
\begin{equation}
C_s(p) \equiv \inf_{\epsilon>0} \limsup_{n\to \infty} \frac{\log M^s_\epsilon(n,p)}{n}.
\end{equation}
It is evident that,
\begin{equation}
C_w(p) \le C_s(p) \le 1-h_\mathrm{B}(p),
\end{equation}
where $h_\mathrm{B}(x) = -x \log x -(1-x)\log(1-x)$ is the {\em binary entropy function}.
This is true because, a strongly-$p$-limited adversary strategy is to flip each symbol with probability $p$, independently. That is, the
adversary can always simulate a binary symmetric channel, whose capacity is $1-h_\mathrm{B}(p)$.
\subsection{Practical limitations to the model and contributions} It is counterintuitive to assume that the adversary,
being memoryless, cannot store the previously transmitted bits, or its own actions, however,
has access to the entire code and can do computations on them. But it should be noted that, the entire computation of the adversary
is done offline, and in each transmission, it just performs according to one of the two options.
Also note that, the adversary knows the time-instance of the transmission. That is, he knows that the $i$th transmission, among the
$n$ possible, is taking place. In that sense the adversary is not completely memoryless.
The main purpose of introducing this
model is to see how weak the adversary can be and still have its capacity dictated by the Plotkin bound.
On the other hand, the concept of such memoryless adversary appears in principle before in literature.
In particular, general classes of restricted adversarial channels were considered in the literature of
{\em arbitrarily varying channels} \cite{ahlswede1970capacity,csiszar1989capacity, csiszar1988capacity} or {\em oblivious channels} \cite{langberg2008oblivious}.
From \cite{guruswami2013optimal} (see also,\cite{ahlswede1978arbitrary}), Thm.~C.1, it is evident that the capacity of weakly-$p$-limited
adversary is $0$ for $p>\frac14$. It is also proved there that, if the adversary can keep a count of how many
bits it has flipped (a log-space channel), then the same fact holds for strongly limited adversaries as well.
In Sec.~\ref{sec:weak}, we present the above fact regarding weakly-limited adversary in a way
that is amenable to our definitions. We then attempt to extend this result to the case of strongly-limited
adversary: what we have forms the main contribution of this paper. In Sec.~\ref{sec:dist} we introduce the important notions of distance distribution of
a code that proves useful in this context. In Sec.~\ref{sec:strong}, we show that the capacity of a strongly-$p$-limited
adversary is strictly separated from the capacity of a BSC($p$). In particular we give an upper bound on $C_s(p)$ that is
strictly below $1-h_\mathrm{B}(p)$ for all $p> \frac14$. Further discussions and concluding remarks
are presented in Sec.~\ref{sec:skew}.
\section{Weakly-limited adversary}\label{sec:weak}
In this small section, we establish the following fact.
\begin{theorem}\label{thm:weak}
$C_w(p) =0$ for $p \ge \frac14$.
\end{theorem}
To prove the theorem, the below lemma, known as the Plotkin bound, is used crucially.
\begin{lemma}[Plotkin Bound]\label{lem:plotkin}
Suppose, $\cC\subseteq \ff_2^n$ is the code and $|\cC|=M$. Randomly and uniformly (with replacement) choose two
codeword $\bfx_1,\bfx_2$ from $\cC$. Then,
\begin{equation}
{\mathbb E} d_\mathrm{H}(\bfx_1,\bfx_2) \le \frac{n}2,
\end{equation}
where $d_\mathrm{H}(\cdot)$ is the Hamming distance.
\end{lemma}
\begin{IEEEproof}
Consider an $M \times n$ matrix with the codewords of $\cC$ as its rows. Suppose, $\lambda_i$ is the
number of $1$s in the $i$th column of the matrix, $i=1,\dots , n$. Then,
$$
\sum_{\bfc_1,\bfc_2 \in \cC} d_\mathrm{H}(\bfc_1,\bfc_2) = 2\sum_{i=1}^n \lambda_i(M-\lambda_i) \le \frac{nM^2}{2}.
$$
Hence,
$
{\mathbb E} d_\mathrm{H}(\bfx_1,\bfx_2) \le \frac{n}2,
$
where, $\bfx_1,\bfx_2$ are two randomly and uniformly chosen codewords.
\end{IEEEproof}
\begin{IEEEproof}[Proof of Theorem \ref{thm:weak}]
We show that there exists an adversary strategy that achieves the claim of the lemma.
In this vein, we use the same adversarial strategy that is used in \cite{guruswami2013optimal,dey2013upper}.
Suppose, $\cC\subset\ff_2^n$ is the code and $|\cC|=M$. The adversary (channel) first choses
a codeword $\bfx = (x_1,x_2,\dots, x_n) \in \ff_2^n$ randomly and uniformly from $\cC$.
Now if $\bfc =(c_1,c_2,\dots,c_n)$ is the transmitted codeword, then,
\[
e_i\equiv f_{\cC}^i(x_i) =
\begin{cases}
0,& \text{ when } x_i = c_i\\
1, & \text{ with probability } \frac12 \text{ when } x_i \ne c_i\\
0, & \text{ with probability } \frac12 \text{ when } x_i \ne c_i.
\end{cases}
\]
Note that, if $\bfc$ is randomly and uniformly chosen from $\cC$, then
\begin{align*}
{\mathbb E} \wt(\bfe) = \sum_{i=1}^n \Pr(e_i =1)& =\frac12\sum_{i=1}^n \Pr(x_i \ne c_i)\\
& = \frac12{\mathbb E} d_\mathrm{H}(\bfx,\bfc) \le \frac{n}{4},
\end{align*}
where, $\bfe =(e_1,\dots,e_n)$. Hence, the adversary is weakly-$\frac14$-limited.
On the other hand, $\Pr(\bfx =\bfc) = \frac1M$. Suppose, $\bfy = \bfx+\bfe$.
At the decoder,
let $\Pr(\bfy \mid \bfc')$, $\bfc' \in \cC$, denote the probability
that $\bfc'$ is transmitted and $\bfy$ is received.
Clearly,
$$
\Pr(\bfy \mid \bfc) =\Pr(\bfy \mid \bfx).
$$
Hence, even with the maximum likelihood decoder will have a probability of error $\ge 1/2-\frac1M$.
Therefore, $C_w(p) =0$ for $p\ge 1/4$.
\end{IEEEproof}
\section{Distance distribution}\label{sec:dist}
To extend Thm. \ref{thm:weak} to the case of strongly-limited adversary, we need to show an adversary strategy,
that, with high probability, keep the number of errors within $pn$. However, for the adversary strategy
of Thm. \ref{thm:weak} to do this, we need the result of Lemma \ref{lem:plotkin} to be stronger, i.e., a high probability
statement.
Let us now introduce some notations that help us cast Lemma \ref{lem:plotkin} as a high-probability result.
The {\em distance distribution} of a code is defined in the following way. Suppose, $\cC \subseteq \ff_2^n$
be a code. Let, for $i=0,1,2,\dots, n$,
\begin{equation}
A_i = \frac1{|\cC|} |\{(\bfc_1,\bfc_2) \in \cC^2: d_\mathrm{H}(\bfc_1,\bfc_2) = i \}|.
\end{equation}
As can be seen, $A_0 =1$.
The dual distance distribution of a code is defined to be, for $i =0,1,\dots, n$,
\begin{equation}
A^{\perp}_i = \frac1{|\cC|}\sum_{j=0}^n K_i(j) A_j,
\end{equation}
where $$K_i(j) = \sum_{k=0}^i (-1)^k \binom{j}{k}\binom{n-j}{i-k}$$
is the Krawtchouk polynomial. Note that, $A^{\perp}_0 = 1$. It is known that
$A^{\perp}_i \ge 0$ for all $i$.
The {\em dual distance} $d^{\perp}$ of the code is defined to be the smallest $i>0$ such that
$A^{\perp}_i$ nonzero.
\bigskip
\begin{lemma}[Pless power moments]\label{lem:pless}
For all $r <d^{\perp}$,
\begin{equation}
\frac{1}{|\cC|} \sum_{i=0}^n (n/2- i)^r A_i = \frac{1}{2^n}\sum_{i=0}^n (n/2 -i)^r \binom{n}{i}.
\end{equation}
\end{lemma}
\begin{IEEEproof}
For a proof of the lemma, see \cite[p.~132]{MS1977}.
\end{IEEEproof}
\bigskip
\begin{lemma}\label{lem:conc}
Suppose, $\cC\subseteq \ff_2^n$ is the code with dual distance greater than $2$, and $|\cC|=M$. Randomly and uniformly (with replacement) choose two
codeword $\bfx_1,\bfx_2$ from $\cC$. Then,
\begin{equation}
\Pr\Big(d_\mathrm{H}(\bfx_1,\bfx_2)< n(1/2+\epsilon)\Big) > 1- \frac{1}{4n\epsilon^2} .
\end{equation}
\end{lemma}
\begin{IEEEproof}
From Lemma \ref{lem:pless}, for any $r < d^{\perp}$,
\begin{align*}
\Pr\Big(d_\mathrm{H}(\bfx_1,\bfx_2)\ge n(1/2+\epsilon)\Big) &\le \frac{{\mathbb E}(d_\mathrm{H}(\bfx_1,\bfx_2) - n/2)^r }{n^r\epsilon^r}\\
&= \frac{\frac{1}{2^n}\sum_{i=0}^n (n/2 -i)^r \binom{n}{i}}{n^r\epsilon^r}.
\end{align*}
In particular, substituting $r=2$ we have,
\begin{align*}
\Pr\Big(d_\mathrm{H}(\bfx_1,\bfx_2)\ge n(1/2+\epsilon)\Big) \le \frac{n/4}{n^2\epsilon^2}
& = \frac{1}{4n\epsilon^2}.
\end{align*}
\end{IEEEproof}
The implication of the above result is following.
For any code $\cC$ with dual distance greater than $2$, there exists a strongly-$p$-limited adversary strategy such
that, probability of error is at least $\frac12-\frac1{|\cC|}$ for all $p\ge \frac14$. The proof follows along the
lines of Thm.~\ref{thm:weak}.
However, this does not mean that the capacity of strongly-$p$-limited adversary becomes
$0$ for $p >\frac14$. There may exist a code with dual distance less than or equal to $2$ that
can reliably transfer information at a nonzero rate for $p >\frac14$. On the other hand, if the dual distance
is that small, then the code must have a skewed or asymmetric distance distribution. In the next section, we
will (formally) see that this fact forces the capacity of the strongly limited adversary to be
strictly below that of binary-symmetric channel\footnote{It is known that the distribution of symbols (and even higher order strings) in the codebook needs to be {\em close} to
the mutual information maximizing input distribution, such as uniform in BSC, for the code to achieve capacity
(see \cite{shamai1997empirical}). However, distance distribution is different than input distribution; and we also want to quantify the
gap to capacity.}.
\section{Strongly-limited adversary}\label{sec:strong}
The main result of the paper concerns the capacity of strongly limited
adversary and is given in the following theorem.
\begin{theorem}\label{thm:strong}
\begin{equation}
C_s(p)\le \begin{cases}
1-h_\mathrm{B}(p), \quad p \le \frac14\\
h_\mathrm{B}(1-3p+4p^2) -h_\mathrm{B}(p), \quad \frac14 < p \le \frac12.
\end{cases}
\end{equation}
\end{theorem}
To show this, we need to show the existence of an apt adversarial strategy.
\subsection{The adversary strategy}
The adversary uses the following strategy.
\begin{itemize}
\item $p \le \frac14$. The adversary just randomly and independently flips every bit with probability
$p$.
\item $p > \frac14$. For the used code $\cC$, the adversary calculates $L_\cC(p,n) = \sum_{w > 2pn} A_w,$
where $A_w$ is the distance distribution of the code. The following two cases may occur.
\begin{enumerate}
\item $\frac{L_\cC(p,n)}{|\cC|} = o(1).$ This case can be
tested \footnote{Indeed, whenever we talk about a code, we mean a code-family, that is indexed by $n$, the length. In this case, the adversary knows this code family. There is a way to
bypass the $o(\cdot)$ notation, that we omit here for clarity.}
if for any absolute constant $\epsilon$, $\frac{L_\cC(p,n)}{|\cC|} <\epsilon$
for sufficiently large $n$.
In this case, the adversary first choses
a codeword $\bfx = (x_1,x_2,\dots, x_n) \in \ff_2^n$ randomly and uniformly from $\cC$.
Now if $\bfc =(c_1,c_2,\dots,c_n)$ is the transmitted codeword, then, errors are introduced in the following way
\[
e_i\equiv f_{\cC}^i(x_i) =
\begin{cases}
0,& \text{ when } x_i = c_i\\
1, & \text{ with Prob. } \frac12 \text{ when } x_i \ne c_i\\
0, & \text{ with Prob. } \frac12 \text{ when } x_i \ne c_i.
\end{cases}
\]
Let, $\bfe = (e_1,e_2,\dots,e_n)$. The received codeword is $\bfc+\bfe$.
\bigskip
\item $\frac{L_\cC(p,n)}{|\cC|} \ge c$ for some absolute constant $c$ for all $n$. In this case,
the adversary just randomly and independently flips every bit with probability
$p$.
\end{enumerate}
\end{itemize}
\subsection{Proof of Thm.~\ref{thm:strong}}
The following lemma will be useful in proving the theorem.
\begin{lemma}[Capacity of constrained input]\label{lem:const}
Let $R^{\ast}(p,\omega)$ denote the supremum of all achievable rates for a code (of length $n$) as $n \to \infty$ such
that:
\begin{enumerate}
\item Each codeword has Hamming weight at most $\omega n$, $\omega \le \frac12$.
\item The average probability of error of using this code over BSC($p$) goes to $0$ as $n \to \infty$.
\end{enumerate}
Then $$
R^{\ast}(p,\omega) = h_\mathrm{B}(\omega\Asterisk p) -h_\mathrm{B}(p), $$
where $\omega \Asterisk p = (1-\omega)p + \omega(1-p)$.
\end{lemma}
\begin{IEEEproof}[Sketch of proof]
To prove this lemma, we calculate the mutual information between the input and output of the BSC($p$),
when the input is i.i.d. Bernoulli($\omega$) random variables. It is not difficult to show that, such random code
must contain almost as large a subset with weight of all codewords less than or equal to $\omega n$. The converse follows from
an application of
Fano's inequality and noting that, asymptotically, $\log\binom{n}{\lambda n} \approx n h_\mathrm{B}(\lambda)$.
\end{IEEEproof}
\bigskip
\begin{IEEEproof}[Proof of Thm. \ref{thm:strong}]
If $p\le \frac14$ then the adversary just simulates the binary symmetric channel.
Below we consider the situation when $p > \frac14$.
In what follows, we treat the two different scenarios for the adversary, based on the adversary strategy sketched above.
Let $\cC$ is the code that is used for transmission and $\{A_w\}$ is the
distance distribution of the code, as usual.
\bigskip
\noindent{\em Case 1:}
Let, $\bfx$ is the codeword adversary has initially chosen. Note that, if $\bfc$ is randomly and uniformly chosen from $\cC$, then,
the random variable $W= d_\mathrm{H}(\bfc,\bfx)$ is distributed according to $\{A_w/|\cC|, w =0, \dots n\}$.
\bigskip
Hence,
\begin{align*}
\Pr\Big(W > 2pn \Big) = o(1).
\end{align*}
Using Chernoff bound,
$$
\Pr\Big(\wt(\bfe)\ge n (p+\epsilon) \Big| d_\mathrm{H}(\bfc,\bfx) \le 2pn \Big) \le e^{-2 n\epsilon^2}.
$$
Hence, for any $\epsilon >0$,
$$
\Pr\Big(\wt(\bfe)< n (p+\epsilon) \Big) > 1- o(1),
$$
which imply that the adversary is strongly-$p$-limited.
Now, just following the arguments of Thm. \ref{thm:weak} we conclude that the code $\cC$ will
result in a probability of error at least $\frac12 -\frac1M$ with this adversary.
Therefore,
If $C_s(p) >0$, then the next case must be satisfied for a code.
\bigskip
\noindent{\em Case 2:}
In this case, there exists absolute constant $0<c<1$ such that,
\begin{equation}
\sum_{w>2p n} A_w \ge c|\cC|.
\end{equation}
For any codeword $\bfx \in \cC$, let $A_w^\bfx, w=0,\dots, n$ be the {\em local weight distribution}, i.e.,
the number of codewords that are at distance $w$ from $\bfx$.
Now as,
$$
\sum_{w>2p n} A_w = \frac{1}{|\cC|} \sum_{\bfx \in \cC} \Big(\sum_{w>2p n} A_w^\bfx \Big),
$$
it is clear that there must exist a codeword $\bfx$ such that
$$
\sum_{w>2p n} A_w^\bfx \ge c|\cC|.
$$
This ensures that, there are
at least $c|\cC|$ codewords that belong within a Hamming ball of radius $n- 2pn = n(1-2p)$.
In particular, consider the ball of radius $n- 2pn$ centered at $\bar{\bfx}$, where $\bar{\bfx}$ is the
complement of $\bfc$ (all zeros are changed to ones, and vice versa). All the codewords of $\cC$ that are
distance more than $2pn$ away from $\bfx$ must belong to this ball; let us call the set of such codewords $\cB \subset \cC$.
Clearly $|\cB| \ge c |\cC|$.
Consider the average probability of error, when $\cB$ is used to transmit a message over a BSC($p$).
Because, the Hamming space is translation invariant, the probability of error of such code is equal to
the probability of error of a code $\hat{\cB}$ that have the Hamming weight of each codeword bounded
by $n(1-2p)$.
But from Lemma \ref{lem:const}, the maximum possible rate for which the probability of error of using $\cB$ in BSC($p$) goes to
$0$ is $R^{\ast}(p, 1-2p)$.
However, if we randomly pick up a codeword from $\cC$, with probability at least $c>0,$ the codeword belong to $\cB$.
Hence $\frac1n \log |\cB|$ must be less than $R^{\ast}(p, 1-2p)$, otherwise the average probability of error
for $\cC$ will be bounded away from $0$. Hence, the rate of $\cC$ is at most
$$
R^{\ast}(p, 1-2p) = h_\mathrm{B}(1-3p+4p^2) -h_\mathrm{B}(p).
$$
\end{IEEEproof}
The capacity of strongly-limited adversary is strictly bounded away from the capacity of BSC.
Indeed, $h_\mathrm{B}(1-3p+4p^2) <1$ for all $\frac14 < p < \frac12$. This is shown in Figure~\ref{fig:upper}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\textwidth]{capacity.eps}
\end{center}
\caption{The upper bound of Thm.~\ref{thm:strong} on the strongly-limited adversary.}
\label{fig:upper}
\end{figure}
\subsection{Erasure Channel}
The entire analysis of the above section can be extended for the case of a memoryless adversarial erasure channel,
where instead of corrupting a symbol, the adversary introduces an erasure.
Recently, an extension (that results in rather nontrivial observations) of the results of \cite{dey2013upper,haviv2011beating}
for the case of erasures have been performed in \cite{bassilycausal}.
We refrain from formally defining a binary-input memoryless adversarial erasure channel; however, that can be done
easily along the lines of the introductory discussions of this paper. For the case of weakly-p-limited adversary the capacity is zero
for all $p \ge \frac12$. On the other hand, we note that, for strongly-p-limited adversarial erasure
channel the capacity is upper bounded by $$(1-p)h_\mathrm{B}(p),$$ for all $p \ge \frac12$. The analysis is similar to
that of this section, and uses the capacity of a constrained input erasure channel as a component of the proof (for example, see Eq.~7.15 of
\cite{cover2012elements}).
\section{A code with skewed distance distribution}\label{sec:skew}
In conclusion we outline a possible route through which an improvement on
the upper bound on $C_s(p)$ might be possible.
From the proof of Thm.~\ref{thm:strong} it is evident that a code $\cC$ that
has nonzero rate can achieve a zero probability of error for the
strongly-$p$-limited adversary only if the distance distribution $\{A_w, w=0, \dots, n\}$ satisfies, for some absolute constant $c>0$,
\begin{equation}\label{eq:skew}
\sum_{w>2p n} A_w \ge c|\cC|.
\end{equation}
From, Delsarte's theory of linear-programming bounds \cite{delsarte1973algebraic}, it is possible
to upper bound the maximum possible size of such code $\cC$. Indeed,
this is given in the following theorem .
\bigskip
\begin{theorem}
Suppose, a code $\cC$ is such that its distance distribution $\{A_w, w=0, \dots, n\}$ satisfies \eqref{eq:skew}
for some $c>0$. Assume there exist a polynomial $f(x)$ of degree at most $n$ with,
\begin{equation}
f(x) = \sum_{k=0}^{n}f_k K_k(x),
\end{equation}
and some $\beta>0$, that satisfy,
\begin{enumerate}
\item $f_0 =1$, $f_k \ge 0$ for $k =1\dots,n$;
\item $f(j) \le c \beta$ for $j = 1,\dots, 2pn$ and
$f(j) \le -(1-c)\beta $ for $j = 2pn+1, \dots, n$.
\end{enumerate}
Then $$ |\cC| \le f(0) -c\beta.$$
\label{thm:skew}
\end{theorem}
\begin{IEEEproof}
We note that, $A_i^\bot \ge 0$ for all $i =0, \dots, n$, a set of linear constraints on the
distance distribution whose sum we want to maximize. Moreover we have the extra linear constraint of
\eqref{eq:skew}. We omit the proof here, but if follows from standard arguments of linear programming bounds for codes.
\end{IEEEproof}
If one could find a polynomial that satisfies the above conditions then
that gives bounds on the capacity of strongly-$p$-limited adversary. Our current approach involves tweaking the
existing polynomials that bound error-correcting codes (i.e., the MRRW polynomials \cite{mceliece1977new})
to construct a polynomial that satisfy the criteria of Thm.~\ref{thm:skew}.
\bibliographystyle{abbrv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 31
|
The new value to set.
InvalidOperationException The DocumentID property is read-only when assigned a value.
You cannot set the DocumentID property if it already has a value.
Inheriting classes uses this method when assigning the identifier of a newly created document to sync the DocumentID property before reloading the created document's properties.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,288
|
August 2017 Newsletter: World Hepatitis Day: A Capitol Celebration Recap!
On July 26th to 28th, Hep B United held our 5th Annual Summit and Advocacy Day in Washington, DC, celebrating five years as a national coalition dedicated to reducing the health disparities associated with hepatitis B. The summit is the largest convening of hepatitis B community leaders from around the country with over 80 partners representing patient advocates, local and state health agencies, national organizations, community-based hepatitis B coalitions, and federal partners. Meeting sessions focused on capacity building strategies and sustaining local hepatitis B coalitions, grant writing, partnering with local and state health agencies, collaborating on data collection, increasing public awareness about hepatitis B through storytelling, and the CDC Know Hepatitis B campaign. This year, we also introduced the Hepatitis B Foundation's " #justB: Real People Sharing Their Stories of Hepatitis B " storytelling campaign at a Congressional reception that featured an exhibit of the 18 stories.
During the Hep B United Advocacy Day, we met with over 35 Congressional representatives to discuss increasing federal resources to address and eliminate hepatitis B. We thank our Congressional champions Senator Mazie K. Hirono, Rep. Grace Meng, Rep. Judy Chu, Rep. Brian Fitzpatrick, and Rep. Charlie Dent for their ongoing support and celebrating World Hepatitis Day with us and OLIVER!
You can re-live the memories through our Storify and on our Facebook album!
During the summit, Hep B United and our CDC partners presented five community leaders with the 2017 Hep B Champion Awards in recognition of their collaborative and successful initiatives to address hepatitis B in their communities:
Cathy Phan, the Health Initiatives Project Manager at HOPE Clinic
Vivian Huang, MD, MPH, the Director of Adult Immunization and Emergency Preparedness for the New York City Department of Health and Mental Hygiene
Hong Liu, PhD, the Executive Director of the Midwest Asian Health Association
Dan-Tam Phan-Hoang, MSc, program manager of HBI-Minnesota
The National Task Force on Hepatitis B for AAPI
#justB August featured story: Kenson - #justB There for Others
The latest #justB story features Kenson!
Kenson learned he had hepatitis B when he was living at home in the Marshall Islands, where treatment was unavailable. He and his wife moved to Hawaii to get treatment. After a successful liver transplant, recovery was challenging, but that did not stop Kenson and his wife from educating others about hepatitis B, challenging myths, and promoting testing and care.
The #justB Storytelling Campaign from the Hepatitis B Foundation highlights the personal stories of people affected by hepatitis B to raise awareness. For more information, please visit the campaign website.
Doctor Knows
Doctor Knows informs us on how Hepatitis B is spread, who should be tested for hep B, and the importance of Hep B testing. The full infographic can be found here. This poster is also available in Traditional Chinese, Vietnamese, and Korean.
Hep B United Coalition Partners Across the Country
Hep B United Partners
Community Partners : Asian American Health Coalition (HOPE Clinic). Asian American Health Initiative. Asian Health Coalition. Asian Pacific Health Foundation. Asian Pacific Community in Action. Asian Services in Action Inc. Center for Pan Asian Community Services (CPACS). Charles B Wang Community Health Center. Chinese American Medical Society - Greater Boston Chapter. CCACC Pan Asian Volunteer Health Clinic. Dallas Fort Worth Hepatitis B Free Project. Hep B Free Las Vegas. Hep B Free Los Angeles. Hep B Project. Hep B United Philadelphia. Hep B United Twin Cities. Hep Free Hawaii. Hepatitis B Coalition of WA. Hepatitis B Initiative of Washington, DC. Hepatitis Education Project. Midwest Asian Health Association. New Jersey Hepatitis B Coalition. North East Medical Services. NYC Hepatitis B Coalition. NYU Center for the Study of Asian American Health (CSAAH). Ohio Asian American Health Coalition. Project Prevention. SF Hep B Free.
National Non-Profit Partners : Hepatitis B Foundation. Association of Asian Pacific Community Health Organizations (AAPCHO). Asian American Network for Cancer Awareness, Research and Training (AANCART). Asian Pacific American Medical Student Association (APAMSA). National Task Force on Hepatitis B. National Viral Hepatitis Roundtable. Asian & Pacific Islander American Health Forum. Immunization Action Coalition.
Federal Partners : US Department of Health and Human Services (HHS). Centers for Disease Control and Prevention, Division of Viral Hepatitis, HHS. HHS Office of Minority Health. White House Initiative on Asian Americans and Pacific Islanders.
Copyright © 2017 Hep B United, All rights reserved.
You are receiving email because you are a partner of the Hep B United coalition or you have opted in for e-newsletter from us.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,447
|
Home / Birmingham Senior Cup / Birmingham Senior Cup Final Details
Birmingham Senior Cup Final Details
Thu Apr 25th at 3:53 pm
The Birmingham County FA have today announced the pricing for the Birmingham Senior Cup Final against Sutton Coldfield Town. The match, to be played at Walsall's Banks's Stadium on Tuesday 30th April, will be priced at £10 for adults & £5 for concessions.
Note: Please remember that the game has an earlier than usual 7.30pm kick off time.
Hednesford Town 5-1 Rushall Olympic Match Preview: Stocksbridge P.S (H)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,001
|
{"url":"https:\/\/brilliant.org\/problems\/lost-in-space\/","text":"# Lost in space\n\nGeometry Level 3\n\nA point $P$ in $\\mathbb{R}^{3}$ has distances of $3,7,8,9$ and $11$ from five of the vertices of a regular octahedron. If the distance from the sixth vertex of the octahedron to $P$ is $\\sqrt{a}$, where $a$ is a positive, square-free integer, then find $a$.\n\n\u00d7","date":"2019-08-20 02:21:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 8, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6165729761123657, \"perplexity\": 323.2862179785384}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027315174.57\/warc\/CC-MAIN-20190820003509-20190820025509-00289.warc.gz\"}"}
| null | null |
Redemption or punishment?
Rina Jimenez-David
Philippine Daily Inquirer
One of the main topics in any Ethics of Journalism class is the right of minors to privacy and confidentiality. Reporters and editors are cautioned against identifying youthful offenders as well as young victims, either by naming them, using their photos or video footage, or providing enough details (such as home addresses, schools or the names of their parents) that would make it possible for anyone to identify the child.
The reason behind this is that a child—or anyone below the legal age of 18—is still young enough to rebuild his or her life, and with guidance and assistance could start anew and move forward without the baggage of an errant youth or tragic past.
Implicit in this is the belief that a person's life history, personality and behavior are not cast in stone. Rather, it is continually evolving, especially in the case of a child who commits a crime or becomes a victim of a crime, who has enough time and is sufficiently malleable to be "rehabilitated," "re-educated" or transformed into a law-abiding, productive and tax-paying citizen.
At least, that is the belief.
But that principle or belief in the possibility of human redemption or reform is increasingly being challenged these days. It isn't just newspaper columnists or radio commentators who complain about youthful offenders, but also local authorities, national officials and even legislators, who now want the amendment or outright repeal of the Juvenile Justice and Welfare Act of 2006, principally authored (and shepherded and championed) in the Senate by Sen. Francis Pangilinan.
For the most part, much of the evidence or arguments against the law have been anecdotal, citing break-ins where the participation of a child is betrayed by a small footprint or handprint. Some report that adult criminals have taken advantage of the law and now recruit and use underage cohorts to carry out their dirty deeds, including robbery and acting as drug mules, believing that, if caught, the young criminals would go scot-free or get mere slaps on the wrist.
Earlier this month, officials in Cebu called for the amendment of the Juvenile Justice Law after a pregnant woman and her five-year-old daughter were killed by the woman's 16-year-old son. Several provincial officials weighed in, one of them saying that teenagers are already capable of committing crimes (nobody said they weren't) and should be punished.
Last January, Interior Secretary Jesse Robredo likewise called for the law's amendment. He cited "intelligence reports" from the PNP which said that "the youth are getting bolder and braver in committing offenses such as theft and robbery because of their knowledge that they will eventually be set free or turned over to the Department of Social Welfare and Development." Foremost among the suggested amendments (cited in an editorial in this paper) is the lowering of the age of criminal responsibility from 15 years to nine years old. Another senator, the youthful Chiz Escudero, has called for the "suspension" of the implementation of the law, citing the perceived upsurge in the number of crimes committed by minors.
However, Pangilinan says the "main problem" with the Law (RA 9344) is that "even its key stakeholders do not understand the law, and therefore are not able to implement it properly."
Contrary to common belief, Pangilinan says that the law does not "exempt" children or youths from taking responsibility for their actions. What it does, though, is ensure that "youth offenders (are not) judged or incarcerated as adults."
In 2005, according to a study by Unicef (one of a broad coalition of child-welfare groups that worked for the passage of the law), some 4,000 Filipino children were in prison. The great majority of them were in jail for minor crimes, most notably theft or burglary. I remember Pangilinan telling me at the time about two boys he met in a local prison who had spent about a year behind bars for stealing used wires, whose value totaled about P400.
Before the law's passage, many child offenders (even before they were brought to trial) were mixed in with adults. Jails served as virtual "universities" for crime, where children learned criminal behavior and were corrupted by their initiation into the criminal underworld. Some children were deliberately kept in the dark about their rights, oftentimes by police or jail wardens who used their young "wards" as gofers, indentured servants or worse, sex toys.
When critics say that youth offenders can go "scot-free" without going through the criminal justice system, they are probably referring to the policy of "diversion" enshrined in the law. Youthful offenders (depending on the crime they are charged with) are "diverted" from the courts and are instead subject to different interventions, from confinement in a juvenile facility, turnover to the DSWD for supervision, and even to community-based reconciliation processes.
One of the main principles in the Juvenile Justice Law is "restorative justice," which has been defined as a way of "resolving conflicts with the maximum involvement of the victim, the offender and the community." It offers "reparation for the victim, reconciliation of the offender, the offended and the community, with a reassurance to the offender that he or she could be reintegrated with the community." In that way is the community's sense of safety "restored," with potential criminals undergoing rehabilitation.
The emphasis, it should be noted, is on reparation and reconciliation, and is entirely in line with our social and philosophical belief in the possibility of renewal, and not just on punitive incarceration that destroys all hope for the future.
View original post on Inquirer.net
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,771
|
You've seen the list of Beer Week festivities—the lineup is awesome—but let's focus on this weekend's fun. Snag a ticket to these Traverse City Beer Week events and enjoy every moment (and sip).
Enjoy complimentary samples of Workshop Brewing Company's newest barrel-aged releases: Friday Shaker, Barrel Aged Spirit Level, Barrel Full O' Monkeys and Barrel Aged Reaper.
German beers and German food at 7 Monks Taproom from 5–10 p.m.
Free tastings of Right Brain brews (and swag) up front at Teetotallers. Like what you taste? Head back to the speakeasy for complimentary snacks and 50 percent off your beers.
Registration includes a one-hour vinyasa flow class plus a pint of beer at The Workshop Brewing Company. Yoga instruction provided by Mary Macey of Yen Yoga and Fitness with DJ Ras Marco D providing the tunes. Bring your own mat. $15 in advance; $20 at the door.
Don't have a DD? No problem. Check out this sweet designated driver service instead of getting behind the wheel. You can thank me later.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,448
|
Q: Firebase blank html page when deployed , but works on local host I am currently developing a firbase app. I am having an issue where whenever I run my firbaseapp on local machine it works perfectly but when deployed to firebase server I get a blank screen. Any thoughts on how to solve this problem?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,522
|
Canon EOS RT är en systemkamera för film som tillverkades mellan åren 1989 och 1992. Till skillnad från andra systemkameror för film så fälls spegeln inte upp när man tar en bild. Spegeln är halvtransparent så därför kan man ha en slutarfördröjning så kort som 8 millisekunder. Nackdelarna med detta spegelsystem innebär att man förlorar ett halvt bländarsteg och att sökaren måste tillverkas av så att den är extremt ljusstark så att det lilla ljus som kommer upp mattskivan ger en ljus sökarbild, detta gör kameran dyrare än kameror som är tillverkade på vanligt sätt.
Källor
Canon EOS-kameror
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,480
|
{"url":"http:\/\/www.texdev.net\/category\/latex\/siunitx\/","text":"# Some TeX Developments\n\nCoding in the TeX world\n\n## siunitx: Planning for v2.6\n\nI try to keep my `siunitx` package ticking over, even though it\u2019s now pretty feature-rich. There are a couple of issues currently on the list for v2.6, and I\u2019m planning to tackle those over the Christmas holidays. So now is a good time to log any feature requests you have for `siunitx`.\n\nWritten by Joseph Wright\n\nOctober 27th, 2012 at 6:37 pm\n\nPosted in LaTeX,siunitx\n\n## siunitx: v2.5 and beyond\n\nAnyone who watches the BitBucket site for siunitx development will have noticed that I\u2019ve been adding a few new features. As I\u2019ve done for every release in the 2.x series, new options means a new minor revision, and so these will all be in v2.5. I\u2019ve also revised some of the behaviour concerning math mode, so there are now very few options which automatically assume math mode.\n\nLooking beyond v2.5, I have some bigger changes I\u2019d like to make to siunitx. When I initially developed the package, it was very much a mixture of things that seemed like a good idea. The work for version 2 meant a lot of changes, and a lot more order. However, I\u2019ve learnt more about units, LaTeX and programming since then, and that means that there are more changes to think about.\n\nThe internal structure is quite good, but I need to work on some parts of the code again. For users, of course, that won\u2019t show up, but it is important to me. It\u2019s also not so straight-forward: the `.dtx` is about 17 000 lines long! However, there are also some issues at the user level. In particular, I think I\u2019ve offered too many options in some areas, for example font selection. Revising those will alter behaviour, but it will also improve performance and the clarity of some edge cases. However, that is not such easy work and will take a while. I\u2019ve got lots of other TeX commitments (plus of course a life beyond LaTeX), so these changes will wait a while yet. So once v2.5 is finalised I\u2019d expect to have little change in siunitx for some time: probably until at least the autumn, and quite possibly the end of the year.\n\nWritten by Joseph Wright\n\nApril 8th, 2012 at 6:15 pm\n\nPosted in LaTeX,siunitx\n\n## siunitx v2.4 beta\n\nDevelopment of the next release of siunitx has gone quite smoothly: I\u2019ve added a few new features, and there is now nothing outstanding for v2.4. So it is time to ask for some volunteers to test the code.\n\nIn terms of new features, I have added the a choice of rounding modes modes the ability to compress down exponents in ranges and lists, both long-standing feature requests. In response to a recent TeX.sx question, siunitx can now also turn exponents into unit prefixes. At a lower level, I\u2019ve also altered some of the options internally so fewer of the assume math mode.\n\nTo test, please download the ready to install TDS-style .zip file and install it locally. You should then be good to go. Feedback as a bug report or by e-mail welcome, as always. Assuming there are no problems, I\u2019d expect to upload to CTAN by the end of the month.\n\nWritten by Joseph Wright\n\nNovember 9th, 2011 at 6:49 pm\n\nPosted in siunitx\n\n## Which siunitx options to set globally?\n\nOn the TeX.SX site recently, there was some discussion about locally over-riding the `round-mode = places` setting in my siunitx package. One thing this highlights for me is the need to think about which settings to apply globally.\n\nSome siunitx settings are about consistency of appearance, and seem to apply naturally to entire documents. A classic example would be `output-decimal-marker`: if you are using `,` as a decimal marker, it should apply everywhere!\n\nHowever, this is not so clear-cut for many of the options related to number-manipulation. The rounding options in particular are really intended for the case where you have some auto-generated data (say a long list from an instrument), and the real accuracy is not as great as the apparent precision. Instruments are great at providing lots of numbers, but it takes a bit of human thought to decide how many of these are really relevant. So for these cases, setting an appropriate rounding scheme is perfectly sensible.\n\nOn the other hand, for a number you\u2019ve typed in yourself I\u2019d hope that you\u2019ve done the thinking part when the number is typed, so rounding by the computer is not needed. That suggests to me that most of the time rounding should not be set as a global option.\n\nOf course, it will depend on the exact nature of the document in question. If all of the data in a document is in tables, all of which need rounding, then there is a performance gain from setting the rounding once globally. So the best I can say, guidance-wise, is \u2018think about your document\u2019!\n\nWritten by Joseph Wright\n\nJuly 17th, 2011 at 4:08 pm\n\nPosted in LaTeX,siunitx\n\n## siunitx v2.3: consolidation\n\nI\u2019m making a start on the next release of siunitx: v2.3. There are a number of issues in the database targeting this release, and these are mainly about dealing with things behind the scenes. Some options need revision, and I need to improve the table code somewhat. However, I doubt that there will be much to excite users. That\u2019s not necessarily a bad thing: there seem to be a lot of siunitx users, and I don\u2019t want to break the code! Of course, if there is a particular issue that needs addressing then the usual rule applies: make a case to me and I\u2019ll see what I can do.\n\nWritten by Joseph Wright\n\nMay 30th, 2011 at 6:36 pm\n\nPosted in siunitx\n\n## siunitx v2.2 released\n\nAs I detailed a little while ago, I\u2019ve been working on v2.2 of siunitx. I\u2019ve now released the latest version, v2.2, to CTAN. There are a number of small changes, introducing new features, but I thought I would highlight a few.\n\nA long-standing feature request has been to be able to use the cancel package to show how units cancel out. This is useful for teaching, although it\u2019s not of course part of the usual typesetting of units for publication. It turns out not to be too hard to allow this, so that you can now use input such as\n\n`\\si[per-mode = fraction]{\\cancel\\kg\\m\\per\\s\\cancel\\kg}`\n\nand have it come out properly. At the same time, I\u2019ve made it possible to highlight particular units\n\n`\\si{\\highlight{green}\\square\\metre\\candela\\second}`\n\nagain for teaching-related purposes.\n\nA second long-standing request is to be able to parse uncertainties given in the form\n\n`\\num{1.23 +- 0.15}`\n\nwhich was something more of challenge, but again is now working properly. So you can get the same output from the above and from\n\n`\\num{1.23(15)}.`\n\nA final highlight is the new `\\tablenum` macro. This is needed for aligning numbers inside `\\multicolumn` and `\\multirow`, which otherwise does not work. (At a technical level, both `\\multicolumn` and `\\mutirow` use the `\\omit` primitive, and so the code inserted by the `S` column is not used. The `\\tablenum` macro effectively makes the same approach available as a stand-alone function.)\n\nWritten by Joseph Wright\n\nApril 14th, 2011 at 8:24 am\n\nPosted in LaTeX,siunitx\n\nTagged with\n\n## Sorting issues for consideration for siunitx v2.2\n\nI\u2019ve been leaving `siunitx` alone for a while, concentrating on bug fixes in the v2.1 branch. The\u00a0 list of issues has continued to grow, and I\u2019m now getting some organisation done before starting on items for v2.2. Some of these are more likely to get tackled, some rather less likely, but it\u2019s best to get everything logged! If you look at the closed issues, you\u2019ll see that part of getting organised is closing some bugs where I don\u2019t feel that it is appropriate to take action. What I would say is that if you\u2019ve got something you\u2019d like considering, put it into the database. I do try to keep up with ideas by e-mail or from forums, but do forget some.\n\nOn particular thing I want to think about is the naming is some options and macros. I\u2019ve had some discussion with Marcus Foster from CSIRO Information Management & Technology about `siunitx`, and he\u2019s pointed out various errors on my part. Some of those, in the documentation, have been fixed. At the code level, he pointed out that what `\\SI` prints are properly called quantities, and that units are separated by products. So I\u2019m thinking of some reasonably radical renaming of macros and options (with the old ones retained, of course!). Feedback on these ideas would be welcome. At the same time, he\u2019s not at all keen on the \u2018qualifiers\u2019 concept, but on that I think users would not be happy if I removed it!\n\nWritten by Joseph Wright\n\nMarch 20th, 2011 at 10:57 am\n\nPosted in LaTeX,siunitx\n\n## Installing achemso and siunitx\n\nA question that comes up from time to time is how to install one or other of my packages, usually either achemso or siunitx. While both are essentially standard LaTeX packages (no weird files or binaries needed), there are still soem stumbling blocks that cause issues. So I thought a few notes by be useful here.\n\n## Installing as part of an up to date TeX system\n\nBy far the easiest way to install my LaTeX packages is to get them as part an up to date TeX system. Both MikTeX 2.9 and TeX Live 2010 include all of my general packages. MiKTeX is of course Windows-only, but TeX Live can be installed on Windows, Mac OS X and Linux. After installation, doing an on-line update should grab all of the latest packages from CTAN. Both MiKTeX and TeX Live include graphical update programs, so this is not such a difficult process nowadays.\n\nMac users may well prefer MacTeX over plain TeX Live, but MacTeX is built on top of TeX Live and so the same ideas apply. You can install either TeX Live or MacTeX and get the same basic functionality.\n\nFor Linux users, it\u2019s worth noting that popular Linux distributions tend to include old versions of TeX Live (or even teTeX), rather than TeX Live 2010. So if you want an up-to-date TeX system you\u2019ll be better off ignoring your Linux package manager and grabbing TeX Live directly.\n\nOne thing to do if you update your TeX system is to check any locally-installed files you might have (see the next section for more about local installation). These will be in `~\/texmf` on Linux, `~\/Library\/texmf` on a Mac and (probably) `%USERPROFILE%\\texmf` on Windows. One problem I see from time to time is that users of achemso have installed some of the BibTeX styles locally, then update the main package and all sorts of things go wrong. So do check carefully on any local files: they might be outdated by a new TeX system.\n\n## Installing using the TDS zip files\n\nThe method above is fine if you are happy installing an entirely new TeX system, but if all you need is access to one of my packages then it is probably over-kill. For these users, I provide ready-to-install zip files on CTAN. For achemso, you need achemso.tds.zip, while for siunitx users you probably need\n\nThe idea with these files is that I have set them up with documentation, ready to use LaTeX styles and all of the support files. All that needs to happen with them is to unzip them inside your local TeX directory and tell TeX about them.\n\nWhere the files should go depends a little on your operating system. The local directory (folder) is usually `~\/texmf` on Linux, `~\/Library\/texmf` on a Mac and (probably) `%USERPROFILE%\\texmf` on Windows. Here, `~` and `%USERPROFILE%` represent your home directory (folder). So on my Windows 7 PC, I have a folder\n\n`C:\\Users\\joseph\\texmf`\n\nwhile on my Mac there is one at\n\n`\/Users\/joseph\/Library\/texmf`\n\nWhichever system you use, copy the appropriate zip files there and unzip. The result should be a structure which looks like\n\n```texmf\/tex\/latex\/achemso\/achemso.sty\n...\ntexmf\/tex\/latex\/siunitx\/siunitx.sty```\n\nand so on. Of course, the exact structure will depend on which packages you install! What is important for installing siunitx is to also install expl3 and xpackages. If the versions do not match then trouble will not be far away.\n\nTo tell TeX about the new files, you need to run the program `texhash`. There is a graphical interface for this in both MiKTeX (Update File Name Database) and TeX Live. I find it easiest just to start a Command Prompt\/Terminal and type\n\n`texhash`\n\n[For users with recent versions of TeX Live (2009 and 2010, I think), running `texhash` is actually not needed. However, it will not do any harm so you may as well run it.)\n\n## Installing from the dtx file\n\nThe traditional method to install a package is to unpack it from the dtx source. I\u2019ve got to say that I only recommend this for experienced LaTeX users. While both achemso and siunitx are designed to be easy to unpack, life is more complex for expl3 and xpackages. So I\u2019d strongly recommed using the TDS zip files unless you know a bit more about LaTeX!\n\nWritten by Joseph Wright\n\nDecember 27th, 2010 at 9:36 pm\n\nPosted in achemso,LaTeX,siunitx\n\nTagged with , ,\n\n## siunitx: testing for v2.1 release\n\nAnyone who follows the BitBucket repository for siunitx will have seen that I\u2019ve made a lot of commits over recent days. If you look at the list of open issues, you\u2019ll also see that it\u2019s got smaller and that some have been moved from targeting v2.1 to targeting v2.2. I\u2019ve been aiming to get v2.1 finished this month, and I\u2019ve been working out what I can and can\u2019t do in that time frame.\n\nThe result is that the code on BitBucket now is what I\u2019ll release as v2.1, baring any remaining bug fixes. The idea is to have some new features, but not so many that I\u2019ll have introduced lots of new bugs. I hope that I\u2019ve got the balance about right, and that there has been enough time for testing the new additions to the package. If all looks okay by the weekend I will be updating CTAN at the weekend. If you\u2019d like to test it before then, either grab the code from BitBucket or, if you use TeX Live, try the TLcontrib installation method.\n\nI\u2019m hoping to work on v2.2 for a release early in the new year, probably late January. There are already a few issues on the list for v2.2, but that may alter if there is a good case made for something else. Of course, I\u2019ll also have to avoid breaking anything!\n\nWritten by Joseph Wright\n\nOctober 27th, 2010 at 6:45 pm\n\nPosted in LaTeX,siunitx\n\nTagged with\n\n## Testing versions of siunitx v2.1 on TLcontrib\n\nI\u2019m working on the list of issues for siunitx v2.1. As I do, I hope that the code is staying usable at all times! The list is getting shorter (finally), so I\u2019m hoping to get something released around the end of the month.\u00a0 One thing that I need for that is testing. My recent post about TLcontrib mentioned this as a route for testing packages prior to release. So I\u2019m taking advantage, and sending snapshots of siunitx to TLcontrib each time I add a new feature. So if you want to help to test things out, then you can run\n\n`tlmgr --repository http:\/\/tlcontrib.metatex.org\/2010 update siunitx`","date":"2013-05-25 02:06:45","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8231856226921082, \"perplexity\": 1469.421506730185}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368705318091\/warc\/CC-MAIN-20130516115518-00073-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
\section{Conclusion and Future Work}
In this paper, we propose a novel qu\textbf{E}stion and answer guided \textbf{D}istractor \textbf{GE}neration(EDGE) framework to automatically generate distractors for multiple choice questions in standard English tests.
In EDGE, we design two modules based on attention and gate strategies to reform the passage and question, which then are combined to decode the distractor.
Experimental results on a large-scale public dataset demonstrate the state-of-the-art performance of EDGE and the effectiveness of two reforming modules.
In future work, we will explore two potential directions.
First, since the beam search ignores the generation diversity, we will explore how to incorporate the prior generated distractor information to guide the generation of successor distractors.
Second, we will work on how to generate the distractors requiring multi-sentence/hop reasoning, which can further improve the plausibility.
\bibliographystyle{coling}
\section{Experiments}
\subsection{Experiment Setup}
\subsubsection{Dataset}
For a fair comparison, we use the distractor generation dataset\footnote{https://github.com/Evan-Gao/Distractor-Generation-RACE} released by \newcite{Gao2018GeneratingDF} as our benchmark.
This dataset is constructed based on RACE~\cite{Lai2017RACELR}, which is collected from the English exams and widely used in the MRC field.
More details about the dataset construction process can be found in \cite{Gao2018GeneratingDF}.
The train/validation/test set contain 96,501/12,089/12,284 examples, respectively.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\linewidth]{count_stats.pdf}
\includegraphics[width=0.45\linewidth]{all_length_dist_type1.pdf}
\caption{The statistics of the evaluation dataset (Left). The word count distribution (Right). The max values are 95th percentile. \label{fig:data_stats}}
\end{figure}
The left sub-figure of Figure~\ref{fig:data_stats} shows the count statistics on this dataset.
We can see that most passages are related to more than two questions.
A question is usually associated with multiple distractors, which proves necessary to conduct the beam search in the testing phase.
The right sub-figure shows the distributions of the lengths of the passages, questions, answers, and distractors.
The distractors and the answers have similar lengths and the questions are slightly longer than them.
Meanwhile, we can see that the median of the distractor lengths is larger than 8.
This suggests the similar word/entity-based methods do not apply to this dataset.
\subsubsection{Model Details}
We use the \textit{GloVe.840B.300d}~\cite{pennington2014glove} as the pre-trained word embeddings (i.e., $d=300$), and the word representations are shared across different components of EDGE.
In the encoding module, we choose the Bi-LSTM as the contextual encoder, the size of the hidden unit is set to 300 (150 for each direction).
Please note that the Bi-LSTM encoder is a plug-in module that can be easily replaced by Transformer~\cite{Vaswani2017AttentionIA}, BERT~\cite{Devlin2019BERTPO} or XLNet~\cite{Yang2019XLNetGA}.
The parameters of the Bi-LSTM are shared among the encoding module and two reforming modules.
According to the 95th percentile values shown in Figure~\ref{fig:data_stats}, we set the maximum lengths of passages, questions, answers, and distractors to be 500, 17, 15, and 15, respectively.
The model is trained with a mini-batch size of 64.
We use Nesterov Accelerated Gradient (NAG) optimizer~\cite{Nesterov1983AMF} with a learning rate of 0.005.
The dropout rate is set to 0.1 to reduce overfitting.
The beam size $n$ is set to 50.
\subsection{Baseline Approaches and Metrics}
The following models are selected as baselines:
\textbf{Basic models:} the basic sequence-to-sequence framework and its variants including (1) \textbf{SS} (Seq2Seq): the basic model that generates a distractor from the passage; (2) \textbf{SEQ} (SS+Enriching Module+Question Initializer): the sequence-to-sequence with the enriching module and the question-initialized decoder in which the initial state is set to the output of the question initializer; and (3) \textbf{SEQA} (SEQ+Attention): the sequence-to-sequence with a decoder same as the distractor generator of the EDGE.
\textbf{HRED (HieRarchical Encoder-Decoder):} the basic framework of \cite{Gao2018GeneratingDF}, which also contains the question initializer and attention mechanism.
\textbf{HSA (HRED+static attention)}~\cite{Gao2018GeneratingDF}: which uses the HRED as the basic architecture and leverages two attention strategies to combine the information of the passage, question, and answer.
\textbf{CHN}~\cite{zhou2019coattention}: which extends HSA with a co-attention mechanism to further strengthen the interaction between the passage and the question. This model achieved state-of-the-art performance previously on this task.
Following the distractor generation work~\cite{Gao2018GeneratingDF} and some question generation works~\cite{Kim2018ImprovingNQ,Chen2019ReinforcementLB}, we use BLEU~\cite{Papineni2001BleuAM} and ROUGE~\cite{Lin2004ROUGEAP} scores as our evaluation metrics.
All hyper-parameters of EDGE and other baselines are selected on the validation set based on the lowest perplexity and the results are reported on the test set.
\subsection{Performance Comparison}
\label{sec:perf}
The experimental results of all models are summarized in Table~\ref{tab:performance}.
Since the dataset in ~\cite{Gao2018GeneratingDF} is slightly different from the public dataset in Github, we not only report the HSA's results from its original paper ~\cite{Gao2018GeneratingDF} but also include the reimplementation results of \cite{Gao2018GeneratingDF} from ~\cite{zhou2019coattention} on the public dataset.
\begin{table*}[h]
\centering
\caption{The performance comparison results. The best results are highlighted bold. HSA* denotes the result reported in ~\cite{Gao2018GeneratingDF} and HSA denotes the result reported in \cite{zhou2019coattention}. \label{tab:performance}}
\resizebox{.8\textwidth}{!}{
\begin{tabular}{l|c|c|c|c|c|c|c}
\toprule[1pt]
1st Distractor & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & ROUGE-1 & ROUGE-2 & ROUGE-L \\
\midrule
SS & 18.76 & 7.84 & 3.82 & 1.91 & 13.46 & 2.73 & 13.60 \\
SEQ & 24.44 & 11.01 & 5.59 & 3.02 & 16.41 & 3.74 & 14.85 \\
SEQA & 28.61 & 15.28 & 9.47 & 6.3 & 18.92 & 5.56 & 15.53 \\
HRED & 27.96 & 14.41 & 9.05 & 6.34 & 14.12 & 3.97 & 14.68 \\
HSA* & 27.32 & 14.69 & 9.29 & 6.47 & 15.69 & 4.42 & 15.12 \\
HSA & 28.18 & 14.57 & 9.19 & 6.43 & 15.74 & 4.02 & 14.89 \\
CHN & 28.65 & 15.15 & 9.77 & 7.01 & 16.22 & 4.34 & 15.39 \\
\textbf{EDGE} & \textbf{33.03} & \textbf{18.12} & \textbf{11.35} & \textbf{7.57} & \textbf{19.63} & \textbf{5.81} & \textbf{19.24} \\
\midrule
\midrule
2nd Distractor & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & ROUGE-1 & ROUGE-2 & ROUGE-L \\
\midrule
SS & 18.19 & 7.15 & 3.24 & 1.51 & 13.12 & 2.35 & 13.33 \\
SEQ & 24.18 & 10.48 & 5.05 & 2.63 & 15.99 & 3.31 & 14.36 \\
SEQA & 28.00 & 14.20 & 8.20 & 5.04 & 18.24 & 4.78 & 14.79 \\
HRED & 27.85 & 13.39 & 7.89 & 5.22 & 15.51 & 3.44 & 14.48 \\
HSA* & 26.56 & 13.14 & 7.58 & 4.85 & 14.72 & 3.52 & 14.15 \\
HSA & 27.85 & 13.41 & 7.87 & 5.17 & 15.35 & 3.40 & 14.41 \\
CHN & 27.29 & 13.57 & 8.19 & 5.51 & 15.82 & 3.76 & 14.85 \\
\textbf{EDGE} & \textbf{32.07} & \textbf{16.75} & \textbf{9.88} & \textbf{6.27} & \textbf{18.53} & \textbf{4.81} & \textbf{18.10} \\
\midrule
\midrule
3rd Distractor & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & ROUGE-1 & ROUGE-2 & ROUGE-L \\
\midrule
SS & 18.60 & 7.38 & 3.33 & 1.56 & 13.33 & 2.40 & 13.41 \\
SEQ & 24.20 & 10.24 & 4.85 & 2.47 & 15.63 & 3.12 & 13.99 \\
SEQA & 27.26 & 13.60 & 7.73 & 4.72 & 17.61 & 4.19 & 15.18 \\
HRED & 26.73 & 12.55 & 7.21 & 4.58 & 15.96 & 3.46 & 14.86 \\
HSA* & 26.92 & 12.88 & 7.12 & 4.32 & 14.97 & 3.41 & 14.36 \\
HSA & 26.93 & 12.62 & 7.25 & 4.59 & 15.80 & 3.35 & 14.72 \\
CHN & 26.64 & 12.67 & 7.42 & 4.88 & 16.14 & 3.44 & 15.08 \\
\textbf{EDGE} & \textbf{31.29} & \textbf{15.94} & \textbf{9.24} & \textbf{5.70} & \textbf{17.83} & \textbf{4.40} & \textbf{17.46} \\
\bottomrule[1pt]
\end{tabular}
}
\end{table*}
There are several observations:
Firstly, the proposed model, EDGE, outperforms all baselines significantly in all metrics and achieves the new state-of-the-art scores on this distractor generation dataset;
Secondly, SEQ and SEQA outperform the basic Seq2Seq which indicates both the question information and the passage information are vital to the distractor generation;
Thirdly, SEQA outperforms HRED which indicates that the co-attention between the question and the passage in the enriching module can help to generate better distractors.
The observation that CHN outperforms HSA also proves the effectiveness of the co-attention mechanism;
Finally, the basic Seq2Seq performs far worse than other models, which indicates that the distractor generation is a challenging task and hard to solve only with simple models.
\subsection{Ablation Analysis}
\begin{table}
\centering
\caption{The ablation study results. We average the BLEU-4 and ROUHE-L over all three generated distractors. Higher scores indicate better performance. \label{tab:ablation}}
\resizebox{.55\columnwidth}{!}{
\begin{tabular}{l|c|c}
\toprule[1pt]
Methods & BLEU-4 & ROUGE-L \\
\midrule
EDGE & 6.51 & 18.27 \\
\midrule
w/o Reforming Passage Module & 5.72 & 17.12 \\
w/o Reforming Question Module & 6.01 & 17.84 \\
w/o Question Initializer & 6.12 & 18.01 \\
w/o Enriching Module & 6.41 & 18.12 \\
w/o Encoding Module & 1.35 & 9.07 \\
\bottomrule[1pt]
\end{tabular}
}
\end{table}
Table~\ref{tab:ablation} shows the experimental results of the ablation study.
We can see that removing the \textit{Reforming Passage Module} or the \textit{Reforming Question Module} leads to the suboptimal results.
This validates the effectiveness of two reforming mechanisms.
Moreover, we find the former module is more important for the overall distractor generation model.
This is probably due to that the reformed passage has a higher impact on the decoding process. Particularly, in each decoding step, the context information from the passage can provide more clues to generate proper distractors than the context information from the question.
We can also observe that the question initializer brings performance gain. This verifies the hypothesis in Section 3.6 that the initial decoder state encoded from the question helps to generate distractors grammatically and semantically consistent with the question.
Removing the encoding or enriching module will also result in a performance drop. This indicates that extracting contextual representations is important for the generation task.
Moreover, the enriching module can further improve the performance by fusing the passage information into the contextual representation of the question.
\subsection{Human Evaluation}
\begin{table}
\centering
\caption{Results of human evaluation. Higher scores indicate better performance. \label{tab:human}}
\resizebox{.5\columnwidth}{!}{
\begin{tabular}{l|c|c|c}
\toprule[1pt]
Methods & Fluency & Coherence & Distracting Ability \\
\midrule
HSA & 7.84 & 5.46 & 3.51 \\
CHN & 8.19 & 5.80 & 4.70 \\
\textbf{EDGE} & \textbf{8.95} & \textbf{7.08} & \textbf{5.50} \\
\bottomrule[1pt]
\end{tabular}
}
\end{table}
We conduct a human evaluation to evaluate the quality of the generated distractors of different models.
We use three metrics designed by ~\newcite{zhou2019coattention} to conduct the evaluation: (1) \textit{Fluency}, which evaluates whether the distractor follows regular English grammar and conforms with human logic and common sense; (2) \textit{Coherence}, which measures whether the key phrases in the distractors are relevant to the passage and the question; (3) \textit{Distracting Ability}, which evaluates how likely a generated distractor will be used by the question composers in real examinations.
We choose the first 100 samples of the test set and the corresponding distractors generated by three models as the input.
We employ five annotators with good English background (at least holding a bachelor's degree) to scores these distractors with three gears (i.e., \textit{Good}, \textit{Fair} or \textit{Bad}) by three metrics, the scores are then projected to 0 - 10.
The results of all models, averaged over all generated distractors, are shown in Table~\ref{tab:human}. We can find that our model performs best in three metrics. This suggests our model is able to generate plausible and useful distractors. This conclusion also aligns with the experimental results of automatic metrics in Section \ref{sec:perf}.
\subsection{Case Study}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{case_single_col.pdf}
\caption{A sample question with the truth distractors and generated distractors. The left sub-figure shows the attention weights in the generator and the gate values in the reforming passage module when decoding the first generated distractor. To enable comparisons among different sentences, we average the attention and gate values of all words in each sentence. Colored sentences are the clues of four options. \label{fig:case}}
\end{figure}
The design of two reforming modules and the generator enables convenient interpretation of the generated distractors.
Take the MCQ in Figure~\ref{fig:case} for example, the \textit{blue} highlighted sentence in the paragraph includes the clue to infer the correct answer. EDGE managed to block this clue by assigning a lower attention weight with the help of the gate layer. In this manner, the clue of the correct answer is prohibited from generating the distractor.
Meanwhile, all the sentences related to three distractors (colored \textit{red} and \textit{orange}) obtain higher gate values, which further help to achieve higher attention weights (highlighted pink in the left part).
Especially, when generating the first distractor, the \textit{red} sentence has the highest attention score, indicating it make the most contributions to the generation.
In summary, the visualization results demonstrate that EDGE provides a good way for the interpretation of the key information of a generated distractor.
\section{Framework Description}
\subsection{Problem Definition}
In this paper, we focus on the automatic distractor generation for MCQs (see Figure~\ref{fig:question_example}). Let $P=\{w^p_t\}^{t=L_p}_{t=1}$ denote the reading passage, which consists of $L_p$ words. Let $Q=\{w^q_t\}^{t=L_q}_{t=1}$ and $A=\{w^a_t\}^{t=L_a}_{t=1}$ denote the question and its correct answer, respectively. $L_q$ and $L_a$ denote the lengths of the question and the answer, respectively. Note that the answer may not be a span of the passage $P$.
\paragraph{Problem Definition.} \textit{Formally, given the reading passage $P$, the question $Q$ and its correct answer $A$ as inputs, an EDGE model $\mathcal{M}$ aims to generate a distractor $D=\{w^d_t\}^{t=L_d}_{t=1}$ about the question, which is defined as finding the best distractor $\overline{D}$ that maximizes the conditional likelihood given $P$, $Q$, and $A$:}
\begin{equation*}
\overline{D}=\underset{D}{\arg \max } \log \Pr(D | P, Q, A)
\end{equation*}
\subsection{Framework Overview}
Inspired by existing question generation works~\cite{Duan2017QuestionGF,Du2017IdentifyingWT,Zhou2017NeuralQG,Kim2018ImprovingNQ}, we employ a sequence-to-sequence based network to generate the distractors.
As shown in Figure~\ref{fig:framework}, our overall framework contains five components.
First, we employ the encoding module to extract the contextual semantic representations for all materials.
Then, we use the attention mechanism to enrich the semantic representations of the question and its answer.
Finally, we design three key components to generate useful distractors.
As mentioned above, the quality of the generated distractor are guaranteed from two aspects:
\begin{itemize}
\item \textit{Incorrectness:} Both the passage and the question contain some parts strongly relevant to the answer, which may disorder the decoder to output the words contained by the answer and further hurt the inherent incorrectness of generated distractors. To guarantee the incorrectness, in our proposed framework, we reform the passage and question by erasing their answer-relevant information before they are fed into the decoder. Based on the gate mechanism, the two reforming modules highlight the distractor-relevant words and constrain the answer-relevant words by measuring the distances between the words and the correct answer.
\item \textit{Plausibility:} To look reasonable, the distractor first should be grammatically and semantically consistent with the question. Otherwise, after reading the question, the students can trivially exclude it. Furthermore, to hinder students from excluding the distractor only by reading the passage, it should also be semantically relevant to the passage. To guarantee the plausibility, in our proposed framework, the distractor generator uses the semantic representation of the reformed question to initialize the generation process and leverages the attention mechanism to obtain the context representation from the reformed passage to guide the output.
\end{itemize}
We will address each component in detail in the following subsections.
\begin{figure*}
\centering \includegraphics[width=.9\linewidth]{framework_fig.pdf}
\caption{The EDGE framework. \label{fig:framework}}
\end{figure*}
\subsection{Encoding and Enriching Module}
In the encoding module, given a passage $P=\{w_t^p\}_{t=1}^{t=L_p}$, a question $Q=\{w_t^q\}_{t=1}^{t=L_q}$ and its answer $A=\{w_t^a\}_{t=1}^{t=L_a}$, we first convert every word $w$ to its $d$-dimensional vector $\mathbf{e}$ via an embedding matrix $\mathbf{E}\in \mathbb{R}^{|V|\times d}$, where $V$ is the vocabulary and $d$ is the dimension of word embedding. Then we use the encoder to extract the contextual representation for each word.
The outputs of the contextual encoder are three matrices: $\mathbf{P}\in \mathbb{R}^{L_p\times d}$, $\mathbf{Q}\in \mathbb{R}^{L_q\times d}$, and $\mathbf{A}\in \mathbb{R}^{L_a\times d}$.
Next, we introduce the attention layer and the fusion layer to enrich the semantic representations of the question and its answer by fusing the passage information.
In the enriching module, we adopt the scaled dot product attention mechanism and the fusion kernel used in recent works~\cite{chen2017neural,mou2016natural} for better semantic understanding.
\begin{equation*}
\mathbf{M}^{q} = \texttt{Attn}(\mathbf{Q}, \mathbf{P})=\textit{softmax}(\frac{\mathbf{Q}\mathbf{P}^\top}{\sqrt{d}})
\end{equation*}
\begin{equation*}
\widetilde{\mathbf{Q}} = \texttt{Fuse}(\mathbf{Q}, \overline{\mathbf{Q}}) = \textit{tanh}([\mathbf{Q}; \overline{\mathbf{Q}}; \mathbf{Q} - \overline{\mathbf{Q}}; \mathbf{Q} \circ \overline{\mathbf{Q}}]\mathbf{W}_f+\mathbf{b}_f) \ \ \ \ \text{where } \overline{\mathbf{Q}} = \mathbf{M}^{q} \mathbf{P}
\end{equation*}
where $\mathbf{W}_f\in \mathbb{R}^{4d\times d}$ and $\mathbf{b}_f\in \mathbb{R}^d$ are the parameters to learn.
$[;]$ denotes the column-wise concatenation. $\circ$ and $-$ denote the element-wise multiplication and subtraction between two matrices, respectively. $\mathbf{M}^{q}\in \mathbb{R}^{L_q\times L_p}$ denotes the attention weight matrix.
For the answer $\mathbf{A}$, we can also obtain the enriched representation $\widetilde{\mathbf{A}}$ via the same attention and fusion process.
\subsection{Reforming Question Module}
As mentioned above, to retain the inherent incorrectness of the distractors, we need to reform the question by constraining the answer-relevant parts.
In this module, we first evaluate the semantic distance between the answer and each word of the question.
Then, the distances are used as the weights to differentiate the useful words from the words strongly relevant to the correct answer.
The reforming process is conducted in the following two steps.
\textbf{Self-Attend Layer.} Firstly, we use this layer to obtain the sentence-level representation $\widetilde{\mathbf{v}}^a\in \mathbb{R}^{1\times d}$ of the answer.
\begin{equation*}
\widetilde{\mathbf{v}}^a = \texttt{SelfAlign}(\widetilde{\mathbf{A}}) = \mathbf{r}^{\top} \widetilde{\mathbf{A}} \ \ \ \ \text{where } \mathbf{r} = softmax(\widetilde{\mathbf{A}}\mathbf{W}_a)
\end{equation*}
where $\textbf{W}_a\in \mathbb{R}^{d\times 1}$ is a trainable parameter.
$\mathbf{r}\in \mathbb{R}^{L_a\times 1}$ denotes the weight vector.
\textbf{Gate Layer.} In this layer, we first use a bilinear layer to measure the distance between each word and the answer.
Then, the distance information is used as gate values to reform each word and gain the reformed question $\dot{\mathbf{Q}}$.
\begin{equation*}
\dot{\textbf{Q}}_{i} = \delta_i \widetilde{\textbf{Q}}_{i}, i\in[1, L_q]\ \ \ \ \text{where } \delta_i = \texttt{Gate}(\widetilde{\textbf{Q}}_{i}, \widetilde{\mathbf{v}}^{a}) = \widetilde{\textbf{Q}}_{i} \textbf{W}_g^q \widetilde{\mathbf{v}}^{a\top} + b_g^q
\end{equation*}
where $\textbf{W}_g^q$ is a trainable bilinear projection matrix and $b_g^q \in \mathbb{R}$ is also a parameter to learn. $\delta_i\in \mathbb{R}$ is the semantic distance between the correct answer and the $i$-th word of the question. $\widetilde{\textbf{Q}}_{i}$ and $\dot{\textbf{Q}}_{i}$ denote the original representation and reformed representation of the $i$-th word in the question, respectively.
\subsection{Reforming Passage Module}
Likewise, the passage also needs to be reformed to erase the impact of the answer.
However, there still are some differences between the two reforming modules' architectures:
\begin{itemize}
\item Some parts of the passage may belong to other questions but have some common words with the answer. These common words may obtain low gate values which further reduce their contributions to the generation. Hence, before passage reforming, we first attend the question information to the answer to constrain it only to affect the words related to its question.
\item To further strengthen the relationship between the generated distractor and the question, we integrate the question information into the reformed passage to further highlight the question-relevant sentences.
\end{itemize}
The reformation process consists of the following four steps.
\textbf{A-Q Attention Layer.} We use the $\texttt{Attn}(\cdot, \cdot)$ and $\texttt{Fuse}(\cdot, \cdot)$ to fuse the question information into the answer. Note that we use the original question but not the reformed question because the former contains complete answer-relevant information.
\begin{equation*}
\hat{\mathbf{A}} = \texttt{Fuse}(\widetilde{\mathbf{A}}, \overline{\mathbf{A}})\ \ \ \ \text{where } \overline{\mathbf{A}} = \texttt{Attn}(\widetilde{\mathbf{A}}, \widetilde{\mathbf{Q}})\widetilde{\mathbf{Q}}
\end{equation*}
\textbf{Self-Attend} \& \textbf{Gate Layer.} As the reforming question module, we first use the $\texttt{SelfAlign}(\cdot)$ to obtain the answer's sentence-level representation $\hat{\mathbf{v}}^a=\texttt{SelfAlign}(\hat{\mathbf{A}})\in \mathbb{R}^{1\times d}$.
We then use another gate layer to obtain the reformed passage $\dot{\mathbf{P}}$.
\begin{equation*}
\dot{\mathbf{P}}_{i} = \texttt{Gate}(\mathbf{P}_{i}, \hat{\mathbf{v}}^{a})\mathbf{P}_{i}, i\in[1, L_p]
\end{equation*}
where $\mathbf{P}_{i}$ and $\dot{\mathbf{P}}_i$ denote the original and reformed representations of the $i$-th passage word, respectively.
\textbf{P-Q Attention Layer.} This layer uses the attention mechanism to fuse the question information as mentioned above.
\begin{equation*}
\widetilde{\mathbf{P}} = \texttt{Fuse}(\dot{\mathbf{P}}, \overline{\mathbf{P}})\ \ \ \ \text{where }\overline{\mathbf{P}} = \texttt{Attn}(\dot{\mathbf{P}}, \dot{\mathbf{Q}})\dot{\mathbf{Q}}
\end{equation*}
\textbf{Re-encoding Layer.} We re-extract the contextual representation by another Bi-LSTM for the reformed passage $\widetilde{\textbf{P}}$.
The final semantic representation of the passage is denoted as $\hat{\mathbf{P}}\in \mathbb{R}^{L_p\times d}$.
\subsection{Question Initializer}
As mentioned above, the generated distractor should be grammatically and semantically consistent with the question. Inspired by \cite{Gao2018GeneratingDF}, we use the question information to initialize the decoding process to enhance the semantic relevance between the distractor and question.
Specifically, we first use a Bi-LSTM~\cite{hochreiter1997long} to re-encode the reformed question $\dot{\textbf{Q}}$.
\begin{equation*}
\overrightarrow{\mathbf{h}^q_i} = \overrightarrow{\textit{LSTM}}(\dot{\textbf{Q}}_{i}, \overrightarrow{\mathbf{h}^q_{i-1}})
\end{equation*}
where $\overrightarrow{\mathbf{h}^q_i}$ is the hidden state of the forward LSTM at time $i$.
We then concatenate the last hidden states of two directions as $\mathbf{h}^q\in \mathbb{R}^{d}$, which then is projected to get the initial state of the decoder $\mathbf{h}_0$.
\begin{equation*}
\mathbf{h}_0 = \mathbf{h}^q \mathbf{W}_p + \mathbf{b}_p\ \ \ \ \text{where }\mathbf{h}^q = [\overrightarrow{\mathbf{h}^q_{L_q}};\overleftarrow{\mathbf{h}^q_{L_q}}]
\end{equation*}
where $\mathbf{W}_p\in \mathbb{R}^{d\times d}$ and $\mathbf{b}_p\in \mathbb{R}^{d}$ are learnable parameters.
\subsection{Distractor Generator}
At the decoder side, we adopt an attention-based LSTM layer.
Specifically, at the first step, we use the output of the reforming question module $\mathbf{h}_0$ as the initial state and use the mean pooling vector of the reformed passage $\hat{\mathbf{P}}$ as the context vector $\mathbf{c}_0\in \mathbb{R}^{d}$.
The first word is set to the special token \textit{[EOS]}.
\begin{equation*}
\mathbf{c}_0 = \texttt{MeanPooling}(\hat{\mathbf{P}}),\ \ \ \mathbf{e}_0 = \mathbf{E}(\textit{[EOS]})
\end{equation*}
Next, for each decoding step $t$, we use the attention mechanism to attend the most relevant words in the reading passage to form the context vector.
\begin{equation}
\label{eq:decode}
\textbf{h}_t = \textit{LSTM}([\mathbf{e}_{t-1}; \mathbf{c}_{t-1}], \mathbf{h}_{t-1}), \ \ \
\textbf{c}_t = \texttt{Attn}(\mathbf{h}_t \mathbf{W}_h, \hat{\mathbf{P}}) \hat{\mathbf{P}}
\end{equation}
where $\textbf{W}_h \in \mathbb{R}^{d\times d}$, projecting the hidden state to the passage context, is the parameter to learn. $\mathbf{e}_{t-1}$ denotes the embedding of the word at the $t-1$-th step.
Moreover, at each step, we concatenate $\textbf{h}_t$ and $\textbf{c}_t$ together and use an MLP layer to predict the word probability distribution.
\begin{equation}
\label{eq:loss}
H_V = \textit{softmax}(\textit{tanh}([\textbf{h}_t; \textbf{c}_t]\textbf{W}_s) \textbf{W}_v + \textbf{b}_v)
\end{equation}
where $\textbf{W}_s\in \mathbb{R}^{2d\times d}$, $\textbf{W}_v \in \mathbb{R}^{d\times |V|}$ and $\textbf{b}_v\in \mathbb{R}^{|V|}$ are learnable parameters. $H_V$ denotes the probabilities of all words in the vocabulary in which the word with the maximum probability is the predicted word at step $t$.
\subsection{Training and Inference}
We train the model by minimizing regular cross-entropy loss:
\begin{equation*}
\mathcal{L}(\theta_{\mathcal{M}}) = -\sum_{\mathcal{D}}\sum_{t}\log \Pr (w_t^d|P,A,Q,w_{<t}^d;\theta_{\mathcal{M}})
\end{equation*}
where $\mathcal{D}$ is the training corpus in which each data sample contains a distractor $D$, a passage $P$, a question $Q$ and an answer $A$. $w_t^d$ is the $t$-th position of the distractor $D$. $\Pr (w_t^d|P,A,Q,w_{<t}^d;\theta)$ is the predicted probability of the $w_t^d$ and can be calculated by the Eq.(\ref{eq:loss}). $\theta_{\mathcal{M}}$ denotes all trainable parameters in EDGE.
During the inference phase, we use a beam search of width $n$ and receive $n$ candidate distractors with decreasing likelihood because an MCQ has several diverse distractors.
Following \newcite{Gao2018GeneratingDF}, we use the Jaccard distance to generate the final multiple diverse distractors from the beam search results.
Specifically, we first select the first candidate distractor with the maximum likelihood from the search results as $D_1^g$.
The second one $D_2^g$ should have a Jaccard distance, larger than 0.5, to $D_1^g$.
Likewise, we select the third one $D_3^g$ which has a restricted distance to both $D_1^g$ and $D_2^g$.
\section{Introduction}
\blfootnote{
%
%
%
%
\hspace{-0.5cm}
This work is licensed under a Creative Commons
Attribution 4.0 International License.
License details:
\url{http://creativecommons.org/licenses/by/4.0/}.
}
Standard test, such as TOFEL and SAT, is an efficient and essential tool to assess knowledge proficiency of a learner~\cite{Ch2018AutomaticMC}. According to testing results,
teachers or ITS (Intelligent Tutoring System) services can develop personalized study plans for different students.
When organizing a standard test, a vital issue is to select a suitable question form.
Among various question forms, multiple choice question (MCQ) is widely adopted in many notable tests, such as GRE, TOFEL and SAT.
MCQs have many advantages including less testing time, more objective and easy on the grader~\cite{Ch2018AutomaticMC}.
A typical MCQ consists of a stem and several candidate answers, among which one is correct, the rest are distractors.
As shown in Figure~\ref{fig:question_example}, in addition to a stem, some tests also include a long reading passage to provide the context of this MCQ.
The quality of an MCQ depends heavily on the quality of the distractors.
If the distractors can not confuse students, the correct answer could be concluded easily. As a result, the discrimination of the question will degrade, and the test will also lose the ability of the assessment.
However, it is a challenging job to design useful and qualified distractors.
Rather than being a trivial wrong answer, the distractor should have the plausibility which confuses learners who did not master the knowledge~\cite{Liang2018DistractorGF,DBLP:conf/cikm/QiuW019}.
A good distractor should be grammatically correct given the question and semantically consistent with the passage context of the question~\cite{Gao2018GeneratingDF}.
Meanwhile, the question composers need to enhance the plausibility of the distractor without hurting its inherent incorrectness.
Otherwise, the distractor easily becomes a definitely wrong answer, further making the question to be sloppy.
Hence, the manual preparation of distractors is time-consuming and costly~\cite{Ha2018AutomaticDS}.
It is an urgent issue to automatically generate useful distractors, which can help to alleviate question composers' workload and relax the restrictions on experience.
It could also be helpful to prepare a large train set to boost the machine reading comprehension (MRC) systems~\cite{Yang2017SemiSupervisedQW}.
In this paper, we focus on automatically generating semantic-rich distractors for MCQs in real-world standard tests, such as RACE~\cite{Lai2017RACELR} which is collected from the English exams for Chinese students from grades 7 to 12.
\begin{figure}[h]
\centering \includegraphics[width=.8\linewidth]{question_examples_single_col.pdf}
\caption{Two examples of multiple choice questions in RACE. Blue choices are the correct answers. \label{fig:question_example}}
\end{figure}
Existing works conduct some attempts on generating short distractors~\cite{Stasaski2017MultipleCQ,Guo2016QuestimatorGK}.
These approaches formulate distractor generation as a similar word selection task.
They leverage the pre-defined ontology or word embeddings to find similar words/entities of the correct answer as the generated distractors.
These word-level or entity-level methods can only generate short distractors and do not apply to semantic-rich and long distractors for RACE-like MCQs.
Recently, generating longer distractors has been explored in a few studies~\cite{zhou2019coattention,Gao2018GeneratingDF}.
For example, ~\newcite{Gao2018GeneratingDF} proposes a sequence-to-sequence based model, which leverages the attention mechanism to automatically generate distractors from the reading passages. However, these methods mainly focus on the relation between the distractor and the passage and fail to comprehensively model the interactions among the passage, question and correct answer which helps to ensure the \textit{incorrectness} of the generated distractors.
To better generate useful distractors, we propose a novel qu\textbf{E}stion and answer guided \textbf{D}istractor \textbf{GE}neration (EDGE) framework.
More specifically, given the passage, the question and the correct answer, we first leverage a contextual encoder to generate the semantic representations for all text materials.
Then we use the attention mechanism to enrich the context of the question and the correct answer.
Next, we break down the distractor's usefulness into two aspects: the incorrectness and plausibility.
Incorrectness is the inherent attribute of the distractor, while plausibility refers to the ability to confuse the students.
We introduce two modules by leveraging the gate layer to guarantee the incorrectness: {\em Reforming Question Module} and {\em Reforming Passage Module}.
We further leverage an attention-based distractor generator, plus the previous two reforming modules, to guarantee the plausibility.
Finally, in the generation stage, we use the beam search to generate several diverse distractors by controlling their distances.
We conduct experiments on a large-scale public distractor generation dataset prepared from RACE.
The experimental results demonstrate the effectiveness of our proposed framework.
Moreover, our method achieves a new state-of-the-art result in the distractor generation task.
\section{Related work}
In recent years, some research efforts have devoted to distractor generation.
Generally, the related work can be classified into the following three categories.
\textbf{Feature-based methods}.
Considerable research efforts~\cite{Liang2018DistractorGF,Sakaguchi2013DiscriminativeAT,Araki2016GeneratingQA} have been devoted to using manual design features, such as POS features and statistic features, to generate the distractors.
However, designing effective features is also labor intensive and hard to scale to various domains. Differently, our work is an end-to-end framework without manual design features.
\textbf{Similar word/entity-based methods}.
Some works~\cite{Stasaski2017MultipleCQ,Guo2016QuestimatorGK,kumar2015automatic,Afzal2014AutomaticGO} focused on finding answer-relevant ontologies or words as the distractors with the help of WordNet and Word2Vec.
For example, ~\newcite{Stasaski2017MultipleCQ} leveraged an educational Biology ontology to conduct the distractor generation.
However, some of these works depend heavily on the well-designed ontology and they can only generate short distractors, which usually only contain one single word or phrase.
\textbf{NN-based methods.}
Recently, neural network based and data-driven solutions emerge.
\newcite{Gao2018GeneratingDF} proposed an end-to-end solution focusing on distractor generation for MCQs in standard English tests.
They employed the hierarchical encoder-decoder network as the base model and used the dynamic attention mechanism to generate the long distractors.
\newcite{zhou2019coattention} further strengthened the interaction between the question and the passage based on the model of \cite{Gao2018GeneratingDF}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,954
|
\section{Introduction}
Remote work is an essential mode of work across industries~\cite{bloom2015does}. Before COVID-19, many companies had already experimented with or implemented various forms of remote work~\cite{brynjolfsson2020covid}, e.g., recruiting remote employees or allowing employees to work from home part-time. During the pandemic, 34.1\% of Americans
switched to working from home~\cite{brynjolfsson2020covid}, and it is estimated that 37\% of jobs in United States can be done remotely~\cite{dingel2020many}. In fact, many large technology companies, such as Quora and Twitter, have announced that they will allow employees to work from home indefinitely~\cite{adam_dangelo_remote_nodate, jennifer_christie_keeping_nodate}.
Central to remote work are remote meetings \cite{olson2000distance, cao2020my}, most commonly experienced through video conferencing tools (e.g., Zoom, Microsoft Teams, Google Meet), which help remote team members stay connected, collaborate and function as an organization, despite physical distance. Therefore, it is critical to understand factors that are associated with remote meeting experiences to better support productive and engaging remote collaborations.
In this paper, we focus on one fundamental experience of remote meetings --- in-meeting multitasking. Information work is often governed by multiple tasks and activities that an individual must remember to perform, often in parallel or in rapid succession, a practice that is called \emph{multitasking}~\cite{Rubinstein2001, Wickens2008, czerwinski2004diary}.
Multitasking behavior is vital, as it is closely linked to one's productivity and well-being~\cite{czerwinski2004diary,madore2020memory} --- increasing the numbers of items to be remembered can wreak havoc with task resumption~\cite{altmannResumption}, attentional focus ~\cite{iqbal2007disruption, mark2004} and prospective memory~\cite{brandimonteProspectiveMemory}. Multitasking during meetings can additionally affect other people and their productivity \cite{iqbal_meeting}.
Prior work has investigated how people engage in multitasking during collaborative activities such as meetings and video chats, both in-person and online \cite{iqbal_meeting, suh-meeting, marlow-meeting, Hembrooke2003TheLA}.
However, these works are mainly based on small-scale qualitative studies, and to the best of our knowledge, no research to date has reported systematic and comprehensive evidence from large populations. As a result, there is little statistical evidence on the meeting and personal context that are associated with people's propensity to multitask, the activities remote attendees engage in while multitasking, and how such remote multitasking behavior may affect workers and groups.
Here, we adopt a mixed-methods approach to systematically understand the context, activities, and consequences of multitasking during remote meetings. Specifically, we analyzed a large-scale dataset collected from from February to May 2020 of U.S. Microsoft employees. The dataset contains anonymized events from major productivity tools: Microsoft Teams for remote meetings, Microsoft Outlook for email services, and OneDrive and Sharepoint for accessing and editing files remotely. Multitasking can involve activities that are unrelated to the meeting (e.g., email communications) and related (e.g., notes taking)~\cite{iqbal_meeting}. Therefore we measured emails sent and files edited during remote meetings as a proxy for multitasking and studied the relationship between multitasking and meeting characteristics through controlled regression analysis. Furthermore, we contextualize the evidence from log analysis with verbatims from a 715-person diary study of Microsoft employees globally, run from mid-April to mid August 2020, exploring their remote meeting experiences during COVID-19.
Our results show that: 1) Multitasking is a common behavior in remote meetings with about 30\% of meetings involving email multitasking. People also reported that multitasking becomes more frequent as meetings move to remote, 2) Intrinsic meeting characteristics, such as meeting size, length, type, and timing, significantly correlate with the extent to which people multitask, e.g., people are more likely to multitask in recurring meetings than in ad hoc meetings, and 3) In-meeting multitasking during remote meetings can lead to both positive (e.g., improve productivity) and negative (e.g., loss of attention) experiences. Our analysis suggests practical ways people can improve remote meeting experiences. For example, to reduce unnecessary scheduled and recurring meetings, keep meetings short, avoid intensive meetings early in the morning, and allow space for positive multitasking. Our work implies that productivity tools can help people better manage their in-meeting attention and decide which meetings or parts of the meetings to attend.
The contributions of this paper are threefold:
\begin{itemize}
\item{We present the first large-scale, empirical study of multitasking behavior during remote meetings, accompanied by rich qualitative evidence. This allows us to understand the factors that correlate with remote meeting multitasking and characterize the motivations and potential consequences of such behavior.}
\item{We discover several key issues in current remote meeting configurations and suggest actionable guidelines for how people may schedule effective remote meetings (Section~\ref{sec:best_practice}).}
\item{Our work points to several concrete design implications for future remote collaboration tools, e.g., support meeting ``focus mode'' and positive multitasking.}
\end{itemize}
\section{Background and Related Work}
Our work is built upon and contributes to the rich HCI literature on remote work and multitasking behavior and its manifestations in meetings. Prior research mostly focused on small-scale in-person studies, whereas our work analyzes a large-scale log dataset accompanied by a large diary study, to systematically reveal multitasking patterns during all-remote meetings.
\subsection{Remote Work}
Remote work has long been an important topic of research across scientific fields. Through a Working-From-Home (WFH) experiment at Ctrip, a 16,000-employee Chinese travel agency, Bloom et al.~\cite{bloom2015does} studied the costs and benefits of remote work, where they showed that WFH led to improved performance but reduced promotion rate. Prior work also investigated other aspects of remote work, such as management of workers~\cite{delle2018missing}, organizational design~\cite{valentine2017flash, retelny2014expert}, communication challenges~\cite{he2014qualitative}, team performance~\cite{lix2020timing}, well-being~\cite{cao2020my,whiting2019did}, and emerging roles~\cite{chen2020understanding} and experience~\cite{cao2020your, cao2020you}. Remote work becomes more ubiquitous after the COVID-19 pandemic~\cite{eriksson2020remote, yang2020work}, and remote meetings have become a central place where people stay connected and collaborate with others~\cite{olson2000distance} --- both DeFilippis et al.~\cite{defilippis2020collaborating} and Yang et al.~\cite{yang2020work} demonstrated a significant increase in remote meetings during the pandemic. Here we contribute to the rich literature of remote work by focusing on how in-meeting multitasking, an artifact of traditional in-person meetings, manifested in all-remote settings.
\subsection{Multitasking in the Workplace}
A large body of prior work has focused on how multitasking impacts attention in the workplace, primarily focusing on the distraction caused to an ongoing task that is interrupted by another activity. Czerwinski et al.~\cite{czerwinski2004diary} employed a diary study to show how information workers switch activities due to interruptions in the workplace, focusing on the difficulty of the continuous switching of context. Iqbal and Horvitz~\cite{iqbal2007disruption} studied how external interruptions cause information workers to enter into a ``chain of distraction'' where stages of preparation, diversion, resumption and recovery can describe the time away from an ongoing task. Gonzalez and Mark~\cite{gonzalez2004constant} reported on how information workers conceptualize and organize basic units of tasks and how switching occurs across these conceptual units --- people were found to spend about 12 minutes in one working unit before switching to another. In the mobile domain, Karlson et al.~\cite{karlson2010mobile} and Park et al.~\cite{park2014analysis} found that tasks on mobile phones become fragmented across devices and they identified challenges that exist in resuming these tasks. While most studies have looked at multitasking results from distractions, there is a gap in prior work that characterizes interactions corresponding to multitasking during remote meetings, as discussed in Section~\ref{sec:mutitasking_meeting}.
\subsection{Factors Associated with Multitasking}
External~\cite{oconaill-frohlich-externalinterruptions, iqbal_meeting, mark-gonzalez-2005, mark-wang-2014-collegestress} and internal~\cite{dabbish-mark-2011, alder2013} interruptions are considered to be the most direct reasons behind multitasking; however there are other indirect factors that are associated with multitasking as well. Past work has shown that personality and organizational environment~\cite{dabbish-mark-2011} are associated with multitasking~\cite{mark-neurotics}. Additionally, previous work~\cite{mark2014bored} has shown that the time of day is associated with workers' focus. On average, people were most focused in their work late-morning (11 a.m.) and mid-afternoon (2-3 p.m.), which is known as the ``double peak day'' of information workers' rhythm of work.
Day of the week also play important role in people's attention level and multitasking behavior. Mark et al.~\cite{mark2014bored} found a relationship between online activity and Mondays, the day when people report being the most bored, but at the same time also the most focused. However, most of the prior studies are focused on in-person work settings, and it is unclear to what extent these patterns are applicable to remote meetings. Also, the existing evidence is mainly based on small-scale data that may not generalize to large organizations. Our study confirms the associations between time, distractions and multitasking in remote meetings through large-scale log analysis and greatly extends prior knowledge by investigating a broader range of meeting characteristics, such as meeting size, type, and length.
\subsection{Multitasking during Meetings}
\label{sec:mutitasking_meeting}
While multitasking during one's own work mostly impacts personal productivity, special consideration of multitasking during meetings is warranted, as this can additionally impact other colleagues and their productivity~\cite{iqbal_meeting}. Past work has looked at how people engage in multitasking both during in-person meetings and presentations~\cite{iqbal_meeting, Hembrooke2003TheLA, barkhuus-laptop}, as well as online collaborative activities, such as remote meetings leveraging subjective feedback or perceptions~\cite{marlow-meeting, suh-meeting}. For example, in educational settings, the use of laptops during a lecture has been shown to have a negative impact on attention, where students tend to engage in activities such as web-surfing or emailing rather than activities related to the lecture~\cite{barkhuus-laptop, Hembrooke2003TheLA}. However, in other studies, people who multitask during in-person meetings report to do so in order to interleave other important activities as they peripherally pay attention to the meeting and engage only when relevant~\cite{iqbal_meeting}. In online settings, both meeting related and personal multitasking are seen as ways people's attention could divert from the actual conversation, though multitasking on a single screen is considered more acceptable than when multitasking is happening on a different screen - often presumed to be unrelated to the meeting~\cite{marlow-meeting}. Similarly, a study on video-chats among teens found that boredom was the main reason why teens would multitask during a video chat, wherein they would engage in scrolling social feeds or play games~\cite{suh-meeting}.
While prior work on meetings and multitasking focus primarily subjective perceptions, no research to the best of our knowledge has looked at large-scale log data accompanied by qualitative evidence on what people are engaging in while attending a meeting and under what conditions people tend to engage in when multitasking. Our analysis of actual interactions can complement subjective perceptions around multitasking motivations, behaviors and potential consequences, and can provide insights into how to conduct meetings for maximal effectiveness.
\section{Method}
To systematically understand multitasking patterns during remote meetings, we proposed the following research questions to guide our research.
\begin{resques}[RQ\ref{resques:volume}] \label{resques:volume}
How much multitasking is happening during remote meetings?
\end{resques}
\begin{resques}[RQ\ref{resques:when}] \label{resques:when}
What factors are associated with multitasking during remote meetings?
\end{resques}
\begin{resques}[RQ\ref{resques:why}] \label{resques:why}
What do people do when multitasking during remote meetings?
\end{resques}
\begin{resques}[RQ\ref{resques:consequence}] \label{resques:consequence}
What are the consequences of multitasking during remote meetings?
\end{resques}
As noted above, we adopt a mixed-methods approach to address these questions: analysis of a large-scale anonymized telemetry dataset coupled with a diary study of people's perceptions and subjective experiences with respect to in-meeting multitasking.
\subsection{Regression Analysis on Large-Scale Telemetry Dataset}
\label{sec:regression_analysis}
\textbf{Data Preparation.} We collected metadata (without any content information) on remote meetings (Microsoft Teams), email usage (Microsoft Outlook), and file edits (Onedrive/Sharepoint) of US employees from Microsoft. The majority of work and communication in Microsoft are carried out through these platforms. We collected four separate week-long snapshots of data from February to May, 2020: 1) February 24-28, which represents a period of pre-COVID, mostly co-located work, 2) March 23-27, which represents the company's transition phase from mostly co-located work to remote work, and 3) April 20-24 and 4) May 18-22, to represent fully remote work periods. While we leveraged all four periods to study work rhythm and derive statistics on multitasking behavior, our regression analysis focused on the snapshot from May 18-22, when employees were fully settled into working from home.
Specifically, for each meeting hosted on Microsoft Teams\footnote{We only included meetings that are longer than two minutes to filter out data noise. We also filtered out meetings lasting longer than 3 hours, which is likely due to the fact people forget to leave the meeting.}, we collected the start and end timestamps of the meeting \footnote{Meeting start and end timestamps was logged on a per-person basis, i.e, the exact timestamp that each person attended and left a meeting. The average standard deviation of meeting duration among people joining the same meeting is about 2.1\% of the maximum meeting duration, indicating that the same meeting generally has similar length across participants.}, meeting size\footnote{Meeting size was logged as the number of all people ever connected to a meeting. Due to aggregated nature of telemetry data, it is impossible to measure the exact maximum concurrent participants.}, and the meeting type (ad hoc, scheduled, recurring or broadcast\footnote{Meetings that have more than 250 attendees.}), and their distributions are presented in Fig.~\ref{fig:meeting_duration}. As shown in Table.~\ref{table:features}, we discretized continuous meeting attributes in the following ways to ensure robustness of the regression analysis. We grouped meeting duration into four categories --- 0-20 mins, 20-40 mins, 40-80 mins, and >80 mins because of the popularity of 30 mins and 60 mins meetings (Fig.~\ref{fig:meeting_duration}). For meeting size, due to its long-tail nature, we split it into five roughly equal sized bins. We considered morning, afternoon, and after hours as three categories for hour of the day in order to align it with common work rhythms in Microsoft.\footnote{We tested different bucketing strategies and it produced robust results.} Furthermore, we tested the meeting attribute correlation, where we find rather weak correlations ($\left|r\right|<0.15$) among different attribute groups. For instance, the correlation between meeting time (e.g., morning) and meeting size (e.g., >10 attendees) is around 0.07. The weak dependency between meeting attributes motivated us to directly include them in the regression analysis rather than trying to cluster various meeting types and study those clusters in the regression. Finally, we note that the telemetry data only records virtual meetings, so February data (before pandemic in U.S.) does not reflect total volume of meetings people attended, but all meetings after the pandemic are recorded.
We focused on two multitasking behavior during remote meetings that can be measured through available telemetry data: email multitasking and file edit multitasking. To enable the analysis, on Microsoft Outlook, we collected the time when people actively send, respond to, or forward an email. On file platforms, we recorded events related to editing productivity-related files, including Excel, Powerpoint, Word, and PDFs. Note that due to the nature of the data, we are not able to differentiate whether the files are related to the meeting or not, thus the measured multitasking file behavior might be related to the meeting (e.g. notes taking).
To ensure people's privacy with the telemetry data, we performed pre-processing to de-identify workers before the data was obtained by researchers. Access to the data was strictly limited to authorized members of the research team who went through extensive privacy and ethical training.
\begin{figure*}[tb]
\begin{subfigure}[Distribution of meeting duration]{
\label{fig:meeting_duration}
\includegraphics[width=0.3\textwidth]{figures/duration.png}}
\end{subfigure}
\begin{subfigure} [Distribution of meeting size]{
\label{fig:meeting_size}
\includegraphics[width=0.3\textwidth]{figures/size.png}}
\end{subfigure}
\begin{subfigure} [Distribution of meeting types]{
\label{fig:meeting_type}
\includegraphics[width=0.3\textwidth]{figures/type.png}}
\end{subfigure}
\caption{Distributions of virtual meeting duration, meeting size, and meeting types measured through our collected telemetry data. We observe two clear peaks surrounding 30 mins and 60 mins in distribution of meeting duration, indicating the popularity of scheduling 30 mins and 60 mins meetings. We observe a long tail distribution of meeting size, where over 20\% of all meetings are one on one meetings. Finally, we observe the majority of meetings are scheduled meetings, followed by recurring meetings, adhoc meetings and broadcast meetings. The data distributions motivate us to discretize features into bins specified in Table 1.}
\label{fig:workrhythm}
\end{figure*}
\textbf{Regression Analysis.} We joined three data sources by unique user identifiers (anonymized), which resulted in 34,524 (meeting, user) records. For each (meeting, user) pair, we labeled it as email multitasking ($Y=1$) if the user was found to have at least one active email action during the meeting, otherwise a non-multitasking label was assigned ($Y=0$). File multitasking was labeled similarly. We conducted a regression analysis to understand the relationships between multitasking and meeting characteristics, while controlling for individual variances since the individual tendency to multitask may confound the estimation. We leveraged stratification \cite{morgan2015counterfactuals} to group all records by worker and used conditional logistic regression to fit the model. The binary meeting indicators used in the regression model are defined in Table.~\ref{table:features}. To account for the correlations between the records from the same meetings, we grouped standard errors at the meeting level. We present regression results in terms of estimated odds ratio and their statistical significance in Fig.~\ref{fig:Regression_email}. We did not find significant correlations between file-related multitasking and meeting characteristics, which could be attributed to the fact that files edited are often related to the meeting (Section~\ref{sec:file_multitasking_qualitative}). An alternative analysis using Generalized Linear Mixed Effects Models~\cite{seabold2010statsmodels} show qualitatively similar results (Appendix~\ref{sec:supp}).
\begin{table*}[t]
\centering
\vspace{-0.1cm}
\caption{Meeting property and discretized bins used for regression analysis. For each meeting property, we create dummy variables and then left one out as baseline. For instance, Friday is left as baseline for the day of the week meeting property, and all effects are relative to meetings on Friday.}
\vspace{-0.2cm}
\scalebox{0.8}{
\begin{tabular}{@{\hspace{0.1cm}}lc@{\hspace{0.1cm}}}
\toprule
\textbf{Meeting Property} & \textbf{Discretized bins} \\
\midrule
Meeting duration & 0-20 mins meetings (baseline), 20-40 mins meetings, 40-80 mins meetings, >80 mins meetings \\
Meeting size & 2 attendees (baseline), 3 attendees, 4-5 attendees, 6-10 attendes, >10 attendees.\\
Meeting types & scheduled, recurring, ad hoc (baseline), broadcast \\
Hour of the day & morning: 7am - noon, afternoon: noon - 7pm, after hours: 7pm - 7am (baseline)\\
Day of the week & Monday, Tuesday, Wednesday, Thursday, Friday (baseline) \\
\bottomrule
\end{tabular}
}
\label{table:features}
\vspace{-0.0cm}
\end{table*}
\textbf{Limitations of Telemetry Data.} While the large-scale telemetry data provides us with a lens onto how people behave during meetings, our emphasis on preserving privacy means that we unfortunately do not have access to all behavioral details. The data was collected in anonymized, aggregated form and does not have sensitive attributes that can potentially reveal the identity of an individual employee or a group in a corporation, such as functional labels, job roles, organizational charts or participant social demographics. Email is perceived as stateless communication in the organization we studied; therefore, fine-grained events e.g., email read was not recorded. Similarly, multitasking behavior in 3rd party platforms and non-digital spaces (e.g., house chores), is not measured due to lack of instrumentation. Finally, telemetry data lacks information on the reasons for and consequences of multitasking. We strive to address these limitations by a complementary diary study where we delve deeper into the trends shown in the telemetry data set.
\subsection{Diary Study}
We complement our quantitative analysis with reports drawn from a diary study of employees from Microsoft reporting their experiences of remote meetings during COVID-19. Diary studies in HCI research \cite{rieman1993diary} have been particularly effective in capturing the nuanced experiences of information workers \cite{czerwinski2004diary,sellen1997paper}.
The diary study data collection ran between mid-April and mid-August 2020. Participants opted-in to the study from bulk email messages sent to internal mailing lists, with rolling recruitment between mid-April and mid-June 2020. Participants were asked to keep a guided diary for two months. Diary reminders were sent as a file via email to participants to set up in their calendars. The diaries consisted of a series of forms embedded in a secure company website. 24 total diary entries were requested, one entry approximately every two working days for two months total. The 24 guided entries were to be filled out in three cycles of eight topics: Physical workspace, Interaction, Productivity, Tools, Multitasking, Types of meetings, Time in meetings, and Approaches to meetings. The three diary entries on Multitasking used the following primary prompt: "What have you noticed about multitasking in recent online meetings?". This was followed by a list of sub-prompts: How and when you multitask; Why you multitask; Managing video and audio; Group expectations around multitasking; Impact of multitasking on productivity; Impact of multitasking on conversations; Features for multitasking; Impact of home life; Suggestions for improvement. Full methodological details are available in a technical report ~\cite{rintel2020methodology}.
849 total participants provided consent and were onboarded, of which 715 completed at least one diary entry. For those who filled out diaries: Gender coverage was 60\% Male, 39\% Female, and 1\% non-binary or preferred not to say. Age coverage was 5\% 18-24, 28\% 25-34, 30\% 35-44, 28\% 45-54, 8\% Above 55, and 1\% Prefer Not To Say. Roles covered were 41\% Business \& Sales, 30\% Development, 11\% Creative, Design, and UX, 11\% Technical Operations, 5\% Administration, and 2\% Research. Participants were drawn from almost all regions of the world, primarily 53\% North America, 20\% Europe, 12\% India, 4\% China, and 4\% South America.
Of 7045 total rows of diary responses, 819 responded specifically to the multitasking prompt\footnote{We did not include responses mentioning "multitasking" under other prompts.} from 413 unique participants. We randomly selected 100 responses (20 per month from April to August). To analyze the responses, we adopted the method of open coding~\cite{corbin2014basics}. Five researchers independently analyzed and coded the first 20\% of the interview transcriptions and met to discuss the codes until they were in complete agreement on the codes needed. Then, one researcher coded the remaining transcriptions but discussed any questions with the four other researchers so as to guarantee consensus on the codes. When these steps were finished, the whole research team met and thoroughly discussed the extracted content classification. Through sub-categorization and constant comparison~\cite{corbin1990grounded}, we consistently revised the emerging themes and the final themes presented in Sec 4, 5, 6 and 7 were developed.
To ensure people's privacy in the diary study, we de-identified data before analysis (a participant key linked demographics to diary entries, and then all diary entries were scrubbed for names, places, and other identifying referents). Similar to the telemetry data set, only authorized researchers have access to the diary data.
\section{RQ0: Volume of Multitasking during Remote Meeting}
Our analysis suggests that multitasking during remote meetings is ubiquitous, and that people find themselves engaging in more multitasking during remote meetings compared to in-person meetings, possibly as a result of a shift in work rhythms, and the lower cost to get noticed.
\textbf{Multitasking intensity over time.}
In our telemetry dataset, from February to May, we find that 31.1\%, 30.9\%, 29.2\%, and 28.9\% meetings involve email multitasking, and 23.7\%, 23.1\%, 24.8\%, and 25.5\% meetings involve file multitasking. Putting these percentages in the societal context of the cognitive efforts and resources that information work costs \cite{spira2005cost}, our results suggest the importance of understanding and potentially intervening toward such behavior.
\textbf{More multitasking as a possible result of work rhythm adaptation.} Fig.~\ref{fig:workrhythm} illustrates the shift of work rhythms, characterized by the distribution of work related actions (emails sent, files edited and meetings attended) within a day, from Feb 2020 to May 2020. The email and file-related rhythms throughout the day remain roughly unchanged, indicating that people worked in a similar fashion on emails and files as they did in co-located settings. Meanwhile, there is a clear increase in the number of remote meetings, compared to pre-COVID-19 period -- note that the telemetry results do not necessarily support the conjecture that people are meeting more since not all in-person meeting frequencies were recorded in the telemetry. However, our diary study results suggest that people do perceive that they have more meetings, and this may be an important underlying cause of multitasking during remote meetings.
\begin{adjustwidth}{1em}{1em}
\emph{"I think this is more of a habit that developed now that folks don't have face to face meetings at all and that the number of meetings has increased so much that it is just more efficient to get the notes and reading out of the way during the meeting than work extra hours end of day or early next day to catch up"} (R498)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"There are so many meetings that there is no time to look at email, or get work done in between. I try and work early in the morning and late at night, but as work flows in during the day and needs response, I find myself more and more just multitasking"} (R179)
\end{adjustwidth}
\begin{figure*}
\centering
\begin{subfigure}[Email Action Daily Rhythm]{
\label{fig:Email_Day}
\includegraphics[width=0.32\textwidth]{figures/Email_Day.png}}
\end{subfigure}
\begin{subfigure} [File Action Daily Rhythm]{
\label{fig:Meeting_Day}
\includegraphics[width=0.32\textwidth]{figures/File_Day.png}}
\end{subfigure}
\begin{subfigure} [Remote Meeting Daily Rhythm]{
\label{fig:POI_revisit_T}
\includegraphics[width=0.32\textwidth]{figures/Meeting_Day.png}}
\end{subfigure}
\caption{Transition of worker work rhythms (i.e., distribution of work related actions over time within a day) from Feb. 2020 (pre-COVID-19) to May 2020 on email, file editing actions and remote meetings. While email and file usage remains stable, there is a clear increase in the volume (over twice as much) of remote meetings after the breakout of the pandemic. Results are normalized using maximum volume in Feburary.}
\label{fig:workrhythm}
\end{figure*}
\textbf{Ease of turning off video and audio may encourage more multitasking.}
In comparison with traditional face-to-face meetings, remote meetings make it much easier for people to stay in the background by turning video off/muting themselves. Given that multitasking has been culturally associated with impoliteness \cite{przybylski2013can}, we assume more multitasking during remote meetings may be caused by the lower probability of getting noticed by others when multitasking. In our diary study responses, we found that turning off the video camera or muting the microphone was closely related to multitasking behavior, as mentioned by many (32\% of responses).
\begin{adjustwidth}{1em}{1em}
\emph{"I typically will not multi-task if I have my video on, because people can definitely tell when you're not paying attention. Sometimes I will choose to turn my video on purely so that I am not tempted to multi-task. If I am an optional participant in a meeting, or I am just listening in, unsure if the agenda really calls for my participation, then I am more likely to keep my video off and openly multi-task until someone says my name."} (R10)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"In general, I have a feeling that in our group the expectation is that participants do not multitask during meetings, but who knows what you are doing if the camera is off. (So yeah, that's when I turn off my camera too.)"} (R14)
\end{adjustwidth}
\section{RQ1: WHAT FACTORS ARE ASSOCIATED WITH MULTITASKING}
\begin{figure*}[tb]
\begin{subfigure} [Meeting duration, >80 mins ($P<0.001$), 40-80 mins ($P<0.001$), 20-40 mins ($P<0.001$)]{
\label{fig:EmailRegression_duration}
\includegraphics[width=0.32\textwidth]{regression_figures/duration.PNG}}
\end{subfigure}
\begin{subfigure} [Meeting size, >10 attendees ($P=0.021$), 6-10 attendees ($P<0.001$), 4-5 attendees ($P<0.001$), 3 attendees ($P=0.021$)]{
\label{fig:EmailRegression_size}
\includegraphics[width=0.34\textwidth]{regression_figures/size.PNG}}
\end{subfigure}
\begin{subfigure} [Meeting type, scheduled ($P=0.012$), recurring ($P<0.001$), broadcast ($P=0.880$) ]{
\label{fig:EmailRegression_type}
\includegraphics[width=0.3\textwidth]{regression_figures/type.PNG}}
\end{subfigure}
\begin{subfigure} [Hour of the day, morning ($P<0.001$), afternoon ($P<0.001$)]{
\label{fig:EmailRegression_hour}
\includegraphics[width=0.32\textwidth]{regression_figures/time.PNG}}
\end{subfigure}
\begin{subfigure}[Day of the Week, Thursday ($P=0.003$), Wednesday ($P=0.003$), Tuesday ($P<0.001$), Monday ($P=0.001$)]{
\label{fig:EmailRegression_Weekday}
\includegraphics[width=0.32\textwidth]{regression_figures/day.PNG}}
\end{subfigure}
\caption{Conditional logistic regression results on the relationship between email multitasking and remote meeting characteristics. We find significant associations between email multitasking and meeting duration, meeting size, meeting types, hour of the day and day of the week.}
\label{fig:Regression_email}
\end{figure*}
\begin{figure*}[tb]
\begin{subfigure} [Meeting duration]{
\label{fig:EmailMultitasking_duration}
\includegraphics[width=0.3\textwidth]{figures/EmailMultitasking_duration}}
\end{subfigure}
\begin{subfigure} [Meeting size]{
\label{fig:EmailMultitasking_size}
\includegraphics[width=0.3\textwidth]{figures/EmailMultitasking_size}}
\begin{subfigure} [Meeting types]{
\label{fig:EmailMultitasking_type}
\includegraphics[width=0.3\textwidth]{figures/EmailMultitasking_type}}
\end{subfigure}
\end{subfigure}
\begin{subfigure} [Hour of the day]{
\label{fig:EmailMultitasking_hour}
\includegraphics[width=0.3\textwidth]{figures/EmailMultitasking_hour}}
\end{subfigure}
\begin{subfigure}[Day of the Week]{
\label{fig:EmailMultitasking_day}
\includegraphics[width=0.3\textwidth]{figures/EmailMultitasking_day.png}}
\end{subfigure}
\caption{Proportion of user-meeting pairs with email multitasking versus meeting characteristics measured by telemetry data. People multitask more in longer meetings, larger meetings, recurring/scheduled meetings, morning meetings and Tuesdays.}
\label{fig:Email_Multitasking}
\end{figure*}
Our analysis suggests that both intrinsic meeting characteristics and external factors are correlated with remote meeting multitasking, as discussed in detail below.
\subsection{Intrinsic Meeting Characteristics}
\textbf{More multitasking happens in large meetings.} As shown in Fig.~\ref{fig:EmailRegression_size} and Fig.~\ref{fig:EmailMultitasking_size}, larger meetings generally involve more multitasking. The odds of email multitasking in 3 attendee meetings, 4-5 attendee meetings, 6-10 attendee meetings, and >10 attendee meetings are 1.12 ($P=0.021$), 1.39 ($P<0.001$), 1.70 ($P<0.001$) and 2.16 ($P<0.001$) times the odds of the meetings with only one or two attendees. This could be explained by the fact that participants need to more actively focus on the meeting conversations when the meetings are small. Our empirical finding is also supported by evidence from the diary study, e.g.,
\begin{adjustwidth}{1em}{1em}
\emph{"If it's a large audience, like a webcast or an internal session on some tech topic, I do multitask more."} (R4)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"In the big meetings, like town halls, I tend to stop and actually listen when something of interest is being said. the rest of the time, I seem to not focus on work at all. In small meetings, I generally don't multitask at all anymore. It takes all of my brainpower to focus on the conversation."} (R182)
\end{adjustwidth}
\textbf{More multitasking happens in long meetings.}
We also observe that more multitasking happens in longer meetings. From telemetry data, we observe that the odds of email multitasking in 20-40 minute meetings, 40-80 minute meetings, and >80 minute meetings are 1.96, 3.22 and 6.21 times the odds of 0-20 minutes meetings ($P<0.001$), as illustrated in Fig.~\ref{fig:EmailMultitasking_duration} and Fig.~\ref{fig:EmailRegression_duration}).
As supported by diary study responses, many people mention that they simply cannot concentrate for a long time and thus engage in multitasking during long meetings.
\begin{adjustwidth}{1em}{1em}
\emph{"Additionally, meetings seem to be longer (e.g., I have a three hour brainstorming session with my team today) and I cannot focus on the meeting that long alone. Then I also tend to work on other tasks from now and then."} (R21)
\end{adjustwidth}
\textbf{Morning meetings involve more multitasking.}
The time schedule of the meeting also plays an important role in multitasking behavior. Our email multitasking data analysis suggests that morning meetings involve more multitasking compared to afternoon and after hours: the odds of multitasking in the morning are 1.86 times the odds of the after hour meeting baseline ($P<0.001$), and the odds of email multitasking in the afternoon are 1.54 times the odds of the after hour meeting baseline ($P<0.001$). We argue that this observation may be closely related to the daily work rhythms of individuals. As demonstrated by prior work \cite{mark2014bored}, in the afternoon people are generally more focused. We also find supporting evidence in the diary study results (6\% of responses)
\begin{adjustwidth}{1em}{1em}
\emph{"I will try to glimpse at email more if a meeting is the first thing I do for work in the morning - so that's schedule-related."} (R621)
\end{adjustwidth}
\textbf{More multitasking happens in recurring and scheduled meetings compared to ad hoc meetings.}
Next, our results suggest that multitasking is more likely to happen in recurring and scheduled meetings compared to ad hoc meetings. We find significant associations between email multitasking and meeting types in our telemetry data. Specifically, the odds of email multitasking in recurring and scheduled meetings is 1.59 ($P<0.001$) and 1.31 ($P=0.012$) times the odds of multitasking in ad hoc meetings. Ad hoc meetings generally involve a specific focus relevant to the specific attendees, while scheduled meetings, especially recurring meetings, are more likely to involve broader information sharing which does apply equally to each attendee.
\begin{adjustwidth}{1em}{1em}
\emph{"I just came off an online call with my larger design sync, it's a 30 min recurring meeting where we share topics as our design teams are in [two cities]. I didn't need to present, I was a listener, so I found myself responding to Teams messages, emails etc as the call was going on."} (R42)
\end{adjustwidth}
\textbf{More multitasking happens Monday through Thursday compared to Friday.}
The telemetry data also demonstrated that Friday has a relatively lower likelihood of multitasking, compared to other days of the week. The odds of email multitasking on Tuesday are 1.35 times the odds of multitasking on Friday ($P<0.001$), followed by Monday (1.23 times, $P=0.001$), Wednesday (1.19 times, $P=0.003$) and Thursday (1.19 times, $P=0.003$). While this pattern corroborates findings from prior work ~\cite{mark2014bored}, we note that the result might not generalize broadly, especially for Fridays, since Microsoft encourages fewer meetings on Friday, so the findings might be company-specific.
\textbf{More multitasking in meetings of lower relevance and engagement.}
People also frequently mentioned (17 \% responses) that they multitask during meetings they find irrelevant or have lack of interest or engagement in, which might be the underlying reason why people multitask more in larger group sizes and longer meetings.
\begin{adjustwidth}{1em}{1em}
\emph{"I tend to multitask more in larger group meetings online. Larger meetings often have topics on the agenda not directly related to what I'm working on day to day so my mind tends to wander when the topic of discussion is irrelevant. I'm definitely aware of trappings of a divided mind. I'm not necessarily productive on these other tasks. So I normally engage in menial work like cleaning and organizing files."} (R11)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"I myself frequently have a web page, source code, or build window open in another window, and I divide my attention - most often when the meeting goes to topics that don't concern me, as most of my meetings do."} (R15)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"When meetings have a lot of topics or don't apply to me, I start multitasking."} (R16)
\end{adjustwidth}
Sometimes people lose their concentration due to high cognitive load under such meetings of low relevance,
\begin{adjustwidth}{1em}{1em}
\emph{"I've noticed that I only multitask when I am tired and it is difficult for me to focus on the ongoing meeting. And I don't do difficult tasks either, at most I am checking if my PR went through or kick off a build or just look at pictures of cats. "} (R14)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"It's really hard to focus, I don't know what people are trying to say or what the actions items after or why we're discussing certain items. When I can't focus or understand what's going on, I tend to check out and look at other things or do something where I feel engaged and useful. "} (R346)
\end{adjustwidth}
\subsection{Extrinsic Factors}
\textbf{People multitask during meetings to catch up on other work.}
Another major reason (39 \% responses) why people multitask is to catch up on their work. Given the increasing number of meetings compared to the in-person work experience, people find they are having a hard time completing all of their work in time.
\begin{adjustwidth}{1em}{1em}
\emph{"It needs to happen or you cant get all your work done "} (R167)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"But these days I am having a lot of meetings, making it hard to find time to get work done. So, feel super tempted to multi task, if not entire day goes in meetings before real work gets started. Another reason to multitask is deadlines... "} (R12)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"My team is often quite bad at sticking to an agenda, so I find myself multitasking as a way to feel like I'm still able to "get things done"while I'm 'sitting in' on these meetings. As a designer, that often means that I'm clicking around in Figma. Increasingly, I am also trying to use meeting time to catch up on context for other meetings. This makes paying attention in any meeting very difficult, but with the volume of meetings and the complexity of the context I feel I need to maintain, I often feel like I have no choice."} (R9)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"Lately (since COVID) I've been forced to multitask during meetings to meet the deadlines that I've been given (and even then, I still don't always make it)... I don't have enough hours in the day to do all of the work that is required of me."} (R346)
\end{adjustwidth}
\textbf{People multitask during meetings due to external distractions.}
We also find people frequently multitask during remote meetings as a result of external distractions - under such situations, people do not purposely multitask, but their attention gets attracted by external sources. Two major classes of distractions are interface design, and the home working environment. As collaboration moves online in remote work, people are interacting with digital tools more than they used to when co-located at work, and people are mentioning that interface designs can be the cause of multitasking behavior, especially pop-ups, e.g.,
\begin{adjustwidth}{1em}{1em}
\emph{"I multi-task in almost every meeting that isn't 1-1 and even in 1-1 meetings it's hard not to multi-task because email and teams chats are popping in. In person for 1-1 i would lock my computer and focus entirely on the person and this is super hard in the WFH setup."} (R502)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"There are a lot of people are multitasking as we are using Teams. Teams is prompting up that someone is trying to get hold of us. This lures us in to checking quickly in to who it is an what they want"} (R664)
\end{adjustwidth}
On the other hand, as pointed out by prior qualitative work~\cite{allen2015effective}, remote work involves more distractions from the home working environment that could lead people to multitasking, e.g.,
\begin{adjustwidth}{1em}{1em}
\emph{"Since COVID19 the multitasking also includes: - answering children's questions - preparing food for children - Helping children with school work - resolving children's disagreements "} (R183)
\end{adjustwidth}
\textbf{People multitask during meetings for anxiety relief.}
Finally, some participants mentioned that anxiety over the COVID-19 pandemic lead them to seek methods for maintaining focus, such as conducting a low cognitive effort non-work activities while monitoring meetings.
\begin{adjustwidth}{1em}{1em}
\emph{"The current situation has increased my general anxiety. This means I have more difficulty focusing on tasks - including meetings. I have been multi-tasking with non-work tasks (i.e. a colouring game on my phone!) quite a bit as I find this actually enables me to focus better on the meeting."} (R5)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"Doing some exercises with my shoulders/back during meetings (with video turned off) is great."} (R13)
\end{adjustwidth}
\section{RQ2: WHAT DO PEOPLE DO WHEN MULTITASKING}
Our analysis further shows that people engage in work-related and non-work-related tasks when multitasking during remote meetings.
\subsection{Work-related tasks}
\label{sec:file_multitasking_qualitative}
Communication with coworkers is one of the most frequently mentioned multitasking behavior, since people generally consider that communication is quick to complete without much need to focus, or "light weight''. In fact, 29 \% of diary study participants mention that they engage in email multitasking, which further confirms and strengthens our motivation to analyze email multitasking using the telemetry data set.
\begin{adjustwidth}{1em}{1em}
\emph{"I might use that time to reply to simple emails that don't require much thinking (so I can also pay attention to the meeting)."} (R170)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"And there are different levels of multitasking, meaning the kinds of multitasking things I will do during meetings can range from writing down a quick reminder to myself on a different topic (takes a few seconds) to triaging email which takes very short attention shifts and I can come back into the proceedings easily, to more cognitively demanding tasks like writing emails, responding to IM threads etc." } (R171)
\end{adjustwidth}
Meanwhile, several diary study participants mentioned that they also frequently check and test scripts that take time to run during remote meetings, which is also rather lightweight.
\begin{adjustwidth}{1em}{1em}
\emph{"There have been situations where I've multitasked in the sense of responding to email or checking on the results of a long running job while in a meeting that is more focused on consumption of information rather than on my own contributions."} (R180)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"Because I have done a lot testing and building scripts. It takes time, so I can work with other stuff while waiting for the result."} (R337)
\end{adjustwidth}
People also mentioned that they engaged in file multitasking, yet these activities are often related to the meeting, e.g., notes taking (R182, R183, R344), checking relevant files (e.g. \emph{"in a meeting discussing aspects of the project I linked to the latest documentation."}, R174), etc., which could be a possible explanation of why there is no significant relationship between file-related multitasking and meeting characteristics.
\subsection{Non-work-related tasks}
Non-work tasks is also an emerging theme in the diary study responses. For instance, checking social media as a break from work.
\begin{adjustwidth}{1em}{1em}
\emph{"I've been multitasking both for personal and professional things - answering emails and chats for work, taking social media and texting breaks for personal."} (R168)
\end{adjustwidth}
Given that working from home environment under COVID-19, people also reported that they engaged in eating, exercise (primarily for anxiety relief and wellness) and other household chores.
\begin{adjustwidth}{1em}{1em}
\emph{"As my kitchen is very close to the desk with my computer, I also get food or drinks from there during the meetings more often."} (R20)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"For me personally, I am more likely to do another task that does not require the computer like managing my to do list, writing down notes, cleaning up my office and desk, eating etc."} (R171)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"It has been beneficial to walk around my house while on meetings (that don't require me to be in front of my computer). I'm able to put dinner in the oven, feed the dogs, etc."} (R668)
\end{adjustwidth}
\section{RQ3: THE CONSEQUENCES OF MULTITASKING}
Finally, we present our findings on the consequences of multitasking during remote meetings. Although multitasking is typically associated with negative outcomes such as decreased task performance ~\cite{monk-interruptionCost, cutrell-interruptionCost}, difficulties in decision making \cite{speier-decisionmakingCost} and negative affect \cite{zijlstra-affect, Bailey2006OnTN}, our participates report that in-meeting multitasking leads to both positive and negative outcomes.
\subsection{Positive Outcomes}
\textbf{Multitasking may help boost productivity.} First, multiple participants (15 \%) mentioned that multitasking helps boost their productivity, which echos prior works on the benefits of multitasking on efficiency \cite{dabbish-interruptionBenefits, maglio-peripheral, rich-multitaskingBenefits}. Given that, under current remote work settings, there are many more remote meetings compared to the pre-COVID-19 period, people explained how multitasking helps them to get work done. Here's a representative response,
\begin{adjustwidth}{1em}{1em}
\emph{"There are some benefits to multi-tasking. I've been able to get more work done. I've been less frustrated by meetings that weren't very useful to me. I haven't made any significant mess ups in a meeting yet when I've done it."} (R330)
\end{adjustwidth}
Multitasking is at its most productive when workers understand that their own and others' attention in a meeting is a spectrum about which they can make active meeting choices \cite{kuzminykh2020low}.
\begin{adjustwidth}{1em}{1em}
\emph{"I find myself multi-tasking in meetings that do not require my attention, but not in meetings that do. In some ways, I may be more productive given the ease of multi-tasking in remote meetings."} (R508)
\end{adjustwidth}
People also mentioned the positive experience with meeting-related multitasking behavior, such as note taking and searching for information, e.g.,
\begin{adjustwidth}{1em}{1em}
\emph{"The type of multitasking that I feel it has a positive impact on my productivity is when I am taking notes during a meeting or when I am navigating the internet to find some relevant information that is being discuss in the meeting."} (R668)
\end{adjustwidth}
\subsection{Negative Outcomes}
\textbf{Multitasking leads to loss of attention/engagement.}
Nevertheless, remote meeting multitasking does cause negative consequences. Among them, the most frequently (36 \%) mentioned negative aspect is loss of attention/engagement, where people lose track of the meeting content (which sometimes is important) due to multitasking activities, as demonstrated by the following cases,
\begin{adjustwidth}{1em}{1em}
\emph{"Its easy to get distracted by multitasking and miss something in the meeting."} (R2)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"I have to channel my concentration on 1 primary task -- whether that be the meeting itself or the side work I'm doing while listening in."} (R346)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"If you leave the meeting window to open a deck or other files, it's hard to get back to the meeting window. Same with chats - if you leave the meeting window to send chats to other people, it's hard to get back to the meeting."} (R344)
\end{adjustwidth}
These observations are well-aligned with previous conclusions on the impact of multitasking behavior on people's attention and task resumption \cite{Bailey2006OnTN, altmannResumption}.
\textbf{Multitasking leads to mental fatigue.} Moreover, we find that remote meeting multitasking behavior does have an impact on well-being: some participants reported that they feel tired after multitasking during remote meetings.
\begin{adjustwidth}{1em}{1em}
\emph{"I tire a bit more with so many meetings and multitasking."} (R72)
\end{adjustwidth}
\textbf{Multitasking can be disrespectful.} One final downside of multitasking is that it has been sometimes regarded as an inappropriate behavior during remote meetings. Some participants explicitly associate multitasking with being impolite.
\begin{adjustwidth}{1em}{1em}
\emph{"I tend to do it less while on video, so they can't tell I'm being rude."} (R173)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"I rarely multitask when on camera--it seems rude."} (R673)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"People are becoming a bit more brazen; it's sometimes unbelievably clear that they aren't paying close attention to the discussion and with the current WFH situation it's hard not to notice eyes straying, backlight (windows) changing, etc. It adds another dimension of rudeness but I also do it to others, sometimes not realizing when I follow the stray notification."} (R343)
\end{adjustwidth}
\begin{adjustwidth}{1em}{1em}
\emph{"I have gotten caught off guard a few times though. Someone will ask me a question and I'll have to ask them to repeat it. They don't seem upset, but I'm still embarrassed/ashamed when it happens. This is mostly due to the fact that I know I have slightly negative feelings toward other people when they get caught in the same situation, or when they ask a question that had been asked and answered recently. I don't want to be one of those people or be thought of as one of those people who is not paying attention."} (R330)
\end{adjustwidth}
\section{Best-practice guidelines for remote meetings}
\label{sec:best_practice}
Remote meetings have become the primary way that people connect and collaborate while working from home. As the number and duration of remote meetings has increased, people appear to have been left with less time to focus on their work and thus have gotten into the habit of multitasking to catch up. This draws participants' attention away from the meeting, and can lead to mental fatigue and disrespectful behaviors. Based on our findings, we propose several practical remote meeting best practices that meeting organizers can use to help people attending the meeting actually attend to the meeting. We note, however, the importance to consider specific worker and corporation contexts when applying these guidelines.
\textbf{Avoid important meetings in the morning.}
As demonstrated by Fig. \ref{fig:workrhythm}, people still adhere to a similar ``double peak'' daily work rhythm~\cite{mcduff2019longitudinal} as they did in pre-COVID, co-located work settings. Through our regression analysis, we find that email multitasking behavior occurs most often in the morning, which coincides with the fact that email actions peak in the morning. Prior evidence also suggests that people are most focused in mid afternoons~\cite{mark2014bored}. Therefore, our results suggest that meeting organizers might avoid scheduling important meetings in the morning, when it is harder for people to concentrate. However, multitasking in the morning is not always bad, as confirmed by our diary study participants (e.g., "balancing home and work"). In fact, scheduling light-weight meetings in the morning may actually help people smoothly transition from ``home'' to ``work'' mode under remote work settings \cite{williams2018supporting,jachimowicz2020between}.
\textbf{Reduce the number of unnecessary meetings.}
Many participants stated that they multitasked because they found that the content of some meetings did not apply to them, especially for information sharing and daily stand up meetings. In the telemetry data analysis, we also found that there was much more email multitasking in recurring and scheduled meetings compared to ad hoc meetings. Given there are so many meetings people now need to attend, meeting organizers should reconsider the necessity of numerous meetings, or the frequency of such meetings, so as to help people better focus. The organizers may also consider sharing information asynchronously. For example, sending out a recording of a presentation for attendees to watch on their own, and then only use the meeting for discussion.
\textbf{Shorten meeting duration and insert breaks.}
In our telemetry analysis, we found that longer meetings are associated with more multitasking behavior, which is also verified by the qualitative evidence. As suggested by prior literature, humans have an upper time limit where they can fully engage and pay attention~\cite{mark2017blocking}, thus we suggest that meeting organizers should shorten the duration of meetings, or insert breaks when meetings have to run long.
\textbf{Encourage active contribution from the appropriate number of attendees.}
Finally, many participants mentioned that they multitasked because they don't have anything to actively contribute to the meeting discussion. They generally muted themselves and turned off their video in such scenarios. If organizers want the full attention of participants in an important meeting, they should encourage participants to actively engage through stimulating interactions, especially if it is a large meeting with a variety of attendees. For meetings where active engagement is particularly important, then the invitee list should be as small as necessary to achieve the right level of engagement from attendees.
\textbf{Allow space for positive multitasking.}
Our findings suggest that multitasking can be positive under remote environment. Therefore, meeting organizers could consider creating personalized meeting agenda so that people are aware of the timing when relevant agenda items come up. Organizers can also implement a convention where video-on implies full attention, and video-off signals multitasking.
\section{Design implications for better support of remote meetings}
We have seen that multitasking in remote meetings is a complex behavior, with both positive and negative aspects. It can help people be more productive, but may also reduce attention, increase fatigue, and appear impolite. While culturally the term multitasking may have a negative connotation \cite{przybylski2013can}, our study adds to a growing body of work that reevaluates differential attention and multitasking in remote work contexts \cite{kuzminykh2020low} and for users of all abilities \cite{das_towards_frth}. Based on our findings, we argue that given the complext nature of remote meeting multitasking, it is important to encourage its positive aspects while reducing its negative implications. In this section we discuss several ways productivity tools might do this.
\textbf{Support a `focus mode' for remote meetings.}
Our analysis shows that pop-ups in current software interfaces during remote meetings distract people from the meeting itself. To alleviate such distractions, we envision future collaboration platforms having a remote meeting `focus mode'. After people choose to enter the mode
(e.g., for a very important meeting), the tool could block all standard pop-up messages, emails, etc., so as to help them concentrate on the meeting itself. The focus mode could also employ a multitasking alert feature. People could give permission to the app to track their behavior in other applications or even their other devices. For non-meeting focused behavior (speaking, screensharing, in meeting parallel chat, etc.), the app could alert people about their multitasking and reflect back the reason (e.g., not being able to absorb all of the information within the meeting). As such, the feature could help people engage in the meeting and avoid unintended loss of attention.
\textbf{Support other types of engagement during remote meetings.}
Some positive multitasking fits the meeting purpose, but technically requires moving outside the meeting window which may increase the risk of being distracted from the meeting. For instance, people often need to work outside the actual meeting window or platform to take notes or work on files, switching back and forth between this work and the meeting. The ability to have more windows, split views, or even a more broadly-defined meeting space could not only reduce the potential for distraction but also improve the shared use of these resources. Given the various reasons for remote meeting multitasking, it is also important to not arbitrarily consider it unacceptable. Meetings could enable attendance along a scale of high to low engagement, to help set attendees' expectations and more closely match actual behaviour~\cite{kuzminykh2020low}.
\textbf{Help people decide which meetings to attend.}
Our study suggests that over-multitasking or negative multitasking is associated with the increasing number of meetings during remote work. Apart from organizational leaders actively reducing the number of meetings, future remote meeting tools could develop a feature for people to self-rank the importance of each meeting, or recommend an importance level assigned to each meeting for each individual based on meeting characteristics (e.g., content, size, attendees, etc.), and add the importance level of each meeting on the person's calendar apps and video conferencing platforms. As such, people will have a better idea for each day which meeting is important and when they should pay special attention in order to not miss key information (thus avoiding multitasking). For less important meetings, the system could recommend ways to catch-up later, alternative ways to attend, or help notify the meeting organizer.
\textbf{Help people skip some parts of the meeting.}
As suggested by our findings, even within the same meeting, certain parts of the meeting may not be so important as other parts for a specific attendee: the attendee may only be interested in a particular section, and works on other things except for that section. If meetings have agendas, one solution would be to for organizers and attendees to flag expected attention per item, and add increasing and decreasing visible attention when relevant.
Future tools can also help better support personalized meeting content importance tracking for attendees. For instance, the system could make use of real time transcriptions of meeting and compare the topic similarity to those that the attendee might be interested in.
\section{Limitations and generalizability}
While our findings build on rich telemetry and diary study data to extend what was previously know about multitasking in remote meetings, it is important to consider them in the context of several limitations.
For one, the data are drawn from one global information technology company (and the telemetry data focus on US workers only), with most participants being information workers. As a result, our findings may not generalize to other worker types under remote settings, or to other cultures. Further, the analyzed data were collected during the COVID-19 pandemic era, but we were not able to distinguish remote work effects from COVID-19 effects (e.g., the impact of remote working from home on people's mental wellbeing) on multitasking behavior. Additionally, while telemetry data is valuable in providing a large-scale, realistic picture of behavior, it does not provide the motivation motivating that behavior \cite{dumais2014understanding}, we are not able to distinguish whether the email and file actions we observed were related to the meeting or not. It is likely that some of the positive multitasking observations may be false positives. Finally, the current diary study does not cover every aspect with regard to remote meeting multitasking, which can be addressed in future work, e.g., references to side-channels, such as Whatsapp, Facebook or other apps that cannot be instrumented through telemetry data analysis and people's perception of multitasking. Nevertheless, we believe that this work presents the most comprehensive analysis of remote meeting multitasking behavior currently available, and could extend to scenarios beyond the narrow-sense of workspace meetings (e.g., real-time distance education that leverages remote meeting tools \cite{chen2020large}).
We leave the above mentioned limitations as future work.
\section{Conclusion}
In this paper, we presented a large-scale and mixed-methods study of multitasking behavior during remote meetings. We analyzed a large-scale telemetry dataset and conducted a longitudinal diary study during COVID-19 period. Our analysis leads to practical guidelines for remote meeting attendees and design implications for productivity tools, both of which can improve remote meeting experiences. Our results point to the importance of multitasking in the consideration of remote meetings, both with respect to their social and technical components. On the one hand, how remote meetings are scheduled and structured are significantly associated with when and to what extent people divide their attentions. On the other hand, multitasking could imply both positive and negative outcomes for individual worker and work group. More future research efforts are needed to address remote meeting experience as it becomes mainstream, and our work provides a foundational and timely understanding of such.
\section{Alternative Regression Analysis}
\label{sec:supp}
To test the robustness of our regression model (Section~\ref{sec:regression_analysis}), we conducted an alternative analysis using Generalized Linear Mixed Effects Models~\cite{seabold2010statsmodels}. Specifically, we estimated random intercepts for workers and approximated the posterior using variational Bayes. The alternative results are shown in Table.~\ref{tbl:alternative_regression}, and they are qualitatively similar to those we present in our main analysis.
\begin{table}[!htbp]
\begin{tabular}{lcc} \hline
Effect & Post. Mean & Post. SD \\ \hline
Day:Monday & 0.2345 & 0.0308 \\
Day:Tuesday & 0.3154 & 0.0277 \\
Day:Wednesday & 0.2052 & 0.0276 \\
Day:Thursday & 0.1940 & 0.0287 \\
Type:broadcast & 0.8271 & 0.1710 \\
Type:recurring & 0.5125 & 0.0191 \\
Type:scheduled & 0.4235 & 0.0199 \\
Size:3 & 0.0591 & 0.0335 \\
Size:4-5 & 0.2152 & 0.0293 \\
Size:6-10 & 0.3656 & 0.0311 \\
Size:\textgreater 10 & 0.5788 & 0.0288 \\
Hour:morning & 0.6053 & 0.0186 \\
Hour:afternoon & 0.3760 & 0.0202 \\
Duration:20-40 min & 0.7863 & 0.0203 \\
Duration:40-80 min & 1.3049 & 0.0265 \\
Duration:\textgreater 80 min & 1.9447 & 0.0699 \\ \hline
\end{tabular}
\caption{Alternative regression results with Generalized Linear Mixed Effects Models~\cite{seabold2010statsmodels}. The model includes random intercepts for workers and is approximated using variational Bayes. The results are qualitatively similar to those of our main analysis. \label{tbl:alternative_regression}}
\end{table}
\end{document}
\endinput
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,011
|
USA Diving's DIVE INTO SERVICE program is a leadership focused mentorship and relationship building program that partners active military service members and veterans with USA Diving athletes to share like experiences as representatives of the United States. As participants in DIVE INTO SERVICE, USA Diving athletes will visit military bases around the world to support and engage members of our military and will have the opportunity to learn leadership and team-building skills through sessions run by military leaders sharing their personal experiences defending our nation. USA Diving athletes will in turn have an opportunity to share their stories of setting both individual and team goals to be the best in the world while representing Team USA.
To make a donation to support this program, please continue below.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,883
|
June 2019 — Monthly Update: The Disease of Addiction
At the Jesus Center, we serve a population whose rate of addiction—as high as 38% according to the National Coalition for the Homeless, a modest estimate in my opinion— is 25% higher than the estimated national average of 10%. Here, my staff and I hear stories from Jesus Center participants about how they first got addicted to drugs or alcohol. Here are some examples of some of the more common stories we hear:
A young woman, kicked out of her apartment by an abusive boyfriend, is assaulted while trying to sleep in a park, and decides to take a hit of meth the next night in order to stay awake and on-guard.
As a twelve-year old child, a participant was introduced to cocaine by his addicted parents and quickly discovered that it numbed the pain of abuse and neglect—and has been addicted for decades since. As one man said while sharing his story, "It's hard to quit when your parents are your drug dealers."
A man in his late thirties, already stressed due to being laid off, begins hearing threatening voices in his head—and discovers that heroin quiets these voices, and calms his feelings of anxiety or depression about his situation.
Whether from childhood abuse, war, or domestic violence, an individual with PTSD resorts to drinking or drugs in order to suppress painful memories, and eventually gets to the place where being sober at all is too much to bear.
What do these stories have in common?
In each case, drug or alcohol use begins as a way for someone to attempt to cope with trauma. The stereotype that an addict is an ignorant, lazy freeloader who doesn't want to contribute to society just doesn't hold. Addiction is not a character flaw: it is a disease. And I can't think of any stories I've heard where drug abuse is not traced back to some form of pain: it is nearly always a symptom of a greater struggle.
We need go no farther into the Bible than Genesis to see that God's word affirms trauma as a cause of substance abuse. Take a moment to imagine this: you and your immediate family are the sole survivors of a devastating flood that has destroyed everything—your home, your entire community, the only landscape you've ever known. How would you cope with all the loss, the change, the survivor's guilt? In Genesis 9:18-28, we read that after the floodwaters recede, Noah plants a vineyard, gets drunk off the wine and passes out naked in his tent, at which point his youngest son Canaan accidentally discovers his unconscious father. Canaan then instructs his brothers, Shem and Japheth, to cover him up without looking at him. Upon realizing that Canaan saw him in such an exposed and vulnerable state, Noah furiously curses him and declares that he hopes Canaan will be the lowest of slaves to his brothers!
As I've been reading this passage this past week, it has spoken to me so much about how humans cope with pain. Even Noah, the one who obeyed God's command to build the ark that would make him and his family the sole survivors of God's destruction, seems to have suffered unbearable trauma as a result of his experiences, and turned to substance abuse to numb his pain before lashing out at the person who witnessed his suffering.
The Bible reminds us that behind every storm of addiction is a gut-wrenching, awe-inspiring story.
How do we heal the addictions of those experiencing homelessness? On one level, the antidote is often medical: whether its nicotine patches, alcohol aversion medications, or pills like methadone that interrupt and reduce cravings in opioid-addicted individuals. On a deeper level, the cure is structural: establishing stable housing, a steady job, and access to adequate medical care. But whether the storm that decimated a person's life came in the form of abuse, mental illness, generational poverty, or a complex mix of all three, we in Butte County have learned firsthand that there is no recovery from a life disaster without a strong community to offer love and support.
Download the Monthly Update
We at the Jesus Center see and experience the devastating affects of addiction regardless of its causes. We are working with our partners to become more equipped to address addiction today and into the future. And each day, we surround every human here with services, support, care, and the love of Christ—the one key that unlocks and releases the deepest of ills. With your help, we can continue to serve as a lighthouse for all who are in the throes of whatever storm pushed them into homelessness.
All News & Updates, Monthly Updates
Written by kwd
View all posts by: kwd
All News & Updates
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,596
|
\section{Introduction}
Capillary forces
can anchor a sufficiently thin elastic solid onto a fluid interface
\cite{Huang07,Gao08,Kumar20}.
Such adsorbed films offer a means to control interfaces by modifying their shape~\cite{Paulsen15}, mechanics~\cite{Vella15,Ripp20}, or permeability~\cite{Kumar18}, or by providing a substrate for physical or chemical patterning~\cite{Reynolds19}.
Crucial to such applications is an understanding of how geometric incompatibilities between a film and an interface are resolved \cite{Hure11,King12,Davidovitch19}.
Here we focus on ultrathin ($\sim 100~\si{\nano m}$) polymer films that strongly resist in-plane stretching yet readily wrinkle, allowing them to conform to a wide range of surface topographies \cite{Paulsen16}.
Such films have given a window into the rich interplay between geometry and mechanics in thin solids \cite{King12,Vella15}, including connections to pattern formation in liquid crystals
\cite{Aharoni17,Tovkach20,Tobasco20}.
Current understanding in this area has been driven primarily by studies on planar sheets \cite{Bico18,Paulsen19}.
Do thin polymer shells exhibit qualitatively different behaviors from planar sheets, or is the response dictated primarily by the \textit{difference} in curvature between the film and the interface, as suggested by recent work \cite{Taffetani17,Bense20}?
More generally, can shells offer new ways to control fluid interfaces, beyond what is possible with planar sheets?
Here we study the deformations of ultrathin axisymmetric shells on curved liquid interfaces using experiment and theory.
Surprisingly, we find that over a wide range of parameters, the underlying liquid simply takes on the intrinsic shape of the shell.
This behavior is distinct from that of planar films, which are inevitably deformed by a curved liquid interface~\cite{King12,Yao13}. The ability to ``sculpt'' a liquid with a polymer shell offers a novel route to controlling the optical properties of an interface.
We form spherical polystyrene shells of Young's modulus $E = 3.4$ GPa and thickness $30<t<631~\si{\nano m}$ by spin coating onto optical lenses with radius of curvature $7< R < 500~\si{\milli m}$~\cite{SM}.
A circular domain of radius $1.8<W<11.4~\si{\milli m}$ is then cut and delivered to a flat air-water interface with surface tension $\gamma=72~\si{\milli\newton/m}$.
The mechanical properties of the shell are set by its stretching and bending moduli, $Y=Et$ and $B=Et^3/[12(1-\nu^2)]$ respectively, and its Poisson ratio $\nu=0.34$.
Our parameters place us in the high bendability regime $\epsilon^{-1}=\gamma W^2/B>10^3$~\cite{King12}: our films buckle under minute compression.
As we will show, their ability to impose their shape on a liquid is rooted in the high cost of stretching, analogous to the rigidity of a stiff mylar balloon rather than the geometric rigidity of shells that underlies the strength of architectural domes \cite{Vella12,Lazarus12}.
In our experiments, we capture the floating shell with a tube as drawn in Fig.~\ref{fig:curved}(a), so that the interface curvature can be varied continuously by injecting air with a syringe.
In the top-view images in Fig.~\ref{fig:curved}(b), we observe a central wrinkled ``core'' that shrinks as the interfacial curvature increases.
We can identify two regions with different curvatures in panels (ii) and (iii), a central core and an outer rim; the core has roughly the same size as the wrinkled region in panel (i).
This distinction disappears in panel (iv), where the curvature seems uniform.
In panel (v), radial wrinkles appear at the edge of the sheet, similar to those observed when a flat sheet is placed on a curved interface~\cite{King12}, suggesting that the liquid interface is more curved than the rest shape of the shell.
\begin{figure*}
\includegraphics[width=0.95\textwidth]{Fig1.pdf}
\caption{\textbf{Sculpting a curved liquid interface.}
(a) Experimental schematic. Adjusting the air pressure in the tube
leads to different shell configurations.
(b) Top-view images of a deformed shell with $t=60$ nm, $W=2.1$ mm, and $R=13.9$ mm, with side-view schematics. Pressure increases from left to right: \textit{i.}\ a flat interface recovers the behavior in Fig.~\ref{fig:flat}; \textit{ii.}\ part of the wrinkled region ``inflates'' to its rest curvature but a wrinkled core remains; \textit{iii.}\ the entire wrinkled region is inflated; \textit{iv.}\ the interface curvature matches the shell curvature; \textit{v.}\ radial wrinkles grow from the outer edge of the shell.
(c)
Mean interfacial curvature $H$ (non-dimensionalized by the intrinsic curvature of the shell $R^{-1}$) versus $r/W$, for a shell with $t=123$~nm, $W=2.2$~mm, and $R=13.9$~mm.
The data span from stage \textit{iii.}~(bottom yellow curve) to stage \textit{iv.}~(top curve).
The center of the shell maintains a constant curvature that is close to its intrinsic value (dashed line: $H=R^{-1}$).
}
\label{fig:curved}
\end{figure*}
To quantify the interface shape
during this process, we view a checkerboard pattern through the interface; tracking the optical distortion of the pattern allows us to deduce the height profile of the interface using a synthetic Schlieren technique~\cite{Moisy2009, SM, Demery21}.
Figure~\ref{fig:curved}(c) shows the measured mean curvature $H$ (averaged azimuthally and non-dimensionalized by $R^{-1}$) versus the fractional distance to the center, $r/W$.
The data are from a range of pressures where no wrinkles are observed [panels (iii) and (iv) in Fig.~\ref{fig:curved}(b)].
As we increase the pressure,
the curvature in the center of the shell remains approximately constant and close to the intrinsic curvature of the shell.
These observations herald the existence of a regime where the shell sculpts the fluid into its rest shape.
\textit{Model.---}
The rest shape of the shell is described by an axisymmetric height function $h(r)=r^2/(2R)$, for $0\leq r\leq W$; our shells have small slope, $W\ll R$.
The shell is placed at the interface of a liquid with density $\rho$, and a pressure drop $P_0$ is imposed across the interface at the edge of the shell, setting the curvature of the interface through the Laplace law.
The stresses in the radial and azimuthal directions, $\sigma_{rr}$ and $\sigma_{\theta\theta}$, and the height $z$ follow the F\"oppl-von K\'arm\'an equations, which read in polar coordinates~\cite{King12, SM}
\begin{align}
\partial_r(r\sigma_{rr})&=\sigma_{\theta\theta},\label{eq:inplane_r}\\
\partial_r (r\sigma_{\theta\theta})&= \sigma_{rr}+\frac{Y}{2}\left(h'^2-z'^2\right),\label{eq:compatibility}\\
z''\sigma_{rr} + \frac{z'}{r}\sigma_{\theta\theta} & = P_0+\rho g z\ ,\label{eq:vertical_force_bal}
\end{align}
where $g$ is the gravitational acceleration.
The first equation is the in-plane force balance in the radial direction.
The second equation is a compatibility condition, which highlights the role of the mismatch between the rest shape and the actual shape of the sheet as a source of stress.
The third equation is the vertical force balance, where we have discarded the bending contribution.
These equations must be supplemented with boundary conditions, provided at $r=0$ by the smoothness of the shape, $z'(0)=0$, the continuity of displacement, $\sigma_{rr}(0)=\sigma_{\theta\theta}(0)$; and at $r=W$ by the radial force balance, $\sigma_{rr}(W)=\gamma$, and the convention $z(W)=0$.
We use tension field theory to predict the shape of our shells~\cite{King12, Mansfield2005}: we impose that the stress field in any direction is positive or zero.
A vanishing stress means that compression is released by small scale features such as wrinkles; we do not describe such features and describe instead the gross shape of the sheet through the height function $z(r)$~\cite{Paulsen15}.
\begin{figure*}[t]
\includegraphics[width=0.99\textwidth]{Fig2.pdf}
\caption{\textbf{Stretching and wrinkling on a flat liquid interface.} (a) Experimental schematic.
(b) Radial (solid line) and azimuthal (dashed line) stress in the sheet from the analytic solution.
(c) Top-view image of an ultrathin shell ($t=112$ nm, $W=6.6$ mm, $R=51.5$ mm) conforming to a flat liquid interface by forming a wrinkled core
of radius $W'$ and an unwrinkled rim. Background subtracted for clarity.
(d) $W'/W$ versus $\alpha$ for shells with $30<t<631$ nm, $13.8<R<500$ mm, and $2.2<W<11.4$ mm on water ($\gamma= 72$ mN/m, filled symbols) or an aqueous solution of sodium dodecyl sulfate ($\gamma = 36$ mN/m, open symbols). Solid line: Theory with no free parameters [Eq.~(\ref{eq:W_prime})].
}
\label{fig:flat}
\end{figure*}
\textit{Flat interface.---} We first consider the situation where no pressure drop is imposed across the interface, $P_0=0$ [Fig.~\ref{fig:flat}(a)].
In this case the sheet remains flat, as $z=0$ solves the vertical force balance [Eq.~(\ref{eq:vertical_force_bal})].
Then, the solution to Eqs.~(\ref{eq:inplane_r}, \ref{eq:compatibility})
depends only on the dimensionless \emph{confinement parameter}~\cite{King12, SM}
\begin{equation}\label{eq:confinement}
\alpha = \frac{YW^2}{2\gamma R^2},
\end{equation}
which compares the tension applied at the edge, $\gamma$, to the stress
that is required to flatten the shell, $YW^2/R^2$.
There is a critical value of the confinement, $\alpha_c=8$, below which the stresses remain positive over the whole sheet [Fig.~\ref{fig:flat}(b), black lines].
On the contrary, above the critical value, the solution
should vanish in a circular region around the center of the sheet, indicating the appearance of small-scale features [Fig.~\ref{fig:flat}(c)].
Inspection of Eqs.~(\ref{eq:inplane_r}, \ref{eq:compatibility}) shows that the stress vanishes in the same region in the two directions: $\sigma_{rr}=0$ and $\sigma_{\theta\theta}=0$ for $r<W'$, so that the boundary condition at $r=0$ has to be replaced by the condition $\sigma_{rr}(W')=0$.
Solving the force balance equations with the new boundary condition for $W'<r<W$ provides the stress field in the sheet [Fig.~\ref{fig:flat}(b), red lines], and the value of $W'$:
\begin{equation}
\label{eq:W_prime}
\frac{W'}{W}=\sqrt{1-\sqrt{\frac{\alpha_c}{\alpha}}}.
\end{equation}
We thus predict a central wrinkled region
whenever $\alpha \geq \alpha_c$, having a size $W'$ that grows continuously with $\alpha$, reaching $W'=W$ in the limit $\alpha\to\infty$ [Fig.~\ref{fig:flat}(d), solid line].
A similar result was obtained for
a neutral scarred zone in a crystalline domain bound to a sphere \cite{Azadi16}.
The sheet remains unwrinkled in a rim of width $L=W-W'$, which becomes independent of
the sheet size
at large confinement:
\begin{equation}\label{eq:rim}
L\sim W\sqrt{\frac{2}{\alpha}}=2R\sqrt{\frac{\gamma}{Y}}.
\end{equation}
Our experiments on a flat bath support these predictions.
Figure \ref{fig:flat}(c) shows a circular region of disordered wrinkles surrounded by an unwrinkled rim.
The radius $W'$ of the wrinkled region is plotted in Fig.~\ref{fig:flat}(d) as a function of the confinement $\alpha$.
The data fall onto Eq.~(\ref{eq:W_prime}) over 4 orders of magnitude in $\alpha$ with no free parameters.
We also find good agreement at a second value of surface tension.
\textit{Curved interface.---}
We turn to the situation where a pressure difference is imposed across the interface.
Once again there are solutions to Eqs.~(\ref{eq:inplane_r}-\ref{eq:vertical_force_bal}) with a wrinkled core or without one.
If there is a wrinkled core with radius $W'$, then the hydrostatic pressure vanishes there: $z(r)=-P_0/(\rho g)$ for $0\leq r\leq W'$,
consistent with the vertical force balance (\ref{eq:vertical_force_bal}) in the absence of stress.
This sets the boundary condition at $r=W'$.
In the unwrinkled portion, we integrate Eqs.~(\ref{eq:inplane_r}-\ref{eq:vertical_force_bal})
numerically using the boundary value problem solver \emph{integrate.solve\_bvp} implemented in SciPy.
Figure~\ref{fig:theo} shows the numerical results
corresponding to the sheet
in Fig.~\ref{fig:curved}(c), which has a confinement $\alpha=73\gg\alpha_c$; we plot the profile of the sheet $z(r)$, its mean curvature $H(r)=[z''(r)+z'(r)/r]/2$ and the radial stress field $\sigma_{rr}(r)$ for different values of the pressure $P_0$.
The top curve of Fig.~\ref{fig:theo}(a) shows that at zero pressure, there is a wrinkled zone in the center and an unwrinkled rim at the edge; this is simply the flat interface case of Fig.~\ref{fig:flat}.
For small positive pressure, the wrinkled region ``inflates'' to the height $z^*=-P_0/(\rho g)$ with wrinkles persisting in the
center where $z=z^*$.
As in the flat case, the radial stress falls to $0$ at the edge of the wrinkled region [Fig.~\ref{fig:theo}(c)].
Remarkably, between the wrinkled region and the outer rim, the profile of the sheet is very close to its shape at rest: $RH\simeq 1$ [Fig.~\ref{fig:theo}(b)].
If the pressure is large enough, the sheet deploys completely: the wrinkles in the center are gone and the sheet is under tension everywhere.
We find that the size of this ``inflatable'' region is close to
that of the wrinkled region when the same shell is
on a flat bath,
so that
the size of the rim on a curved interface is also given by Eq.~(\ref{eq:rim}).
This phenomenology matches the experimental observations (Fig.~\ref{fig:curved}, see~\cite{SM} for a quantitative comparison).
The behavior is very different at small confinement, where there would be no wrinkles on a flat interface: in this case the shell departs significantly from its rest shape on a curved interface~\cite{SM}.
\begin{centering}
\begin{figure}[tb]
\includegraphics[width=0.9\textwidth]{Fig3.pdf}
\caption{\textbf{Numerical solution on a curved interface with $\alpha>\alpha_c$.}
We use $R/W=6.3$, $\gamma/Y=1.7\times 10^{-4}$, $W^2\rho g/Y=1.1\times 10^{-4}$ ($\alpha=73$).
(a) Profiles of shells at different pressures. Dashed lines indicate wrinkled regions. The (partially) inflated regions nearly match the initial shape.
(b) Curvature versus normalized radial position $r/W$.
Dots indicate boundaries between wrinkled and smooth regions.
A curvature $RH=1$ corresponds to the initial shape of the shell.
(c) Radial stress versus normalized radial position $r/W$.
Green curves:
Analytic solution for a flat
interface ($P_0=0$).
}
\label{fig:theo}
\end{figure}
\end{centering}
A key quantity is the minimum pressure $P_\textrm{c}$ needed to inflate the shell completely.
It can be estimated as $P_\textrm{grav}=\rho g\Delta h$, where $\Delta h=W^2/(2R)$ is the initial height of the shell.
For the parameters in Fig.~\ref{fig:theo}, we find $P_\textrm{grav} \simeq 0.20 P_\textrm{Lap}$, where $P_\textrm{Lap} = 2\gamma/R$ is the Laplace pressure required to create a liquid interface of the same curvature.
This estimate is an upper bound due to the flattened rim; the pressure needed to inflate the sheet in our numerical solution is $0.16 P_\textrm{Lap}$.
In the inextensible limit $Y\to \infty$,
the rim disappears and the sheet is perfectly inflated for
$P_\textrm{grav} < P < P_\textrm{Lap}$.
This range of pressures is shown in Fig.~\ref{fig:phase_diag}(a) as a function of the curvature of the shell; there is a wrinkled core for $P<P_\textrm{grav}$ and a wrinkled edge for $P>P_\textrm{Lap}$.
For a finite stretching modulus, the range of the ``inflated region'' increases while the size of the inflated core shrinks [Eq.~(\ref{eq:W_prime})].
Experiments with a flat sheet by King et al.~\cite{King12} correspond to the vertical axis
of Fig.~\ref{fig:phase_diag}(a)
at zero curvature; there only the ``wrinkled edge'' region is accessible.
We validate this theoretical picture by entering the inflated regime in 5 additional experiments spanning a range of curvatures and thickness, all at large confinement.
Each shell inflates to its original shape: the measured curvature in the center of the shell is in agreement with the intrinsic shell curvature [Fig.~\ref{fig:phase_diag}(b)].
\begin{figure}
\begin{center}
\includegraphics[width=0.92\textwidth]{Fig4.pdf}
\end{center}
\caption{
\textbf{Inflated regime.}
(a) Phase diagram in the inextensible limit.
(b)
Mean curvature at the center of the
shell
in the ``inflated'' regime, for shells with a large confinement.
The values are close to the intrinsic shell curvatures, $R^{-1}$ (dashed line), for a variety of shell curvatures and thicknesses.
Repeated symbols are from the same shell at different pressures.
}
\label{fig:phase_diag}
\end{figure}
\textit{Discussion.---}
We have shown how a thin interfacial shell with vanishing bending rigidity behaves qualitatively differently than a planar film.
Namely, a shell may impose its own shape on an interface over a range of pressures, offering a straightforward method to control the equilibrium shape of a fluid.
One advantage of this self-inflating regime is that the deployed shape is robust to perturbations in pressure, unlike a bare liquid interface where the curvature varies continuously with the Laplace pressure.
This property could be useful for optical applications, and it may be achieved with little intervention, which we demonstrate by inflating a shell using an oil droplet floating on water~\cite{SM}.
Although we focused on spheres, our analysis can be generalized to any axisymmetric shell.
When the stretched rim is narrow, its size should depend only on the slope of the shell at the edge, $h'(W)$, since this is the sole aspect of the shape that appears explicitly in the force balance [Eqs.~(\ref{eq:inplane_r}-\ref{eq:vertical_force_bal})].
Writing Eq.~(\ref{eq:rim}) using the slope $h'(W)=W/R$, we find $L\simeq 2W\sqrt{\gamma/Y}/h'(W)$.
This generalization is supported by a detailed analysis of a conical shell on a curved interface~\cite{SM}.
Moreover, our numerical results for a cone
show that the region
that is wrinkled for $P_0=0$ corresponds to the region that inflates to its rest shape
at sufficient pressure, just as it does for a spherical shell~\cite{SM}.
Not all axisymmetric shells
inflate to their rest shape.
The question of which shapes are maintained upon inflation dates back to the optimization of parachutes by Taylor~\cite{Taylor63}. Since then, closed surfaces have received the most attention~\cite{Paulsen94, Pak10}.
Recently, Gorkavyy
reported a condition for an axisymmetric shell to retain its shape upon inflation~\cite{Gorkavyy2010}, although
these calculations are
for a uniform pressure drop across the shell; the condition in the presence of a pressure gradient is as yet unknown.
Whatever this condition may be, our work suggests that it is satisfied for a sphere and a cone.
\begin{acknowledgements}
We are grateful to G.~M.~Grason and F.~Montel for useful discussions, G.~C.~Leggat for help with an early version of the experiment, and S.~Prasch in the Syracuse University Glass Shop for assisting with the tubes.
We thank P.~Damman and D.~Vella for useful comments on the manuscript.
This work was supported by NSF Grants No.~DMR-CAREER-1654102 (Y.T. and J.D.P.) and No.~REU DMR-1460784 (A.R.H.).
J.D.P. gratefully acknowledges support from the ESPCI Paris Total Chair.
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,912
|
Mike Lejeune, has spent over 20 years helping organizations select, engage and empower top tier talent while minimizing turnover through proven retention strategies. As a keynote speaker, facilitator, executive coach and author, Mike understands what's needed to achieve the best results possible.
Additionally, Mike loves to share his knowledge with organizations, corporations and universities alike! To get an idea on the types of leadership topics Mike speaks on, take a look below.
To learn more about a top recruiter's best-kept secrets for finishing 1st, go here. To schedule Mike for a speaking engagement, please visit the contact page.
Mike is one of only a handful of staffing industry speakers I have heard who truly crosses over, who after touching your heart, moves up to your brain and gives you the words that will come out of your mouth and lead you where you're meant to be.
Signature Leadership®: Today's employees become tomorrow's leaders, but will yours be ready?
As an Authorized Partner, Mike is now among a growing number of independent trainers, coaches, and consultants in North America authorized to use Jill Hickman Companies' industry-leading Signature Leadership Curriculum to augment their training and development solutions.
Jill Hickman Companies, headquartered in Houston, Texas, is an established leadership development and consulting business serving clients throughout the world since 1998. The Authorized Partner Network of Jill Hickman Companies is a faith-based community of collaborative professionals dedicated to personal and business growth and development.
Every time I hear Mike speak I am impressed with his uncanny ability to make learning effortless. Through his gift of storytelling I have watched him mesmerize audiences all over the country. If you want a speaker that knows the business, understands how to teach the business, and relates well to any group, you want Mike Lejeune for your next event.
Interested in booking Mike Lejeune for a speaking engagement?
Mike's unique insights are helping me evaluate my connections with my clients in ways I hadn't thought of before. We thank him for the impetus to improve our business. - Dale Samuelsen, Sweetwater Video Productions, Inc.
I would love to get more connected with you on social media. Please click the icons below to get started!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,213
|
\section{Introduction}
Low energy $e^+e^-$ annihilation into hadrons provides a source of
valuable information about the interactions of light quarks. Precise
measurements of the exclusive cross sections as well as the
total cross section appear important for different applications
like, e.g., determination of various QCD parameters like quark masses,
quark and gluon condensates~\cite{qcd},
calculations of the hadronic contributions to the muon
anomalous magnetic moment and running fine structure
constant~\cite{ej}.
The hypothesis of conserved vector current (CVC) and
isospin symmetry relate to each other the isovector part of
$e^{+}e^{-} \rightarrow$ hadrons and corresponding (vector current
induced) hadronic decays of the $\tau$-lepton~\cite{tsai,saku}.
These relations provide a possibility to use an independent
high-statistics data sample from $\tau$ decays
for increasing the precision of the
spectral functions directly measured in $e^+e^-$ annihilation~\cite{adh}.
While this idea appeared fruitful 10 years ago~\cite{adh}, further
increase of the experimental statistics in both $e^+e^-$ and $\tau$
sectors revealed unexpected problems: the $2\pi$ spectral function
determined from $\tau$ decays using CVC was significantly higher than that
obtained from $e^+e^-$, there are indications at a similar deviation
in the four-pion channel~\cite{dehz1,dehz2}.
It is therefore interesting to perform a systematic test of CVC
relations using available experimental information on various
final states. For the vector part of the weak hadronic current
the mass distribution of the produced hadronic system is
\small
\begin{equation}
\frac{d\Gamma}{dq^{2}} =
\frac{G_{F}|V_{\rm ud}|^{2}S_{\rm EW}}{32\pi^{2}m^{3}_{\tau}}(m^{2}_{\tau}
- q^{2})^{2}(m^{2}_{\tau}+2q^{2})v_{1}(q^{2}),
\end{equation}
\normalsize
\noindent
where a spectral function $v_{1}(q^{2})$ is given by the expression
\begin{equation}
v_{1}(q^{2}) =
\frac{q^{2}\sigma^{I=1}_{e^{+}e^{-}}(q^{2})}{4\pi\alpha^{2}},
\end{equation}
\noindent
and $S_{\rm EW}$ is an electroweak
correction equal to 1.0194 according to~\cite{marc}.
The allowed quantum numbers for the hadronic decays channels are:
\begin{equation}
J^{PG} = 1^{-+}, \tau \rightarrow 2n\pi\nu_{\tau},
\omega\pi\nu_{\tau}, \eta\pi\pi\nu_{\tau}, \ldots
\end{equation}
\noindent
After integration
\small
\begin{eqnarray}
\frac{{\cal B}(\tau^{-} \rightarrow X^{-}\nu_{\tau})}
{{\cal B}(\tau^{-} \rightarrow e^{-}\nu_{e}\nu_{\tau})} =
\frac{3|V_{\rm ud}|^{2}S_{\rm EW}}{2\pi\alpha^{2}}\times \nonumber \\
\int_{4m^{2}_{\pi}}^{m^{2}_{\tau}}dq^{2}
\frac{q^{2}}{m^2_{\tau}}(1-\frac{q^{2}}{m^{2}_{\tau}})^{2}
(1+2\frac{q^{2}}{m^{2}_{\tau}})\sigma^{I=1}_{e^{+}e^{-}}(q^{2}).
\end{eqnarray}
\normalsize
Theoretical predictions for the branching ratios of
different $\tau$ decay modes based on CVC were earlier given
by many authors, see the bibliography in Ref.~\cite{eid}.
New comparison of CVC based predictions with experiments on $\tau$
lepton decays was motivated by recent progress
of experiments on $\tau$ decays as well as by updated
information from $e^{+}e^{-}$ annihilation into hadrons,
coming mostly from the BaBar collaboration. In this work we focus
on two particular final states -- $\eta \pi^+ \pi^-$ and
$\eta^{\prime} \pi^+ \pi^-$.
For numerical estimates we will use the value of the electronic
branching
${\cal B}(\tau \rightarrow e\nu_{e}\nu_{\tau})=$ 17.85 $\pm$ 0.05 \%
and $|V_{\rm ud}|^2$=0.9742
recommended by RPP-2008~\cite{pdg}.
\section{$\tau^- \rightarrow \eta\pi^{-}\pi^{0}\nu_{\tau}$}
The reaction $e^{+}e^{-} \rightarrow \eta\pi^{+}\pi^{-}$ was
recently studied by the BaBar collaboration using ISR in a broad
energy range~\cite{bab}. Earlier
measurements were performed at the ND~\cite{nd}, CMD-2~\cite{cmd2}
detectors from 1.25~GeV to 1.4~GeV and at the DM1~\cite{dm1} and DM2~\cite{dm2}
detectors above this energy. Figure~\ref{eta} shows results of various
measurements. In general, they are in fair agreement with each
other within errors although below 1.4~GeV the values of the cross
section from BaBar are somewhat higher than those of previous
experiments. Above this energy the results of BaBar are significantly
higher than those of DM2, whereas they are in good agreement with
DM1. However, the measurement of the latter has much worse accuracy
compared to DM2.
\begin{figure}[htb]
\vspace{9pt}
\includegraphics[width=17pc,height=10pc,scale=0.8]{vol2.eps}
\caption{Cross section of the process
$e^{+}e^{-} \rightarrow \eta\pi^{+}\pi^{-}$}
\label{eta}
\end{figure}
We calculated the branching fraction of $\tau^- \to \eta\pi^-\pi^0\nu_{\tau}$
decay expected from the above mentioned $e^+e^-$ data using the
relation (4).
The direct integration of experimental points in the energy range
from 1.25~GeV to the $\tau$ mass using older data
gives for the branching ratio $(0.132 \pm 0.016)\%$ in agreement with
the previous estimate~\cite{eid}
while the one based on the BaBar data
gives $(0.165 \pm 0.015)\%$, where we took into account the 8\% systematic
uncertainty claimed by the authors~\cite{bab}. Averaging and inflating
the error by a scale factor of 1.50, we arrive at the CVC prediction
of $(0.150 \pm 0.016)\%$ for the energy range
(1.25-1.77)~GeV. Finally,
we add the contribution of the low energy range from 1.0~GeV to 1.25~GeV
(the BaBar data set) to obtain the total CVC expectation of
$(0.155 \pm 0.017)\%$. It can be compared to the results of the measurements
which are shown in Table~\ref{tabeta} and include
older results from CLEO~\cite{cleo1} and ALEPH~\cite{aleph}
as well as the new result of Belle reported at this Workshop~\cite{belle}:
$(0.135 \pm 0.003 \pm 0.007)\%$.
\begin{table*}[htb]
\hspace{81pt}Table 2
\hspace{81pt}Experimental values of ${\cal B}(\tau^- \to \eta\pi^-\pi^0\nu_{\tau})$
\begin{center}
\label{tabeta}
\newcommand{\hphantom{$-$}}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}lll}
\hline
Group & ${\cal B}$, \% & Ref. \\
\hline
CLEO, 1992 & $0.170 \pm 0.020 \pm 0.020$ & \cite{cleo1} \\
ALEPH, 1997 &$0.180 \pm 0.040 \pm 0.020$ & \cite{aleph} \\
Belle, 2008 & $0.135 \pm 0.003 \pm 0.007$ & \cite{belle} \\
\hline
\end{tabular}\\[2pt]
\end{center}
\end{table*}
The average of the experimental results gives
${\cal B}(\tau^- \to \eta\pi^-\pi^0\nu_{\tau})=(0.139 \pm 0.008)\%$,
which is $0.9\sigma$ lower than the prediction above.
It is also interesting to compare our result with earlier
theoretical estimates of this branching fraction, see
Table~\ref{tabetap}.
\begin{table*}[htb]
\hspace{81pt}Table 2
\hspace{81pt}Theoretical predictions for ${\cal B}(\tau^- \to \eta\pi^-\pi^0\nu_{\tau})$
\label{tabetap}
\newcommand{\hphantom{$-$}}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2.45pc}
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{@{}lll}
\hline
Method & ${\cal B}, \%$ & Ref. \\
\hline
$\rho^{\prime}$ & $\sim 0.3$ & \cite{pich87} \\
CVC & $\sim 0.15$ & \cite{gilman} \\
Eff. Lagr. & $0.14^{+0.19}_{-0.10}$ & \cite{eric} \\
Eff. Lagr. & 0.18--0.88 & \cite{kram} \\
CVC & $0.13 \pm 0.02$ & ~\cite{eid} \\
CVC & $0.14 \pm 0.05$ & \cite{nari} \\
CVC + Eff. Lagr. & $\sim 0.19$ & \cite{deck} \\
Eff. Lagr. & $\sim 0.19$ & \cite{li} \\
\hline
\end{tabular}\\[2pt]
\end{center}
\end{table*}
It can be seen that the older predictions based on the $e^+e^-$ data
and CVC agree with the much more accurate result of this work,
which uses more data,
in particular a more precise data sample of BaBar. Other predictions,
which are more theoretically driven and use the low-energy effective
Lagrangian, show a much larger spread of the results.
\section{$\tau^- \rightarrow \eta^{\prime}\pi^{-}\pi^{0}\nu_{\tau}$}
Recently, the BaBar collaboration presented the very first measurement
of the cross section of the process
$e^{+}e^{-} \rightarrow\eta^{\prime}\pi^{+}\pi^{-}$~\cite{bab},
see Fig.~\ref{fig:etap}. The cross section of the process clearly shows
resonant behavior with a maximum slightly above 2~GeV, but its values
at energies below the $\tau$ lepton mass are very small. We fit the
cross section assuming the Breit-Wigner amplitude for production of
three pseudoscalar mesons~\cite{ps3} and obtain
the following resonance parameters (mass, width and cross section at
the peak):
\begin{equation}
M = 2071 \pm 32~{\rm MeV},~~
\end{equation}
\begin{equation}
\Gamma = 214 \pm 76~{\rm MeV},~~
\end{equation}
\begin{equation}
\sigma_0 = 0.223 \pm 0.073 \pm 0.022~{\rm nb}.
\end{equation}
Here the systematic error of $\sigma_0$ is 10\% following the estimate
of the overall systematic error of the cross section in Ref.~\cite{bab}.
Taking into account that we do not estimate systematic uncertainties
on mass and width one can conclude that the resonance parameters are
compatible with those of the $\rho(2150)$~\cite{pdg}. Using (4) we
integrate the optimal curve for the cross section up to the $\tau$
mass and obtain for the branching ratio:
\small
\begin{equation}
{\cal B}(\tau^{-} \rightarrow \eta^{\prime}\pi^{-}\pi^{0}\nu_{\tau})=
(13.4 \pm 9.4 \pm 1.3 \pm 6.1) \times 10^{-6},
\end{equation}
\normalsize
where the first error is statistical (that of the fit), the second is
experimental systematic and the third is the model one estimated by
using the world average values of the $\rho(2150)$ mass and width
and varying them within errors. The obtained result is consistent with
zero and we place the following upper limit at 90\% CL using the
method of Ref.~\cite{fc}:
\begin{equation}
{\cal B}(\tau^{-} \rightarrow \eta^{\prime}\pi^{-}\pi^{0}\nu_{\tau})<
3.2 \times 10^{-5},
\end{equation}
which is two times more restrictive than the upper limit based on the
only existing measurement from CLEO~\cite{cleo2}:
\begin{equation}
{\cal B}(\tau^{-} \rightarrow \eta\prime\pi^{+}\pi^{-}\nu_{\tau})<
8 \times 10^{-5},
\end{equation}
but still an order of magnitude higher than a theoretical estimate
${\cal B}(\tau^{-} \rightarrow \eta^{\prime}\pi^{-}\pi^{0}\nu_{\tau})
\approx 4.4 \times 10^{-6}$ based on the chiral Lagrangian~\cite{li}.
\begin{figure}[htb]
\vspace{7pt}
\includegraphics[width=17pc,height=10pc,scale=0.6]{vol4.eps}
\caption {Cross section of the process
$e^{+}e^{-} \rightarrow \eta^{\prime}\pi^{+}\pi^{-}$}
\label{fig:etap}
\end{figure}
\section{Conclusions}
Using available data from $e^+e^-$ annihilation and CVC we obtained
the following results for the $\tau$ lepton branching fractions:
\begin{itemize}
\item
for the $\eta \pi^-\pi^0\nu_{\tau}$ the expected branching
is $(0.155 \pm 0.017)\%$ compatible with the world average of
$(0.139 \pm 0.008)\%$;
\item
for the $\eta^{\prime} \pi^-\pi^0\nu_{\tau}$ the upper limit is
$< 3.2 \times 10^{-5}$ or 2.5 times smaller than the
experimental upper limit of $< 8 \times 10^{-5}$, both at 90\% CL.
\end{itemize}
We are grateful to D.A.~Epifanov and E.P.~Solodov for useful
discussions. This work was supported in part by grants RFBR 06-02-16156,
RFBR 08-02-13516, RFBR 08-02-91969, INTAS/05-1000008-8328, PST.CLG.980342
and DFG GZ RUS 113/769/0-2.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,180
|
Q: ERROR: could not stat file "XX.csv": Unknown error I run this command:
COPY XXX FROM 'D:/XXX.csv' WITH (FORMAT CSV, HEADER TRUE, NULL 'NULL')
In Windows 7, it successfully imports CSV files of less than 1GB.
If the file is more then 1GB big, I get an "unknown error".
[Code: 0, SQL State: XX000] ERROR: could not stat file "'D:/XXX.csv' Unknown error
How can I fix this issue?
A: For anyone else who googled this Postgres error message after attempting to work with a >1gb file in Postgres 11, I can confirm that @亚军吴's answer above is spot-on. It is indeed a size issue.
I tried a different approach, though, than @亚军吴's and @Loren's: I simply uninstalled Postgres 11 and installed the stable version of Postgres 10.7. (I'm on Windows 10, by the way, in case that matters.)
I re-ran the original code that had prompted the error and voila, a few minutes later I'd filled in a new table with data from a medium-ish-size csv file (~3gb). I initially tried to use CSVSplitter, per @Loren, which was working fine until I got close to running out of storage space on my machine. (Thanks, Battlefield 5.)
In my case, there isn't anything in PGSQL 11 that I was relying on that wasn't in version 10.7, so I think this could be a good solution for anyone else who runs into this problem. Thanks everyone above for contributing, especially to the OP for posting this in the first place. I cured a huge, huge headache!
A: This has been fixed in commit bed90759f in PostgreSQL v14.
The file limit for the error is actually 4 GB.
The fix was too invasive to be backported, so you can only upgrade to avoid the problem. Once the fix has had some field testing, you could lobby the pgsql-hackers mailing list to get it backported.
A: With pgAdmin and AWS, I used CSVSplitter to split into files less than 1GB. Lame, but worked. pgAdmin import appends to the existing table. (Changed escape character from ' to " in order to avoid error due to unquoted text in the source file. Typically I apply quotes in LibreOffice, but these files were too big to open.)
A: It seems this is not a database problem, but a problem of psql / pgadmin. The workaround is using an admin software from the previous psql versions:
*
*Use the existing PostgreSQL 11 database
*Install psql or pgadmin from the PostgreSQL 10 installation and use it to upload the file (with the command shown in the question)
Hope this helps anyone coming across the same problem.
A: You can work around this by piping the file through a program. For example I just used this to copy from a 24GB file on Windows 10 and PostgreSQL 11.
copy t(c,d) from program 'cmd /c "type x:\path\to\file.txt"' with (format text);
This copies the text file file.txt into the table t, columns c and d.
The trick here is to run cmd in a single command mode, with /c and telling it to type out the file in question.
A: https://github.com/MIT-LCP/mimic-code/issues/493
alistairewj commented Nov 3, 2018 • ►
edited
Okay, the could not stat file "CHARTEVENTS.csv": Unknown error is actually a bug in PostgreSQL 11. Under the hood it makes a call to fstat() to make sure the file is not a directory, and unfortunately fstat() is a 32-bit program which can't handle large files like chartevents. I tested the build on Windows with PostgreSQL 10.5 and I didn't get this error so I think it's fairly new.
The best workaround is to keep the files compressed (i.e. keep them as .csv.gz files) and use 7zip to load in the data directly from compressed files. In testing this seemed to still work. There is a pretty detailed tutorial on how to do this here: https://mimic.physionet.org/tutorials/install-mimic-locally-windows/
The brief version of above is that you keep the .csv.gz files, you add the 7zip binary to your windows environment path, and then you call the postgres_load_data_7zip.sql file to load in the data. You can use the postgres_checks.sql file after everything to make sure you loaded in all the data correctly.
edit: For your later error, where you are using this 7zip approach, I'm not sure why it's not loading. Try redownloading just the ADMISSIONS.csv.gz file and seeing if it still throws you that same error. Maybe there is a new version of 7zip which requires me to update the script or something!
A: Add two lines to your CSV file: One at the begining and one at the end:
COPY XXX FROM STDIN WITH (FORMAT CSV, HEADER TRUE, NULL 'NULL');
<here are the lines your file already contains>
\.
Don't forget another newline after the \. line. Then call
psql -h hostname -d dbname -U username -f 'D:/XXX.csv'
A: This is what worked for me:
\COPY member_data.lab_result FROM PROGRAM 'gzip -dcf lab_result.dat.gz' WITH (FORMAT 'csv', DELIMITER '|', QUOTE '`')
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,503
|
Der Hedemora SK ist schwedischer Eishockeyklub aus Hedemora, der 1930 gegründet wurde. Die Mannschaft spielt in der Division 2.
Geschichte
Der Hedemora SK nahm seit der Saison 1999/2000 regelmäßig am Spielbetrieb der drittklassigen Division 1 teil. Zwischenzeitlich ist sie aber in die Division 2 abgestiegen.
Bekannte ehemalige Spieler
Adam Boqvist
Jesper Boqvist
Daniel Widing
Weblinks
Offizielle Website (schwedisch)
Schwedischer Eishockeyclub
Sport (Dalarnas län)
Gemeinde Hedemora
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,640
|
Heroic pet owner ends up in hospital after saving puppy from savage dog attack
Police are investigating the incident
Chloe Parkman
Sign up for our Newsletter to get the latest and breaking local news for the day
A pet owner has endured a trip to hospital after a savage dog attack in Sidmouth.
Craig Smith and his partner Marie Lawrence were out walking their 18-month old boxer and 10-month-old Boston terrier, Manu, along The Esplanade yesterday (March 7).
Blissfully enjoying their stroll, the pet owners said they suddenly found themselves wrestling a stranger's Staffie in order to save Manu's life.
Marie said: ''Suddenly, without any warning a Staffie moved around the front of its owner and lunged at Manu.
''Craig dived on to the floor to try and get Manu's head out of the Staffie's mouth.
''There was blood everywhere, Manu was screaming, I was screaming.
''The Staffie owner showed no concern for me, my partner or our dog, he just shouted at us and walked off.
''I am so disgusted, there was blood pouring from my partner's hand.
''My dog was bleeding. If Craig hadn't intervened I am certain Manu would be dead.''
Marie says that not only did Craig need to go to accident and emergency, but they have also been stung with a vet bill of more than £250 as Manu was severely injured, including bleeding behind his eye, bruising to both eyes and cuts and grazes.
Marie - who has had pets her whole life - says they were walking along with Manu on a one-metre long lead when the incident took place.
She said: ''As we were walking I saw the man coming along the other way with his Staffie, which was on a lead.
''We are fairly new to the area so we never let our dogs off the lead.
''The Staffie was on the opposite side of its owner so there was a gap between Manu and his Staffie.''
The Staffie, who was also on a lead, began to make its way around its owner and without any warning lunged and had Manu's entire head in its mouth.
Marie said: ''Manu was screaming.
''I was screaming and crying.
10-month old Manu suffered injuries to his eyes (Image: Marie Lawrence)
''The dog owner was shouting at Craig telling him to get off.
''There was blood everywhere.''
Despite the horrifying attack, Marie says that the owner showed no concern at all.
She said: ''When I said to him why have you not got a muzzle on your dog he just pointed at me and said 'its your dog's ******** fault.''
After grappling with the Staffie and receiving several cuts to his hand, Craig heroically managed to retrieve Manu from the dog's mouth.
Dog attack in Sidmouth (Image: Marie Lawrence)
Marie said: ''It was such a strong well built dog.
''I don't blame the breed, I've met some lovely Staffies.
''He just had no control over his own dog.
''My dog couldn't get to his dog, it was his dog that approached mine.
''Manu is now on a load of painkillers and anti-inflammatory twice a day.''
Manu has cuts and grazes all down his neck following the attack (Image: Marie Lawrence)
Despite witnessing the event, Marie says that the owner of the other dog walked off, forcing her to ring the police.
She says that when the police arrived they took Craig to A&E in Exeter to check over his injuries.
Craig has around three puncture wounds to his hand, which have now been bandaged up.
''He is on antibiotics to make sure he doesn't get an infection as well all co-codamol.
''He needs to have it re bandaged soon, he is in quite a lot of pain.
''He is a personal trainer by trade so the injury has also affected his work.
Police officers took Craig to A&E after the attack (Image: Marie Lawrence)
''I am just so disgusted that he showed no concern for me, my partner or our dog.
''The lack of morality and dignity.
''A human and a tiny puppy are both bleeding because of his dog, and he didn't even ask if we were okay or offer to pay for the vet bill.''
Shortly after the attack, Marie says a veterinary nurse came running over to help.
You can stay up-to-date on the top news near you with DevonLive's FREE newsletters – find out more about our range of daily and weekly bulletins and sign up here or enter your email address at the top of the page.
She said: ''It was so terrifying, I don't know how Craig stuck his hand in there.
''The local people in Sidmouth were so fantastic and lovely and they really helped us.
''One person ran after the man and got his photograph.''
Following the vet's advice, Marie says she has taken Manu out for a walk today (March 8) however he was very 'wary'.
''We won't go to the beach again now.
''Lead or no lead that man had no control when it attacked our dog.
The man showed no remorse following the incident (Image: Marie Lawrence)
''If that was a child that went near it, I dread to think what would have happened. ''
Officers from Devon and Cornwall Police are aware of the incident and are investigating it.
A spokesperson for Devon and Cornwall Police said: ''Police are investigating a report that a dog attacked a man and his puppy while walking along the Esplanade in Sidmouth yesterday afternoon, Sunday , March 7, at around 3.30pm.
''The dog reportedly bit the puppy and then the owner of the puppy's hand when he intervened. A man in his 30s sustained a cut to his hand in the incident.
''The owner of the offending dog was reportedly unapologetic and walked away from the scene.
''The owner of the dog, is described as being approximately 50–60 years old with grey and black hair and was wearing blue jeans, a green hooded jumper with black and white trainers. The dog was a brindle coloured Staffordshire bull terrier-type dog.''
Anyone with any information is asked to contact police on 101 pr via email at 101@dc.police.uk and quoting crime reference CR/017909/21.
Devon and Cornwall Police
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,013
|
import unittest
import httpreplay
class TestHttp(unittest.TestCase):
def test_request(self):
r = httpreplay.interpret_http("POST / HTTP/1.0\r\nConnection: close\r\n\r\n1234", True)
self.assertEqual(r.body, "1234")
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,119
|
/* bigBedLabel.c - Label things in big beds . */
/* Copyright (C) 2018 The Regents of the University of California
* See README in this or parent directory for licensing information. */
#include "common.h"
#include "bPlusTree.h"
#include "bbiFile.h"
#include "bigBed.h"
#include "hgFind.h"
#include "trix.h"
#include "trackHub.h"
#include "hubConnect.h"
#include "hdb.h"
#include "errCatch.h"
#include "hui.h"
#include "obscure.h"
void bigBedLabelCalculateFields(struct cart *cart, struct trackDb *tdb, struct bbiFile *bbi, struct slInt **labelColumns )
/* Figure out which fields are available to label a bigBed track. */
{
//struct bbiFile *bbi = fetchBbiForTrack(track);
struct asObject *as = bigBedAsOrDefault(bbi);
struct slPair *labelList = buildFieldList(tdb, "labelFields", as);
if (labelList == NULL)
{
// There is no labelFields entry in the trackDb.
// If there is a name, use it by default, otherwise no label by default
if (bbi->fieldCount > 3)
slAddHead(labelColumns, slIntNew(3));
}
else if (sameString(labelList->name, "none"))
return; // no label
else
{
// what has the user said to use as a label
// we need to check parents as well as this tdb
char cartVar[1024];
struct hashEl *labelEl = NULL;
struct trackDb *cartTdb = tdb;
while ( labelEl == NULL)
{
safef(cartVar, sizeof cartVar, "%s.label", cartTdb->track);
labelEl = cartFindPrefix(cart, cartVar);
if ((labelEl != NULL) || (cartTdb->parent == NULL))
break;
cartTdb = cartTdb->parent;
}
// fill hash with fields that should be used for labels
// first turn on all the fields that are in defaultLabelFields
struct hash *onHash = newHash(4);
struct slPair *defaultLabelList = buildFieldList(tdb, "defaultLabelFields", as);
if (defaultLabelList != NULL)
{
for(; defaultLabelList; defaultLabelList = defaultLabelList->next)
hashStore(onHash, defaultLabelList->name);
}
else
// no default list, use first entry in labelFields as default
hashStore(onHash, labelList->name);
// use cart variables to tweak the default-on hash
for(; labelEl; labelEl = labelEl->next)
{
/* the field name is after the <trackName>.label string */
char *fieldName = &labelEl->name[strlen(cartVar) + 1];
if (sameString((char *)labelEl->val, "1"))
hashStore(onHash, fieldName);
else if (sameString((char *)labelEl->val, "0"))
hashRemove(onHash, fieldName);
}
struct slPair *thisLabel = labelList;
for(; thisLabel; thisLabel = thisLabel->next)
{
if (hashLookup(onHash, thisLabel->name))
{
// put this column number in the list of columns to use to make label
slAddHead(labelColumns, slIntNew(ptToInt(thisLabel->val)));
}
}
slReverse(labelColumns);
}
}
char *bigBedMakeLabel(struct trackDb *tdb, struct slInt *labelColumns, struct bigBedInterval *bb, char *chromName)
// Build a label for a bigBedTrack from the requested label fields.
{
char *labelSeparator = stripEnclosingDoubleQuotes(trackDbSettingClosestToHome(tdb, "labelSeparator"));
if (labelSeparator == NULL)
labelSeparator = "/";
char *restFields[256];
if (bb->rest != NULL)
chopTabs(cloneString(bb->rest), restFields);
struct dyString *dy = newDyString(128);
boolean firstTime = TRUE;
struct slInt *labelInt = labelColumns;
for(; labelInt; labelInt = labelInt->next)
{
if (!firstTime)
dyStringAppend(dy, labelSeparator);
switch(labelInt->val)
{
case 0:
dyStringAppend(dy, chromName);
break;
case 1:
dyStringPrintf(dy, "%d", bb->start);
break;
case 2:
dyStringPrintf(dy, "%d", bb->end);
break;
default:
assert(bb->rest != NULL);
dyStringPrintf(dy, "%s", restFields[labelInt->val - 3]);
break;
}
firstTime = FALSE;
}
return dyStringCannibalize(&dy);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,338
|
\section{Introduction}
Let $G(V,E)$ be a simple undirected graph, where the cardinality of $V$ and $E$ are respectively denoted by $n$ and $m$. A {\em path} (also referred to as a simple path) in a graph $G$ is a sequence of $t\geq 1$ distinct vertices, $(v_1, v_2,\dots, v_t)$ such that for every $1\leq i\leq t-1$, $(v_i, v_{i+1})\in E$. These edges $(v_i, v_{i+1})$ are referred to as the path edges. The size of a path is the number of vertices in the sequence. A path of size one has no edges.
A {\em path cover} of $G$ is a set of paths such that every vertex in $V$ belongs to {\em at least} one of the paths, whereas a {\em path partition} of $G$ is a set of paths such that every vertex in $V$ belongs to {\em exactly} one path. (In some of the related literature the term {\em path cover} is used in the sense that we refer to here as {\em path partition}.)
Given a path partition, we refer to each path in the set as a {\em component}. The cardinality of a path partition refers to the number of components in it.
\pgfdeclarelayer{background}
\pgfsetlayers{background,main}
{\em The path partition number} of $G$, denoted by $\pi_{p}(G)$, is the minimum cardinality among all path partitions of $G$. The problem of finding a path partition of minimum cardinality is called {\em the path partition problem}. This problem is NP-hard \cite{Hartmanis82}, since it contains the Hamiltonian path problem as a special case. There are polynomial-time algorithms for the path partition problem for some families of graphs, including (among others)
forests \cite{skupien1974path}, interval graphs \cite{HUNG2011648}, circular arc graphs, bipartite permutation graphs \cite{srikant1993optimal}, block graphs \cite{pak1999optimal}, and cographs \cite{chang19962}.
An interesting theoretical question is to provide upper bounds on $\pi_{p} (G)$ that apply to large families of graphs. Ore~\cite{ore1961arc} proved that given a graph $G$ with $n$ vertices, if $\pi_{p}(G)\geq 2$ then $\pi_{p}(G) \leq n- \sigma_2(G)$, where $\sigma_2(G)$ is the minimum sum of degrees of two non-adjacent vertices. This theorem implies that a graph has a Hamiltonian path if $\sigma_2(G)\geq n-1$.
This theorem was generalized by Noorvash have expanded the \cite{noorvash1975} who found a relation between the number of edges in a graph and $\pi_{p}(G)$. Another early contribution to bound the path partition number is the Gallai-Milgram Theorem \cite{gallai1960verallgemeinerung}, which states that the independence number $\alpha(G)$ is an upper bound for this number. Namely, $\pi_{p} (G) \leq \alpha(G)$. This holds not only for undirected graphs, but also for a corresponding notion of path partitions for directed graphs. The minimum degree in $G$, denoted by $\delta(G)$ can also provide lower bounds on the path partition number. By the classical theorem of Dirac \cite{dirac1952some}, if $\delta(G) \geq n/2$ then $G$ has an Hamiltonian cycle.
It is worth mentioning that there are graphs where the path partition number can be very large. For example, the star graph on $n$ vertices, requires at least $n-2$ components in any path partition.
Our work relates to the following conjecture of Magnant and Martin \cite{MagnantM09}.
\begin{con}{Partition number conjecture:} \label{con1} For every $d$-regular graph $G$, \[ \pi_{p} (G)\leq \frac{n}{d+1}\]
\end{con}
Megnant and Martin proved their conjecture for all $d\leq 5$.
The upper bound in the conjecture is tight for a
graph containing $\frac{n}{d+1}$ copies of cliques on $d+1$ vertices ($K_{d+1}$).
However, for connected regular graphs, some improvement is possible. For connected cubic graphs, Reed \cite{Reed96} provided a sharper bound $ \pi_{p} (G)\leq \ceil{\frac{n}{9}}$. In the same paper, Reed conjectured that every $2$-connected $3$-regular graph has a path partition with at most $\lceil \frac{n}{10}\rceil$ components. This has been recently confirmed by Yu \cite{yu2018covering}. For every $d \geq 4$, there are connected graphs (even 2-connected) for which the path partition number of $G$ is at least $\frac{n(d-3)}{d^2+1}$. In particular, for $d\geq 13$, there are connected graphs that require $\frac{n}{d+4}$ components in any path partition.
See examples in \cite{yu2018covering,suil2010balloons,suil2011matching}.
It is worth mentioning that almost all $d$-regular
graphs are Hamiltonian for $d \geq 3$ \cite{robinson1994almost}. For a broader literary review and more information see \cite{manuel2018revisiting}.
\subsection{Related work}
The set of edges in a path partition is also referred to as a {\em linear forest}. Given a graph $G(V,E)$, its {\em linear arboricity}, denoted by $la(G)$, is the minimum number of disjoint linear forests in $G$ whose union is all $E$. This notion was introduced by Harary in \cite{harary1970covering}. The following conjecture, known as the linear arboricity conjecture, was raised in \cite{Akiyama1980}:
\begin{con}\label{con3}
For every $\Delta$ and every graph $G$ of maximum degree $\Delta$, \[la(G)\leq\ceil*{\frac{\Delta+1}{2}}\]
\end{con}
Every graph of maximum degree $\Delta$ can be embedded in some $d$ regular graph for $d=\Delta$ (this may require adding vertices). For $d$-regular graphs $ la(G)\geq \ceil*{\frac{d+1}{2}}$. This is because each linear forest $G$ can use only two of the edges for any vertex. Hence if $d$ is odd, then there are at least $\frac{d+1}{2} $ linear forests. For the case where $d$ is even, at least one of the vertices is an end-vertex of a path in a linear forest, giving us at least $\frac{d}{2} +1 $ linear forests. Therefore, the linear arboricity conjecture is equivalent to proving that $ la(G)=\ceil*{\frac{d+1}{2}}$ for every $d$-regular graph $G$.
As $G$ has $\frac{nd}{2}$ edges, and each one of them is in at least one linear forest, this implies that given $la(G)$ there is a linear forest with at least $\frac{\frac{nd}{2}}{la(G)}$ edges. Note that the cardinality of a path partition is exactly $n$ minus the number of edges in it. Hence
\begin{equation}\label{eq:linear}
\pi_p(G)\leq n- \frac{\frac{nd}{2}}{la(G)}
\end{equation}
\begin{pro}
\label{pro:odd}
For $d$ odd, \hyperref[con3] {\text{Conjecture} \ref{con3}} implies \hyperref[con1] {\text{Conjecture} \ref{con1}}.
\end{pro}
\begin{proof}
Assuming \hyperref[con3] {\text{Conjecture} \ref{con3}} to be true, for odd $d$ we have that $la(G)=\frac{d+1}{2}$. Plugging it to inequality (\ref{eq:linear}) implies $\pi_p(G)\leq n- \frac{\frac{nd}{2}}{\frac{d+1}{2}} = \frac{n}{d+1}$.
\end{proof}
We state a relaxed version of \hyperref[con1] {\text{Conjecture} \ref{con1}}:
\begin{con}{The relaxed partition number conjecture:}
\label{con2} For any $d$-regular graph $G$,
\[ \pi_{p} (G)= O\left(\frac{n}{d+1}\right)\]
\end{con}
\begin{pro}
\label{pro:even}
For $d$ even, \hyperref[con3] {\text{Conjecture} \ref{con3}} implies \hyperref[con2] {\text{Conjecture} \ref{con2}}.
\end{pro}
\begin{proof}
Assuming \hyperref[con3] {\text{Conjecture} \ref{con3}} to be true, for even $d$ we have that $la(G)=\frac{d+2}{2}$. Plugging it to inequality (\ref{eq:linear}) implies $\pi_p(G)\leq n- \frac{\frac{nd}{2}}{\frac{d+2}{2}} = \frac{2n}{d+2}$.
\end{proof}
The linear arboricity conjecture has been proved in the special cases of $d = 3,4,5,6,8$ and $10$ \cite{Akiyama1980,akiyama1981covering,enomoto1984linear,guldan1986linear}. (Remark: these results imply \hyperref[con1] {\text{Conjecture} \ref{con1}} for $d = 3,5$, by \hyperref[pro:odd] {\text{Proposition} \ref{pro:odd}}.)
The linear arboricity conjecture was shown to be asymptotically correct as $d \to \infty$. Alon \cite{alon1988linear} showed that for every $d$-regular graph $G$, $la(G) \leq \frac{d}{2} + O\left(\frac{d\log \log d}{\log d}\right)$. This result was subsequently improved to $la(G) \leq \frac{d}{2} + O\left(d^{\frac{2}{3}} (\log d)^{\frac{1}{3}}\right)$ \cite{alon2004probabilistic}, and to $la(G)\leq \frac{d}{2} + O\left(d ^{\frac{2}{3}-c}\right)$, for some constant $ c>0$ \cite{ferber2019towards}.
Plugging the best result on the linear arboricity conjecture to inequality (\ref{eq:linear}) gives the following asymptotic bounds on the path partition number of regular graphs.
$$\pi_p(G) \leq O\left( \frac{n}{ d^{\frac{1}{3}+c} }\right) $$
The linear arboricity can be thought of as an integer programming problem, where for each linear forest $F_i$ of $G$ we chose a weight $\alpha_i\in \{0,1\}$, and the goal is to minimize $\sum_{i} \alpha_i$ under the constraint that for every $e\in E$, $\sum_{i|e\in F_i} \alpha_i \geq 1$. The relaxed version of this problem, where we are allowed to pick a linear forest fractionally ($0\leq \alpha_i\leq 1$) , is called the {\em fractional linear arboricity} and is denoted by $fla(G)$.
Given $fla(G)$ for some $\alpha_i^*$ we get that:
$$|E(G)| \leq \sum_{e\in E} \sum_{i|e\in F_i}\alpha_i^*=\sum_{i}\sum_{e|e\in F_i} \alpha_i^*= \sum_{i} \alpha_i^* |F_i|\leq \max_{i}|F_i|\cdot fla(G)$$
Hence the number of edges in the linear forest of maximal size is at least $\frac{\frac{nd}{2}}{fla(G)}$. Therefore, the following inequality holds:
\begin{equation}\label{eq:fraclinear}
\pi_p(G)\leq n- \frac{\frac{nd}{2}}{fla(G)}
\end{equation}
Feige, Ravi and Singh~\cite{feige2014short} proved that $fla(G)= \frac{d}{2} + O(\sqrt{d})$, and deduced from inequality (\ref{eq:fraclinear}) that $$\pi_p(G) \leq O\left( \frac{n}{ \sqrt{d}}\right)$$
They also proved that if \hyperref[con2] {\text{Conjecture} \ref{con2}} holds, then every $n$-vertex $d$-regular graph has a tour of length $(1 + O(\frac{1}{d}))n$ visiting all its vertices.
\subsection{Our contribution }
The main purpose of the paper is proving \hyperref[con1] {\text{Conjecture} \ref{con1}} for $d=6$.
Our main result is the following theorem:
\begin{mdframed}[hidealllines=true,backgroundcolor=gray!25]
\vspace{-5pt}
\begin{theorem} \label{thm:main}
Every $6-$regular graph with $n$ vertices, has a path partition whose cardinality is at most $\frac{n}{7}$. Equivalently, the average size of a component in this path partition is at least $7$.
\end{theorem}
\end{mdframed}
In passing, we also show the following theorem for $d=5$.
\begin{mdframed}[hidealllines=true,backgroundcolor=gray!25]
\vspace{-5pt}
\begin{theorem} \label{thm:main5}
Every $5$-regular graph with $n$ vertices and no $K_6$ (clique of size $6$), has a path partition whose cardinality is at most $\frac{n}{6+\frac{1}{3}}$. Equivalently, the average size of a component in this path partition is at least $6+\frac{1}{3}$.
\end{theorem}
\end{mdframed}
\section{Proof overview}\label{sec:overview}
Given a path partition for a graph $G(V,E)$ we distinguish between three types of components.
\nomenclature[a]{\em Path partition}{A path partition (also referred to as a linear forest)
of a graph $G$ is a set of paths such that every vertex in $V$ belongs to exactly one path.
\begin{enumerate}\label{components}
\item Cycles. A component of size $t \ge 3$ is referred to as a {\em cycle component} if the induced graph on the vertices of the component contains a spanning cycle.
\item Isolated vertices. A component of size $1$.
\item Paths. All the rest of the components are referred to as {\em path components}.
\end{enumerate}\label{can}
Given a 6-regular graph $G(V,E)$, a path partition is said to be {\em canonical} if it satisfies the following properties:
\begin{enumerate}
\item It has the smallest number of components.
\item Conditioned on the first property, it has the largest number of cycles.
\item It has no isolated vertices.
\end{enumerate}
\hyperref[lem:canonical]{\text{Lemma} \ref{lem:canonical}} (extension of the work of~\cite{MagnantM09}, see Section~\ref{sec:canonical}) shows that given a partition satisfying the first two properties, one can extend it to satisfy the third property as well. Thus, every $6$-regular graph has a canonical path partition as defined above. We wish to show that the average size of a component in a canonical path partition is at least~7. We first introduce some notation that will assist in explaining the main ideas in our proof.
Given a canonical path partition, we partition the set of edges $E$ into three sets:
\begin{enumerate}
\item Path edges $E_P$: this set includes all edges of the path partition that belong to components that are paths.
\item Cycle edges $E_C$: this set includes all edges (whether part of the path partition or not) both whose endpoints are in a cycle component. Observe that in a canonical path partition there is no edge that has its endpoints in different cycles, as then the two cycles can be replaced by one component.
\item Free edges $E_F$. These are all the remaining edges. None of them is part of the path partition, and the endpoints of a free edge either lie in two different components (provided that not both of them are cycles), or within the same path component.
\end{enumerate}
We partition the set of vertices $V$ into five disjoint classes. A vertex belongs to the first applicable class:
\nomenclature[e4]{{\em balanced edges}}{A free edge with one vertices in $V_1$ and the other is in $V_2$ (see definition bellow).
\nomenclature[v0]{$V_1$}{end-vertices of paths, and vertices of cycles.
\nomenclature[v0]{$V_2$}{path vertices that are connected by a free edge to a vertex in $V_1$.
\nomenclature[v3]{$V_3$}{path vertices that are connected by two path edges to vertices in $V_2$.
\nomenclature[v5]{$V_4$}{path vertices that are connected by exactly one path edge to a vertex in $V_2$.
\nomenclature[v6]{$V_5$}{the remaining vertices.
\begin{enumerate}
\item $V_1$: end-vertices of paths, and vertices of cycles.
Observe that in a canonical path partition, there is no free edge that has its end-vertices both in $V_1$, as then either the two components can be replaced by one component, or a path components can be made into a cycle. Hence the graph $G(V_1,E_F)$ (containing only vertices from $V_1$ and their induced free edges) is an independent set.
\item $V_2$: path vertices that are connected by a free edge to a vertex in $V_1$.
\item $V_3$: path vertices that are connected by two path edges to vertices in $V_2$.
\item $V_4$: path vertices that are connected by exactly one path edge to a vertex in $V_2$.
\item $V_5$: the remaining vertices.
\end{enumerate}
A free edge is {\em balanced}
if one of its end-vertices is in $V_1$ and the other is in $V_2$. Observe that all the free edges incident to $V_1$ are balanced.
Balanced edges play a key role in our analysis. We now provide some intuition of how they may be used, and what are the issues that need to be handled. Suppose, for simplicity, that the canonical path partition contains $p$ paths and no cycles. This implies that $|V_1| = 2p$. As every vertex in $V_1$ is incident with exactly~5 balanced edges, we have that the number of balanced edges is $10p$. As every vertex in $V_2$ is incident with at most~4 balanced edges, we have that $|V_2| \ge \frac{5}{2}p$. It follows that $n=|V| \ge |V_1| + |V_2| \ge \frac{9p}{2}$. This falls short of the bound of $n \ge 7p$ that we would like to prove. Here are two alternatives, each of which by itself suffices in order to fill in the missing gap.
\begin{enumerate}
\item Show that on average, every vertex in $V_2$ is incident with only two balanced edges, rather than four.
\item Show that $|V_3 \cup V_4 \cup V_5| \ge \frac{5}{2}p$.
\end{enumerate}
Our proof will not show that either of the alternatives holds, but rather will show that a ``convex combination" of these alternatives hold. That is, in every canonical path partition, some of the $V_2$ vertices have fewer than four balanced edges. In addition there are vertices of types $V_3 \cup V_4 \cup V_5$, and the combination of these two aspects has a quantitative effect that is sufficiently large so as to conclude that $n \ge 7p$.
\addfigure A key observation in this proof plan (made also in~\cite{MagnantM09}) is that if two vertices $u,v \in V_2$ are neighbors in a path $P$, then this severely limits the number of balanced edges that they can consume.
For example, it cannot be that $u$ has a balanced edge to an end-vertex of path $P_u$ and $v$ has a balanced edge to an end-vertex of path $P_v\neq P_u$, because then the path partition is not canonical: path $P$ can be eliminated by appending one part of it to $P_u$ and the other part to $P_v$. (See Figure \ref{example:1}.)
Based on the above observation, we may infer (in some approximate sense that is made rigorous in our proof) that in a canonical path partition, a vertex from $V_2$ is either incident with only two balanced edges, or is not a path-neighbor of any other vertex from $V_2$. The former case corresponds to alternative~1, whereas the latter case is related to alternative~2 in the following sense: if no two vertices in $V_2$ are path-neighbors, {\em and in addition all paths have even size},
then necessarily $|V_3 \cup V_4 \cup V_5| \ge |V_2| \ge \frac{5}{2}p$, proving alternative~2.
However, the canonical path partition might contain paths of odd size, and then the argument above does not suffice. For example, a path of size~3 may have its middle vertex $v$ belong to $V_2$, the vertex $v$ might be incident with four balanced edges, and yet the path contributes no vertex to $V_3 \cup V_4 \cup V_5$. To compensate for this, we must have other paths in which there are strictly more vertices in $V_3 \cup V_4 \cup V_5$ than in $V_2$. Observe that indeed there must be paths of size greater than~3, because if all paths are of size~3 we have that $|V_2| \le \frac{1}{2}|V_1|$, contradicting the fact that $|V_2| \ge \frac{5}{4}|V_1|$. Observe also that in every path $P$ we have that $|V_3 \cap P| \le |V_2 \cap P|$, and hence for path $P$ to compensate for a path of size~3, the set $(V_4 \cup V_5) \cap P$ must be nonempty.
How can we infer that $V_4 \cup V_5$ has substantial size (where the interpretation of substantial depends on the extent to which alternative~1 fails to hold)? The key is to consider the set $E_F(V_3)$ of free edges incident with vertices from $V_3$. These edges cannot be incident with vertices from $V_1$ (as their free edges are incident with vertices from $V_2$). If edges from $E_F(V_3)$ are incident with vertices from $V_2$, this brings us closer to alternative~1 (as this reduces for a vertex in $V_2$ the number of balanced free edges that are incident with it), hence assume for simplicity that this does not happen either. We shall show that there cannot be many free edges joining two vertices from $V_3$, as otherwise the path partition is not canonical. There are several cases to analyze here.
\vspace{5 mm}
\addfiguretwo One such case replaces four paths $P_1, P_2, P_3, P_4$ in the path partition by only three paths, by making use of a balanced edge between $V_1 \cap P_1$ and $V_2 \cap P_2$, a free edge between $V_3 \cap P_2$ and $V_3 \cap P_3$, and a balanced edge between $V_2 \cap P_3$ and $V_1 \cap P_4$. This process removes one path edge in $P_2$ and one path edge in $P_3$. (See Figure \ref{example:2}).
Given that free edges incident with $V_3$ only rarely have their other endpoint in $V_1 \cup V_2 \cup V_3$, we can infer that $V_4 \cup V_5$ has substantial size.
\vspace{10pt}
The overview provides intuition of why one may hope that the theorem is true. The following section explains how to turn this intuition into a rigorous proof, and moreover, keep the complexity of the proof manageable.
\subsection{Proof of \hyperref[thm:main]{\text{Theorem} \ref{thm:main}} }
In order to make the proof rigorous, we shall use an accounting method, in which each vertex counts as $1$ point, and points (and also fractions of points) are transferable among vertices. We shall design a collection of point transfer rules, and will prove that using these rules every component ends up with at least~7 points. This implies that the average size of a component is at least~7.
For the purpose of defining the point transfer rules, we refine the classification of vertices introduced earlier.
\begin{itemize}
\item $V_2$ is partitioned into two subclasses depending on whether it has path neighbors in $V_2$:
\begin{itemize}
\item $V_2^a$ (``a" for ``alone") for those vertices in $V_2$ that do not have a path neighbor in $V_2$.
\item $V_2^b$ (``b" for ``both") for those vertices in $V_2$ that have a path neighbor in $V_2$.
\end{itemize}
\item We will identify certain subsets of $V_2$:
\begin{itemize}
\item A vertex $u\in V_2$
is called {\em moderate} if it has at least two balanced edges, where at least one of them is to a path.
\item A vertex $u\in V_2$
is called {\em heavy} if it has at least three balanced edges to paths. Observe that every heavy vertex is also moderate.
\end{itemize}
\item A vertex $u \in V_3$ will be called {\em dangerous}
if one of its path neighbors is heavy and the other is moderate.
\end{itemize}
\nomenclature[v4]{{\em dangerous} $V_3$ }{vertices in $V_3$ that one of its path neighbors is a heavy and the other is moderate.
\nomenclature[v0]{$V_2^a$}{vertices in $V_2$ that do not have a path neighbor in $V_2$.
\nomenclature[v0]{$V_2^b$}{vertices in $V_2$ that have a path neighbor in $V_2$.
\nomenclature[v1]{{\em heavy} $V_2$ }{vertices in $V_2$ that have at least three balanced edges to paths.
\nomenclature[v2]{{\em modereate} $V_2$ }{vertices in $V_2$ that have at least two balanced edges where at least one of them is to a path.
Every vertex starts with $1$ point, and transfers points using the following point transfer rules.
\begin{mdframed}[linecolor=black!40,
outerlinewidth=1pt,
roundcorner=.5em,
innertopmargin=1.3ex,
innerbottommargin=.5\baselineskip,
innerrightmargin=1em,
innerleftmargin=0.4em,
backgroundcolor=blue!10,
shadow=true,
shadowsize=6,
shadowcolor=black!20,
frametitle={\Large Transfer rules:},
frametitlebackgroundcolor=cyan!40,
frametitlerulewidth=14pt
]
\begin{enumerate}[leftmargin=40pt,label=\textbf{Rule \arabic*
]
\item From $V_2$ to $V_1\cap \mathcal{C}$. $v\in V_2$ transfers $\frac{1}{i}$ points to $u\in V_1\cap \mathcal{C}$ , if $u$ is a vertex of a cycle of size $i \le 6$ and $(u,v)$ is a balanced edge. \label{rule:v2_cycle}
\item From $V_2$ to $V_1\cap \mathcal{P}$. $v\in V_2$ transfers $\frac{2}{3}$ points to $u\in V_1$ if $u$ is an end-vertex of a path and $(u,v)$ is a balanced edge. \label{rule:v2_path}
\item From $V_2^a$ to dangerous $V_3$. $v\in V_2^a $ transfers $\frac{1}{6}$ points to $u\in V_3$, if $u$ is dangerous and $(u,v)$ is a free edge. \label{rule:v2a_dangerous}
\item From $V_2^b\cup V_4$ to dangerous $V_3$. $v\in V_2^b \cup V_4$ with exactly one path neighbor in $V_2$ transfers $\frac{1}{12}$ points to $u\in V_3$, if $u$ is dangerous and $(u,v)$ is a free edge. \label{rule:v4_dangerous}
\item From $V_5$ to $V \setminus V_5$. $v\in V_5$ transfers $\frac{1}{4}$ points to $u\in (V \setminus V_5)$ if $(u,v)$ is a free edge. \label{rule:v5_rest}
\end{enumerate}
\end{mdframed}
We note that due to the transfer rules, a vertex may end up with a negative number of points.
The number of points of a component in a path partition is the sum of points that its vertices have. The point transfer rules immediately imply the following propositions.
\begin{pro}
\label{pro:easy7}
Let $C$ be a cycle in a canonical path partition. Then after applying the transfer rules $C$ has at least~7 points.
\end{pro}
\begin{proofing}{Proposition \ref{pro:easy7}}
{According to the point transfer rules, a vertex of a cycle cannot transfer points to other vertices (because all its vertices are in $V_1$). Let $i$ denote the size of the cycle. If $i \ge 7$ then we are done. If $i < 7$ (and necessarily $i \ge 3$), then each vertex has at least $7-i$ balanced edges connecting it to vertices in $V_2$, because the degree of each vertex is~6. Each balanced edge contributes to the vertex $\frac{1}{i}$ points (by \ref{rule:v2_cycle}),hence, each vertex on the cycle has at least $1 + (7-i)\frac{1}{i}$ points. Hence the cycle has at least $i\cdot(1 + (7-i)\frac{1}{i}) = 7$ points, as desired.}
\end{proofing}
\begin{pro}
\label{pro:easyouter}
Let $P$ be a path in a canonical path partition. Then after applying the transfer rules the end-vertices of $P$ have $7+ \frac{5}{3}$ points.
\end{pro}
\begin{proofing}{Proposition \ref{pro:easyouter}}
{In a canonical path partition, there are no isolated vertices. Hence $P$ has~10 balanced edges incident with its two end-vertices. Applying \ref{rule:v2_path} together with the two starting points, sums up to $2 + 10\cdot \frac{2}{3}=7+ \frac{5}{3}$ points. }
\end{proofing}
\begin{pro}\label{pro:easypoints}
The only vertices that can end up with negative points are those from $V_2$. In particular:
\begin{enumerate}
\item For vertices in $V_2$:
\begin{enumerate}
\item A vertex in $V_2$ has at least $-\frac{5}{3}$ point.\label{item:v2}
\item A vertex in $V_2$ that among its balanced edges, has no edge to a path, ends up with at least $-\frac{1}{3}$ points. \label{item:v2_4cycles}
\item A vertex in $V_2$ that has among its balanced edges, one edge to a path, ends up with at least $-\frac{2}{3}$ points. \label{item:v2_1path3cycles}
\item A vertex in $V_2$ that has among its balanced edges, two balanced edges to paths, ends up with at least $-1$ points.\label{item:v2_2path2cycles}
\item A vertex in $V_2^a$ that has only one balanced edge, ends up with at least $-\frac{1}{6}$ points.\label{item:v2_1path0cycles}
\end{enumerate}
\item For vertices in $V_3$.
\begin{enumerate}
\item A vertex in $V_3$ has at least $1$ point.\label{item:v3}
\item A vertex in $V_3$ that has all its frees to $V_5$ or $V_2^a$ ends up with at least $\frac{5}{3}$ points.\label{item:v3_good}
\end{enumerate}
\item Every vertex in $V_4$ has at least $\frac{2}{3}$ points.\label{item:v4}
\item Every vertex in $V_5$ has a non-negative number of points.\label{item:v5}
\end{enumerate}
\end{pro}
\begin{proofing}{Proposition \ref{pro:easypoints}}
{
Each vertex starts with $1$ point.
\begin{enumerate}
\item \begin{enumerate}
\item A vertex in $V_2$ has four free edges and may transfer at most $\frac{2}{3}$ points on each free edge, by \ref{rule:v2_path}. Hence it has at least $1 - 4\cdot \frac{2}{3} = -\frac{5}{3}$ points.
\item An edge to a cycle can transfer at most $\frac{1}{3}$ points by $\ref{rule:v2_cycle}$. Hence if $x$ has no balanced edges to paths it will end with at least $1 -4\cdot\frac{1}{3}\geq -\frac{1}{3}$ points.
\item If among its balanced edges $x\in V_2$ has exactly one edge to a path, then on this edge, $x$ transfers $\frac{2}{3}$ points (by \ref{rule:v2_path}). The remaining three free edges might be balanced edges to cycles of size three, transferring $3\cdot \frac{1}{3}$ points (by \ref{rule:v2_cycle}). Hence the number of points on $x$ is at least $1-\frac{2}{3}-3\cdot\frac{1}{3}=-\frac{2}{3}$.
\item If among its balanced edges $x\in V_2$ has two balanced edges to paths, then these edges can transfer at most $2 \cdot \frac{2}{3}$ points (by \ref{rule:v2_path}). The remaining free edges can be balanced edges to cycles, and may transfer at most $2 \cdot \frac{1}{3}$ points (by \ref{rule:v2_cycle}). Hence, on $x$ there are at least $1- 3\cdot \frac{2}{3}\geq -1$ points.
\item An edge that is not balanced can transfer at most $\frac{1}{6}$ points from a vertex in $V_2^a$ by $\ref{rule:v2a_dangerous}$. Hence if $x\in V_2^a$ and $x$ has only one balanced edge, it has at least $1-\frac{2}{3} -3\cdot\frac{1}{6}\geq -\frac{1}{6}$ points.
\end{enumerate}
\item \begin{enumerate}
\item Vertices in $V_3$ can only receive points, hence they have at least $1$ point.
\item If $x\in V_3$ has all its frees to $V_5$ or $V_2^a$ then there are two options to consider.
\begin{itemize}
\item If $x$ has an edge that is incident to $V_5$, then by \ref{rule:v5_rest} it receives $\frac{1}{4}$ points.
\item If $x$ has an edge that is incident to $V_2^a$, then by \ref{rule:v2a_dangerous} it receives $\frac{1}{6}$ points.
\end{itemize}
Hence $x$ receives at least $\frac{1}{6}$ points on each free edge and hence ends up with at least $1+4\cdot\frac{1}{6}= \frac{5}{3}$ points.
\end{enumerate}
\item Vertices in $V_4$ have four free edges, and by \ref{rule:v4_dangerous} they may transfer no more than $\frac{1}{12}$ points on each free edge. Hence vertices in $V_4$ have at least $\frac{2}{3}$ points.
\item Vertices in $V_5$ have four free edges, and may transfer at most $\frac{1}{4}$ points on each free edge (by \ref{rule:v5_rest}).
\end{enumerate}
}
\end{proofing}
We now consider the internal vertices of a path, and show that in total they have at least $-\frac{5}{3}$ points. This combined with \hyperref[pro:easyouter]{\textbf{Proposition }\ref{pro:easyouter}} will imply that the path has at least~7 points.
We divide each path $P$ in the canonical partition into disjoint blocks. The blocks need not include all vertices of $P$. We scan the vertices of the path $P$ from left to right. The first block starts at the first vertex from $V_2$ that we encounter (if there is no such vertex, then $P$ has no blocks). Thereafter, we create a sequence $B_1B_2.....B_m$ of blocks as follows. Each block $B_i$ is of the form $B_i=X_iP_i$. Here $X_i$ is a nonempty sequence of consecutive vertices from $V_2$ that is maximal (cannot be extended neither to the left nor to the right). $P_i$ contains those vertices that follow $X_i$, but {\em excluding the vertices that are in $V_5$}. Excluding these vertices is done so as to simplify the presentation. All lower bounds proved on the number of points of $P_i$ without vertices of $V_5$ also hold when vertices of $V_5$ are added back, due to item \ref{item:v5} of \hyperref[pro:easypoints]{\textbf{Proposition} \ref{pro:easypoints}}. $P_i$ ends when either a vertex from $V_2$ is reached (and then this vertex starts a new block), or when the path ends (and then the end vertex from $V_1$ is not included in $X_i$). Hence, in a block $B_i$ that is not last ($i\neq m$), $P_i$ is nonempty and can only be of one of the following two forms: either a single vertex from $V_3$, or a pair of vertices from $V_4$ (because if there were vertices from $V_5$, they were discarded). However, in the last block, $B_m$, it could be that $P_m$ is empty or a single vertex from $V_4$.
We distinguish between four kind of blocks:
\begin{enumerate}
\item $B_i$ is of Kind $1$ if $|X_i|=1$ and $|P_i|=1$ and $P_i=u\in V_3$.
\item $B_i$ is of Kind $2$ if $|X_i|=1$ and $|P_i|=2$ (hence vertices of $P_i$ are from $V_4$).
\item $B_i$ is of Kind $3$ if $|X_i|>1$ and $|P_i|\neq 0$.
\item $B_i$ is of Kind $4$ if it is the last block of the path. ($|P_i|=0$, or $|P_i|=1$ and $P_i=u\in V_4$.)
\end{enumerate}
Considering this definition of blocks, we state two key lemmas. The proofs of these lemmas show that if the premises of the lemma do not hold, then the path partition is not canonical. The arguments proving this involve a fairly complicated case analysis, and hence the proofs of the lemmas are deferred to after the proof of the main theorem.
\begin{restatable}{lemma}{lem:blockalone}
\label{lem:blockalone}
Let $\mathcal{S}$ be a canonical path partition. Let $P$ be a path in $\mathcal{S}$.
Every block $B_i$ in $P$ has at least $-\frac{5}{3}$ points.
\end{restatable}
\begin{restatable}{lemma}{lem:twoblocks}
\label{lem:twoblocks}
Let $\mathcal{S}$ be a canonical path partition. Let $P$ be a path in $\mathcal{S}$. Any sequence of two blocks in $P$, $B_iB_{i+1}$ satisfies at least one of the following three options: \begin{enumerate}
\item The number points on $B_i$ is non-negative.
\item The number points on $B_iB_{i+1}$ is non-negative.
\item The number points on $B_iB_{i+1}$ is at least $-1$, and $B_{i+1}$ is of Kind $4$.
\end{enumerate}
\end{restatable}
Now we can prove \hyperref[thm:main]{\textbf{Theorem} \ref{thm:main}}:
\begin{proofing}{Theorem \ref{thm:main}}
{Consider a canonical path partition $\mathcal{S}$. We show that applying the set of transfer rules to $\mathcal{S}$ leads to a situation where every component in $\mathcal{S}$ has at least $7$ points. \hyperref[pro:easy7]{\textbf{Proposition} \ref{pro:easy7}} implies that we only need to handle paths. For a path $P$ we know from \hyperref[pro:easyouter]{\textbf{Proposition} \ref{pro:easyouter}} that it receives $7+\frac{5}{3}$ points on its end-vertices. Hence we need to show that the internal vertices have at least $-\frac{5}{3}$ points. All internal vertices up to the beginning of the first block contribute a non-negative number of points (by \hyperref[pro:easypoints]{\textbf{Proposition} \ref{pro:easypoints}}). Thereafter,
apply options~1 or~2 of \hyperref[lem:twoblocks]{\textbf{Lemma} \ref{lem:twoblocks}} as long as possible, removing blocks with a non-negative contribution to the number of points. Eventually, we are left with either no block (and we are done), one block (and then we can apply \hyperref[lem:blockalone]{\textbf{Lemma} \ref{lem:blockalone}}) or two blocks (with option~3 of \hyperref[lem:twoblocks]{\textbf{Lemma} \ref{lem:twoblocks}} applicable). In any case, we get that for the sequence of blocks we lose at most $\frac{5}{3}$ points, which leaves the path with at least $7$ points at the end.
}
\end{proofing}
\section{Proofs of \hyperref[lem:blockalone]{\text{Lemma} \ref{lem:blockalone}} and \hyperref[lem:twoblocks]{\text{Lemma} \ref{lem:twoblocks}}} \label{sec:key_lemmas}
In this section we prove the two key lemmas, \hyperref[lem:blockalone]{\textbf{Lemma} \ref{lem:blockalone}} and \hyperref[lem:twoblocks]{\textbf{Lemma} \ref{lem:twoblocks}}.
In order to prove these two lemmas, we state a set of lemmas that we will prove in a later section.
\begin{lemma}\label{lem:dangerous}
Let $\mathcal{S}$ be a canonical path partition. Let $P$ be a path in $\mathcal{S}$.
Let $x_1v_3$ ($v_3x_1$, respectively) be two consecutive vertices in $P$, where $x_1$ is heavy and $v_3$ is dangerous. If $(v_3,u)\in E_F$ is a free edge and $u$ has a path neighbor in $V_2$ ($u\in V_4\cup V_2^b\cup V_3 $), then:
\begin{enumerate}
\item $u$ is on the same path as $v_3$.\label{itm:1}
\item $u$ is on the right (left, respectively) of $v_3$ and has only one path neighbor in $V_2$ (hence $u\in V_4\cup V_2^b $), which we denote by $x_2$. $x_2$ is right (left, respectively) of $u$ in the path.\label{itm:2}
\item $x_2$ has exactly one balanced edge among its four free edges. This edge is incident to the end-vertex of the path that is closer to $x_2$ than to $u$. We will denote this end-vertex by $o_2$ ($o_1$, respectively).\label{itm:3}
\item $y_1\in V_2$ which is the other path neighbor of $v_3$ is moderate with exactly one balanced edge to $o_2$ ($o_1$, respectively). The rest of its free edges may be balanced edges to cycles.\label{itm:4}
\end{enumerate}
\end{lemma}
\setlength{\intextsep}{12pt
\begin{figure}[H]
\begin{subfigure}[t]{0.5\columnwidth} \resizebox{\linewidth}{!}
{\dangerouslemma}
\caption{$x_1v_3$ } \label{fig1:subfig1}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}[t]{0.5\columnwidth} \resizebox{\linewidth}{!}
{\dangerouslemmatwo}
\caption{$v_3x_1$} \label{fig1:subfig2}
\end{subfigure}
\caption{The two cases of \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} }\label{fig1}
\end{figure}
\begin{lemma}\label{lem:kadjacent}
Let $\mathcal{S}$ be a canonical path partition and let $P$ be a path in $\mathcal{S}$. Let $X_i$ be a part of some block $B_i=X_iP_i$ in path $P$ such that $|X_i|=k>1$.
Then $X_i$ ends with at least $\frac{1}{3}\cdot k-\frac{4}{3}$ points.
\end{lemma}
\begin{lemma}\label{lem:special_case}
Let $\mathcal{S}$ be a canonical path partition and let $P$ be a path in $\mathcal{S}$. Let $B_iB_{i+1}$ be two consecutive blocks in path $P$ such that $B_i$ is of Kind $1$ ($X_i=x_i$) and $|X_{i+1}|=2$.
If $x_i$ is heavy then $X_{i+1}$ ends with at least $-\frac{1}{3}$ points.
\end{lemma}
\subsection{Proof of \hyperref[lem:blockalone]{\text{Lemma} \ref{lem:blockalone}} }
\begin{proofing}{Lemma \ref{lem:blockalone}}{
Let $B_i=X_iP_i$ be a block of a path. By \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}, every vertex in $P_i$ contributes a non-negative number of points, and hence $P_i$ contributes a non-negative number of points. As to $X_i$, there are two cases to consider.
\begin{enumerate}
\item If $|X_i| = 1$ then the vertex $x_i$ that makes up $X_i$ has at least $-\frac{5}{3}$ points by item \ref{item:v2} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}.
\item If $|X_i|=k >1$ then by \hyperref[lem:kadjacent]{\text{Lemma} \ref{lem:kadjacent}} the number of points is at least $\frac{k}{3}-\frac{4}{3}\geq -\frac{2}{3}$.
\end{enumerate}
}
\end{proofing}
\subsection{Proof of \hyperref[lem:twoblocks]{\text{Lemma} \ref{lem:twoblocks}}}
We state two observations that prove some special cases of \hyperref[lem:twoblocks]{\text{Lemma} \ref{lem:twoblocks}}.
\setcounter{lemma}{2}
\begin{obs}\label{obs:kindtwoblock}
If block $B_i$ is of Kind $2$ then the number of points on $B_i$ is non-negative.
\end{obs}
\begin{proofing}{Observation \ref{obs:kindtwoblock}}
{Let $B_i=X_iP_i$ be a block of Kind $2$.
Let $X_i=x_i$ and $P_i=v_4^av_4^b$, where $v_4^a,v_4^b\in V_4$.
By \ref{rule:v4_dangerous} of the points transfer rules, vertices from $V_4$ can transfer points among free edges to {\em dangerous} vertices. Hence we split the proof into two cases (see Figure \ref{fig2}):
\begin{enumerate}[leftmargin=40pt,label=(\roman*)]
\item If $v_4^a$ has no free edge to a dangerous vertex then $v_4^a$ ends up with $1$ point. As to the rest of the block, from items \ref{item:v2} and \ref{item:v4} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}, $x_i$ and $v_4^b$ have at least $-\frac{5}{3}$ and $\frac{2}{3} $ points respectively. Hence in total we get that $X_iP_i$ ends up with at least $ -\frac{5}{3}+\frac{2}{3}+1\geq 0 $ points.
\item If $v_4^a$ has a free edge that is incident to a dangerous vertex $v_3$ then by \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} (case as in Figure \ref{fig1:subfig2}), $v_3$ must be on the same path as $v_4^a$ and must be on the \textbf{right} of it and $x_i$ will have exactly one balanced edge among its four free edges. Hence by \ref{item:v2_1path0cycles} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}, $x_i$ ends up with at least $-\frac{1}{6}$ points. By item \ref{item:v4} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}} each one of the vertices of $P_i$ has at least $\frac{2}{3}$ points and hence $X_iP_i$ will have at least $-\frac{1}{6}+\frac{4}{3}>1$ points.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.45\columnwidth} \centering \resizebox{0.9\linewidth}{!}
{\blockkindtwo{$-\frac{5}{3}$}{$+1$}{$+\frac{2}{3}$}}
\caption{$v_4^a$ has no free edge to a dangerous vertex } \label{fig2:subfig1}
\end{subfigure}
\hspace{2mm}
\begin{subfigure}[t]{0.5\columnwidth}
\centering \resizebox{1.05\linewidth}{!}
{\blockkindtwo{$-\frac{1}{6}$}{$+\frac{2}{3}$}{$+\frac{2}{3}$}}
\caption{$v_4^a$ has a free edge to a dangerous vertex $v_3$} \label{fig2:subfig2}
\end{subfigure}
\caption{\hyperref[obs:kindtwoblock]{\text{Ovservation} \ref{obs:kindtwoblock}} - Block of Kind $2$}\label{fig2}
\end{figure}
\end{enumerate}
}
\end{proofing}
\begin{obs}\label{obs:kindthreeblock}
If block $B_i$ is of Kind $3$ then number of points on $B_i$ is non-negative. (In fact, at least $\frac{1}{3}$.)
\end{obs}
\begin{proofing}{Observation \ref{obs:kindthreeblock}}
{Let $B_i=X_iP_i$ be of Kind $3$ where $|X_i|=k > 1$. By \hyperref[lem:kadjacent]{\text{Lemma} \ref{lem:kadjacent}} the number of points on $X_i$ is at least $ \frac{k}{3}-\frac{4}{3}\geq -\frac{2}{3}$. As to $P_i$, there are two cases (see Figure \ref{fig3}).
\begin{enumerate}[label=(\roman*)]
\item If $|P_i| = 1$ then the vertex of $P_i$ is in $V_3$, and by item \ref{item:v3} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}} it has at least $1$ point.
\item If $|P_i|=2$ then the two vertices of $P_i$ are from $V_4$. By item \ref{item:v4} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}} each one of them has at least $\frac{2}{3}$ points.
\begin{figure}[H]{
\centering
\begin{subfigure}[t]{0.3\columnwidth}
\centering
\resizebox{0.6\linewidth}{!}
{\blockkindthree{$2$}}
\caption{$|P_i| = 1$ } \label{fig3:subfig1}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering
\resizebox{0.8\linewidth}{!}
{\blockkindthree{$1$}}
\caption{$|P_i| = 2$} \label{fig3:subfig2}
\end{subfigure}
\caption{\hyperref[obs:kindthreeblock]{\text{Observation} \ref{obs:kindthreeblock}} - Block of Kind $3$}\label{fig3}
}
\end{figure}
\end{enumerate}
In either case, the total number of points on $B_i$ is at least $\frac{1}{3}$.
}
\end{proofing}
Using the above two observations we now prove \hyperref[lem:twoblocks]{\text{Lemma} \ref{lem:twoblocks}}.
\begin{proofing}{Lemma \ref{lem:twoblocks}}
{
By \hyperref[obs:kindthreeblock]{\text{Observation} \ref{obs:kindthreeblock}} and \hyperref[obs:kindtwoblock]{\text{Observation} \ref{obs:kindtwoblock}} we get that the only case to deal with is when $B_i$ is of Kind $1$. In this case it suffices to show that if the number of points on $B_i$ is negative, then the number of points on $B_iX_{i+1}$ is at least $-1$. This suffices by the following two cases. If $B_{i+1}$ is of kind $4$, the number of points on $B_iB_{i+1}$ is at least as the number of points on $B_iX_{i+1}$, proving option~$3$ in the lemma. If $B_{i+1}$ is not of kind $4$, then $P_{i+1}$ is not empty and has at least one point (as in the proof of \hyperref[obs:kindthreeblock]{\text{Observation} \ref{obs:kindthreeblock}}), implying that the number of points on $B_iB_{i+1}$ is non-negative.
In order to prove the sufficient condition above, we now split the proof into cases according to $X_i$.
Let $X_i=x_i$ and $P_i=v_i$, where $v_i\in V_3$.
\begin{enumerate}
\item If $x_i$ is not heavy then among its four free edges it has at most two balanced edges to paths. Hence by item \ref{item:v2_2path2cycles} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}, it has at least $-1$ points. By item \ref{item:v3} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}, $v_i$ has at least $1$ point. Hence the number of points on $B_i$ is non-negative. (See Figure \ref{fig4}.)
\begin{figure}[H]{
\centering
\resizebox{0.2\linewidth}{!}
{\blockkindone{$-1$}{$+1$}}
\caption{Block of Kind $1$ when $x_i$ is not heavy}\label{fig4}
}
\end{figure}
\item If $x_i$ is heavy then from item \ref{item:v2} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}} $x_i$ has at least $-\frac{5}{3}$ points. We now consider two cases depending on the size of $X_{i+1}$.
\begin{enumerate}
\item If $|X_{i+1}|=1$, let $x_{i+1}$ be the vertex that makes up $X_{i+1}$. We split this case into three cases. (See Figure \ref{fig5}.)
\begin{enumerate}
\item If $x_{i+1}$ is moderate and has at least two balanced edges to paths, then $v_i$ (which is dangerous) has all its free edges to $V_5$ or $V_2^a$. This is because if $v_i$ has an edge that is incident to a vertex that has a path neighbor in $V_2$, then by \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} on $x_{i}v_i$, $x_{i+1}$ cannot have more than one balanced edge to a path, contradiction.
Hence, by item \ref{item:v3_good} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}, $v_i$ ends with at least $\frac{5}{3}$ points. Thus, $B_i$ is non-negative.
\item If $x_{i+1}$ is moderate and has exactly one balanced edge to paths, then by item \ref{item:v2_1path3cycles} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}} we get that the number of points on $x_{i+1}$ is at least $-\frac{2}{3}$. As to the free edges incident with $v_i$, there are three options.
\begin{itemize}
\item If $v_i$ has a free edge that is incident to a vertex that has a path neighbor in $V_2$ then by \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} on $x_{i}v_i$ it must be incident to $V_4\cup V_2^b$. Hence by \ref{rule:v4_dangerous} it receives $\frac{1}{12}$ points.
\item If $v_i$ has an edge that is incident to $V_2^a$, then by \ref{rule:v2a_dangerous} it receives $\frac{1}{6}$ points.
\item If $v_i$ has an edge that is incident to $V_5$, then by \ref{rule:v5_rest} it receives $\frac{1}{4}$ points.
\end{itemize}
Therefore, $v_i$ receives at least $\frac{1}{12}$ points on each free edge and hence has at least $1+4\cdot\frac{1}{12}= \frac{4}{3}$ points. Thus, the number of points on $B_iX_{i+1}$ is at least $-\frac{5}{3}+\frac{4}{3}-\frac{2}{3}\ge -1$.
If $x_i$ is not heavy then among its four free edges it has at most two balanced edges to paths.
\item If $x_{i+1}$ is not moderate then there are two options. Either its all balanced edges are to cycles or it has exactly one balanced edge, and this edge is incident to a path.
\begin{itemize}
\item If $x_{i+1}$ has no balanced edges to paths, then it ends up with at least $-\frac{1}{3}$ points by item \ref{item:v2_4cycles} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}}.
\item If $x_{i+1}$ has exactly one balanced edge then by item \ref{item:v2_1path0cycles} of \hyperref[pro:easypoints]{\text{Proposition} \ref{pro:easypoints}} it ends up with at least $-\frac{1}{6}$ points.
\end{itemize}
Therefore, the number of points on $B_iX_{i+1}$ is at least $-\frac{5}{3}+1-\frac{1}{3}\geq -1$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.25\columnwidth}
\centering
\resizebox{0.6\linewidth}{!}{\blockkindoneheavy{a}}
\caption{\small If $x_{i+1}$ is moderate and has at least two balanced edges to paths }\label{fig5:subfig1}
\end{subfigure}
\hspace{7mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering
\resizebox{0.8\linewidth}{!}
{\blockkindoneheavy{b}}
\caption{\small If $x_{i+1}$ is moderate and has only one balanced edge to a path} \label{fig5:subfig2}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering
\resizebox{0.8\linewidth}{!}
{\blockkindoneheavy{c}}
\caption{If $x_{i+1}$ is not moderate } \label{fig5:subfig3}
\end{subfigure}
\caption{\small $x_i$ is heavy and $X_{i+1}$ is of size $1$}\label{fig5}
\end{figure}
\end{enumerate}
\item If $|X_{i+1}|=k > 1$, then we show that for any $k\geq 2$, $X_{i+1}$ has at least $-\frac{1}{3}$ points. Taking it together with the fact that on $B_i$ there are at least $-\frac{5}{3}+1\geq -\frac{2}{3}$ points, we get that on $B_iX_{i+1}$ there are at least $-\frac{2}{3}-\frac{1}{3}\geq -1$ points. (see Figure \ref{figheavy}.)
\begin{enumerate}
\item If $k=2$ then because $x_i$ is heavy, \hyperref[lem:special_case]{\text{Lemma} \ref{lem:special_case}} implies that $X_{i+1}$ has at least $-\frac{1}{3}$ points.
\item If $k>2$ then by \hyperref[lem:kadjacent]{\text{Lemma} \ref{lem:kadjacent}}, $X_{i+1}$ has at least $ \frac{1}{3}\cdot k-\frac{4}{3}\geq -\frac{1}{3}$ points for $k\geq 3$.
\end{enumerate}
\begin{figure}[H]
\centering
\resizebox{0.2\linewidth}{!}
{\blockkindoneend}
\caption{$x_i$ is heavy and $X_{i+1}$ is not of size $1$}\label{figheavy}
\end{figure}
\end{enumerate}
\end{enumerate}
}
\end{proofing}
\section{Properties of a canonical path partition} \label{sec:canonical}
We first prove that indeed a canonical path partition as defined on page~\pageref{can} exists. The following Lemma is based on a similar lemma that appears in~\cite{MagnantM09}.
\setcounter{lemma}{5}
\begin{lemma}\label{lem:canonical}
For $d>0$, every $d$-regular graph $G$ has a canonical path partition.
\end{lemma}
\begin{proofing} {Lemma \ref{lem:canonical}}{
We show that given that Properties 1 and 2 of canonical path partitions hold, Property 3 can be enforced to hold as well.
Let $\mathcal{S}$ be a path partition satisfying Properties $1$ and $2$ with as few isolated vertices as possible. We refer to such an $\mathcal{S}$ as a pseudo-canonical path partition. Our goal is to show that a pseudo-canonical path partition has no isolated vertices, making it a canonical path partition.
Assume towards contradiction that there is a vertex $v$ that forms a component in $\mathcal{S}$. We define a collection $\mathcal{F}$ of components of $\mathcal{S}$ by the following process. We add to $\mathcal{F}$ all those components that are incident to $v$ by free edges. These components must be paths, by Property $1$. Now recursively, in an arbitrary order, for each such path $P$, add to $\mathcal{F}$ all paths that are incident to an end-vertex of $P$ (if they were not added previously).
This process must end because the graph is finite.
\begin{obs}\label{sizethree}
Every path that is added to $\mathcal{F}$ is of size three.
\end{obs}
\begin{proofing}{Observation \ref{sizethree}}
{
Let $v$ be an isolated vertex in a pseudo-canonical path partition. Note that $v$ cannot be incident to a vertex on a cycle or to an end-vertex of a path because of Property $1$. Hence if $v$ is incident to a vertex $y$, then $y$ must be an internal vertex of a path $P$. Moreover, $P$ cannot have more than one internal vertex, because then we could make two paths out of $v$ and $P$, with at least two vertices each, contradicting the minimality of the number of isolated vertices. Hence $P$ must be a path of size three. Let $P=\{x,y,z\}$. We claim that every free edge of an end point of $P$ (say, $x$) is incident only to vertices that are internal vertices of paths of size three in $\mathcal{S}$. For $x$ one can easily create a new path partition, in which $x$ is isolated, by adding the edge $(v,y)$. This simple transformation will create a path partition that has the same number of components, cycles and isolated vertices as in $\mathcal{S}$, and hence it too is pseudo-canonical. As a result, repeating this process recursively we add to $\mathcal{F}$ only paths of size three.
}
\end{proofing}
We denote by $O$ the end-vertices of the paths in $\mathcal{F}$ and by $I$ the inner vertices.
Every path in $\mathcal{F}$ has two vertices in $O$ and one vertex in $I$. Hence, we get that $|I|\leq \frac{|O|}{2} \implies |I|< |O|$.
By Properties~1 and~2, there are no edges between vertices of $O$. Moreover, when the process of constructing $\mathcal{F}$ ends, $O$ has no neighbor outside $\mathcal{F}$. Hence, all $d$ neighbors of vertices in $O$ are in $I$. On the other hand, any vertex in $I$ can be incident to at most $d$ vertices of $O$. We get that $d|O|\leq d|I|\implies |O|\leq |I|$, and this is a contradiction.
}
\end{proofing}
\subsection{Properties of $V_3$ and the proof of \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}} \label{subsec:pro3}
In this section we prove \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}. We show that if the premises of the lemma do not hold, then the path partition is not canonical. To prove \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} we will use the following scheme. We start with a canonical path partition $\mathcal{S}$ and temporarily create from $\mathcal{S}$ a new path partition $\mathcal{S}_0$ with one more component. Then we show that we can create from $\mathcal{S}_0$ a new path partition that either has fewer components than $\mathcal{S}$, or has the same number of components as $\mathcal{S}$, but more cycles. Both of these two cases imply that $\mathcal{S}$ is not canonical.
To simplify the presentation, some components in $S_0$ will be referred to as {\em pseudo-paths}. These are components that are treated as paths but might actually be cycles, and their nature becomes apparent only in later stages. Hence $S_0$ will have components that are cycles, components that are paths, and components that are pseudo-paths.
We say that a path partition $\mathcal{S}_0$ is {\em derived} from $\mathcal{S}$ if: \label{def:derived_path}
\begin{itemize}
\item $\mathcal{S}_0$ is a path partition of size $|\mathcal{S}_0|=|\mathcal{S}|+1$
\item $\mathcal{S}$ and $\mathcal{S}_0$ have the same cycle components.
\item All the end-vertices of paths in $\mathcal{S}$ are end-vertices of either paths or pseudo-paths in $\mathcal{S}_0$.
\item The two new end-vertices (of paths or pseudo-paths) in $\mathcal{S}_0$ (that are not end-vertices in $\mathcal{S}$) are in $V_2$ of $\mathcal{S}$.
\end{itemize}
\begin{lemma}\label{lem:derived}
Let $\mathcal{S}_0$ be a path partition that is derived from $\mathcal{S}$. If one of the new end-vertices of $\mathcal{S}_0$ is heavy in $\mathcal{S}$, then $\mathcal{S}$ is not canonical.
\end{lemma}
\begin{proofing}{Lemma \ref{lem:derived}}
{ Let $\mathcal{S}_0$ be a path partition that is derived from $\mathcal{S}$. Let $x_1$ and $x_2$ be the new end-vertices of $\mathcal{S}_0$, where $x_1,x_2$ are in $V_2$ in $\mathcal{S}$, and moreover, $x_1$ is heavy in $\mathcal{S}$.
Let $(x_2,o_{x_2})\in E$ be a balanced edge incident to $x_2$ (there must be at least one such edge). Note that $o_{x_2}$ is in $V_1$ of $\mathcal{S}$, and hence it is either a cycle vertex of $\mathcal{S}_0$, or an end vertex of either a path or a pseudo-path of $\mathcal{S}_0$, but $o_{x_2} \not= x_1$.
One should consider the following cases (see Figure \ref{fig11}):
\begin{enumerate}[leftmargin=40pt, label= \textbf{Case \arabic*}]
\item If $x_1$ and $x_2$ are the end-vertices of the same path or pseudo-path, $Q_1$, in $\mathcal{S}_0$, then $o_{x_2}$ is on a component $P_{x_2}\neq Q_{1}$ (which can be a cycle). As $x_1$ is heavy (has at least three balanced edges to paths), there must be a balanced edge $(x_1, o_{x_1})\in E$ where $o_{x_1}$ is on
$P_{x_1}\neq P_{x_2}$. We then concatenate $Q_1$, $P_{x_2}$ and $P_{x_1}$ into one path, making a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1. (See Figure \ref{fig11:subfig1}.)
\item If $x_1$ and $x_2$ are on two different paths or pseudo-paths, $Q_1=o_1P_1x_1$ and $Q_2=o_2P_2x_2$ in $\mathcal{S}_0$, then there are three cases to consider.\label{case2} (See Figure \ref{fig11:subfig2}.)
\begin{enumerate}
\item If $o_{x_2}$ is an end-vertex of $Q_1$ or $Q_2$, then because $x_1$ is heavy (has at least three balanced edges to paths), there must be a balanced edge connecting $x_1$ to an end-vertex of some path $P_{x_1}\neq Q_1,Q_2$. (See Figure \ref{fig11:subfig2a}).
\begin{enumerate}
\item If $o_{x_2}=o_2$ is on $Q_2$,
then the pseudo-path $Q_2$ is a cycle. Concatenate $Q_1$ and $P_{x_1}$ into one path leads to a new path partition of size $|\mathcal{S}|$ that contains one more cycle than $\mathcal{S}$ (because $P_{x_1}$ is not a cycle), contradicting Property~2. (See blue in \ref{fig11:subfig2a}.)
\item If $o_{x_2}=o_1$ on $Q_1$,
we concatenate $Q_2$, $Q_1$ and $P_{x_1}$ into one path, making a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1. (See green in Figure \ref{fig11:subfig2a}.)
\end{enumerate}
\item If $o_{x_2}$ is on a cycle component $C$, we consider a balanced edge $(x_1, o_{x_1})\in E$ where $o_{x_1}$ is an end vertex of a path $P_{x_1}$ (such an edge must exist because $x_1$ is heavy and consequently has at least three balanced edges to paths). We concatenate $Q_1$ and $P_{x_1}$ into one path and $C$ and $Q_2$ into anther path, making a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1. (See Figure \ref{fig11:subfig2b}.)
\item If $o_{x_2}$ is on a path $P_{x_2}\neq Q_{1},Q_2$,
then consider a balanced edge $(x_1, o_{x_1})\in E$ incident to $x_1$, with $o_{x_1}\neq o_{x_2}$ being an end-vertex of a path (such an edge must exist because $x_1$ is heavy). There are three cases to consider. (See Figure \ref{fig11:subfig2c}.)
\begin{enumerate}
\item If $o_{x_1}$ is an end-vertex of $Q_1$ or $Q_2$ then we treat in a way similar to Case 2(a). (See blue in Figure \ref{fig11:subfig2c}.)
\item If $o_{x_1}$ is on the other side of $P_{x_2}$ then we create one path out of $Q_1$, $P_{x_2}$ and $Q_2$, making a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1.(See green in Figure \ref{fig11:subfig2c}.)
\item If $o_{x_1}$ is on a path $P_{x_1}\neq P_{x_2}$, then create one path out of $Q_1$ and $P_{x_1}$, and one path out of $Q_2$ and $P_{x_2}$, making a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1. (See red in Figure \ref{fig11:subfig2c}.)
\end{enumerate}
\end{enumerate}
\end{enumerate}
\begin{figure}[H]
\centering
\begin{subfigure}{0.4\textwidth}
\centering
\resizebox{\linewidth}{!}{\caseexternaltwo}
\caption{Case 1 - $x_1$ and $x_2$ are on the same path or pseudo-path $Q_1$.}
\label{fig11:subfig1}
\end{subfigure
\vspace{10mm}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}{0.27\textwidth}
\renewcommand\thesubfigure{\roman{subfigure} a}
\centering
\resizebox{\linewidth}{!}{\caseexternala}
\caption{\centering $o_{x_2}$ is an end-vertex of $Q_1$ or $Q_2$}
\label{fig11:subfig2a}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}{0.27\textwidth}
\addtocounter{subfigure}{-1}
\renewcommand\thesubfigure{\roman{subfigure} b}
\centering
\resizebox{\linewidth}{!}{\caseexternalb}
\caption{\centering $o_{x_2}$ is on a cycle $C$ }
\label{fig11:subfig2b}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}{0.27\textwidth}
\addtocounter{subfigure}{-1}
\renewcommand\thesubfigure{\roman{subfigure} c}
\centering
\resizebox{\linewidth}{!}{\caseexternalc}
\caption{\centering $o_{x_2}$ is on a component $P_{x_2}\neq Q_{1},Q_2$}
\label{fig11:subfig2c}
\end{subfigure}
\addtocounter{subfigure}{-1}
\caption{Case 2 - $x_1$ and $x_2$ are on different paths or pseudo-paths. } \label{fig11:subfig2}
\end{subfigure}
\caption{\hyperref[lem:derived]{\text{Lemma} \ref{lem:derived}} where $x_1$ is heavy }
\label{fig11}
\end{figure}
}
\end{proofing}
The following observations consider situations where a dangerous vertex, $v_3$, has a free edge to a vertex $u$. All of them are proven easily using \hyperref[lem:derived]{\text{Lemma} \ref{lem:derived}}.
\setcounter{lemma}{3}
\begin{obs}\label{different}
Let $\mathcal{S}$ be a canonical path partition. Let $(u,v_3)\in E_F$ be a free edge, such that $v_3$ is dangerous. If $u$ is not on the same path as $v_3$, then $u$ has no path neighbor in $V_2$.
\end{obs}
\begin{proofing}{Observation \ref{different}}
{Let $\mathcal{S}$ be a canonical path partition. Let $(u,v_3)\in E_F$ be a free edge, such that $v_3$ is dangerous and $u$ is not on the same path as $v_3$. Assume towards contradiction that $u$ has a path neighbor $x_2\in V_2$. Let $P_1=P_1'x_1v_3y_1P_1^{''}$ be the path that $v_3$ lies on ($x_1$ is the heavy neighbor of $v_3$). Let $P_2=P_2'x_2uP_2^{''}$ be the path that $u$ lies on ($x_2\in V_2$). (See Figure \ref{fig7:subfig1}.) We can then create from $P_1$ and $P_2$ the following three paths: $Q_1= P_1'x_1$, $Q_2=P_2'x_2$ and $Q_3=P_2^{''}uv_3y_1P_1^{''}$. From the union of $\{Q_1,Q_2,Q_3\}$ and $\mathcal{S}\backslash \{P_1, P_2\}$ we create a path partition $\mathcal{S}_0$. (See Figure \ref{fig7:subfig2}.) Note that $\mathcal{S}_0$ is a derived path partition from $\mathcal{S}$ (as defined on page~\pageref{def:derived_path}). In this derived path partition, $x_1$, which is one of the ``new" end vertices, is heavy. Hence by \hyperref[lem:derived]{\text{Lemma} \ref{lem:derived}}, $\mathcal{S}$ is not canonical, which is a contradiction to $\mathcal{S}$ being canonical.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.4\columnwidth} \resizebox{\linewidth}{!}
{ \difpath{$x_1$}{$v_3$}{$u$}{$x_2$}}
\caption{Before } \label{fig7:subfig1}
\end{subfigure}
\hspace{15mm}
\begin{subfigure}[t]{0.4\columnwidth} \resizebox{\linewidth}{!}
{\difshiled{$x_1$}{$x_2$}{$u$}{$v_3$}}
\caption{After} \label{fig7:subfig2}
\end{subfigure}
\caption{\hyperref[different]{\text{Observation} \ref{different}}} \label{fig7}
\end{figure}
}
\end{proofing}
\begin{obs}\label{samesides}
Let $P= P_1x_1v_3y_1P_2x_2uP_3$ ($x_1,x_2$ are on the left side) be a path in a canonical path partition $\mathcal{S}$, where $(v_3,u)\in E_F$ and $x_1$ is heavy. Then $x_2\not \in V_2$. Likewise, if the premises hold for $P= P_1y_1v_3x_1P_2ux_2P_3$ ($x_1,x_2$ are on the right side), then $x_2\not \in V_2$.
\end{obs}
\begin{proofing}{Observation \ref{samesides}}
{Let $P= P_1x_1v_3y_1P_2x_2uP_3$, where $(v_3,u)\in E_F$ and $x_1$ is heavy. (see Figure \ref{fig8}). Assume towards contradiction that $x_2\in V_2$. We can then create the following two paths from $P$: $Q_1=P_3uv_3P_2x_2$ and $Q_2=P_1x_1$. From the union of $\{Q_1,Q_2\}$ and $\mathcal{S}\backslash P$ we create a path partition $\mathcal{S}_0$. Note that $\mathcal{S}_0$ is a derived path partition from $\mathcal{S}$.
In this derived path partition, $x_1$, which is one of the ``new" end vertices, is heavy. Hence by \hyperref[lem:derived]{\text{Lemma} \ref{lem:derived}}, $\mathcal{S}$ is not canonical, contradicting the premises of the lemma.
The case $P= P_1y_1v_3x_1P_2ux_2P_3$ is handled in an analogous way.
\begin{figure}[!h]
\centering
\scalebox{.8}{\samesides{$x_1$}{$v_3$}{$x_2$}{$u$}}
\caption{\hyperref[samesides]{\text{Observation} \ref{samesides}}}\label{fig8}
\end{figure}
}
\end{proofing}
\begin{obs}\label{between}
Let $P= P_1ux_2P_2x_1v_3y_1P_3$ ($x_1,x_2$ are between) be a path in a canonical path partition $\mathcal{S}$, where $(v_3,u)\in E_F$ and $x_1$ is heavy. Then $x_2\not\in V_2$.
\end{obs}
\begin{proofing}{Observation \ref{between}}
{ Let $P= P_1ux_2P_2x_1v_3y_1P_3$ be a path in a canonical path partition $\mathcal{S}$, where $(v_3,u)\in E_F$ and $x_1$ is heavy (see Figure \ref{fig9}). Assume towards contradiction that $x_2\in V_2$. We can then create the following two paths from $P$: $Q_1=x_1P_2x_2$, $Q_2=P_1uv_3y_1P_3$. From the union of $\{Q_1,Q_2\}$ and $\mathcal{S}\backslash P$ we create a path partition $\mathcal{S}_0$. Note that $\mathcal{S}_0$ is a derived path partition from $\mathcal{S}$.
In this derived path partition, $x_1$, which is one of the ``new" end vertices, is heavy. Hence by \hyperref[lem:derived]{\text{Lemma} \ref{lem:derived}}, $\mathcal{S}$ is not canonical, which is a contradiction.
\begin{figure}[H]
\centering
\scalebox{.8}{ \between
}
\caption{\hyperref[between]{\text{Observation} \ref{between}}}\label{fig9}
\end{figure}
}
\end{proofing}
Now we can prove \hyperref[lem:dangerous]{\text{Observation} \ref{lem:dangerous}}.
\begin{proofing}{Lemma \ref{lem:dangerous}}
{Let $\mathcal{S}$ be a canonical path partition. Let $P$ be a path in $\mathcal{S}$. Let $x_1v_3$ be two consecutive vertices in $P$ where $x_1$ is heavy and $v_3$ is dangerous. (The case where the order is $v_3x_1$, can be handled in a similar way by reversing the order of all vertices in $P$.) Let $(v_3,u)\in E_F$ be a free edge that is incident to a vertex $u$ which has a path neighbor in $V_2$ (hence $u\in V_4\cup V_2^b\cup V_3$).
\hyperref[different]{\text{Observation} \ref{different}} implies that $u$ is on the same path as $v_3$. This proves item \ref{itm:1} in \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}. Thereafter, the combination of \hyperref[samesides]{\text{Observation} \ref{samesides}} and \hyperref[between]{\text{Observation} \ref{between}} exclude all cases except for the following: $u$ is on the right of $v_3$ and has only one path neighbor in $V_2$, where this path neighbor (that we denote by $x_2$) is on the left of $u$. Consequently, path $P$ can be represented as $P= o_1P_1x_1v_3y_1P_2ux_2P_3o_2$. This proves item \ref{itm:2} in \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}.
\begin{figure}[H]
\centering
\scalebox{.8}{ \oppositesides
}
\caption{\hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}}\label{fig10}
\end{figure}
The following proposition proves item \ref{itm:3} in \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}.
\begin{prop}\label{nine}
There is only one balanced edge that is incident to $x_2$. The other end-vertex of this edge is $o_2$.
\end{prop}
\begin{proofing}{Proposition \ref{nine}}
{
Let $o_{x_2}\in V_1$ be a vertex that is incident to $x_2$ through a balanced edge. There are some cases to consider:
\begin{enumerate}
\item If $o_{x_2} = o_1$, then we create from $P$ the path $P'=o_2P_3x_2o_1P_1x_1v_{3}uP_2y_1$ in which $y_1$ is an end-vertex of $P'$. Because $y_1$ is moderate, then it has at least two balanced edges. Hence, by the Pigeonhole principle, $y_1$ must be adjacent either to a component $P_{y_1}\neq P$ (that can be a cycle) or to $o_2$.
\begin{itemize}
\item
If $y_1$ is incident to a component $P_{y_1}$, then we can create one path out of the two component $P_{y_1}$ and $P'$, contradicting Property ~1.
\item If $y_1$ is incident to $o_2$, then $P'$ can be made into a cycle, contradicting Property ~2.
\end{itemize}
\item If $o_{x_2}$ lies on a cycle $C_1$, there are three cases we have to exclude. Recall that $y_1$ is moderate and therefore $y_1$ has at least one balanced edge that is incident to a vertex of a path. We denote this vertex by $o_{y_1}$ and the path by $P_{y_1}.$
\begin{itemize}
\item If $o_{y_1} = o_2$, then we create from $P$ the path $o_1P_1x_1v_3uP_2y_1o_2P_3x_2C_1$, contradicting Property $1$.
\item If $o_{y_1}= o_1$, then we use the fact that $x_1$ is heavy and
hence must have at least one edge to an external path $P_{x_1}$. We then create $o_2P_3x_2C_1$ and $P_{x_1}x_1P_1o_1y_1v_3uP_2$, contradicting Property $1$.
\item If $o_{y_1}$ lies on a path $P_{y_1}\neq P$, then we can create $o_1P_1x_1v_3uP_2y_1P_{y_1}$ and $o_2P_3x_2C_1$, two components out of three, contradicting Property $1$.
\end{itemize}
\item If $o_{x_2}$ is an end-vertex of path $P_{x_2}\neq P$, then we create a path partition with one more cycle and the same number of components, contradicting Property $2$. The fact that $x_1$ is heavy and hence has three balanced edges to paths, implies that one of them must be incident to $o_{x_1}\neq o_{x_2},o_1$. The options are as follows:
\begin{itemize}
\item If $o_{x_1} = o_2$, then we create from $P$ and $P_{x_2}$ one path $o_1P_1x_1o_2P_3x_2P_{x_2}$ and one cycle $v_3y_1P_2u$, making a path partition with the same number of components and one more cycle, contradicting Property $2$.
\item If $o_{x_1}$ lies on the path $P_{x_2}$ (but is not $o_{x_2}$), then by connecting $x_1$ to the end-vertex of $P_{x_2}$ we create from $P$ and $P_{x_2}$ one path $o_1P_1x_1P_{x_2}x_2P_3o_2$ and one cycle $v_3y_1P_2u$, contradicting Property $2$.
\item If $o_{x_1}$ is on a path $P_{x_1}\neq P,P_{x_2}$ then we create from these three paths two paths $o_2P_3x_2P_{x_2}$, $P_{x_1}x_1P_1o_1$ and one cycle $v_3y_1P_2u$, contradicting Property $2$.
\end{itemize}
\end{enumerate}
Hence we conclude that $o_{x_2} = o_2$.
}
\end{proofing}
The following proposition proves item \ref{itm:4} in \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}.
\begin{prop}\label{nineone}
Given that $x_2$ has an edge to $o_2$, then only one of $y_1's$ balanced edges is incident to a vertex of a path, and the other end-vertex of this edge is $o_2$.
\end{prop}
\begin{proofing}{Proposition \ref{nineone}}
{ Let $o_{y_1}\in V_1$ be a path vertex that is incident to $y_2$ by a balanced edge. There are two cases to exclude:
\begin{enumerate}
\item $x_1$ is heavy and hence must have at least one edge that is incident to an end-vertex of a path $P_{x_1}\neq P$. If $o_{y_1}=o_1$ then we can create from $P$ and $P_{x_1}$ the path $v_3uP_2y_1o_1P_1x_1P_{x_1}$ and the cycle $x_2P_3o_2$, contradicting Property $2$ \\
\item If $o_{y_1}$ is an end-vertex of an external path $P_{y_1}$ then from $P$ and $P_{y_1}$ we create $o_1P_1x_1v_3uP_2y_1P_{y_1}$ and the cycle $x_2P_3o_2$, contradicting Property $2$.
\end{enumerate}
Hence $o_{y_1} = o_2$.
}
\end{proofing}
}
\end{proofing}
\subsection{Properties of $V_2$, and proofs of \hyperref[lem:kadjacent]{\text{Lemma} \ref{lem:kadjacent}} and \hyperref[lem:special_case]{\text{Lemma} \ref{lem:special_case}}} \label{subsec:pro2}
\subsubsection{Proof of \hyperref[lem:kadjacent]{\text{Lemma} \ref{lem:kadjacent}}} \label{subsub:lemma4}
\hyperref[lem:kadjacent]{\text{Lemma} \ref{lem:kadjacent}} is a result of the following two lemmas.
\setcounter{lemma}{7}
\begin{lemma}\label{lem:none_to_dangerous}
Let $P=o_1P_1XP_2o_2$ be a path in a canonical path partition where $X = x_1x_2...x_k$ is a sequence of vertices in $V_2^b$. If $X$ has no free edges to dangerous vertices then it can transfer at most $(2+k)\cdot\frac{2}{3}$ points. Consequently, $X$ ends with at least $\frac{1}{3}\cdot k-\frac{4}{3}$ points.
\end{lemma}
\begin{lemma}\label{lem:has_to_dangerous}
Let $P=o_1P_1XP_2o_2$ be a path in a canonical path partition where $X = x_1x_2...x_k$ is a sequence of vertices in $V_2^b$. If there is a vertex $x_i\in X$ that has an edge to a dangerous vertex then $X$ transfers at most $(k+1)\cdot\frac{2}{3} +\frac{5}{12}$ points. Consequently, $X$ ends with at least $\frac{1}{3}\cdot k-\frac{13}{12}$ points.
\end{lemma}
\setcounter{lemma}{8}
At first we refine our definitions for vertices in $V_2$.
\begin{itemize}
\item We say that $x\in V_2$ \textbf{ goes} to $y$ if the edges $(x,y)\in E$ is a balanced edge.
\begin{itemize}
\item We say that $x\in V_2$ \textbf{goes} a path, if $x$ goes to $y\in V_1$ and $y$ is an end vertex of a path.
\item We say that $x\in V_2$ \textbf{goes} a cycle, if $x$ goes to $y\in V_1$ and $y$ is lies on a cycle.
\end{itemize}
\item A vertex $y\in V_2$ in some path $P\in \mathcal{S}$ is an \text{\em inner} if it \textbf{goes} to one of the end-vertices of $P$.
\item A pair of vertices $x_1,x_2\in V_2$ in some path $ P=o_1P_1x_1P_2x_2P_3o_2\in \mathcal{S}$ will be {\em crossing-inners} if $x_1$ \textbf{goes} to $o_2$ and $x_2$ \textbf{goes} to $o_1$.
\item A pair of vertices $x_1,x_2\in V_2$ in some path $ P=o_1P_1x_1P_2x_2P_3o_2\in \mathcal{S}$ will be {\em splitting-inners} if $x_1$ \textbf{goes} to $o_1$ and $x_2$ \textbf{goes} to $o_2$.
\end{itemize}
Given a set $X \subset V_2$ we use the following notation:
\begin{itemize}
\item $N_b(X)$ - The set of all vertices in $V_1$ that $X$ \textbf{goes} to.
\item $b(X)$ - the total points that vertices of $X$ transfer by the transition rules using balanced edges (by \ref{rule:v2_cycle} and \ref{rule:v2_path}).
\end{itemize}
Hence, in this new notation \hyperref[lem:none_to_dangerous]{\text{Lemma }\ref{lem:none_to_dangerous}} states that $\displaystyle b(X)=\sum_{j=1}^{k}b(x_j)\leq \frac{2}{3}\cdot(2+k)$.
In order to prove the lemma we use some observations on path neighbor vertices in $V_2^b$. In the following observations we consider a path $P=o_1P_1x_1x_2P_2o_2$ in a canonical path partition, in which $x_1,x_2\in V_2^b$ are path neighbors. In some of the observations, we will prove things for $x_1$. These observations will imply the same results for $x_2$, but in a symmetric fashion.
\begin{obs}\label{crossinginacceptor}
There are no path neighbors that are crossing-inners in a canonical path partition.
\end{obs}
\begin{figure}[H]
\centering
\scalebox{.8}{\crossing
}
\caption{\hyperref[crossinginacceptor]{\text{Observation} \ref{crossinginacceptor}} - Crossing-inners}\label{fig12}
\end{figure}
\begin{proofing}{Observation \ref{crossinginacceptor}}
{ Let $P=o_1P_1x_1x_2P_2o_2$ where $x_1,x_2$ are crossing-inners. Then we create from $P$ the cycle $o_1P_1x_1o_2P_2x_2$, contradicting the assumption that $P$ is a path and not a cycle.
}
\end{proofing}
\begin{obs}\label{inacceptor1}
If $x_1$ \textbf{goes} to $o_{2}$, then $N_b(x_2)= \{o_2\}$.
\end{obs}
\begin{proofing}{Observation \ref{inacceptor1}}
{
Let $P=o_1P_1x_1x_2P_2o_2$ and $(x_1,o_2)\in E$. Let $o_{x_2}$ be a vertex that $x_2$ \textbf{goes} to. We will show that $o_{x_2} = o_2$. Here too, there are some cases to consider.
\begin{enumerate}[label=\roman*]
\item If $o_{x_2}= o_1$, then $x_1$ and $x_2$ are crossing-inners and this contradicts \hyperref[crossinginacceptor]{\text{Observation} \ref{crossinginacceptor}}.
\item If $o_{x_2}$ lies on a component $P_{x_2}\neq P$, we will create $o_1P_1x_1o_2P_2x_2P_{x_2}$, making a path partition with less components, contradicting Property $1$.
\end{enumerate}
Hence $o_{x_2}= o_2$.
\begin{figure}[H]
\begin{subfigure}[t]{0.4\columnwidth} \centering \resizebox{\linewidth}{!}
{ \crossing}
\caption{$o_{x_2} = o_1$ } \label{fig13:subfig1}
\end{subfigure}
\hspace{20mm}
\begin{subfigure}[t]{0.4\columnwidth} \centering \resizebox{\linewidth}{!}
{\inacceptor}
\caption{$P_{x_2} \neq P$ } \label{fig13:subfig2}
\end{subfigure}
\caption{\hyperref[inacceptor1]{\text{Observation} \ref{inacceptor1}}}\label{fig13}
\end{figure}
}
\end{proofing}
\begin{obs}\label{inacceptor2}
If $x_1$ \textbf{goes} to $o_{1}$, then $x_2$ can \textbf{go} either to end-vertices of $P$, to vertices of cycles, or to both.
\end{obs}
\begin{proofing}{Observation \ref{inacceptor2}}
{
Let $P=o_1P_1x_1x_2P_2o_2$, and suppose that $x_1$ \textbf{goes} to $o_1$. Assume in contradiction that $x_2$ \textbf{goes} to a vertex on an external path $P_{x_2}\neq P$. Then a path $o_2P_2x_2P_{x_2}$ and a cycle $x_1P_1o_1$ will be created, making a path partition with the same number of components and one more cycle, contradicting Property $2$.
}
\end{proofing}
\begin{obs}\label{cycleout}
If $x_1$ \textbf{goes} to a cycle $C_1$, then $x_2$ \textbf{goes} to $o_2$ or to a vertex on $C_1$ or to both.
\end{obs}
\begin{proofing}{Observation \ref{cycleout}}
{
Let $P=o_1P_1x_1x_2P_2o_2$ and suppose that $x_1$ \textbf{goes} to the vertex $o_{x_1}$ which lies on a cycle $C_1$. Let $o_{x_2}$ be the vertex $x_2$ \textbf{goes} to. We consider the following cases:
\begin{enumerate}
\item If $o_{x_2} = o_1$, then applying \hyperref[inacceptor1]{\text{Observation} \ref{inacceptor1}} on $x_2$ implies that $N_b(x_1)= \{o_2\}$, which is a contradiction.
\item If $o_{x_2}$ is on $P_{x_2} \neq C_1 $, then we can create two paths $o_2P_2x_2P_{x_2}$ and $o_1P_1x_1C_1$ making a new path partition of size $|\mathcal{S}|-1$, contradiction to Property~1.
\end{enumerate}
}
\end{proofing}
\begin{obs}\label{pathout}
If $x_1$ \textbf{goes} to a vertex $o_{x_1}$, which is an end-vertex of a path $P_{x_1}\neq P$, then $N_b(x_2)=\{o_{x_1}\}$.
\end{obs}
\begin{proofing}{Observation \ref{pathout}}
{
Let $P=o_1P_1x_1x_2P_2o_2$ and suppose that $x_1$ \textbf{goes} to $o_{x_1}$, which is an end-vertex of the path $P_{x_1}\neq P$. Let $o_{x_2}$ be a vertex that $x_2$ \textbf{goes} to. We will show that $o_{x_1}=o_{x_2}$ by excluding all other cases.
\begin{enumerate}
\item If $o_{x_2}=o_2$, then we create a cycle $x_2P_2o_2$ and a path $o_1P_1x_1P_{x_1}$, making a new path partition of size $|\mathcal{S}|$ with one more cycle (because $P_{x_1}$ is not a cycle), contradicting Property~2.
\item If $o_{x_2}=o_1$ then applying \hyperref[inacceptor1]{\text{Observation} \ref{inacceptor1}} on $x_2$ implies that $N_b(x_1)= \{o_2\}$, which is a contradiction.
\item If $o_{x_2}$ is the other end-vertex of path $P_{x_1}$, then we concatenate $P$ and $P_{x_1}$ into one path $o_1P_2x_2P_{x_1}x_1P_1o_1$, contradicting Property~1.
\item If $o_{x_2}$ is on $P_{x_2} \neq P_{x_1}$ (it can be a cycle), we then create two paths $o_2P_2x_2P_{x_2}$, $o_1P_1x_1P_{x_1}$, making a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1.
\end{enumerate}
}
\end{proofing}
Now we will generalize our observations for any $k\geq 2$ vertices.
In the following observations $P=o_1P_1XP_2o_2$ is a path in a canonical path partition where $X=x_1x_2...x_k$ is a sequence of $k\geq 2$ vertices in $V_2^b$.
\begin{obs}\label{pathoutk}
If $x_i$ \textbf{goes} to a vertex $o_{x_i}$ that is an end-vertex of a path $P_{x_i}\neq P$, then $N_b(X)=\{o_{x_i}\}$. Consequently, $b(X)=k\cdot\frac{2}{3}$.
\end{obs}
\begin{proofing}{Observation \ref{pathoutk}} {
For $o_{x_i}$ as above, \hyperref[pathout]{\text{Observation} \ref{pathout}} implies that the two path-neighbors of $x_i$ must \textbf{go} to $o_{x_i}$, which in turn implies that $x_i$ goes only to $o_{x_i}$. By induction we get that $N_b(X)=\{o_{x_i}\}$. Hence $b(X)=k\cdot\frac{2}{3}$.
}
\end{proofing}
\begin{obs}\label{pathink}
If $x_i$ \textbf{goes} to $o_2$ then $\forall j>i, N_b(x_j)=\{o_2\}$.
If $x_i$ \textbf{goes} to $o_1$ then $\forall j<i, N_b(x_j)=\{o_1\}$.
\end{obs}
\begin{proofing}{Observation \ref{pathink}} {
If $x_i$ goes to $o_2$ then from \hyperref[inacceptor1]{\text{Observation} \ref{inacceptor1}} $N_b(x_{i+1})=\{o_2\}$ and by induction $\forall j>i, N_b(x_j)=\{o_2\}$.
Likewise, if $x_i$ \textbf{goes} to $o_1$, then $\forall j<i, N_b(x_j)=\{o_1\}$.
}
\end{proofing}
\begin{obs}\label{cycleoutk}
If $x_i$ \textbf{goes} to a cycle, then:
\begin{enumerate}
\item $\forall j>i$, $x_j$ \textbf{goes} to the same cycle as $x_i$, or \textbf{goes} to $o_2$, or \textbf{goes} to both.
\item $\forall j<i$, $x_j$ \textbf{goes} to the same cycle as $x_i$, or \textbf{goes} to $o_1$, or \textbf{goes} to both.
\end{enumerate}
\end{obs}
\begin{proofing}{Observation \ref{cycleoutk}} {
Let $o_{x_i}$ be a vertex on a cycle that $x_i$ \textbf{goes} to. \hyperref[cycleout]{\text{Observation} \ref{cycleout}} $x_{i+1}$ implies then that $x_{i+1}$ must \textbf{go} to the same cycle, or \textbf{go} to $o_2$. By induction, using either \hyperref[cycleout]{\text{Observation} \ref{cycleout}} or \hyperref[inacceptor1]{\text{Observation} \ref{inacceptor1}} we get that $\forall j>i$, $x_j$ must \textbf{go} either to the same cycle, or to $o_2$.
A similar proof can be given to $x_j$ with $j<i$.
}
\end{proofing}
Let $X'\subseteq X$ be a sequence of adjacent nodes in $X$ that all \textbf{go} to the same cycle $C$.
The following two observations show that on average, the number of points each vertex in $X'$ transfers to $C$ is at most $\frac{1}{2}$.
\begin{obs}\label{samecycle2}
If $X'= x_1, x_2$ \textbf{go} to the same cycle $C$, then they cannot \textbf{go} to neighboring vertices along the cycle. Moreover, $X'$ transfer to $C$ (by \ref{rule:v2_cycle}) at most one point ($2\cdot\frac{1}{2}$).
\end{obs}
\begin{proofing}{Observation \ref{samecycle2}}
{
Assume that $X'= x_1, x_2$ \textbf{go} to vertices $o_{x_1}$ and $o_{x_2}$, that lie on the same cycle, $C$.
If $(o_{x_2},o_{x_1})\in E$ is an edge that is part of the cycle, we open the cycle into a path with end-vertices $o_{x_1}$ and $o_{x_2}$. Thereafter, we crate the path $o_2P_2x_2Cx_1P_1o_1$ out of $P$ and $C$. Consequently, we have a new path partition of size $|\mathcal{S}|-1$, contradicting Property~1.
Let $c$ be the size of cycle $C$.
\begin{enumerate}
\item If $c=3$ then only one vertex in $C$ can have an edges incident to $X'$, because any two vertices on $C$ are neighbors.
Hence the number of edges that can transfer points from $X'$ to $C$ is at most $2$. Consequently the number of points transferred to $C$ is at most $\frac{2}{3}$.
\item If $c=4$ then at most two vertices in $C$ can have edges incident to $X'$. Hence the number of edges that can transfer points from $X'$ to $C$ is at most $4$. Consequently the number of points transferred to $C$ is at most $1$.
\item If $c=5$ then at most three vertices in $C$ have edges incident to $X'$, and the number of edges that can transfer points from $X'$ to $C$ is at most $4$. Consequently the number of points transferred to $C$ is at most $\frac{4}{5}$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.3\columnwidth} \centering \resizebox{0.8\linewidth}{!}
{\fivecycletwo}
\caption{\centering two vertices in $C$ have edges incident to $X'$ } \label{fig100:subfig1}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering \resizebox{0.8\linewidth}{!}
{\fivecyclethree}
\caption{\centering three vertices in $C$ have edges incident to $X'$} \label{fig100:subfig2}
\end{subfigure}
\caption{The two cases of $c=5$ }\label{fig100}
\end{figure}
\item If $c=6$ then at most four vertices in $C$ have edges incident to $X'$, and the number of edges that can transfer points from $X'$ to $C$ is at most $6$. Consequently the number of points transferred to $C$ is at most $1$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.3\columnwidth} \centering \resizebox{0.8\linewidth}{!}
{\sixcyclethree}
\caption{\centering three vertices in $C$ have edges incident to $X'$ } \label{fig201:subfig1}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering \resizebox{0.8\linewidth}{!}
{\sixcyclefour}
\caption{\centering four vertices in $C$ have edges incident to $X'$ } \label{fig201:subfig2}
\end{subfigure}
\caption{The two cases of $c=6$ }\label{fig201}
\end{figure}
\end{enumerate}
}
\end{proofing}
\begin{obs}\label{samecycle3}
If $X'= x_1,x_2,x_3$ in $X$ \textbf{go} to the same cycle $C$, then $X'$ transfers to $C$ (by \ref{rule:v2_cycle}) at most $\frac{3}{2}$ points ($3\cdot\frac{1}{2}$).
\end{obs}
\begin{proofing}{Observation \ref{samecycle3}}
{
Suppose that $X'= x_1,x_2,x_3$ \textbf{go} to the same cycle $C$.
Let $c$ be the size of cycle $C$. By \hyperref[samecycle2]{\text{Observation} \ref{samecycle2}} any two adjacent vertices among $x_1,x_2,x_3$ cannot \textbf{go} to neighboring vertices along the cycle.
\begin{enumerate}
\item If $c=3$ then only one vertex in $C$ can have an edges incident to $X'$, because any two vertices on $C$ are neighbors. Hence the number of edges that can transfer points from $X'$ to $C$ is at most $3$. Consequently, the number of points transferred to $C$ is at most $1$.
\item If $c=4$ then at most two vertices in $C$ can have edges incident to $X'$. Hence the number of edges that can transfer points from $X'$ to $C$ is at most $6$. Consequently, the number of points transferred to $C$ is at most $\frac{6}{4}$.
\item If $c=5$ then at most three vertices in $C$ have edges incident to $X'$. Hence the number of edges that can transfer points from $X'$ to $C$ is at most $7$. Consequently, the number of points transferred to $C$ is at most $\frac{7}{5}$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.3\columnwidth} \centering \resizebox{0.8\linewidth}{!}
{\fivecycletwob}
\caption{\centering two vertices in $C$ have edges incident to $X'$ } \label{fig101:subfig1}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering \resizebox{0.8\linewidth}{!}
{\fivecyclethreeb}
\caption{\centering three vertices in $C$ have edges incident to $X'$} \label{fig101subfig2}
\end{subfigure}
\caption{The two cases of $c=5$ }\label{fig101}
\end{figure}
\item If $c=6$ then at most four vertex in $C$ have edges incident to $X'$. Therefore, the number of edges that can transfer points from $X'$ to $C$ is at most $9$. Consequently, the number of points transferred to $C$ is at most $\frac{9}{6}$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.3\columnwidth} \centering \resizebox{0.8\linewidth}{!}
{\sixcyclethreeb}
\caption{\centering three vertices in $C$ have edges incident to $X'$ } \label{fig301:subfig1}
\end{subfigure}
\hspace{10mm}
\begin{subfigure}[t]{0.3\columnwidth}
\centering \resizebox{0.8\linewidth}{!}
{\sixcyclefourb}
\caption{\centering four vertices in $C$ have edges incident to $X'$} \label{fig301:subfig2}
\end{subfigure}
\caption{The two cases of $c=6$ }\label{fig301}
\end{figure}
\end{enumerate}
}
\end{proofing}
\begin{obs}\label{samecyclek}
Let $X'\subseteq X$ be all the vertices in $X$ that \textbf{go} to vertices on cycles. If $|X'|>1$, then all vertices in $X'$ are adjacent and \textbf{go} to the same cycle, and the the number of points $X'$ transfers to $C$ is at most $\frac{1}{2}\cdot|X'|$. Consequently, $b(X')\leq |X'|\cdot\frac{1}{2}+2\cdot\frac{2}{3}$.
\end{obs}
\begin{proofing}{Observation \ref{samecyclek}} {
Consider $X'\subseteq X$ to be all the vertices in $X$ that \textbf{go} to vertices on cycles. We prove, that if $|X'|=2$, the vertices of $X'$ will be adjacent and \textbf {go} to the same cycle. Let $x_i$ and $x_j$ be the two vertices that make $X'$ where $i<j$. Assume towards contradiction that $j\neq i+1$, hence there is $i<l<j$ ,$x_l$ that does not \textbf{go} to any cycles. Applying \hyperref[cycleoutk]{\text{Observation} \ref{cycleoutk}} on $x_i$ we get that $x_l$ must \textbf{go} to $o_2$, and only to it. Applying \hyperref[cycleoutk]{\text{Observation} \ref{cycleoutk}} on $x_j$ implies that $x_l$ must go to $o_1$, which is contradiction.
Hence $j=i+1$ and moreover both vertices \textbf{go} to the same cycle. In the same way, a simple induction will prove that for any size of $|X'|\geq 2$, the vertices of $X'$ are adjacent and all have edges to the same cycle.
From \hyperref[samecycle2]{\text{Observation} \ref{samecycle2}} and \hyperref[samecycle3]{\text{Observation} \ref{samecycle3}} we get that $X'$ transfers at most $\frac{1}{2}\cdot|X'|$
points to $C$.
Let $x_i$ be the first vertex from left in $X'$ and $x_j$ be the last vertex from left. $x_i$ can \textbf{go} to $o_1$ and $x_j$ can \textbf{go} to $o_2$, in addition to \textbf{going} to $C$. Hence in total $b(X')\leq |X'|\cdot\frac{1}{2}+2\cdot\frac{2}{3}$.
}
\end{proofing}
Now we prove \hyperref[lem:none_to_dangerous]{\text{Lemma} \ref{lem:none_to_dangerous}}.
\begin{proofing}{Lemma \ref{lem:none_to_dangerous}}
{
Let $P=o_1P_1XP_2o_2$ be a path in a canonical path partition where $X = x_1x_2...x_k$ is a sequence of vertices in $V_2^b$. According to this lemma, $X$ has no free edges incident to dangerous vertices. There are four cases we need to consider (see Figure \ref{fig16}):
\begin{enumerate}
\item If there is $i$ such that $x_i$ \textbf{goes} to a path $P_{x_i}\neq P$, then from \hyperref[pathoutk]{\text{Observation} \ref{pathoutk}} we get that $b(X) = k\cdot\frac{2}{3}$. (See Figure \ref{fig16:subfig1}.)
\item If all vertices in $X$ \textbf{go} to $o_1$ or $o_2$ then we will show that there can be only one vertex in $X$ that has two balanced edges.
Assume in contradiction, that there are two vertices, $x_i$ and $x_j$ in which each one of them \textbf{goes} to $o_1$ and $o_2$. W.l.o.g suppose that $i<j$. Hence
by \hyperref[pathink]{\text{Observation} \ref{pathink}} and the fact that $x_i$ \textbf{goes} to $o_2$ then $x_j$ cannot \textbf{go} to $o_1$ - contradiction. Hence $b(X)= (k+1)\cdot\frac{2}{3}$. (See Figure \ref{fig16:subfig2}.)\label{item2:lemma}
\item If there is exactly one vertex $x_i\in X$ that \textbf{goes} to a cycle. (See Figure \ref{fig16:subfig3}.) From \hyperref[cycleoutk]{\text{Observation} \ref{cycleoutk}} we know that all $j<i$ must \textbf{go} to $o_1$ and all $j>i$ must \textbf{go} to $o_2$. Moreover, $x_i$ can \textbf{go} to $o_1$, $o_2$ as well and to two cycles of size three. Hence $b(X) = b(x_i)+ (k-1)\cdot\frac{2}{3}\leq 2\cdot\frac{2}{3}+2\cdot\frac{1}{3} +(k-1)\cdot\frac{2}{3} = (k+2)\cdot\frac{2}{3}$. \label{item3:lemma}
\item Let $X'\subseteq X$ be the set of all the vertices in $X$ that \textbf{go} to cycles. (See Figure \ref{fig16:subfig4}.) If $|X'|\geq 2$ then from \hyperref[samecyclek]{\text{Observation} \ref{samecyclek}} it must be that all vertices in $X'$ are adjacent and $b(X')\leq|X'|\cdot\frac{1}{2}+2\cdot\frac{2}{3}$. Let $x_i$ be the first in $X'$ and $x_j$ be the last one.
Then from \hyperref[cycleoutk]{\text{Observation} \ref{cycleoutk}} for all $l<i$ $x_l$, must \textbf{go} to $o_1$ and for all $l>j$ $x_l$, must go to $o_2$.
Hence each vertex in this group will transfer $\frac{2}{3}$ points. Hence we can conclude that $b(X) =(k-|X'|)\cdot\frac{2}{3}+ |X'|\cdot\frac{1}{2}+2\cdot\frac{2}{3} = (k+2)\cdot\frac{2}{3} -\frac{1}{6}\cdot|X'|$. $|X'|\geq 2$ and hence $b(X) \leq (k+2)\cdot\frac{2}{3} -\frac{1}{3}$.\label{item4:lemma}
\end{enumerate}
Hence we can conclude that in all cases $b(X)\leq (k+2)\cdot\frac{2}{3}$. Consequently, $X$ ends with at least $\frac{1}{3}\cdot k-\frac{4}{3}$ points.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.45\columnwidth} \centering \resizebox{\linewidth}{!}
{\kouterpath}
\caption{$x_i$ goes to an external path } \label{fig16:subfig1}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[t]{0.45\columnwidth} \centering \resizebox{\linewidth}{!}
{\knonouter}
\caption{All vertices in $X$ go to $o_1$ or $o_2$} \label{fig16:subfig2}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[t]{0.45\columnwidth} \centering \resizebox{\linewidth}{!}
{\ksequence}
\caption{only $x_i$ goes to a vertex on a cycle} \label{fig16:subfig3}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[t]{0.45\columnwidth} \centering \resizebox{\linewidth}{!}
{\koutercycles}
\caption{At least two nodes in $X$ go to cycles} \label{fig16:subfig4}
\end{subfigure}
\caption{\hyperref[lem:none_to_dangerous]{\text{Lemma} \ref{lem:none_to_dangerous}}} \label{fig16}
\end{figure}
}
\end{proofing}
To prove \hyperref[lem:has_to_dangerous]{\text{Lemma} \ref{lem:has_to_dangerous}} we will need some observations.
\refstepcounter{lemma}
\begin{obs}\label{onlyone}
Let $X$ be $k$ adjacent vertices $x_1,x_2....x_k \in V_2^b$ on a path $P=o_1P_1x_1x_2...x_kP_2o_2$. If there is $x_i$ that has an edge to a dangerous vertex, then $i\in \{1,k\}$. Moreover,
\begin{itemize}
\item if $x_1$ has an edge to a dangerous vertex, then $N_b(X \setminus \{x_1\})= \{o_2\}$.
\item if $x_k$ has an edge to a dangerous vertex, then $N_b(X \setminus \{x_k\})= \{o_1\}$.
\end{itemize}
\end{obs}
\begin{proofing}{Observation \ref{onlyone}}
{Let $X$ be $k$ adjacent vertices $x_1,x_2....x_k \in V_2^b$ on a path $P=o_1P_1x_1x_2...x_kP_2o_2$. If $x_i$ has an edge that is incident to a dangerous vertex $v_3$, then \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} implies that $v_3$ is on $P$ and that $x_i$ cannot have both its neighbors in $V_2$. Hence $i\in \{1,k\}$. W.l.o.g assume that $x_1$ has an edge to dangerous vertex. \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}} implies that $N_b(x_2)= \{o_2\}$.
Using the fact that $x_3\in V_2$ we get from \hyperref[inacceptor1]{\text{Observation} \ref{inacceptor1}} that $N_b(x_3)= \{o_2\}$. With an easy induction we get that $N_b(X \setminus \{x_1\})= \{o_2\}$. The same proof will work for $x_k$.
}
\end{proofing}
\begin{obs}\label{only}
Let $X$ be two adjacent vertices $x_1,x_2\in V_2^b$ on a path $P=o_1P_1x_1x_2P_2o_2$.
If both of them have an edge to a dangerous vertex, then $N_b(x_2)= \{o_2\}$ and $N_b(x_1)= \{o_1\}$. Consequently, $X$ transfers at most $2\cdot\frac{2}{3}+\frac{1}{2}$ points.
\end{obs}
\begin{proofing}{Observation \ref{only}}
{ Let $X$ be two adjacent vertices $x_1,x_2\in V_2^b$ on a path $P=o_1P_1x_1x_2P_2o_2$. If both of them have an edge to a dangerous vertex, then $N_b(x_2)= \{o_2\}$ and $N_b(x_1)= \{o_1\}$. (This follows from \hyperref[lem:dangerous]{\text{Lemma} \ref{lem:dangerous}}.) According to $\ref{rule:v4_dangerous}$ an edge that is not balanced can transfer at most $\frac{1}{12}$ points.
Hence, in the case above, $X$ will transfer at most $2\cdot(\frac{2}{3}+3\cdot\frac{1}{12})=2\cdot\frac{2}{3}+\frac{1}{2}$ points.
}
\end{proofing}
\begin{proofing}{Lemma \ref{lem:has_to_dangerous}} { Let $\mathcal{S}$ be a canonical path partition. Let $X$ be $k\geq 2$ adjacent vertices $x_1,x_2....x_k \in V_2^b$ on a path $P=o_1P_1x_1x_2...x_kP_2o_2$. Suppose that there is $i$ such that $x_i$ has an edge to a dangerous vertex. We prove that $X$ transfers at most $(k+1)\cdot\frac{2}{3} +\frac{5}{12}$ points. We break the proof into two cases:
\begin{enumerate}
\item If there is exactly one $x_i$ which has an edge to a dangerous vertex, then by \hyperref[onlyone]{\text{Observation} \ref{onlyone}} we can assume w.l.o.g that $x_1$ is the only vertex that has an edge to a dangerous vertex and $N_b(X \setminus \{x_1\})= \{o_2\}$. Hence for every $2\leq i\leq k$, $x_i$ will transfer $\frac{2}{3}$ points. In particular, $x_2$ \textbf{goes} to $o_2$ and hence \hyperref[pathout]{\text{Observation} \ref{pathout}} implies $x_1$ cannot \textbf{go} to an end-vertex of an external path. Hence, $x_1$ can have its remaining three free edges either to cycles, to ends vertices of $P$ ($o_1$ or $o_2$) or to dangerous vertices. If $x_1$ has an edge to $o_2$, $o_1$ and to a cycle of size three it will transfer $\frac{1}{12}+2\cdot\frac{2}{3}+\frac{1}{3}$ points. Any other case will lead $x_1$ to transfer less points.
Hence we get that $X$ will transfer at most
$\frac{1}{12}+2\cdot\frac{2}{3}+\frac{1}{3}+(k-1)\cdot\frac{2}{3}$
$=(k+1)\cdot\frac{2}{3} +\frac{1}{3}+\frac{1}{12}=(k+1)\cdot\frac{2}{3} +\frac{5}{12}$ points.
\item If there are at least two vertices in $X$ that have an edge to dangerous vertices, then we show that it must be that $k=2$. Assume in contradiction that $k>2$.
By \hyperref[onlyone]{\text{Observation} \ref{onlyone}} and because $k>2$, $x_2\neq x_k$ and $N_b(x_2)= \{o_1\}$ and $N_b(x_2)= \{o_2\}$ leading to a contradiction. Hence
we can assume that $k=2$ and both vertices have edges to dangerous vertices. By \hyperref[only]{\text{Observation} \ref{only}}, $X$ transfers at most $2\cdot\frac{2}{3} +\frac{1}{2}$ points.
\end{enumerate}
Hence we can conclude that in all cases, $X$ transfers at most $(k+1)\cdot\frac{2}{3} +\frac{5}{12}$ points. Consequently, $X$ ends with at least $\frac{1}{3}\cdot k-\frac{13}{12}$ points.
}
\end{proofing}
\subsubsection{Proof of \hyperref[lem:special_case]{\text{Lemma} \ref{lem:special_case}}} \label{subsub:lemma5}
\refstepcounter{lemma}
\setcounter{lemma}{5}
We prove now the following observations for the specific case we need to handle.
\begin{obs}\label{onlyonetocycles}
Let $P=o_1P_1x_1x_2P_2o_2$ be a path, where $x_1$ and $x_2$ are in $V_2$ and have no free edges to dangerous vertices. If $x_1$ and $x_2$ transfer more than $\frac{7}{3}$ points ($b(x_2)+b(x_1) >\frac{7}{3}$), then $x_2$ and $x_1$ are splitting-inners.
\end{obs}
\begin{proofing}{Observation \ref{onlyonetocycles}}
{ Let $P=o_1P_1x_1x_2P_2o_2$ where $x_1$ and $x_2$ are in $V_2$ and have no free edges to dangerous vertices. According to the premises of the lemma $b(x_2)+b(x_1)> \frac{7}{3}$. The proof of \hyperref[lem:none_to_dangerous]{\text{Lemma} \ref{lem:none_to_dangerous}} implies that we must be in case similar to case \ref{item3:lemma} of the proof, because in all other cases $b(x_1)+b(x_2)\leq \frac{7}{3}$. Hence we are in the case where either $x_1$ or $x_2$ go to a vertex on a cycle. W.l.o.g we will assume that it is $x_1$. \hyperref[cycleout]{\text{Observation} \ref{cycleout}} implies that $x_2$ must go to $o_2$, and only to $o_2$. Hence $b(x_2)=\frac{2}{3}$.
Consequently, $b(x_1) > \frac{5}{3}$. Since any balanced edge to a cycle transfers at most $\frac{1}{3}$ points, $x_1$ must have at least two balanced edges to paths. Moreover, $x_1$ cannot go to vertices that lie on other paths, by \hyperref[inacceptor2]{\text{Observation} \ref{inacceptor2}}. Hence $x_1$ must have an edge to $o_1$, making $x_1$ and $x_2$ splitting-inners.
}
\end{proofing}
\begin{obs}\label{splittinginacceptors}
Let $P=o_1P_1x_1x_2P_2o_2$ be a path. If $x_1$ and $x_2$ are adjacent splitting-inners, then no vertex $u$ on $ P_1$ or on $P_2$ is heavy.
\end{obs}
\begin{proofing} {Observation \ref{splittinginacceptors}}{
Let $P=o_1P_1x_1x_2P_2o_2$ be a path where $x_1$ and $x_2$ are splitting inners. Assume towards contradiction that $P_1= P_1^{'}uP_1^{''}$ where $u$ is heavy. Therefore, $u$ must go to a vertex on a path $P_{u}\neq P$. We will create a cycle component $x_2P_2o_2$, and the path $P_1^{''}x_1o_1P_1^{'}uP_u$, making a path partition with the same number of components and one more cycle, contradicting Property $2$.
\begin{figure}[H]
\centering
\resizebox{0.4\textwidth}{!}
\splittinginacceptors
}
\caption{splitting-inners}
\end{figure}
}
\end{proofing}
\begin{proofing}{Lemma \ref{lem:special_case}}
{ Let $\mathcal{S}$ be a canonical path partition and let $P$ be a path in $\mathcal{S}$. Let $B_iB_{i+1}$ be two consecutive blocks in path $P$ such that $B_i$ is of kind $1$ ($X_i=x_i$) and $|X_{i+1}|=2$ ($X_{i+1}=x_{i+1}^1x_{i+1}^2$). Suppose that $x_i$ is heavy. There are three cases we need to handle:
\begin{enumerate}
\item If all free edges of $X_{i+1}$ are balanced edges then we show that $b(X_{i+1})\leq \frac{7}{3}$ and hence transfers at most $\frac{7}{3}$ points. Assume in contradiction that $b(X_{i+1})>\frac{7}{3}$. Hence, by \hyperref[onlyonetocycles]{\text{Observation} \ref{onlyonetocycles}}, $x_{i+1}^1$ and $x_{i+1}^2$ are splitting-inners. Therefore, from \hyperref[splittinginacceptors]{\text{Observation} \ref{splittinginacceptors}} $x_i$ is not heavy, contradiction.
\item If both $x_{i+1}^1$ and $x_{i+1}^2$ have an edge to a dangerous vertex, then by \hyperref[only]{\text{Observation} \ref{only}} they transfer at most $2\cdot\frac{2}{3}+\frac{1}{2}$ points.
\item If there is only one vertex that has an edge to a dangerous vertex, then w.l.o.g assume it is $x_{i+1}^1$. (The same proof will apply for $x_{i+1}^2$.)
Let $o_1$ be the left end-vertex of path $P$ and $o_2$ be the right one. \hyperref[onlyone]{\text{Observation} \ref{onlyone}} implies that $N_b(x_{i+1}^2)= \{o_2\}$. As a result, there is no other edge through which $x_{i+1}^2$ can transfer points, and hence $x_{i+1}^2$ will transfer $\frac{2}{3}$ points. On the other hand, $x_{i+1}^1$ cannot have an edge to $o_1$ because then they will be splitting inners, contradicting the fact that $x_i$ is heavy (\hyperref[splittinginacceptors]{\text{Observation} \ref{splittinginacceptors}}).
\hyperref[inacceptor2]{\text{Observation} \ref{inacceptor2}} implies that
the remaining three balanced edges incident to $x_{i+1}^1$ are incident to cycles, to $o_2$ or to dangerous vertices. On an edge to $o_1$, $x_{i+1}^1$ transfers $\frac{2}{3}$ points by \ref{rule:v2_path}. A balanced edge to a cycle will transfer at most $\frac{1}{3}$ points by \ref{rule:v2_cycle}. A free edge that is not balanced can transfer at most $\frac{1}{12}$ points by \ref{rule:v4_dangerous}.
Hence the worst case is if the three edges incident to $x_{i+1}^1$ are to $o_2$ and to two cycles of size three.
Hence in total $x_{i+1}^1$ will transfer at most $\frac{1}{12}+\frac{2}{3}+2\cdot\frac{1}{3}=2\cdot\frac{2}{3}+\frac{1}{12}$ points.
Hence $X_{i+1}$ will transfer at most $2\cdot\frac{2}{3}+\frac{1}{12}+\frac{2}{3}=3\cdot\frac{2}{3}+\frac{1}{12}$ points.
\end{enumerate}
Hence in all cases, $X_{i+1}$ will transfer at most $\frac{7}{3}$ points. Consequently, as $X_{i+1}$ starts with two points, it has at least $-\frac{1}{3}$ points.
}
\end{proofing}
\section{Proof of \hyperref[thm:main5]{\text{Theorem} \ref{thm:main5}} }
To prove \hyperref[thm:main5]{\text{Theorem} \ref{thm:main5}}
we modify the point transfer rules in the following way: \ref{rule:v2_path} and \ref{rule:v5_rest} remain unchanged,
\ref{rule:v4_dangerous} and \ref{rule:v2a_dangerous} are deleted, and \ref{rule:v2_cycle} is changed such that the points that are transferred to cycles on each edge is multiplied by $ \frac{4}{3}$ (hence $\frac{1}{i}\cdot\frac{4}{3}$). \label{rule:v2_cycle_changed}
For completeness we provide here the changed point transfer rules:
\begin{mdframed}[linecolor=black!40,
outerlinewidth=1pt,
roundcorner=.5em,
innertopmargin=1.3ex,
innerbottommargin=.5\baselineskip,
innerrightmargin=1em,
innerleftmargin=0.4em,
backgroundcolor=blue!10,
shadow=true,
shadowsize=6,
shadowcolor=black!20,
frametitle={\Large Changed Transfer rules:},
frametitlebackgroundcolor=cyan!40,
frametitlerulewidth=14pt
]
\begin{description}[leftmargin=100pt
]
\item[\namedlabel{changed_rule:v2_cycle}{Changed Rule 1}] From $V_2$ to $V_1\cap \mathcal{C}$. $v\in V_2$ transfers $\frac{1}{i}\cdot\frac{4}{3}$ points to $u\in V_1\cap \mathcal{C}$ , if $u$ is a vertex of a cycle of size $i \le 6$ and $(u,v)$ is a balanced edge.
\item [\namedlabel{changed_rule:v2_path}{Changed Rule 2}] From $V_2$ to $V_1\cap \mathcal{P}$. $v\in V_2$ transfers $\frac{2}{3}$ points to $u\in V_1$ if $u$ is an end-vertex of a path and $(u,v)$ is a balanced edge.
\item [\namedlabel{changed_rule:v5_rest}{Changed Rule 5}] From $V_5$ to $V \setminus V_5$. $v\in V_5$ transfers $\frac{1}{4}$ points to $u\in (V \setminus V_5)$ if $(u,v)$ is a free edge.
\end{description}
\end{mdframed}
This change gives us the following propositions:
\begin{pro}
\label{pro:easy6}
Let $C$ be a cycle in a canonical path partition. Then after applying the transfer rules $C$ has at least~$6+\frac{1}{3}$ points.
\end{pro}
\begin{proofing}{Proposition \ref{pro:easy6}}
{
Let $i$ denote the size of the cycle. If $i \ge 7$ then we are done, because according to the point transfer rules, a vertex of a cycle can only receive points. If $i=6$, then there are at least two balanced edges incident to the cycle, as there are no $K_6$ in the graph. Hence cycles of size $6$ ends with at least $6+\frac{4}{9}$ points. If $i \leq 5$ (and necessarily $i \ge 3$), then each vertex has at least $6-i$ balanced edges connecting it to vertices in $V_2$, because the degree of each vertex is~5. Each balanced edge contributes to the vertex $\frac{1}{i}\cdot \frac{4}{3}$ points (by \ref{changed_rule:v2_cycle}), hence, each vertex on the cycle has at least $1 + (6-i)\frac{1}{i}\cdot \frac{4}{3}$ points. Hence the cycle has at least $i\cdot(1 + (6-i)\frac{1}{i}\cdot \frac{4}{3}) = 8-\frac{i}{3}$ points. Hence, $i\leq 5$ there are at least $6+\frac{1}{3}$ points as desired.
}
\end{proofing}
\begin{pro}
\label{pro:easyouter6}
Let $P$ be a path in a canonical path partition. Then after applying the transfer rules the end-vertices of $P$ have $6+ \frac{4}{3}$ points.
\end{pro}
\begin{proofing}{Proposition \ref{pro:easyouter6}}
{In a canonical path partition, there are no isolated vertices. Hence $P$ has~8 balanced edges incident with its two end-vertices. Applying \ref{changed_rule:v2_path} together with the two starting points, sums up to $2 + 8\cdot \frac{2}{3}=6+ \frac{4}{3}$ points. }
\end{proofing}
\begin{pro}\label{pro:easypoints5}
The only nodes that can end up with a negative number of points are those from $V_2$. In particular:
\begin{enumerate}
\item Every node in $V_2$ has at least $-1$ points.
\item Every node in $V_4$ or $V_3$ has at least one point.
\item Every node in $V_5$ has a non-negative number of points.
\end{enumerate}
\end{pro}
\begin{proofing}{Proposition \ref{pro:easypoints5}}
{ Each vertex starts with one point.
\begin{enumerate}
\item Vertices in $V_2$ have at most three balanced edges, and may transfer at most $\frac{2}{3}$ points on each balanced edge, hence they have at least $-1$ points.
\item Vertices in $V_3\cup V_4$ can only receive points, hence they have at least one point.
\item Vertices in $V_5$ have three free edges, and may transfer at most $\frac{1}{4}$ points on each free edge.
\end{enumerate}
}
\end{proofing}
For completeness we state the following proposition, that is identical to \hyperref[lem:kadjacent]{\textbf{Lemma} \ref{lem:kadjacent}} .
\begin{pro}
\label{pro:lemmafour}
Let $X$ be a sequence of $k\geq 2$ vertices in $V_2^b$, then $X$ ends with at least $\frac{1}{3}\cdot k-\frac{4}{3}$ points.
\end{pro}
\begin{proofing}{Proposition \ref{pro:lemmafour}}
{
We prove this, mimicking the proof of \hyperref[lem:kadjacent]{\textbf{Lemma} \ref{lem:kadjacent}} with a change in the following places:
\begin{itemize}
\item \hyperref[samecycle2]{\text{Changed Observation} \ref{samecycle2}} - the same proof works, but \ref{changed_rule:v2_cycle} will imply that $X'$ transfers to $C$ up to $\frac{4}{3}$ points.
\item \hyperref[samecycle3]{\text{Changed Observation} \ref{samecycle3}}- the same proof works, but \ref{changed_rule:v2_cycle} will imply that $X'$ transfers to $C$ up to $2$ points.
\item \hyperref[samecyclek]{\text{Changed Observation} \ref{samecyclek}} - the same is true and by \hyperref[samecycle2]{\text{Changed Observation} \ref{samecycle2}} and \hyperref[samecycle3]{\text{Changed Observation} \ref{samecycle3}} we get that $b(X')\leq |X'|\cdot\frac{2}{3}+2\cdot\frac{2}{3}$.
\item Changed Case \ref{item3:lemma} in the proof of \hyperref[lem:none_to_dangerous]{\text{Lemma} \ref{lem:none_to_dangerous}} - now $x_i$ has only three free edges and hence can \textbf{go} only to one cycle (instead of two) in addition to $o_1$ and $o_2$. The points $x_i$ will transfer to a cycle is strictly less than $\frac{2}{3}$. Hence, $b(X)< (k+2)\cdot\frac{2}{3}$.
\item Changed Case \ref{item4:lemma} in the proof of \hyperref[lem:none_to_dangerous]{\text{Lemma} \ref{lem:none_to_dangerous}} - by \hyperref[samecyclek]{\text{Changed Observation} \ref{samecyclek}} then
$b(X')\leq|X'|\cdot\frac{2}{3}+2\cdot\frac{2}{3}$, and hence $b(X) = (k+2)\cdot\frac{2}{3}$.
\end{itemize}
All the rest of the proof remains the same. Hence $X$ ends with at least $\frac{1}{3}\cdot k-\frac{4}{3}$ points.
}
\end{proofing}
\begin{pro}
\label{pro:intenal}
Let $P$ be a path in a canonical path partition. The number of points on internal nodes in $P$ is at least $-1$.
\end{pro}
\begin{proofing}{Proposition \ref{pro:intenal}}
{
We show that except for blocks of kind $4$ (the last block) that may end with $-1$ points, the number of points on all blocks is non-negative.
Because all internal vertices up to the beginning of the first block contribute a non-negative number of points (by item $3$ in \hyperref[pro:easypoints5]{\textbf{Proposition} \ref{pro:easypoints5}}), we have that the internal vertices of a path will have at least $-1$ points.
By item $1$ in \hyperref[pro:easypoints5]{\textbf{Proposition} \ref{pro:easypoints5}} a vertex in $V_2$ has at least $-1$ point , hence blocks of kind $1$ and $2$ are not a problem because they are followed by one or two vertices with at least one point (item $2$ in \hyperref[pro:easypoints5]{\textbf{Proposition} \ref{pro:easypoints5}}). Hence the number of points on any block of kind $1$ or $2$ is non-negative.
For blocks of kind $3$,
\hyperref[pro:lemmafour]{\textbf{Proposition} \ref{pro:lemmafour}} ensures that the number of points on $k\geq 2$ vertices in $V_2^b$ is at least $-1$ (even $-\frac{2}{3}$).
}
\end{proofing}
Now we prove the theorem:
\begin{proofing}{Theorem \ref{thm:main5}}
{Consider a canonical path partition $\mathcal{S}$. We show that applying the set of transfer rules to $\mathcal{S}$ leads to a situation where every component in $\mathcal{S}$ has at least $6+\frac{1}{3}$ points. \hyperref[pro:easy6]{\textbf{Proposition} \ref{pro:easy6}} implies that we only need to handle paths. For a path \hyperref[pro:intenal]{\textbf{Proposition} \ref{pro:intenal}} combined with \hyperref[pro:easyouter6]{\textbf{Proposition }\ref{pro:easyouter6}} will imply that the path has at least $6+\frac{1}{3}$ points, which proves the theorem.
}
\end{proofing}
\section*{Acknowledgements}
Work supported in part by the Israel Science Foundation (grant No. 1388/16).
\bibliographystyle{alpha}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,421
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.