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{"url":"http:\/\/pisarenko.net\/blog\/2013\/07\/10\/replacing-preview-glass-on-canon-500d\/","text":"Material things break all the time - one slightly wrong move and your beloved possession is cracked, damaged or shattered. My camera is no exception. Another SLR has slipped from a hand of my friend and landed on the corner of the preview screen of my Canon 500D causing an ugly crack:\n\nMany folks like to buy new things whenever old ones break or pay a lot for repair. I prefer to fix as much as I can myself. It\u2019s fun and very much in-line with the growing stoic inside of me.\n\nIs it possible to fix this problem yourself?\n\nCertainly. And the process is not complicated. All you need is a couple of household tools and the actual replacement screen which you can order from eBay.\n\n## Tools and supplies that I have used\n\nI expect you already have a hair dryer and something sharp (like a knife or a blade). Here\u2019s what I have used:\n\nLeatherman Wave knife\n\nRemington hair dryer\n\nWe will also need a small suction cup with a handle\/head. I did not have any so I had to order one through eBay. I have paid 12$(with shipping to Switzerland) for a set of 6 (size 1.5 inches). 6 is the smallest quantity I found. You can search for \u201csuctioncups4u\u201d on eBay to find the seller. A replacement glass can be found on eBay too. Search for \u201c500D outer screen window glass cover\u201d. I have paid 12$ (with shipping).\n\n## Step-by-step instructions\n\n1. Heat up the sides of the preview glass for about 10-40 seconds on each edge to weaken the glue that holds the glass.\n\n2. Carefully slide a sharp knife or blade underneath the glass and slowly push it along the side.\n\n3. Put the suction cup on the glass and pull gently:\n\n4. Remove the broken glass:\n\n5. Remove residue glue:\n\n6. Take off the protective paper from the replacement glass:\n\n7. Carefully drop the new glass. Note that it\u2019s not symmetrical so if it does not immediately fit rotate the glass by 180 degrees.\n\nCamera fixed! Total cost of materials is 24\\$. The whole process took about 10 minutes. The same instructions apply equally well to some other SLRs. A YouTube video I found for Canon 5D confirms this: Replacing a scratched LCD cover on a Canon EOS 5D SLR\n\nWith the camera fixed I am off to the next adventure. Good luck!","date":"2019-04-26 06:11:11","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\": 0.2512776255607605, \"perplexity\": 1799.3102675237842}, \"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-2019-18\/segments\/1555578760477.95\/warc\/CC-MAIN-20190426053538-20190426075538-00425.warc.gz\"}"}
| null | null |
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Keep your users up-to-date on the status of their purchase with order history. WooCommerce mobile app lets them review the details of any order they have made in the past.
With WooCommerce mobile app purchases are made with speed and ease. Control anonymity of the buyers by letting them check out as guests or forcing them to login. Present various shipping and payment options pulled directly from your WooCommerce store.
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We constantly keep working on making WooCommerce mobile app even better. New features and improvements are getting rolled out on a regular basis. We care about our community and are happy to hear your feedback.
Collect and interpret data from your store with the help of Google's Enhanced Ecommerce tracking and Facebook Analytics. Being informed is the key to making smart decisions.
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Products, categories, shipping and payment methods are streamed from your WooCommerce store into the mobile app.
The app is styled to match your company's brand and identity. Aspects such as colors, icons and labels can be modified to give your app a unique look.
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Do you have special requirements like adding a nice looking animation or matching some cool UI design? Maybe you'd like to add another page to the app? Or maybe even build a completely distinct mobile shopping experience. MobileFront development team has got you covered. We can introduce endless customizations to WooCommerce mobile app to fulfill any of your wishes.
MobileFront development services is the most cost effective and quick way to create a unique m-commerce app. We are reusing the existing WooCommerce mobile app instead of building a new app from scratch. The lets us significantly cut the development time and minimize time to market.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,287
|
La casa de tothom és una comèdia en tres actes, original de Josep Morató, estrenada al Gran Teatre Espanyol, ubicat al Paral·lel de Barcelona, la nit del 20 de desembre de 1912, per la companyia del Sindicat d'Autors Dramàtics Catalans.
L'acció té lloc a la sala menjador d'una torre de Sant Gervasi, de Barcelona, a l'època de l'estrena.
Repartiment de l'estrena
Laura: Emília Baró
Senyora Francisca: Dolors Pla
Glòria: Ramona Mestres
Don Ramon: Antoni Piera
Cisonet: Carles Capdevila
Don Lluís: Rafael Bardem
Minguet: Domènec Aymerich.
Magí: Vicent Daroqui
Don Enric: Miquel Sirvent
Albert: Avel·lí Galceran
Director d'escena: Antoni Piera
Referències
Obres de teatre en català
1912 a Catalunya
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,149
|
GadgetsRumors suggest that OPPO may launch fast charging SuperVOOC...
GadgetsNewsSMARTPHONES
Rumors suggest that OPPO may launch fast charging SuperVOOC 3.0 with 80W
Rumors: OPPO may launch fast charging SuperVOOC 3.0 with 80W
For a long time, the main problem with modern cell phones has been the battery. Gradually, manufacturers are increasing the energy capacity of their devices, but that was not enough.
Therefore, fast charging technologies, which use high power to reduce the recharge time of the devices. Among them is OPPO's SuperVOOC 3.0, one of the fastest solutions on the market, whose version 2.0, currently in force, reaches an impressive 65W.
The Chinese woman does not seem to be satisfied, however, and may already be preparing version 3.0 of her fast charging system, which could raise battery limits even further. As leaker Really Assen points out in a post on Weibo, the manufacturer's technology should reach 80W of power.
With gains of just over 18% over its predecessor, the novelty was supposed to be able to fully recharge the brand's smartphones in about 15 minutes. For comparison, SuperVOOC 2.0 charges cell phones with 10V and 6.5A, and the Gizmochina website speculates that the new protocol should use 20V and 4A.
OPPO may launch fast charging SuperVOOC 3.0 with 80W
Also according to the rumor, SuperVOOC 3.0 should arrive only in 2021, but chances are that we will see its arrival even before the end of 2020. However, it is worth noting that the news must be considered carefully. Although possible, the fact that we have only one comment does not give credit to the information, and the novelty may be false after all.
It is worth remembering that Xiaomi also seems to be planning to achieve impressive charging speeds, according to a leak released this week. In it, a video shows what appears to be a company charger with no less than 120W of power. This, however, would not be the first time that we would see such power: Vivo unveiled in June last year its iQOO 5G, which also brought 120W of charging. The device was able to fill your tank in an unbelievable 13 minutes.
Android 11: first beta launches with news in notifications, media controls and more
Movies you only find on Amazon Prime Video
Over 1000 NPCs in Cyberpunk 2077 will have unique daily routines
Samsung may launch Fold 2 later this year with UTG and cheaper variant in 2021
Sony will announce PlayStation 5 games at an event on Thursday (11)
Previous articleAndroid 11: first beta launches with news in notifications, media controls and more
Next articleApple plans to discontinue the iTunes U app by the end of 2021
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,593
|
June 19, 2019 10:15AM EDT
UN: Expert Seeks Criminal Probe of Khashoggi Murder
Top Rights Body Should Monitor Saudi Abuses
Candles lit by activists protesting the killing of Saudi journalist Jamal Khashoggi are placed outside Saudi Arabia's Consulate in Istanbul.
© 2018 Lefteris Pitarakis/AP Photo
(Geneva) – The United Nations expert's call for justice for the murder of the Saudi journalist Jamal Khashoggi should lead to the creation of a comprehensive system to monitor Saudi rights abuses, Human Rights Watch said.
Agnes Callamard, the special rapporteur for extrajudicial executions, noted on June 19, 2019, in releasing the findings of her investigation into the October 2018 killing, that there is evidence that responsibility for Khashoggi's murder extends beyond the 11 individuals currently on trial for the murder in Saudi Arabia, and that the mission to execute Khashoggi required "significant government coordination, resources and finances." The special rapporteur determined that there is credible evidence warranting further investigation of high-level Saudi officials, including Crown Prince Mohammad bin Salman, for their role in the murder.
"The Saudi leadership has little incentive to change its behavior unless it knows that it will be held to account for its systemic abuse and repression," said John Fisher, Geneva director at Human Rights Watch. "It is paramount for members of the UN Human Rights Council to address the intolerance of dissent and climate of impunity that made Khashoggi's brutal and brazen murder possible with a system to ensure ongoing scrutiny of Saudi Arabia's rights abuses."
In response to this and other egregious Saudi abuses, UN member countries should support targeted sanctions on members of the Saudi leadership responsible for ongoing human rights violations, the UN secretary-general should initiate a full criminal investigation as recommended by the special rapporteur, and the Human Rights Council should establish a mechanism to monitor and report on human rights violations in Saudi Arabia.
While the report did not find definitive evidence linking bin Salman to the murder, it noted that he "had played an essential role in a campaign of repressing dissidents" and that experts found it "inconceivable" that such a large-scale operation could be implemented without the crown prince being aware that a "mission of a criminal nature, directed at Mr. Khashoggi, was being launched." The report called for further investigation into the role of top Saudi officials, including the crown prince.
The Callamard investigation, opened in January, followed the failure of UN Secretary-General Antonio Guterres to establish a broader UN investigation under his auspices. While Callamard's report focused on Khashoggi's murder, it also highlighted the widespread rights violations in Saudi Arabia. Further monitoring is needed to address the underlying repression and abuse of dissidents and activists out of which the murder took place, Human Rights Watch said.
After a series of denials, Saudi Arabia acknowledged that government agents murdered Khashoggi in October 2018 at the country's Istanbul consulate. But the Saudi authorities have provided little information about the trial of all people accused of his murder and appear to be shielding high-level current and former officials implicated in the murder from additional scrutiny.
Khashoggi's murder took place amid successive waves of arrests of Saudi dissidents, clerics, journalists, intellectuals, businesspeople, royal family members, and women's rights activists after bin Salman became crown prince in July 2017. Authorities have subjected many to unfair trials, and some have alleged that authorities tortured them in detention.
Human Rights Watch has previously called for individual sanctions against bin Salman over the Saudi-led coalition's indiscriminate bombing and unlawful blockading of essential goods to Yemen's civilian population. The government run by bin Salman's responsibility for continued major human rights violations amounting to crimes only strengthens the case for sanctions against top Saudi leaders.
In May 2018, just weeks before Saudi authorities lifted the ban on women driving on June 24, they launched a widespread arrest campaign against the women's rights movement, detaining nearly 20 people. It brought 11 women to trial in March. At least four of the women said that they were tortured in detention.
The charges against the women on trial include speaking about women's rights to international journalists, diplomats, and international human rights organizations. Eight women have been temporarily released while still facing trial, but at least five women's rights activists and one man associated with them remain detained. Loujain al-Hathloul is among those who is on trial but remains in prison. Two other prominent women's rights activists, Samar Badawi and Nassima al-Sadah, remain detained without charge or trial.
On April 23, 2019, Saudi Arabia announced the mass execution of 37 men in various parts of the country. At least 33 were from the country's minority Shia community. They had been convicted following unfair trials for various alleged crimes, including protest-related offenses, espionage, and terrorism.
Beginning in mid-2018, Saudi prosecutors began seeking the death penalty against peaceful dissidents on charges related solely to their alleged activism or political affiliations. Those on trial in capital cases include several prominent clerics, intellectuals, and academics detained in September 2017. Among the clerics is Salman al-Awda, who is on trial solely for alleged ties to the Muslim Brotherhood and opposition to Saudi government policies.
One detainee, Murtaja al-Qurairis, faced a capital trial for protest-related crimes and violent acts even though he allegedly committed some of the offenses while he was just 10 or 11 years old. He was arrested when he was 13. On June 15, 2019, most likely as a result of international pressure, Saudi authorities told Reuters that al-Qurairis had received a 12-year sentence and could be released by 2022.
Saudi Arabia's discriminatory male guardianship system remains intact despite government pledges to abolish it. Under this system, adult women must obtain permission from a male guardian – usually a husband, father, brother, or son – to travel abroad, obtain a passport, marry, or be discharged from prison. Women may be required to provide guardian consent to work or to receive health care.
In today's report, the special rapporteur noted that to avoid repetition, Saudi Arabia should release all individuals imprisoned for the peaceful expression of their opinion and belief; independently investigate all allegations of torture and lethal use of force in formal and informal places of detention; and independently investigate all allegations of enforced disappearances and make public the whereabouts of individuals disappeared.
In March, 36 countries at the Human Rights Council endorsed a joint statement delivered by Iceland calling on Saudi Arabia to improve its human rights record. The statement condemned Khashoggi's murder, urged an end to Saudi Arabia's use of counterterrorism regulations to target dissidents and human rights activists, and called for the release of the Saudi women's rights activists. As a member of the Human Rights Council, Saudi Arabia should be subject to increased scrutiny to ensure compliance with membership obligations.
"Human Rights Council member countries should move to end the climate of impunity around Saudi Arabia's escalating repression," Fisher said. "Given the level of abuses at home and abroad, amounting to crimes, Saudi Arabia's leadership should face sanctions until they end the abuse. A comprehensive UN monitoring mechanism would be a major step toward accountability and justice."
Yemen: Attack on Saudi Airport Apparent War Crime
Saudi Arabia/UAE: Join Landmine Ban Treaty
Syria: US Coalition Should Address Civilian Harm
First Public 'Condolence' Payment Provides Way Forward
Lebanon: Syrian Refugee Shelters Demolished
Coercive Measures Intensify Pressures to Return to Syria
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,155
|
How becoming a Covid long-hauler made me rethink disability
Áine Kelly-Costello | Guest writer
Despite having lived with a disability her whole life, it took becoming one of the unlucky few who experience symptoms long after they should have 'recovered' from Covid-19 that forced Áine Kelly-Costello to reckon with a different kind of disability experience.
I was born with a congenital condition with a fancy name that designates a diagnosis: I'm blind. However, my encounters with the medical establishment around disability/impairment have otherwise been almost non-existent. Apart from not fulfilling their primary function, my eyes have fortunately been perfectly healthy, I haven't even needed to get fitted for glasses and to my knowledge I have no other underlying medical conditions.
Then, a bewildering pandemic came along, in March I got sick with strongly suspected Covid, and I'll mark six months experiencing illness on September 21.
So, despite living for 25 years as someone whose access and inclusion tends to be marginalised, attempting to manage an invisible, morphing post-viral illness that is sticking around for a while has been a reckoning with a kind of disability experience that I ignorantly had not given great consideration to before.
Where I lived, in Sweden, I couldn't get tested back in March, so I have no virus diagnosis. But say I did? My symptoms are only now about to cross the threshold of having lasted six months or longer. Our most authoritative international disability mechanism, the UN Convention on the rights of Persons with Disabilities, recognises that disability is an evolving concept, but also states that it is "long-term". So, from a medical and legal point of view, I'm not disabled by Covid, at least, not yet.
To me, disability is much more fluid. I feel disabled by my body on days where fatigue makes eating a chore, and where stringing together enough focus to send a quick email – let alone tackle my thesis – is like wading through mud. I have wonderful family support and can financially afford to spend time recuperating. But even on days when I'm not experiencing additional symptoms like extreme emotions, fever, pain and inflammation, I still feel the pressure of a world where I have been taught that productivity is prized.
I feel disabled by the fact that when I say I'm free on a certain day to join a Zoom with my colleagues or just chat to a friend, it often comes with the caveat that, if possible, I will move it if I get a flare up and I think my body can't handle it. I know I only succeed in asking for this adjustment at all because I have spent enough time reading the generous wisdom passed down by people with other chronic illnesses to know that the best thing we can do is honour and listen to our bodies. And still, I feel discomfort and shame, because I know I live in a world that values planning, which isn't used to making room for prioritising our bodies and our minds and that lives by the 24-hour clock. Even with fabulously understanding friends and colleagues, with every new person I explain to, I have to remind myself that making space for my body should not be embarrassing. Rather than an inconvenience, I am someone who wants to bring the people I am honest with into a type of accountability that is about necessary pacing and self-care before it is about clocks.
Compared to many Covid long-haulers who were previously non-disabled, I perhaps more willingly embrace a terminology around disability because I have already been immersed in many of its rich communities and creative, resilient ways of thinking. A lot of long-haulers may understandably associate disability specifically with getting a medical diagnosis of chronic illness. And let's face it, none of us actually want our symptoms to be chronic, even where we are actively seeking medical validation of their existence. If the actual symptoms and complications weren't deterrent enough, notions of disability as brokenness, undesirability, burden and victim, built up over a lifetime of marginalising and oppressing disabled people, are also well baked into us all.
And yet, when I think of the Covid patient-led research group that in six weeks produced a preliminary paper detailing long-haul experiences, when I think of the 60 people with long Covid who dedicated themselves to talking to the WHO after struggling for mainstream recognition for months, when I think of all of us being there for each other through our uncertainty and illness, I am fiercely proud of the grit and resilience and persistence these communities have shown. I am proud and thankful that, through support groups like this one, I have found more courage to speak out than I otherwise would have, such as in writing this piece. These groups assure me that my wild, disjointed, discombobulating set of symptoms and experiences are valid, that some involve my brain but that does not mean they are imaginary, that even as our experiences fail to exist in many official stats and records, that we are here and we count.
This is the kind of validation the Disability Pride movement seeks to create. For three years in a row, starting in 2017, I put fingers to keyboard to try to unpick some of both the untapped potential and messy complexity of Disability Pride. I have called for imagining collective pride, I have interviewed Disability Pride organisers and I have shared an example of my own struggle disclosing disability and a vision forward.
I had no intention to write yet another piece this year, until I got Covid-19.
I'm so glad I have found long Covid groups to add to the disability communities I was already part of, because I'm sure my reckoning with new ways of experiencing disability is far from over. As long as we are there for each other, uplifting each other and affirming that we count, I'm a lot less worried about whether or not we call it Disability Pride.
Áine Kelly-Costello is a disabled campaigner and writer from Aotearoa currently based in Norway. She has helped organise Disability Pride Week among other disability community-building initiatives.
Five ideas to fix NZ's completely pathetic Covid QR scanning record
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Siouxsie Wiles: What the new, more infectious strains of Covid-19 mean for us
Siouxsie Wiles
Sorry: the first 12 days of 2021 prove the apocalypse is coming
Emily Writes
Parents Editor
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 439
|
package org.thymeleaf.inline;
import java.io.StringWriter;
import java.util.Arrays;
import java.util.Collections;
import java.util.List;
import java.util.Map;
import com.fasterxml.jackson.annotation.JsonAutoDetect;
import org.junit.Assert;
import org.junit.Test;
import org.thymeleaf.TemplateEngine;
import org.thymeleaf.context.Context;
import org.thymeleaf.templateresolver.ITemplateResolver;
import org.thymeleaf.templateresolver.StringTemplateResolver;
import org.thymeleaf.util.DateUtils;
public class ScriptInlineTest {
private static void testInlineResult(final String script, final String expectedResult) {
testInlineResult(script, expectedResult, Collections.<String, Object>emptyMap());
}
private static void testInlineResult(final String script, final String expectedResult, final String variableName, final Object variableValue) {
testInlineResult(script, expectedResult, Collections.singletonMap(variableName, variableValue));
}
private static void testInlineResult(final String script, final String expectedResult, final Map<String,Object> variables) {
final String completeScript =
"<script th:inline=\"javascript\">\n/*<![CDATA[ */\n" +
script +
"\n/* ]]> */\n</script>";
final ITemplateResolver templateResolver = new StringTemplateResolver();
final TemplateEngine templateEngine = new TemplateEngine();
templateEngine.setTemplateResolver(templateResolver);
final Context ctx = new Context();
ctx.setVariables(variables);
final StringWriter stringWriter = new StringWriter();
templateEngine.process(completeScript, ctx, stringWriter);
final String result = stringWriter.toString();
final String extractedResult =
result.substring(0, result.indexOf("\n/* ]]> */\n</script>")).substring(24);
Assert.assertEquals(expectedResult, extractedResult);
}
@Test
public void testDateInline() throws Exception {
testInlineResult(
"[[${a}]]",
"\"something\"",
"a", "something");
testInlineResult(
" /*[[${a}]]*/ 'prototype';",
" \"something\";",
"a", "something");
final java.util.Calendar calendar1 =
DateUtils.create(Integer.valueOf(2013),Integer.valueOf(01),Integer.valueOf(01),Integer.valueOf(14),Integer.valueOf(30));
final java.util.Date date1 = calendar1.getTime();
final java.sql.Date dateSql1 = new java.sql.Date(date1.getTime());
testInlineResult(
" /*[[${calendar1}]]*/ 'prototype';",
// Calendar should be inlined as ISO6801 date string literal
// e.g. '2013-01-01T14:30:00.000+01:00'
" \"" + DateUtils.formatISO(calendar1) + "\";",
"calendar1", calendar1);
testInlineResult(
" /*[[${date1}]]*/ 'prototype';",
// Date should be inlined as ISO6801 date string literal
// e.g. '2013-01-01T14:30:00.000+01:00'
" \"" + DateUtils.formatISO(date1) + "\";",
"date1", date1);
}
@Test
public void testObjectInline() throws Exception {
final SomeObjectA obj01 = new SomeObjectA();
obj01.one = "value number one";
obj01.two = 1231;
obj01.three = 1231.12f;
obj01.four = true;
testInlineResult(
" /*[[${obj01}]]*/ 'whatever';",
" {\"one\":\"value number one\"};",
"obj01", obj01);
final SomeObjectB obj02 = new SomeObjectB();
obj02.one = "value number one";
obj02.two = 1231;
obj02.three = 1231.12f;
obj02.four = true;
testInlineResult(
" /*[[${obj02}]]*/ 'whatever';",
" {\"one\":\"value number one\",\"two\":1231,\"three\":1231.12,\"four\":true};",
"obj02", obj02);
}
@Test
public void testArrayInline() throws Exception {
final String[] array01 = new String[] { "hello", "goodbye" };
testInlineResult(
" /*[[${array01}]]*/ 'whatever';",
" [\"hello\",\"goodbye\"];",
"array01", array01);
}
@Test
public void testCollectionInline() throws Exception {
final List<String> list01 = Arrays.asList(new String[] { "hello", "goodbye" });
testInlineResult(
" /*[[${list01}]]*/ 'whatever';",
" [\"hello\",\"goodbye\"];",
"list01", list01);
}
public static class SomeObjectA {
private String one;
private int two;
private float three;
private boolean four;
public String getOne() {
return one;
}
}
@JsonAutoDetect(fieldVisibility = JsonAutoDetect.Visibility.ANY)
public static class SomeObjectB {
private String one;
private int two;
private float three;
private boolean four;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 131
|
Typical Learning and Development Sector challenges..
Mary Gober International (MGI) help organisations actively and rapidly improve performance. With bases in the UK, Australia, the Middle East and the United States, they work with clients around the world to deliver tailored solutions inspired by the work of Mary Gober. "MGI were using Excel spreadsheets to drive many core processes & reporting within the business.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,640
|
{"url":"http:\/\/mathoverflow.net\/questions\/3241\/when-is-a-locally-convex-topological-vector-space-normal-or-paracompact","text":"# When is a locally convex topological vector space normal or paracompact?\n\nAll locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is paracompact or normal.\n\nSome background: I have a particular space in mind. It is a directed colimit (union) over an uncountable family of nuclear Frechet spaces. It is complete but not metrisable nor separable nor nuclear. I have a very concrete description of it and can describe the bounded (and compact) subsets. This space is fairly badly behaved in other ways so I'm half anticipating a negative result, thus answers along the lines of \"If you can find a subset that looks like X then it can't be normal\" could be just what I'm looking for.\n\nSo in particular, I'm interested in the general question. To forestall a couple of \"easy\" answers: as my space is not Frechet, it is not metrisable so I can't directly use theorems on metrisability (however, as it is a colimit there may be some scope for indirect use). And, of course, paracompact would imply normality since it is completely regular.\n\nTo forestall another possible comment, I'm not going to tell you what the particular space is. Partly because I'm more interested in the general situation, this space is merely focussing my attention on the question, and partly because it's a more useful question if it's general. A bit of intelligent searching would reveal what space it is anyway so it's no great hardship.\n\nEdit: There is a simple example of a space that would be very interesting to know about: the sum (i.e. coproduct) of an uncountable number of copies of $\\mathbb{R}$, or even more specifically $\\sum_{\\mathbb{R}} \\mathbb{R}$. This isn't the specific space that I'm interested in, but is close enough that I think that an answer either way for this space will tell me what to do for my space.\n\n-\n...sum of an uncountable number of copies of R... with what topology? \u2013\u00a0 Gerald Edgar Nov 12 '09 at 13:48\nColimit topology: strongest locally convex topology so that the inclusions of all finite subsums are continuous. \u2013\u00a0 Loop Space Nov 13 '09 at 10:28\n\nThanks to your other question, I was on a LCTVS kick. I did find one general criterion that implies that a locally convex space is paracompact. According to the Encyclopedia of Mathematics, if it is Montel (which means that it is barrelled and the Heine-Borel theorem holds true for it), then it is paracompact. Although this criterion is important, it is of no use to your specific question.\n\nI thought I had a proof for half of your question, which I wrote up as the first version of this answer, but I made a mistake and proved something different. My thinking is based on the fact that the normality axiom for a topological space is equivalent to the Tietze extension theorem. (Tietze extension follows from normality. In the other direction, if $A$ and $B$ are the two closed sets, you obtain disjoint open neighborhoods from a continuous function that is 0 on $A$ and 1 on $B$.) However, in my argument I conflated the locally convex direct sum of spaces with the topological direct sum. For a countable direct sum of copies of $\\mathbb{R}$, they are the same topology, and they agree with the box topology. But Waelbroeck, LNM 230 points out that they are different in the uncountable case.\n\nLet $\\alpha$ be an ordinal, for instance an ordinal of cardinality $2^{\\aleph_0}$. Then $\\mathbb{R}^\\alpha$ in the topological direct sum topology satisfies Tietze extension. Let $A \\subset \\mathbb{R}^\\alpha$ be a closed set and let $f:A \\to \\mathbb{R}$ be a continuous function. For $\\beta < \\alpha$, let $A_\\beta$ be the intersection of $A$ and with $\\mathbb{R}^\\beta$. Suppose that $\\alpha = \\beta+1$ is a successor ordinal. If $\\alpha$ is finite, then the conclusion is standard. Otherwise, by induction, there is an extension $f_\\beta$ of $f$ to $\\mathbb{R}^\\beta$. Moreover, by induction in a different sense, we have already proved that $\\mathbb{R}^{\\beta+1}$ is normal, since $\\beta$ and $\\beta+1$ have the same cardinality. So there exists an extension $f_\\alpha$ to $\\mathbb{R}^\\alpha$. If instead $\\alpha$ is a limit ordinal, then the extensions all the way up to $\\alpha$ work just because they work; that's the behavior of topological direct limits.\n\nHaving failed to normality for the locally convex direct sum, I can't say much about paracompactness either. :-) However, there is an interesting result called the Michael selection theorem which seems to do for paracompactness what the Tietze theorem does for normality. If the Tietze theorem is useful for your spaces, then maybe the Michael selection theorem is too.\n\n-\nThis is going to take me a while to parse. First question: you say that the direct limit for direct sums in LCTVS is the box topology. Do you have a reference for that? Is it peculiar to having a limit ordinal? I seem to be able to construct a neighbourhood using an infinite simplex that doesn't contain a box, but I may well be wrong. \u2013\u00a0 Loop Space Dec 14 '09 at 18:56\nWell, geez. I need to review whether or not it is finer than the box topology. If so, that is a mistake already, but not the essential point. What I really may need, whether or not it is the box topology, is that this l.c. inductive limit happens to equal the topological inductive limit. I construct a Tietze function by transfinite induction. I should also simplify and reword the argument a bit, but first let me review whether or not it works. \u2013\u00a0 Greg Kuperberg Dec 14 '09 at 19:34\nIt's been a while since I first worked on this problem and I'd forgotten some of the subtleties. In particular, reading your answer again I recall that I'm interested in both the inductive LCTVS topology and the inductive topology (over inclusions of finite dimensional subspaces) so your answer on the latter is still very useful even though it's not directly an answer of the stated question! \u2013\u00a0 Loop Space Dec 16 '09 at 8:59\n\nA paper dealing with this (and other) questions for the weak topology of a Banach space ... H. H. Corson, \"The weak toplology of a Banach space\" Trans. Amer. Math. Soc. 101 (1961) 1--15.\n\nIn that case:\n\nX is paracompact iff X is Lindelof.\n\nIf X^n is normal for all n, then X is real-compact.\n\n-\nI'll take a look. My space is not Lindelof, but then it's not a Banach space with the weak topology either. Still, there may be ideas that still apply. Thanks. \u2013\u00a0 Loop Space Oct 30 '09 at 8:45","date":"2015-05-30 18:48:14","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\": 0.8751852512359619, \"perplexity\": 220.1153072117215}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"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-2015-22\/segments\/1432207932596.84\/warc\/CC-MAIN-20150521113212-00284-ip-10-180-206-219.ec2.internal.warc.gz\"}"}
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Q: bash scripting in os x -- making networksetup changes I'm making a script for geektool so that I can always see what my DNS setting's are on my desktop. It works as far as outputting the correct commands, however I'm getting a
command not found...
here's my script
#!/bin/bash
IFS='
'
getdns='networksetup -getdnsservers'
# [-getdnsservers networkservice]
setdns='networksetup -setdnsservers'
# usage [-setdnsservers networkservice dns1 [dns2] [...]]
services=(`networksetup -listallnetworkservices`)
for i in ${services[*]}; do
# original setup, which returns the correct command
# [[ ! $i =~ "*" || $1 =~ "W" ]] && echo $getdns \"$i\"
# command that fails when attempting to execute string...
# # [[ ! $i =~ "*" || $1 =~ "W" ]] && echo `$getdns \"$i\"`
# my attempt to concatenate the strings
[[ ! $i =~ "*" || $1 =~ "W" ]] && cmd="$getdns '$i'" ;echo `$cmd`
done
Here's an example output:
/Users/user/bin/netd3: line 15: networksetup -getdnsservers 'Wi-Fi': command not found
I can run which networksetup which returns /usr/sbin/networksetup, but it's not running.
What am I missing?
A: Your script is trying to execute the string networksetup -getdnsservers 'Wi-Fi' as a single command.
Use
eval $cmd
A: You could also add the following at the top of your script, under #!/bin/bash:
PATH=/bin:/usr/bin:/sbin:/usr/sbin export PATH
A: Storing a command in a variable as you're doing is problematic (see BashFAQ #50: I'm trying to put a command in a variable, but the complex cases always fail!). The biggest problem is that quotes are evaluated before variables are substituted, so embedding quotes in variables (e.g. the single-quotes in cmd="$getdns '$i'") doesn't do what you expect it to.
In your particular case, there's an additional problem: you set IFS to just linefeed, so when the shell expands $cmd it splits it into "words" by looking for linefeeds -- there aren't any, so it mistakes what're supposed to be arguments (-getdnsservers and 'Wi-Fi') for part of the command itself. That is, it doesn't run the command "networksetup" with arguments "-getdnsservers" and
"'Wi-Fi'"; instead it tries to run "networksetup -getdnsservers 'Wi-Fi'" as the command.
I'd make several recommendations to clean this up. First, after getting the list of services, set IFS back to its usual value. This'll also mean you need to use proper quoting in the for loop (use double-quotes, and [@] instead of [*]). You also need to clean up the list to remove the header and "*" on disabled services. Finally, I prefer $() to backquotes, so I'll go ahead and change that:
saveIFS="$IFS"
IFS='
'
services=( $(networksetup -listallnetworkservices | sed '/An asterisk ([*]) denotes that a network service is disabled./d; s/^[*]//') )
IFS="$saveIFS"
for i in "${services[@]}"; do
...
Now, for the commands themselves, the best option depends on why you're putting them in variables in the first place. The simplest, most direct option, is to just use the commands directly:
[[ ! $i =~ "*" || $1 =~ "W" ]] && networksetup -getdnsservers "$i"
If you want to use a shorthand for the command, use a function rather than a variable:
getdns() {
networksetup -getdnsservers "$@"
}
...
[[ ! $i =~ "*" || $1 =~ "W" ]] && getdns "$i"
If you need to store the entire command (including arguments) before executing it, use an array:
[[ ! $i =~ "*" || $1 =~ "W" ]] && cmd=(networksetup -getdnsservers "$i") && "$cmd[@]"
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 5,703
|
{"url":"https:\/\/brilliant.org\/discussions\/thread\/both-counfused-now-frac-271828314159ed-up\/","text":"\u00d7\n\n# Both counfused, now $$\\frac {2.71828...}{3.14159...}$$ed up!\n\nWhat is $$\\pi^{e}$$?\n\nNote by Bryan Lee Shi Yang\n1\u00a0year, 10\u00a0months ago\n\nSort by:\n\n22.4591577184 \u00b7 1\u00a0year, 9\u00a0months ago\n\n$$e$$=$$Euler's$$ $$Number$$ =$$2.71828......$$ \u00b7 1\u00a0year, 9\u00a0months ago\n\nDo you want to find it out mathematically ? Using some function of some sort ? \u00b7 1\u00a0year, 9\u00a0months ago\n\nOf course. Just don't know how to do it. \u00b7 1\u00a0year, 9\u00a0months ago\n\nYes? \u00b7 1\u00a0year, 9\u00a0months ago","date":"2017-01-19 00:10:42","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\": 0.8498806953430176, \"perplexity\": 10070.381076170752}, \"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-2017-04\/segments\/1484560280410.21\/warc\/CC-MAIN-20170116095120-00523-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
Q: tracd login Error I am trying to use tracd but i cannot get past the authentication part. I created a digest file based on their wiki description:
admin:ITC:98e7d6e0a7506cd5558067794aed9cce
PS C:\Users\22193\AppData\Roaming\Python\Scripts> .\tracd.exe --port 8000 auth="Test_Trac,D:\TracEnv\conf\login2.digest,
ITC" D:\TracEnv
Server starting in PID 5948.
Serving on 0.0.0.0:8000 view at http://127.0.0.1:8000/
Using HTTP/1.1 protocol version
127.0.0.1 - - [28/Apr/2016 17:37:25] "GET /TracEnv HTTP/1.1" 200 -
127.0.0.1 - - [28/Apr/2016 17:37:25] "GET /TracEnv/chrome/site/your_project_logo.png HTTP/1.1" 404 -
127.0.0.1 - - [28/Apr/2016 17:37:26] "GET /TracEnv/chrome/site/your_project_logo.png HTTP/1.1" 404 -
127.0.0.1 - - [28/Apr/2016 17:37:27] "GET /TracEnv/login HTTP/1.1" 500 -
127.0.0.1 - - [28/Apr/2016 17:37:27] "GET /TracEnv/chrome/site/your_project_logo.png HTTP/1.1" 404 -
When i click on login i still get an authentication error. I used certutil in windows to generate the digest. What should be the first argument in the --auth option. In the documentation it mentions base project directory, but i did not create any separate project directory, i only created an environment (D:\TracEnv). What could be the problem here?
Update: I now get an error:
PS C:\Users\22193\AppData\Roaming\Python\Scripts> .\tracd.exe -s --port 8000 --auth="TracEnv,D:\TracEnv\conf\login2.htdi
gest,ITC" D:\TracEnv
Warning: found no users in realm: ITC
Server starting in PID 3296.
Serving on 0.0.0.0:8000 view at http://127.0.0.1:8000/
When i click on login it prompts for username and password but i cannot login with the user name and password i used to create the digest file. i think this is beacuse of no users in realm. How do i add users in realm?
A: The first argument should be TracEnv. Add the -s switch if you wish to access the Trac instance at http://localhost:8000 rather than http://localhost:8000/TracEnv.
It would be helpful if you specified the wiki documentation you are following. I assume you are following TracStandalone.
A: The solution that worked for me is generating the digest file using htdigest.py instead of using Windows certutil. Even though the contents of both files (one generated by certutil and one by htdigest.py) when viewed in notepad++ are the same, tracd doesn't seem to work with the file generated by certutil. Maybe i'm missing something here but i couldn't figure out why this happens.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,173
|
O Clube Atlético Montenegro é um clube brasileiro de futebol, com sede na cidade de Paranapanema, estado de São Paulo. Conhecido como "Águia do Vale", o clube foi fundado em 13/12/1999 e teve uma breve passagem pelo Campeonato Paulista da Série B, promovido pela Federação Paulista de Futebol.
Montenegro
Clubes de futebol fundados em 1999
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 647
|
\section{Acknowledgement}
\label{sec:acknowledgement}
This work was supported by DOE Grant DE-SC020374, and DOE Quantum Information Science Enabled Discovery (QuantISED) for High Energy Physics (KA2401032). This work was completed in part with resources provided by the University of Massachusetts' Green High Performance Computing Cluster (GHPCC). We thank Thomas Langford from Yale University and Jeffrey S. Nico from National Institute of Standards and Technology for useful discussion. W. G. acknowledges the support from the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida. P.K.P. would also like to thanks Chandan Ghosh from University of Massachusetts, Amherst for assistance in improving the triggering logic.
\section{Conclusions}
\label{sec:conclusions}
We have successfully built and characterized a large area backing detector for keV scale neutrons whose neutron detection efficiency matches simulation. We also demonstrated that some degree of position information is retained by looking at the degree of asymmetry in the light received in both channels. The ambient background rate of our large scale backing detector is 10$\,$Hz, thus the probability of having an accidental coincidence in a window of 10$\,\mu$s, which is the n-capture time window for keV scale neutrons, is negligible. We plan to construct more such backing detectors, which shall be employed to perform low energy nuclear recoil calibrations of detectors for dark matter and coherent elastic neutrino nucleus scattering.
\section{DesignStudies}
\label{sec:Design Studies}
The following section describes the goals of the neutron backing detector, which have dictated the various design choices made.
\begin{itemize}
\item \textbf{Maximum neutron capture efficiency}: Employing a pulsed neutron source ~\cite{CHERNIKOVA201474} to perform \textit{in-situ} calibrations of DM detectors implies that the calibration setup would be limited by statistics due to the limited number of pulses available from a pulsed neutron source. This motivates a large-area neutron detector with a high capture efficiency per neutron.
\item \textbf{Maximum angular resolution:} Angular resolution is the second most severe limitation to the planned neutron scattering experiments, limiting the ultimate energy precision of the calibration. Angular resolution is challenging in capture based detectors because the moderation process takes place over $\sim$10~cm length scales. A further degradation in angular measurement occurs when a neutron initially scatters in a backing detector at one angle but escapes and captures in a backing detector at another \emph{incorrect} angle. One can think of this as neutron cross-talk. The dimensions and spacing of the rings must be optimized to maximize both the angular resolution and minimize this neutron cross talk effect while retaining capture efficiency.
\item \textbf{Minimum neutron capture timescale}: Because the planned calibrations are dependent on a coincidence between the pulsed neutron source and the detection of a scattered neutron, the capture timescale would ideally be similar to or less than the source's pulse duration (in our case, 1~$\mu$s). The design goal is to minimize the neutron capture time so that the effectiveness of the coincidence timing technique is maximized. Achieving the 1~$\mu$s capture time goal is challenging at keV energies. \ce{^{10}B}-doped liquid scintillator can achieve the goal, but with a downside of significant gamma rates and poor gamma rejection via PSD.
\item \textbf{Minimum contamination from background gammas:} The pulsed neutron source will produce a simultaneous pulse of gammas, many of which are produced by neutron capture on various shielding or incidental materials. The interaction rates of these gammas in the active scintillating material must be minimized if the neutron capture signal is to be observed. Additionally, it is highly beneficial if the scintillation process distinguishes between gamma scatters and neutron captures. Also, this difference in gamma or neutron signals is benefited by maximum light detection efficiency.
\end{itemize}
Based on the above constraints, we designed a ring shaped backing detector where the scintillators were sandwiched between moderating hydrogenous materials (Fig.~\ref{fig:backing_detector_crossection}). A very thin scintillator of thickness 0.5$\,$mm was chosen in order to minimize gamma interactions. Two layers of scintillating material were used to increase the capture efficiency. Backgrounds originating in the WLS fibers themselves were reduced by using two arrays of fibers to read out the capture signals, requiring coincidence between the two channels read out by separate fiber arrays and SiPMs. The WLS fibers in the inner part of the detector ring were attached to one of two SiPMs. This comprised the \textit{inner channel}. Similarly, the outer ring WLS fibers and their corresponding SiPM comprise the \textit{outer channel}. The chosen scintillator was EJ-426(ZnS:Ag/\ce{^{6}Li}F)~\cite{wilhelmDevelopmentOperation6LiF2017}, a 95\%-enriched \ce{^{6}Li} solid-state scintillator material in the form of \ce{^{6}Li}F dispersed in a ZnS:(Ag) scintillator matrix. The neutron is detected via the n-capture signal on \ce{^{6}Li}, n-capture reaction \ce{^{6}Li}(n, $\alpha$)T produces alpha($\alpha$) and tritium(T) where the kinetic energy of the $\alpha$ particle is 2.05 MeV, the kinetic energy of T is 2.73 MeV. ZnS(Ag) scintillator matrix has a light yield of about 50,000 Photons/MeV~\cite{yehuda-zadaOptimization6LiFZnS2018}, thus a large light output of 1.6$\,\times\,$10$^5$ photons~\cite{osovizkyDesignUltrathinCold2018} is produces per n-capture signal in the ZnS(Ag) scintillator. ZnS(Ag) also exhibits Pulse Shape Discrimination (PSD) for additional rejection power. The main downside of ZnS(Ag)/\ce{^{6}Li}F is its poor light transmittance to its own scintillation light due to its structure as a matrix of multiple powdered crystals. In order to ensure neutron capture events from all depths within the ZnS(Ag) are detectable, a high light detection efficiency is still motivated, as will be discussed later.
After selecting the active scintillating material, the next step was to optimize the moderator dimensions so they could efficiently moderate keV scale neutrons to the meV scale required for capture. GEANT4~\cite{AGOSTINELLI2003250} was used for neutron simulations and a custom physics list was created. The Shielding Physics list~\cite{G4_physics} was chosen to simulate neutrons from the thermal range up to an energy of 20$\,$MeV. For thermal energy, G4NeutronThermalScattering was used with a thermal treatment of hydrogen.
4$\,$eV was set as both the maximum energy for thermal scattering and the minimum energy for elastic scattering. There has been good agreement between MCNP and GEANT4 using the thermal scattering data that has been studied by different groups~\cite{vanderendeUseGEANT4Vs2016}.
\subsection{Moderator Layers and Thicknesses}
\label{sec:sim_thicknesses}
HDPE and acrylic were both used as moderators in the detector due to their high hydrogen content and low cost. Simulations were performed to optimize the thickness of these two materials. Optimum thicknesses were found between the extreme of too-thick (the neutron will capture on hydrogen, before reaching the \ce{^{6}Li}-doped scintillator) and too-thin (the neutron will fail to moderate before reaching the scintillator). In addition to finding the combination of HDPE and acrylic thicknesses which maximize capture efficiency, the capture timescale was studied, and some compromise was made between the conflicting goals of high capture efficiency and high timing resolution. Timing resolution is benefited by thinner moderating materials.
Thus we must find the optimal combination of capture efficiency and capture time.
Results of these thickness-varying simulations are shown in Fig.~\ref{fig:optimised graph}, assuming a 2~keV neutron energy incident perpendicular to the moderating layers. The simulated geometry consisted of a sandwich geometry of HDPE-Acrylic-HDPE layers. The thickness of each layer was varied in the simulations. Based on these simulations, the thickness of each HDPE layer was set to 0.5-inch (12.7~mm) and the thickness of the central acrylic layer was set to 1.0-inch (25.4~mm), with a capture efficiency of 27\% at 2keV. This is slightly lower efficiency than the maximum achievable (at slightly increased acrylic thickness) but was chosen to reduce the mean neutron capture time to 17$\,\mu$s.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figures/Efficiency_Capture_Time.png}
\caption{All the dimensions in the legends are in cm. (Left) Shows various capture efficiencies for different thickness of central acrylic layer and HDPE thicknesses. (Right) Shows the corresponding mean capture times for various configurations. The grey dots signify the selected configurations. The selected configuration of the moderator consists of two HDPE layers each 0.5$\,$inch (12.7$\,$mm) thick and a central acrylic layer with a thickness of 1.0$\,$inch (25.4$\,$mm), which together produce a neutron capture efficiency of 27\% at 2$\,$keV and a capture time of roughly 17$\,\mu$s. }\label{fig:optimised graph}
\end{figure}
\subsection{Ring Dimensions and Spacing}
\label{sec:sim_light}
After optimizing the acrylic and HDPE thickness for capture efficiency and timing, other aspects of the ring geometry were optimized for angular resolution and minimizing neutron crosstalk between rings. As previously discussed, a neutron will diffuse radially during thermalization before capture, both limiting the position resolution of the detector and risking neutron capture in an adjacent detector.
To estimate the rough scale of the radial diffusion length for keV neutrons, a histogram of radial distance of the n-capture position was made using the data from the previous simulation. And it was found that most neutrons gets captured after diffusing radially a distance of 8$\,$cm. Thus a ring detector with radial thickness of 15$\,$cm.
However, increasing the radial thickness of the detector also increases the probability that neutrons are captured in the wrong ring in the assembly (neutron crosstalk). To study this effect, the full geometry consisting of a small target material, a series of neutron detector rings and a central monochromatic collimated neutron beam shown in Fig.~\ref{fig:NeutronCrossTalk_effect} was simulated. It was observed that neutron crosstalk adds only a minor contamination as long as the spacing between rings is significantly larger than the radial thickness of each ring. In other words, the solid angle coverage of another ring as viewed from the primary ring can be kept low, while the solid angle coverage of the collection of rings is kept collectively high. The simulations showed that crosstalk events safely consisted of less than 10\% of all neutron capture events, requiring no further fine-tuning or optimization of the geometry.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/Neutron_Crosstalk_3.pdf}
\caption{(Top) Shows the geometry of the setup that was simulated, array of six backing detectors with the target material and a pencil beam of 24$\,$keV neutrons. (Bottom) Shows the degree of crosstalk resulting from the configuration in the Top. The scattered 24$\,$keV neutron from the target is supposed to be detected by the second ring, but it scatters off the second ring and gets detected in the first ring instead.}\label{fig:NeutronCrossTalk_effect}
\end{figure}
From this study, we also estimated the range of scattering angles that are accessible, 10$^{\circ}$ - 60$^{\circ}$ of scattering angle. Due to the relatively large radial thickness of the detector, careful study of the geometry was necessary in order to achieve the desired angular resolution. Finally, to get a holistic picture of the n-capture efficiency of the backing detector across all energy scales, simulations were performed and the results are shown in Fig.~\ref{fig:efficiency_to_all_energy_scale}. The thermal and DT neutrons would be the main background sources in a NR-calibration experiment based on a filtered neutron source generated with a DT neutron generator. Consequently, the tapering of n-capture efficiency to lower values at the thermal and DT neutron energies is promising.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/Efficency_All_EnergyRange.png}
\caption{Two sets of simulations were performed here: one with neutrons normally incident to the backing detector the other with neutrons isotropically incident to it. The region of interest shows 2$\,$keV and 24$\,$keV neutrons from a Sc and Fe filtered neutron source. Thermal and DT neutrons, the main backgrounds in the NR calibration experiment, are also highlighted.
}\label{fig:efficiency_to_all_energy_scale}
\end{figure}
\subsection{Light Collection}
\label{sec:sim_light_collection}
An efficient optical system is necessary in order to increase the light collection efficiency of the detector (to ensure all neutron captures on \ce{^{6}Li} were above threshold, and to maximize the utility of PSD). After the scintillation photons escape the ZnS:(Ag) matrix, they undergo multiple reflections off the acrylic walls aided by a highly reflective Tyvek layer. Then, a large fraction of these photons will be captured within a parallel array of WLS fibers. We selected a 1.5$\,$mm diameter fiber from Kurray: Y-11 multicladding non S-type~\cite{abreuOptimisationScintillationLight2018}. These fibers were placed in grooves made on the inner surfaces of the two acrylic sheets. Finally, these WLS fibers were connected to a Hamamatsu S13360 series SiPM~\cite{otteCharacterizationThreeHigh2017} via a connector interface, to be discussed later. The choice of WLS fibers and SiPM was made to ensure a maximum overlap between both the emission spectrum of the \ce{^{6}Li}F/ZnS(Ag) scintillator and the absorption spectrum of the WLS fibers, and the emission spectrum of the WLS fibers and the quantum efficiency spectrum of the SiPM. See Fig.~\ref{fig:ANTS2input}.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/ANTS2_input.png}
\caption{(Left) The optical properties of EJ-426 as a function of wavelength, adapted from~\cite{wilhelmDevelopmentOperation6LiF2017}. (Right) The normalized plot of emission and absorption spectra of WLS fibers and the emission spectrum of EJ-426 (\ce{^{6}Li}F-Zns(Ag)) scintillator. Also included is the quantum efficiency of the SiPM. }\label{fig:ANTS2input}
\end{figure}
A Monte Carlo (MC) simulation was performed based on the study by Pino et.al~\cite{pinoStudyThermalNeutron2015}, to estimate the number of photons needed to perform PSD. The n-capture signal from the EJ-426 (\ce{^{6}Li}F-Zns(Ag)) scintillator is of the order of a few $\mu$s, and the gamma signal is of the order of a few ns. Both are highlighted in the left plot of Fig.~\ref{fig:EJ426 Photon Pulse}. This long tail feature in the n-capture signal helps to discriminate between the ER and n-capture events even though the SiPM has a high dark count at room temperature. From the MC simulation it was concluded that capturing roughly 100 photons will be sufficient to discriminate between ER and neutron capture events at room temperature (Fig.~\ref{fig:EJ426 Photon Pulse}, right plot). Next, the number of WLS fibers was determined by performing an optical simulation using ANTS2~\cite{morozovANTS2PackageSimulation2016}. The ability to perform PSD dictated the number of WLS fibers needed. Several inputs to the optical simulation were provided, most important of which were the absorption, transmission and reflection coefficients of the EJ-426 layer (Fig.~\ref{fig:ANTS2input}). The wavelength shifting ability of the WLS fibers was also modeled in the simulations. The refractive indices of various materials as well as their other optical properties were imported from the XCOM: Photon Cross Sections Database~\cite{curtis.supleenist.govXCOMPhotonCross2009}. The simulation suggested that a total of 16 fibers were needed to have $\gtrsim\,$100 photons detected by the SiPMs.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figures/Photon_rate.png}
\caption{(Left) Typical pulse shapes for a neutron signal (blue) of different sizes and for a gamma signal (magenta), based on~\cite{pinoStudyThermalNeutron2015}. The gamma pulse signal is very short as compared to the neutron signal. (Right) Result of a simple Monte Carlo performed to estimate the lowest number of photons that need to be detected for PSD.}\label{fig:EJ426 Photon Pulse}
\end{figure}
\section{PrototypeFabricationAndInitialTesting}
\label{sec:prototype}
\subsection{Fabrication, Assembly and Readout}
\label{sec:prototype_fabrication}
The 16 fibers were grouped into an inner channel and an outer channel. Each channel was then mated to two different SiPMs. The central acrylic layer of 24$\,$mm (optimized thickness was 25.4$\,$mm) was cut into two halves, and each channel was embedded in grooves made in the acrylic. Next, a custom clamping strategy was designed and built to closely pack the fibers together before mating them to the 6$\,$mm$\,\times\,$6$\,$mm SiPM face (see Fig.~\ref{fig:clamp_fiber_SiPM}). First, the fibers from the acrylic are arranged into a 4$\,\times\,$4 grid with the use of a \textit{grid plate} that helps in orientating the fibers. Next, the fibers are bent inside a tee shaped piece of aluminum, which has an upward slanted groove. Finally, at the other end of the aluminum tee, set screws are used to drive \textit{pressing plates} that hold the fibers in the grid formation. The fibers are then placed through a \textit{connector interface}, which has chamfered edges to provide extra directionality for the outward protruding fibers so that mating with the SiPM is easier. The connector interface also functions as a holder for the SiPM. Once the fibers are fed into the connector interface, they are polished and mated to the SiPM using optical glue. A final copper part was used to secure the SiPM in place and allow the optional use of Peltier coolers. A couple of inner channel fibers suffered slight cracks as a result of this securing process.
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figures/fiber_clamp_assembly_labelled.pdf}
\caption{\textbf{1)} Grooved acrylic. \textbf{2)} Grooves. \textbf{3)} HDPE for neutron moderation. \textbf{4)} Grid plate to orient the fibers. \textbf{5)} Tee shaped aluminum slot with upward slanted groove. \textbf{6 \& 7)} Pressing plates. \textbf{8)} Connector interface with chamfered edges. \textbf{9)} SiPM. \textbf{10)} Copper support.
}
\label{fig:clamp_fiber_SiPM}
\end{figure}
The top and bottom flat surfaces of the acrylic are optically clear, and the grooved portion has a semi mill-finished surface. It was decided that there would be no air gap between the WLS fibers and the grooves in the acrylic. Thus the space between the fibers and grooves was filled with optical glue, and care was taken to minimize the creation of air bubbles as they could act as photon sinks in the system. The different steps of the assembly process are shown below in Fig.~\ref{fig:Fabrication_process}.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/Fabrication_process.png}
\caption{
\textbf{1)} Grooved acrylic with all fibers glued into the groove using RTV-615 silicone glue. \textbf{2)} The 4$\times$4 matrix of the fibers coming out of the clamp. \textbf{3)} \ce{^{6}Li}F-Zns(Ag) scintillator attached to the acrylic via optical glue. \textbf{4)} Tyvek is wrapped around the detector for diffusive reflection of light \textbf{5)} A final layer of thick black opaque paper wraps the detector to make it light tight. \textbf{6)} Fully assembled Backing Detector along with its support structure.
}
\label{fig:Fabrication_process}
\end{figure}
Next we shall discuss the readout strategy of the SiPM. First, the signal from the SiPM was amplified using an off-the-shelf charge sensitive preamplifier from Cremat (CR-113), which has a gain of 1.3$\,$mV/pC. Appropriately biased resistors and capacitors were then connected to the preamplifier circuit. The preamplifier signal was then passed to a shaping amplifier from an ORTEC 671 Spectroscopy Amplifier. An approximate gain of 20 was applied, and the unipolar gaussian output pulse with a pulse width of 500$\,$ns was passed to the Pice676 DAQ. The bipolar output pulse was passed down to build an OR trigger circuit using the signals from both the inner and outer channels, and this OR trigger was used as an external trigger for the Pice676 DAQ. Each triggered event was recorded while operating the Pice676 DAQ at a sampling frequency of 3$\,$MHz, and each sample was 6$\,$µs pre-trigger and 14$\,$µs post-trigger in duration. A custom-made high-pass filter with cutoff frequency 13$\,$kHz was also introduced so as to reduce the unwanted low frequency noise from the recorded unipolar gaussian pulse signal.
\subsection{Analysis}
\label{sec:prototype_SPE}
The experimental data were analyzed to measure the neutron detection efficiency of the neutron backing detector. The primary goal of the analysis process is to clearly distinguish the neutron population from the gamma population. The \ce{^{6}Li}F-ZnS(Ag) has an intrinsic PSD quality based on pulse area as reported in \cite{pinoStudyThermalNeutron2015}. The pulse shape discrimination technique explored in our work is based on the ratio of Delayed Scintillation to that of Total Scintillation. Delayed Scintillation is defined to be the area under the pulse between 1$\,\mu$s and 6$\,\mu$s of the pulse and the total scintillation is defined to be the area under the pulse between -1$\,\mu$s and 6$\,\mu$s, with time zero being the maximum amplitude of the pulse. Since an external hardware trigger was used for this work, a dedicated pulse finder method was not implemented in our analysis. A few pulses are highlighted here in Fig.~\ref{fig:different_types_Pulses} for illustration.
\begin{equation}
\text{PSD}=\dfrac{\text{Delayed Scintillation}}{\text{Total Scintillation} }
\end{equation}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/Different_pulse_types.png}
\caption{
This figure shows different types of pulses that are observed with the backing detector. (Left) Neutron-like pulse, scintillation signal caused due to n-capture are relatively slower.
(Right) Gamma-like pulse, most likely caused by gammas striking the WLS fibers, signals caused due to gamma pulses are relatively faster.
}\label{fig:different_types_Pulses}
\end{figure}
Signal size was converted to the units of detected photons by dividing the raw pulse area by the measured single photoelectron (SPE) area. Because of a high dark rate at room temperature, the SPE measurement was performed at 194$\,$K using dry ice. The SPE size is independent of temperature \emph{if} the over-voltage set point (voltage above breakdown voltage) is kept constant~\cite{Nagai:2016vym}. The breakdown voltage of the SiPM when cold was inferred using the SPE size at different bias voltages (Fig.~\ref{fig:SPE}), and found to be 48$\,\pm\,$0.7$\,$V. At an over-voltage of 3.1$\,$V, the SPE size when cold was found to be 450$\,\pm\,$10$\,$mV*$\mu$s. We then operated the SiPMs at an identical 3.1$\,$V over-voltage at room temperature and assumed the same SPE area. Note: During actual data-taking, the gain of the amplifier was scaled down by a factor of 40 assuming a linear response of the amplifier circuit. SPE size was also reduced by the same factor.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/SPE_gaussian_linear.png}
\caption{ Measurement was done at dry-ice temperature.
(Left) SPE sizes at different bias voltages is plotted. SPE value was estimated by performing a gaussian fit at different bias-voltage.
(Right) SPE size with varying bias voltage is plotted, from which a linear extrapolation was made to figure out the breakdown voltage of the SiPM. Given this breakdown voltage, an over voltage of 3.1$\,$V was applied and the SPE size was measured. The measured SPE size was assumed to be the SPE size at room temperature.
}
\label{fig:SPE}
\end{figure}
\subsubsection{Backgrounds}
\label{sec:prototype_backgrounds}
Before describing the final efficiency of the neutron detector, various types of background pulses that need to be rejected are discussed below:
\begin{itemize}
\item Fast Pulse (similar to right panel of Fig.~\ref{fig:different_types_Pulses}): These are purely gamma events as their rate gets higher when gamma sources such as \ce{^{137}Cs} and \ce{^{152}Eu} were placed near the detector. Due to thinness of \ce{^{6}Li}F-ZnS(Ag) scintillator, it scintillates mostly due to n-capture signal and is blind to most of the gammas. However, a fast pulse can be produced if an ambient gamma ray strikes the WLS fiber causing the fiber to scintillate. Another cause of these fast pulses is when a high energy gamma from a cosmic ray interacts with the scintillator and deposits considerable energy, though it is very unlikely. The observed rate of such Fast Pulse events is about 10$\,$Hz in ambient room environment.
\item Slow Pulse in absence of a neutron source (similar to left panel of Fig.~\ref{fig:different_types_Pulses}): The cause of these backgrounds are likely due to the presence of radioactive materials inside the neutron detector that can decay by the emission of an alpha, which mimics a neutron capture signal to some degree. The rate that has been reported earlier~\cite{kozlovLargeAreaDetector2018a} is quite low (4.88$\times10^{-6}\,cm^{-2}\,sec^{-1}$) as compared to our observation. The rate of Slow Pulse events observed in our backing detector is 0.5$\,$Hz (3.4$\times10^{-4}\,cm^{-2}\,sec^{-1}$). Another possibility to create a Slow Pulse is due to the capture of ambient neutrons in the environment, but that is also highly unlikely since the \ce{^{6}Li}F-Zns(Ag) layer is covered with HDPE plastic, and any ambient neutrons in the room environment would most likely get captured in the HDPE.
\end{itemize}
\subsubsection{Cuts}
After defining the Pulse Shape Discrimination variable, finding the true signal size in units of detected photons and having an understanding of the various types of backgrounds, we can finally compare the observed neutron detection efficiency to simulation after applying some data cuts. To highlight the applied data cuts, a plot of PSD parameter vs log$_{10}$(signal size) is shown in Fig.~\ref{fig:PSD_cuts} for a particular source-detector configuration. One can clearly see the two distinct populations; the slower and the faster. Separation of these two populations in this 2d space was done by performing a combination of linear cuts. The clear separation is a signature that almost all the n-capture signals are being detected without the loss of any light signal. Next we will characterize the level of agreement of the optical model with experiment
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/Cuts_photoh_detected.pdf}
\caption{
The 2-d histogram in the plane of log$_{10}$ of photons detected vs the delayed scintillation obtained after analysis of data from (Panel 1) Background dataset. (Panel 2) from Configuration \textbf{1} dataset (refer Table~\ref{tab:different_geometrical_configuration_table}). The slower and faster populations are clearly distinct in this space. Neutron-like signals are the slower signals and the gamma-like signals are the faster signals. To get the number of n-capture signals, data cuts shown in black dotted line were applied.
(Panel 3) Shows the photons detected by the detector in presence of the neutron source and in absence of the neutron source. (before and after the data cuts were applied)
}
\label{fig:PSD_cuts}
\vspace{-0.5em}
\end{figure}
\subsection{Light collection}
\label{sec:prototype_light}
In this section we evaluate the light collection efficiency of the setup. The number of photons collected from our experimental setup and the simulation was compared. The total light collected matches the simulation if the photons collected from simulation were scaled up by a factor of five. The following are some of the reasons due to which discrepancy can be seen:
\begin{itemize}
\item\textbf{Imperfect tuning of optical parameters}:
Many optical parameters were taken as inputs to the optical simulation, such as reflectivity of Tyvek sheets, reflectivity of EJ-426-HD and the absorption coefficient of the WLS fibers, which may vary in some percent level with each specimen. An extensive tuning of various optical parameters in our simulation setup was not performed.
\item\textbf{Optical crosstalk}:
Given the large size of the light signal, there is a chance that this might cause optical crosstalk among the various pixels of the SiPM and enhance the size of our signal.~\cite{masudaSuppressionOpticalCrosstalk2021}
\end{itemize}
The end goal of the backing detector is to tag keV scale neutrons efficiently; we are not concerned with the amount of energy that is deposited by each n-capture event. Thus correct simulation of light propagation is of greater relative importance. To quantify the agreement of the light propagation model with experiments, we compared the \textit{asymmetry} of the light signals in both the inner and outer channels which is defined as \ref{eq:2}. The right panel of figure~[\ref{fig:Photons_detected}] shows the asymmetry of the light signal, which is in good agreement with simulation. The outer channel has larger amounts of fiber (due to the larger radius of curvature) and thus more photons are captured in it, which is the reason the right peak is larger than the left peak. As discussed earlier, a few of the fibers in the inner channels were slightly damaged. This is reflected in the skewness of the asymmetry towards the outer channel in the data obtained from the experiments. The right panel of figure~[\ref{fig:Photons_detected}] also signifies that some degree of radial information is retained as the capture events happening at the outer edge of the detector will deposit most of the light in the outer channel and vice versa.
\begin{equation}
\text{Asymmetry} = \dfrac{\text{Outer channel - Inner channel}}{\text{Outer channel + Inner channel}} \label{eq:2}
\end{equation}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/Assymetry_Photons_detected.png}
\caption{
(Left) Compares the number of photons detected by the backing detector with the simulation. A factor of 5 discrepancy between the number of photons detected and simulation was found. Possible reasons for this are discussed in the main text.
(Right) Asymmetry of the signal matches with optical model, validating our model. Some degree of angular information is also retained in the signal. If needed, a distinction between a capture event happening in the inner channel vs outer channel can be made.
}\label{fig:Photons_detected}
\end{figure}
\subsection{Efficiency}
\label{sec:prototype_efficiency}
The absence of low-energy neutron sources at our immediate disposal makes it challenging to measure the neutron detection efficiency of the backing detector for 2$\,$keV and 24$\,$keV neutrons. Validations were instead performed using a non-monoenergetic \ce{^{252}Cf} source in various source-detector configurations as described in Fig.~\ref{fig:different_configuration_image} and Table~\ref{tab:different_geometrical_configuration_table}. For each configuration, the capture rate in a Geant4 simulation was compared with experimental observation. The simulations used the same physics lists as mentioned earlier. The \ce{^{252}Cf} source had a neutron yield of 80 neutrons/s.
We define a simple figure of merit: the ratio of observed counts (after background-subtraction) to expected counts given by the simulation. As seen in Fig.~\ref{fig:Final efficiency}, this relative ratio ranges from 0.78 to 1.03 depending on the configuration (where 1.0 would signify perfect agreement). This level of agreement is consistent with expectations from the literature.~\cite{gressierInternationalKeyComparison2014}
\begin{table}
\label{tab:different_geometrical_configuration_table}
\centering
\vskip 3mm
\begin{tabular}{ |p{6cm}|p{6cm}| }
\hline
\multicolumn{2}{|c|}{List of various geometrical configurations} \\
\hline
Configuration Number & Description \\
\hline
Configuration \textbf{1} & Source is kept inside a \\
& lead pig.\\
\hline
Configuration \textbf{2(b)} and \textbf{2(c)} & Source is kept on an \\
& Al tee, whose height is varied.\\
\hline
Configuration \textbf{3} & Source is kept directly \\
& on the detector.\\
\hline
Configuration \textbf{4} & Source is kept in a bucket \\
& of water, which is placed\\
& at the center of the detector. \\
\hline
\end{tabular}
\caption{Shows the different configurations, each labeled with a different number, and a short description of the setup. Refer to Fig.~\ref{fig:different_configuration_image}}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/Cf252_configuration_3.png}
\caption{Simulated Geometry of different configurations. The red dot represents the source position. The blue cylinder represents the bucket of water that was placed at the center of the backing detector (black colored cylindrical object).
\textbf{a)} Configuration 1
\textbf{b, c)} Configuration 2, which comprises two configurations \textbf{b)} Short Al Tee \textbf{c)} Long Al Tee
\textbf{d)} Configuration 3
\textbf{e.1)} Configuration 4
\textbf{e.2)} Actual setup.
}
\label{fig:different_configuration_image}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/Final_efficency.png}
\caption{The ratio of observed to expected n-capture signals from the set-up simulated using GEANT4. The solid black lines with caps represent statistical uncertainty. And the gray patch represents systematic uncertainty. Systematic uncertainty were computed using the effect of position error uncertainties, from the source-target detector configuration and density uncertainties, assuming a uncertainty of 2\% for the density of LiF-ZnS(Ag) Scintillator and the 3\% uncertainty in the activity of the \ce{^{252}Cf} source,a s informed by the manufacturer.}
\label{fig:Final efficiency}
\vspace{-0.5em}
\end{figure}
\section{Introduction} \label{sec:intro}
Many dark matter (DM) models predict that scattering in target materials will be dominated by coherent scattering with the nucleus. Recent advancements in detector technologies and theoretical understanding of DM~\cite{alkhatibLightDarkMatter2020,barakSENSEIDirectDetectionResults2020,ibeMigdalEffectDark2018,kouvarisProbingSubGeVDark2017,kahn2021searches} have pointed out the possibility of low-mass DM in a mass range from keV to GeV, below the long-standing $>$GeV focus of the field. Numerous detector technologies have been proposed to look for sub-GeV DM~\cite{PhysRevD.96.016026,hertelPathDirectDetection2019,KNAPEN2018386}. In this DM mass regime, the energy deposited by DM in a target nucleus is $\mathcal{O}$(eV) or lower. While detector technologies are rapidly advancing to these low nuclear recoil (NR) thresholds, \textit{calibration} methods capable of producing and tagging $\mathcal{O}$(eV) NR are lagging. To perform such low-energy NR \textit{in-situ} calibrations via coincidence techniques, one benefits from a source of monoenergetic neutrons with keV energies and a backing detector capable of measuring the scattering angle of such neutrons in the target. Here we focus on this backing detector portion of a future $\mathcal{O}$(eV) NR calibration setup.
Neutrons of the keV energy scale are particularly challenging to detect because their energy is typically too small to efficiently cause detectable scintillation from scattering and also typically too large to be efficiently or quickly captured in capture-based neutron detectors. A significant amount of moderation is required before neutron capture can occur. The drawbacks of this moderate-then-capture detection method are the long capture time (roughly 10$\,\mu$s) and the large radial diffusion distance during moderation (roughly 10$\,$cm)~\cite{YEN2000476}.
Many DM sensor technologies under consideration in this sub-GeV regime rely on phonon propagation and are therefore relatively slow in response. The risk of pile-up therefore limits the maximum calibration rate and implies NR calibrations of such technologies are naturally statistics-limited. The neutron backing detector must then have a maximized detection efficiency, meaning both a large solid angle coverage and a large detection efficiency per solid angle. While detectors with the ability to tag keV neutrons do exist, most of them are either prohibitively expensive per unit area (implying only small angular coverages are practical), or are inefficient (meaning a large fraction of neutrons escape before capture). Table~[\ref{tab:my_label}] shows a few representative neutron detector options in this energy regime. The long-term goal of $\mathcal{O}$(eV) NR calibration would benefit from a high-efficiency capture-based backing detector technology that is also inexpensive (enabling high angular coverage).
\begin{table}[]
\centering
\begin{tabular}{ |p{4cm}|p{2cm}|p{3cm}|p{4cm}| }
\hline
\multicolumn{4}{|c|}{Neutron Detector for $\sim\mathcal{O}$(keV) neutrons } \\
\hline
Detector & Max Efficiency & Geometry & Notes\\
\hline
\ce{^{6}Li}I(Eu) crystal~\cite{barbeauDesignCharacterizationNeutron2007a} & 29\% at 24keV&2.5 cm radius x 7.5 cm long &high efficiency, small surface area, high price\\
\hline
GS20 glass~\cite{kleinNeutronResonanceTransmission2020a} & - & 2.54 cm radius × 0.5 cm long& small surface area, high price \\
\hline
\ce{^{10}B}-loaded liquid scintillator~\cite{YEN2000476} &71\% at 1keV & 43 cm radius x 4 cm long &high efficiency, large surface area, extremely costly \\
\hline
\end{tabular}
\caption{This table shows the currently used neutron detectors to detect keV-scale neutrons }
\label{tab:my_label}
\end{table}
The principle of detection of our neutron backing detector is as follows: A keV-scale neutron that has first scattered in a target material is then moderated in the hydrogen-rich backing detector materials (HDPE and acrylic). The moderated neutron is then captured in a thin layer of \ce{^{6}Li}-containing ZnS(Ag) scintillator. The ZnS(Ag) scintillation light propagates into the clear acrylic and is captured in an array of embedded wavelength-shifting (WLS) fibers. Finally, the concentrated light within the fibers is detected using commercial silicon photomultipliers (SiPMs). This article is organized as follows: section~[\ref{sec:Design Studies}] discusses the designing studies, section~[\ref{sec:prototype}] is focused on the assembly and characterization of the detector, and finally we conclude in section~[\ref{sec:conclusions}].
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{figures/Backing_detector_crossection_2.pdf}
\caption{(Top) Shows the schematic representation of the calibration setup with the arrays of backing detectors. (Bottom) Shows the cross-sectional view of the backing detector. The central acrylic layer is divided into two parts, and the fibers were embedded on the grooves made on these parts.}\label{fig:backing_detector_crossection}
\end{figure}
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"redpajama_set_name": "RedPajamaArXiv"
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X-Ray eyes and inhuman appearance.
After being abandoned by her parents, the young mutant Maggie was taken in by the daughters of photographer Phil Sheldon. When the Sentinels were first unleashed, she fled so as to not get the family in trouble.
Maggie ended up in New Guinea, where she was accepted by it's people and in turn taught their children. She later returned to New York to visit Sheldon on his deathbed, and worked with the rest of the family to posthumously finish his second book.
Kurt Busiek originally planned the mutant child to be a green boy. Alex Ross proposed giving the child a more bizarre appearance to contrast with Angel's more attractive mutation on the cover. Using a deformed little girl from Weird Science #20 as a reference, Kurt eventually decided to make the entire character a girl as well.
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"redpajama_set_name": "RedPajamaC4"
}
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For a recent overview of the computing resources, see the slides from the 2014 computing seminar.
Our primary point of contact for desktop IT support is the Faculty of Science IT Support Team. You can submit a request for support here. Requests for support for the Unix network, including Fedora desktops, may be submitted by email to gripe@fas.sfu.ca. See the Gripe Helpdesk page for more information on submitting a gripe.
The departmental Unix computers are part of FASnet, a computer network maintained by the IT Services Research Computing Group. Our access to FASnet is controlled by membership to the departmental mailing lists stat-fac, stat-grad and stat-visitors-sessionals. If you are a member of one of these lists, you have access to the following login and compute servers. Users new to Unix might consult a beginners Unix tutorial or a tutorial specifically on the Unix shell.
from the Unix command line loads the module containing R version 3.3.1. You can now type R from the Unix command line to start R.
Login, or "terminal" servers are computers you can access using an ssh client, such as PuTTY. Use these login servers to copy files to and from the network, access the compute servers, etc.
FASnet users have access to the Colony Cluster for high-performance computing. For the most up-to-date information on the Colony Cluster, please visit the SFU HPC Wiki. There is also a getting started with the cluster page with some examples of potential interest for statistics.
SFU students, staff and faculty may download a copy of SAS, JMP, Splus, SPSS PASW, and more for free from the SFU IT Services Software Download Page.
Stat Lab information and Minitab handbooks.
WestGrid, a high performance computing (HPC), collaboration and visualization infrastructure across western Canada.
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"redpajama_set_name": "RedPajamaC4"
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Q: Remove blank space in dataTable primefaces I have a simple primefaces DataTable with Expansion row that display some data.
How can I remove the blank space between the parent table and the nested one ?(as shown in the image below) ?
I have tried adding tableStyle="table-layout:auto;width:100%;left-padding:0px" to my dataTable but no result. Can you please help me to resolve this little issue ?
FYI : I am using Primefaces 5.1
Thanks,
A: UPDATED
Try changing the style of the MAIN table (not the nested table):
tableStyle="padding:0;"
If that takes away the padding for the nested table, then change the padding for the main table back to an acceptable value (lets assume 4), and then on the nested table try:
tableStyle="margin:-4px;"
...and see if that does it for you.
It looks like the default padding is that the -left,-right is one value, and -top,-bottom is a different value. If so just try to replicate those values as negative numbers using margin- on the inner table.
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"redpajama_set_name": "RedPajamaStackExchange"
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Five Reasons You'll Love It - October 24, 2019
Midsommar - Five Reasons You'll Love It
Ari Aster's 2018 directorial debut Hereditary was a revelation. It was a movie that everyone, whether they were horror aficionados or not, was intrigued by. And, when that intrigue led to sitting down to watch it, it was a movie that scared the hell out of everyone who saw it.
Now though, he needs a follow-up. And, for that, he takes us away from the dark dampness of Hereditary's suburban Utah and out into the bright Swedish sunshine.
His new film Midsommar gathers an excellent young cast, which is led by Jack Reynor, Florence Pugh, Will Poulter, Vilhelm Blomgren, William Jackson Harper, Ellora Torchia and Archie Madekwe.
Reynor and Pugh play Christian and Dani, a young couple who travel to Sweden to visit their friend's rural hometown for its fabled mid-summer festival.
The trip begins as an idyllic retreat, but quickly devolves into an increasingly violent and bizarre competition at the hands of a pagan cult.
Aster is on writing and directing duty here.
Midsommar hit DVD shelves on Monday (October 28th), but we caught an early preview and have put together five reasons why you'll love it...
It's very stylish...
All the praise Aster received for Hereditary has put a spring in his step. This is a very confidently directed film. The transitions are swish, the camera angles are experimental and the lucid cinematography all add up to a stylish, but thoroughly unsettling watch.
But very, very out there...
Aster has worked hard to give this film a trippy, psychedelic feel. It's seductive and troubling to witness. That said, when the gore comes, it is as grisly as anything you'll see on screen this year.
Florence Pugh continues to wow...
Florence Pugh seems to be on a mission to prove that she can just about anything. She's on DVD this week commanding a big-hearted wrestling comedy, and has wowed in tortuous drama Lady Macbeth and offbeat coming of age story The Falling. Now she can add horror star to her list. As Dani, she's the portal to all the story's madness and she commands the film superbly.
The supporting cast are superbly chosen...
Jack Reynor's is great as Dani's feckless boyfriend Christian, as is Vilhelm Blomgren, who plays local boy Pelle. Will Poulter and William Jackson Harper are on great form too. Only Harper is an American native, but all their accents are flawless too.
It's a slow burn, but you're never bored...
At 147 minutes, this is not a short, sharp shock. This is a languid, elegant film, which builds slowly and gradually. Don't get us wrong, it's profoundly unsettling from the get-go, but to earn the madness of the last half hour, you need plenty of room to get there.
Midsommar is released on DVD and Blu-Ray on Monday (October 28th).
Midsommar Ari Aster
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John Cena leads new cast additions to Fast and Furious 9 as production gets underway
Daniel Craig leads the dark first trailer for Rian Johnson's murder mystery Knives Out
Harry Styles, Ansel Elgort, Miles Teller among candidates to play Elvis in Baz Luhrmann's new biopic
Will Smith and Tom Holland voice the slapstick new trailer for Spies In Disguise
Title and first trailer for Jumanji: Welcome To The Jungle sequel unveiled
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,194
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Q: Sendmail name resolution problem I was asked to have a look at an old RedHat server (with old as in uname -a giving Linux server 2.4.20-27.7 #1 Thu Dec 11 15:04:48 EST 2003 i686 unknown) which is having problem with sendmail. The server was set up 2003 and hasn't really been touched since then from what I've learned. After a power failure it needed a fsck to boot, and since then the users aren't getting their mail.
I had a look at /var/log/maillog, and there are tons of lines like these:
Aug 22 21:26:22 server sendmail[12250]: p7KIujl05665: to=<id5367@demons.murgent.com>, delay=2+00:22:16, xdelay=00:00:20, mailer=esmtp, pri=4369005, relay=demons.murgent.com., dsn=4.0.0, stat=Deferred: Name server: demons.murgent.com.: host name lookup failure
Aug 22 21:27:22 server sendmail[12250]: p7KHujo05650: to=<apache@sweclo-web02.driften.net>, delay=2+00:27:53, xdelay=00:00:20, mailer=esmtp, pri=4404312, relay=sweclo-web02.driften.net., dsn=4.0.0, stat=Deferred: Name server: sweclo-web02.driften.net.: host name lookup failure
Aug 22 21:27:43 server sendmail[12435]: p7MJNuk12435: SYSERR: putoutmsg ([190.242.41.83]): error on output channel sending "250 2.1.5 <user@domain.com>... Recipient ok (will queue)": Connection reset by [190.242.41.83]
Aug 22 21:27:43 server sendmail[12435]: p7MJNuk12435: lost input channel from [190.242.41.83] to MTA after rcpt
Aug 22 21:27:43 server sendmail[12435]: p7MJNuk12435: from=<no-reply.1@nyc.gov>, size=0, class=0, nrcpts=0, proto=ESMTP, daemon=MTA, relay=[190.242.41.83]
Aug 22 21:28:22 server sendmail[12250]: p7KIujm05665: to=<noreply@cgsociety.org>, delay=1+23:39:41, xdelay=00:00:20, mailer=esmtp, pri=4413757, relay=cgsociety.org., dsn=4.0.0, stat=Deferred: Name server: cgsociety.org.: host name lookup failure
However, name resolution works from the command line with every utility I've tried (ping, host, dig...). The server is also running a named, but it seems to have been shifted over to using another name server at some point (/etc/resolv.conf has the server IP listed, but commented out, and instead points at the router, which forwards to the ISP's DNS servers). Does sendmail have some internal way of doing name resolution?
I have never looked at a sendmail.cf file before today (what has been seen cannot be unseen) but couldn't make much out of it. It did not seem to mention name resolution. Any ideas what I should check?
EDIT: The requested config files:
resolv.conf: (192.168.0.25 is the server, 192.168.0.1 is the gateway/router)
# nameserver 192.168.0.25
nameserver 192.168.0.1
named.conf:
// generated by named-bootconf.pl
options {
directory "/var/named";
/*
* If there is a firewall between you and nameservers you want
* to talk to, you might need to uncomment the query-source
* directive below. Previous versions of BIND always asked
* questions using port 53, but BIND 8.1 uses an unprivileged
* port by default.
*/
// query-source address * port 53;
};
//
// a caching only nameserver config
//
zone "." IN {
type hint;
file "named.ca";
};
zone "localhost" IN {
type master;
file "hosts.domain.com";
allow-update { none; };
};
zone "0.168.192.in-addr.arpa" IN {
type master;
file "db.192.168.0";
allow-update { none; };
};
key "key" {
algorithm hmac-md5;
secret "secret-key-edited-out";
};
EDIT 2: I rearranged the resolv.conf file to fallback to the server itself on failure, and now it is slowly but surely (700 MHz Celeron, woo!) processing the mail queue. I'm not certain for how long it has been commented out, but maybe someone else has been asked to have a look recently... Anyway, why would it only work when using it's own DNS?
A: This may be of help.
https://stackoverflow.com/questions/43970/configuring-sendmail-behind-a-firewall
In short:
Updated sendmail.mc with:
define(`confSERVICE_SWITCH_FILE',`/etc/mail/service.switch')dnl
And then configure the mail.switch file:
# cat /etc/mail/service.switch
hosts files
EDIT: let's see the output of resolv.conf. Also, can we get the output of named.conf as well?
EDIT2: It looks like this machine has it own master DNS server with specific zone records in "hosts.domain.com" that were resolving prior to the reboot. I would imagine that if you look at that zone file, you'll see that the domains in that zone file match the domains that sendmail could not resolve. Of course, considering that name server was commented out in /etc/resolv.conf, it is not really likely. But just case, uncomment that line and see if sendmail will resolve the domains.
A: I've run into issues with sendmail and incorrect name resolution before...
The problem was that my "public" ip address wasn't assigned to any interface on my sendmail box. Sendmail would try & do a resolution of the domain in my emails to direct to the proper mail server... and would re-try to forward the incomming messages to it's publically NAT'd address. The only fix was to setup a bind server locally, and give it entries that resolve to the private address on that box.
I am just now reading Rilindo's answer... and that sounds like a better solution. I might have to try that one out sometime.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,112
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FROM php:7.0-apache
LABEL maintainer Daniel Palumbo <daniel.palumbo@heig-vd.ch>
LABEL maintainer Christopher Meier <christopher.meier@heig-vd.ch>
# We use the image default apache configuration.
# We copy the web site files.
COPY ./httpdoc/ /var/www/html
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,284
|
Isn't this whimsical snowman cute! He's from the North Coast Creations Let it Snow stamp set (shown below), that was released last month. He's perfect for your holiday cards and home decor projects, he would also make fabulous holiday thank you cards!
I had some ornament blanks left over that I purchased last year, they are just perfect for this image! I stamped the snowman in Black Archival on the ornament blank, then heat set. Once the ornament was cool to the touch, I colored the image with STAEDTLER Lumocolor Markers and added a bit of Spica Glitter Gel Pen to the snowman image. I edged the ornament with a Gold Leafing Pen and tied on Silver Cording and a Vintage Red Seam Binding Bow.
what a cute idea for ornaments!!! tfs!
Such a cute ornament and wonderful idea, Lisa! The gold edging is perfect.
What an awesome idea! Such a cute ornament!!
Lisa, this is such an adorable ornament! Love that snowman!
Adorable ornament!!! Love the gold trim. The colors are so cheery!!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,420
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\section{INTRODUCTION}
Domain structure is an important ingredient in functionality of ferroelectric
materials. Among others, it has impact on their nonlinear optical properties,
dielectric permittivity and polarization switching phenomena. Since domain
boundaries in ferroelectric perovskite materials can simultaneously play the
role of the ferroelectric and ferroelastic walls, such domain walls also
strongly influence the electromechanical material properties: they facilitate
switching of spontaneous polarization and spontaneous deformation, thus giving
rise to a large extrinsic contribution to e.g. piezoelectric constants, which
makes ferroelectric materials extremely attractive for applications. The domain
structure also provides additional degree of freedom for tuning of material
properties. In general, further development of domain engineering strategies
requires deeper understanding of the physics of ferroelectric domain wall
itself.
The Ginzburg-Landau-Devonshire (GLD) theory provides a feasible
tool for such a purpose. Landau-Devonshire model describes
phase-transition properties of single-domain crystal using a
limited number of parameters, which are determined experimentally
(or recently also using ab-initio methods). Introduction of
Ginzburg gradient term to the free energy functional enables
addressing nonhomogeneous multi-domain ferroelectric state. The
GLD model was previously used for computation of domain wall
properties in ferroelectric materials (e.g.
Refs.\,\onlinecite{art_cao_cross_prb_1991,
art_ishibashi_salje_2002, art_huang_jiang_hu_liu_JPCM_1996,
art_hlinka_marton_prb_2006,
art_erhart_cao_fousek_ferroelectrics_2001}) and in phase-field
computer modeling of domain formation and
evolution.\cite{art_nambu_sagala_1994, art_hu_chen_1998,
art_ahluwalia_2003, art_marton_hlinka_2006} The GLD model can be
regarded as a bridge model covering length-scales inaccessible by
ab-initio and micro-mechanical models.
BaTiO$_3$ represents a typical ferroelectric material which
undergoes a sequence of phase transitions from high-temperature
paraelectric cubic $Pm3m$ ($O_h$) to the ferroelectric tetragonal
$P4mm$ ($C_{4v}$), orthorhombic $Amm2$ ($C_{2v}$) and rhombohedral
$R3m$ ($C_{3v}$) phase. Energetically equivalent directions of
spontaneous polarization vector, identifying possible
ferroelectric domain states in a particular ferroelectric phase,
are displayed in Fig.\,\ref{fig_spontanni_stavy}. Domain
boundaries separating two domain states are characterized by the
rotational angle needed to match spontaneous polarizations on both
sides of the boundary. For example, the boundary separating
domains with mutually perpendicular spontaneous polarization is
commonly called 90$^\circ$ wall, while the one between
antiparallel spontaneous polarization regions is called
180$^\circ$ wall. Other angles are possible in orthorhombic and
rhombohedral phases of BaTiO$_3$, where the spontaneous
polarization is oriented along the cubic face diagonals and body
diagonals, respectively.
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton1.eps}}
\caption{Directions of spontaneous polarization in the (a)
tetragonal, (b) orthorhombic and (c) rhombohedral phase. Angles
between polarization direction '0' and its symmetry equivalent
ones are indicated.}\label{fig_spontanni_stavy}
\end{figure}
The aim of this paper is to calculate basic characteristics of all
electrically neutral and mechanically compatible domain walls in
all ferroelectric phases of BaTiO$_3$. For a better comparison of
domain wall properties like their thickness or energy density, we
employ an Ising-like approximation leading to previously proposed
analytically solvable one-dimensional
solutions.\cite{art_cao_cross_prb_1991} The paper is organized as
follows. In Section II. we give an overview of the different kinds
of mechanically compatible domain walls in the three ferroelectric
phases. It follows from general theory about macroscopic
mechanical compatibility of adjacent domain
states.\cite{art_fousek_janovec_jap_1969,janovecIT} The GLD
parameters used for calculation of the domain wall properties in
BaTiO$_3$ are the same as in our preceding
work,\cite{art_hlinka_marton_prb_2006} but for the sake of
convenience, the definition and the GLD model and its parameters
are resumed in Section III. Section IV. is devoted to the
description of the computational scheme and approximations applied
here to solve analytically the Euler-Lagrange equations. The main
result of our study - systematic numerical evaluation of
thicknesses, energies, polarization profiles and other properties
for different domain walls, is presented in Section V. Sections
VI. and VII. are devoted to the discussion of validity of used
approximations and the final conclusion, respectively.
\section{\label{sec_domain_walls}MECHANICALLY COMPATIBLE DOMAIN WALLS IN BARIUM TITANATE}
The energy-degeneracy of different directions of spontaneous polarization leads
to the appearance of ferroelectric domain structure. Individual domains are
separated by domain walls, where the polarization changes from one state to
another. Here, only planar domain walls are considered. Orientations of
mechanically compatible domain walls are determined by the equation for
mechanically compatible interfaces separating two domains with the strain
tensors $e_{ij}(-\infty)$ and $e_{ij}(\infty)$ :
\begin{eqnarray
\sum_{m,n=1}^3[e_{mn}(\infty)-e_{mn}(-\infty)]x_m x_n=0~.
\end{eqnarray
Systematic analysis of this equation using symmetry arguments has been done e.g.
in Refs.\,\onlinecite{art_fousek_janovec_jap_1969, art_fousek_czp_1971,
janovecIT}. In general, the number $N$ of mechanically compatible domain walls
separating two particular domain states can have only one of the three values:
$N=0$, $N=2$ or $N=\infty$. In case of $N=2$ there exist two mutually
perpendicular domain walls. Each of them is either a crystallographic
($W_f$-type) wall or non-crystallographic ($S$-type) wall. Orientation of the
$W_f$-wall is fixed by symmetry of the crystal, while orientation of the
$S$-wall is determined by components of the strain tensor in adjacent domains
(and its orientation can be therefore dependent on temperature). For $N=\infty$
there exists infinite number of wall orientations, some of them may be preferred
energetically.
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton2.eps}}
\caption{Set of mechanically compatible and electrically neutral
domain walls in the three ferroelectric phases of BaTiO$_3$. In the
case of 180$^\circ$ domain walls, where the orientation is not
determined by symmetry, walls with the most important
crystallographic orientations are displayed.}\label{fig_plochy}
\end{figure}
Further, the electrically neutral domain walls will be
considered.\cite{art_fousek_janovec_jap_1969} It implies that the difference
${\bf P}(\infty)-{\bf P}(-\infty)$ between the spontaneous polarizations in the
adjacent domains is perpendicular to the unit vector ${\bf s}$, normal to the
domain wall: \begin{equation} \label{eq:neutral} \left( {\bf P}(\infty)-{\bf
P}(-\infty) \right)\cdot{\bf s} = 0\,. \end{equation} We also define a unit
vector ${\bf r} \parallel ({\bf P}(\infty)-{\bf P}(-\infty))$, which identifies
the component of the spontaneous polarization which reverses when crossing the
wall. Then the charge neutrality condition (\ref{eq:neutral}) can be expressed
as ${\bf r}\cdot{\bf s} = 0$. Finally, let us introduce a third base vector
${\bf t} = {\bf r}\times {\bf s}$, which complements the symmetry-adapted
orthonormal coordinate system (r, s, t).
BaTiO$_3$ symmetry allows a variety of domain
walls.\cite{art_fousek_czp_1971} Ferroelectric walls of BaTiO$_3$
can be divided in two groups - the
non-ferroelastic\cite{janovecIT} walls separating domains with
antiparallel polarization ($e_{mn}(\infty)-e_{mn}(-\infty)=0$,
$N=\infty$) and the ferroelastic walls with other than 180$^\circ$
between polarization in the adjacent domain states ($N=2$). The
${\bf r}\cdot{\bf s} = 0$ condition implies that the neutral
non-ferroelastic walls are parallel to the spontaneous
polarization, and the neutral ferroelastic walls realize a
"head-to-tail" junction. The set of plausible neutral and
mechanically compatible domain wall types are schematically shown
in Fig.\,\ref{fig_plochy}. Domain walls are labeled by a symbol
composed of the letter specifying the ferroelectric phase (T, O or
R staying for the tetragonal, orthorhombic or rhombohedral,
resp.), number indicating the polarization rotation angle (180,
120, 109, 90, 71 or 60 degrees) and, if needed, the orientation of
the domain wall normal with respect to the parent pseudo-cubic
reference structure.
\begin{table*}
\caption{Cartesian components of switching vectors ${\bf r}$, domain wall
normals ${\bf s}$, and boundary conditions for polarization and strain in
adjacent domain states for the inspected domain walls. Vector ${\bf s}_{\rm
O60}$ is defined in Eqn.\,(\ref{eqn_O60rst_coord}), $P_0$ stands for magnitude
of spontaneous polarization. As usual, spontaneous quantities are those
minimizing GLD functional. Numerical values used in this work are given in
Sec\.\,\ref{sec_results}.}\label{tab_boundary_conditions}
\begin{tabular}{lcccccc}
\hline\hline\
Wall &{\bf r} &{\bf s}&${{\bf P}(-\infty)}/{P_0}$ &${{\bf P}(\infty)}/{P_0}$ &${\bf e}(-\infty)$ &${\bf e}(\infty)$\\
\hline
T180\{001\} &$(1,0,0)$ &$(0,0,1)$ &$(1,0,0)$ &$(-1,0,0)$ &$(e_\parallel,e_\perp,e_\perp,0,0,0)$ &$(e_\parallel,e_\perp,e_\perp,0,0,0)$\\
T180\{011\} &$(1,0,0)$ &$(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$ &$(1,0,0)$ &$(-1,0,0)$ &$(e_\parallel,e_\perp,e_\perp,0,0,0)$ &$(e_\parallel,e_\perp,e_\perp,0,0,0)$\\
T90 &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0)$ &$(1,0,0)$&$(0,-1,0)$&$(e_\parallel,e_\perp,e_\perp,0,0,0)$ &$(e_\perp,e_\parallel,e_\perp,0,0,0)$\\
O180\{1$\bar{1}$0\} &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0)$ &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(\frac{-1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0)$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$\\
O180\{001\} &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(0,0,1)$ &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(\frac{-1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0)$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$\\
O90 &$(0,1,0)$ &$(1,0,0)$ &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0)$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,-2e_{\rm b})$\\
O60 &$(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$ &${\bf s}_{\rm O60}$ &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}})$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$ &$(e_{\rm c},e_{\rm a},e_{\rm a},-2e_{\rm b},0,0)$\\
O120 &$(\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}},\frac{-1}{\sqrt{6}})$ &$(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$ &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(0,\frac{-1}{\sqrt{2}},\frac{1}{\sqrt{2}})$ &$(e_{\rm a},e_{\rm a},e_{\rm c},0,0,2e_{\rm b})$ &$(e_{\rm c},e_{\rm a},e_{\rm a},-2e_{\rm b},0,0)$\\
R180\{1$\bar{1}$0\} &$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0)$ &$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},2e_{\rm b},2e_{\rm b},2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},2e_{\rm b},2e_{\rm b},2e_{\rm b})$\\
R180\{$\bar{2}$11\} &$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(\frac{-2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}})$ &$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},2e_{\rm b},2e_{\rm b},2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},2e_{\rm b},2e_{\rm b},2e_{\rm b})$\\
R109 &$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ &$(0,0,1)$ &$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},2e_{\rm b},2e_{\rm b},2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},-2e_{\rm b},-2e_{\rm b},2e_{\rm b})$\\
R71 &$(0,1,0)$ &$(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$ &$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},2e_{\rm b},2e_{\rm b},2e_{\rm b})$ &$(e_{\rm a},e_{\rm a},e_{\rm a},-2e_{\rm b},2e_{\rm b},-2e_{\rm b})$\\
\hline\hline
\end{tabular}
\end{table*}
Our choice of the base vectors ${\bf r}, {\bf s}$ and of the
spontaneous polarization and strain components in the adjacent
domain pairs for each domain wall type shown in
Fig.\,\ref{fig_plochy} are summarized in
Table\,\ref{tab_boundary_conditions}. Base vectors coincide with
special crystallographical directions, except for the O60 wall
where the ${\bf s}$ and ${\bf t}$ vectors depend on the
orthorhombic spontaneous strain (see
Table\,\ref{tab_boundary_conditions}) as
follows:\cite{art_erhart_cao_fousek_ferroelectrics_2001}
\begin{eqnarray}
{\bf r}_{\rm O60}&=&\left(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}\right)\nonumber\\
{\bf s}_{\rm O60}&=&\left(\frac{e_{\rm a}-e_{\rm c}}{D_1},\frac{2e_{\rm b}}{D_1},\frac{e_{\rm c}-e_{\rm a}}{D_1}\right)\nonumber\\
{\bf t}_{\rm O60}&=&\left(\frac{-e_{\rm b}}{D_2},\frac{e_{\rm
a}-e_{\rm c}}{D_2},\frac{e_{\rm
b}}{D_2}\right)~~,\label{eqn_O60rst_coord}
\end{eqnarray}
with $D_1=\sqrt{2}D_2$, $D_2=\sqrt{(e_{\rm a}-e_{\rm c})^2+2e_{\rm
b}^2}$ and with $e_{\rm a}, e_{\rm b}$, and $e_{\rm c}$ defined in
Table\,I.
Although only the neutral walls are discussed in the following, the
Fig.\,\ref{fig_plochy} is actually helpful in enumeration of all possible
mechanically compatible domain wall species in BaTiO$_3$. In principle,
mechanical compatibility allows 180$^\circ$ $W_\infty$-type domain walls with an
arbitrary orientation of the domain wall in all three ferroelectric phases
(T180, O180, R180). Obviously, they are electrically neutral only if the domain
wall normal is parallel with the spontaneous polarization. Ferroelastic walls
exists in mutually perpendicular pairs. In the tetragonal phase, there exist
90$^\circ$ $W_f$-type domain walls (T90), either charged (head-to-head or
tail-to-tail) or neutral (head-to-tail). The orthorhombic phase is more complex.
In the case of 60$^\circ$ angle between polarization directions, the N=2 pair is
formed by a charged $W_f$-type wall and neutral $S$-type wall. The case of
120$^\circ$ angle is similar but $W_f$-wall is neutral and $S$-wall is charged.
In addition, there are again charged or neutral 90$^\circ$ $W_f$-walls (O90).
The rhombohedral phase has pairs of charged and neutral $W_f$-type domain walls
with the angle between polarizations either 109$^\circ$ or 71$^\circ$ (R109 or
R71, resp.). Since only neutral walls are discussed here, the $S$-type domain
wall will be referred to as O60 and $W_f$ wall as O120.
\section{\label{sec_energy-expansion}GLD MODEL FOR BARIUM TITANATE}
Calculations presented in this paper are based on the GLD model with
anisotropic gradient terms, reviewed in
Ref.\,\onlinecite{art_hlinka_marton_prb_2006}. The free energy $F$
is expressed in terms of polarization and strain field taken for
primary and secondary order-parameter, resp.:
\begin{eqnarray}\label{eqn_total_potential}
F\left[\{ P_i, P_{i,j}, e_{ij} \}\right]=\int f {\rm d{\bf r}}~,
\end{eqnarray}
where the free energy density $f$ consists of Landau,
gradient, elastic and electrostriction part
\begin{eqnarray}
f=f_{\rm L}^{(e)}\{P_i\}+f_{\rm C}\{P_i,e_{ij}\}+f_{\rm
q}\{P_i,e_{ij}\}+f_{\rm
G}\{P_{i,j}\}.
\end{eqnarray}
The Landau potential considered here is expanded up to the sixth
order in components of polarization for the cubic symmetry
($O_h^1$):
\begin{eqnarray}
f_{\rm L}^{\rm (e)}=&&\alpha_{1}\left(P_1^2+P_2^2+P_3^2\right) \nonumber\\
&&+ \alpha_{11}^{\rm (e)}\left(P_1^4+P_2^4+P_3^4\right) \nonumber\\
&&+ \alpha_{12}^{\rm (e)}\left(P_1^2P_2^2+P_2^2P_3^2+P_1^2P_3^2\right) \nonumber\\
&&+ \alpha_{111}\left(P_1^6+P_2^6+P_3^6\right) \nonumber\\
&&+ \alpha_{112}(P_1^4(P_2^2+P_3^2)+P_2^4(P_1^2+P_3^2)\nonumber\\
&&\,\,\,\,\,+ P_3^4(P_1^2+P_2^2)).\nonumber\\
&&+ \alpha_{123}P_1^2P_2^2P_3^2\label{eqn_landauexpansion}
\end{eqnarray}
with the three temperature-dependent coefficients
$\alpha_{1},\alpha_{11}$ and $\alpha_{111}$, as in
Ref.\,\onlinecite{art_bell_jap_2000}. This expansion produces the
6 equivalent domain states in the tetragonal phase, 12 in the
orthorhombic, and 8 in the rhombohedral phase (see
Fig.\,\ref{fig_spontanni_stavy}).
Dependence of the free energy on the strain is encountered by including
elastic and linear-quadratic electrostriction functionals $F_{\rm C}$ and
$F_{\rm q}$, resp. Their corresponding free energy densities are
\begin{eqnarray}
f_{\rm C}=\frac{1}{2}e_{\rho}C_{\rho
\sigma}e_{\sigma}\label{eqn_elastic_energy}
\end{eqnarray}
and
\begin{eqnarray}
f_{\rm q}=-q_{ijkl}e_{ij}P_k P_l~,\label{eqn_electrostriction_energy}
\end{eqnarray}
where $e_{ij}=\frac{1}{2}\left(\partial u_i/\partial x_j+\partial u_j/\partial
x_i\right)$ and $C_{\alpha\beta}$, $q_{\alpha\beta}$ are components of
elastic and electrostriction tensor in Voigt notation, $C_{11}=C_{1111}$,
$C_{12}=C_{1122}$, $C_{44}=C_{1212}$, $q_{11}=q_{1111}$, $q_{12}=q_{1122}$, but
$q_{44}=2q_{1122}$.
The elastic and electrostriction terms result in re-normalization of
the bar expansion coefficients $\alpha_{11}^{(e)}$ and
$\alpha_{12}^{(e)}$ when minimizing the free energy with respect to
strains (in the homogeneous sample). The bar $\alpha_{11}^{(e)},\
\alpha_{12}^{(e)}$ and the relaxed $\alpha_{11},\ \alpha_{12}$
coefficients are related as:\cite{art_hlinka_marton_prb_2006}
\begin{eqnarray}
\alpha_{11}^{(e)}&=&\alpha_{11}+\frac{1}{6}\left[\frac{\hat{q}_{11}^2}{\hat{C}_{11}}+2\frac{\hat{q}_{22}^2}{\hat{C}_{22}}\right]\nonumber\\
\alpha_{12}^{(e)}&=&\alpha_{12}+\frac{1}{6}\left[2\frac{\hat{q}_{11}^2}{\hat{C}_{11}}-2\frac{\hat{q}_{22}^2}{\hat{C}_{22}}+3\frac{q_{44}^2}{C_{44}}\right]\label{eqn_alpha_clamped}
\end{eqnarray}
with
\begin{eqnarray}
\hat{C}_{11}&=&C_{11}+2C_{12}\nonumber\\
\hat{C}_{12}&=&C_{11}-C_{12}\nonumber\\
\hat{q}_{11}&=&q_{11}+2q_{12}\nonumber\\
\hat{q}_{12}&=&q_{11}-q_{12}~.
\end{eqnarray}
The Ginzburg gradient term $f_{\rm G}$ is considered in the form
\begin{eqnarray}
f_{\rm G}&=&\frac{1}{2}G_{11}(P_{1,1}^2+P_{2,2}^2+P_{3,3}^2)\nonumber\\
&&+ G_{12}(P_{1,1}P_{2,2}+P_{2,2}P_{3,3}+P_{1,1}P_{3,3})\nonumber\\
&&+
\frac{1}{2}G_{44}(\left(P_{1,2}+P_{2,1}\right)^2+\left(P_{2,3}+P_{3,2}\right)^2\nonumber\\
&&\,\,\,\,\,+\left(P_{3,1}+P_{1,3}\right)^2)~.\label{eqn_gradient-direct}
\end{eqnarray}
It was pointed out\cite{art_hlinka_marton_prb_2006} that the tensor of gradient
constants of BaTiO$_3$ is highly anisotropic with fundamental consequences
on predicted domain wall
properties. Up to now, the isotropic gradient tensor $G_{i,j}$ was mostly
employed in the computations.
The material-specific coefficients in the model are assumed being constant,
except for the three Landau potential coefficients
\begin{eqnarray
\alpha_{1}&=&3.34\times10^5(T-381)\nonumber\\
\alpha_{11}&=&4.69\times10^6(T-393)-2.02\times10^8\nonumber\\
\alpha_{111}&=&-5.52\times10^7(T-393)+2.76\times10^9~,
\end{eqnarray
where $T$ is absolute temperature.\cite{art_bell_jap_2000} The phase transitions
occur in this model at the temperatures 392.3\,K (C$\rightarrow$T), 282.5\,K
(T$\rightarrow$O), and 201.8\,K (O$\rightarrow$R).\cite{art_bell_jap_2000} All
phase transitions are of the first order, the local minima corresponding to the
tetragonal, orthorhombic and rhombohedral phase exist for this Landau potential
between 237\,K and 393\,K, between 104\,K and 303\,K, and below 256\,K,
respectively. Full set of temperature independent parameters of the GLD model
reads:\cite{art_hlinka_marton_prb_2006,note_q44}
$\alpha_{12}=3.230\times\rm10^8\,J m^5 C^{-4}$,
$\alpha_{112}=4.470\times\rm10^9\,J m^{9}C^{-6}$,
$\alpha_{123}=4.910\times\rm10^9\,J m^{9}C^{-6}$,
$G_{11}=51\times\rm10^{-11}\,J m^3 C^{-2}$,
$G_{12}=-2\times\rm10^{-11}\,Jm^3C^{-2}$,
$G_{44}=2\times\rm10^{-11}\,J m^3 C^{-2}$,
$q_{11}=14.20\times\rm10^9\,J m C^{-2}$,
$q_{12}=-0.74\times\rm10^9\,J m C^{-2}$,
$q_{44}=1.57\times\rm10^9\,J m C^{-2}$,
$C_{11}=27.50\times\rm10^{10}\,J m^{-3}$,
$C_{12}=17.90\times\rm10^{10}\,J m^{-3}$ and
$C_{44}=5.43\times\rm10^{10}\,J m^{-3}$.
\section{\label{sec_one-dimensional-model} STRAIGHT POLARIZATION PATH APPROXIMATION}
Let us consider a single mechanically compatible and electrically
neutral domain wall in a perfect infinite stress-free crystal.
Within the continuum GLD theory, such domain wall is associated
with a planar kink solution of the Euler-Lagrange equations of the
GLD functional for polarization vector and the strain
tensor.\cite{art_zhirnov_1958, art_ishibashi_salje_2002,
art_cao_cross_prb_1991, art_hlinka_marton_prb_2006} Domain wall
type is specified by selection of the wall normal $\bf s$ and by
the two domain states at $s = -\infty$ and $+\infty$. This ideal
geometry implies that the polarization and strain vary only along
normal to the wall ${\bf s}$ and domain wall can be thus
considered as a trajectory in order-parameter space.
Even if the domain wall is neutral, in its central part the local electric
charges can occur due to the position-dependent polarization. The additional
assumption $\nabla \cdot {\bf {\rm P}}=0$ ensures the absence of charges in the
whole domain wall. Then the local electric field is zero and the electrostatic
contribution vanishes. Such strictly charge-free
solutions\cite{art_zhirnov_1958} were found to be an excellent approximation for
ideal dielectric materials.\cite{art_hlinka_marton_prb_2006} The condition
$\nabla \cdot {\bf {\rm P}}=0$ implies that the polarization vector variation is
restricted to a plane perpendicular to $\bf s$ (the trajectory is constrained to
$P_{\rm s}=P_{\rm s}(\pm\infty)$ plane).
In fact, the polarization trajectories representing the T180 and T90 walls
calculated under $\nabla \cdot {\bf {\rm P}}=0$ constraint were found to be the
straight lines connecting the boundary values.\cite{art_zhirnov_1958,
art_ishibashi_salje_2002, art_cao_cross_prb_1991, art_hlinka_marton_prb_2006}
This greatly simplifies the algebra and the solution of the variational problem
can be found analytically. Therefore, we decided to impose the condition of a
direct, straight polarization trajectory for the variational problem of all
domain wall species of BaTiO$_3$. This condition, further referred as straight
polarization path (SPP) approximation, implies that both $P_{\rm s}$ and $P_{\rm
t}$ polarization components are constant across the wall. We shall come back to
the meaning and possible drawbacks of this approximation in Section VI.
Polarization and strain in the mechanically compatible and
electrically neutral SPP walls can be cast in the form: ${\bf
P}=[P_{\rm r}(s), P_{\rm s}(\pm\infty), P_{\rm t}(\pm\infty)]$ and
$e=[e_{\rm rr}(\pm\infty), e_{\rm ss}(s), e_{\rm tt}(\pm\infty),
2e_{\rm st}(s), 2e_{\rm rt}(\pm\infty), 2e_{\rm rs}(s)]$,
respectively. The $s$-dependent strain components are calculated
from the mechanical equilibrium condition
\begin{eqnarray}
\sum_{j=1}^3\frac{\partial \sigma_{ij}}{\partial
x_j}=\sum_{j=1}^3\frac{\partial}{\partial x_j}\left(\frac{\partial
f}{\partial e_{ij}}\right)=0~.\label{eqn_mechanical-equilibrium}
\end{eqnarray}
Boundary conditions for stress and the
fact that all quantities vary only along direction $s$ imply
\begin{eqnarray}
\frac{\partial f_{\rm Cq}}{\partial
e_{ij}}+C=0\label{eqn_mechanical-equilibrium-constant}
\end{eqnarray}
for $ij\in\{\rm ss,st,rs\}$ with the integration constant $C$ to be determined from
boundary conditions.
The Euler-Lagrange equation for polarization reduces to
\begin{eqnarray}
\frac{\partial}{\partial s}\frac{\partial f}{\partial P_{{\rm
r},{\rm s}}}-\frac{\partial f}{\partial P_{\rm r}}=0~.
\end{eqnarray}
where the strain components from
Eqn.\,(\ref{eqn_mechanical-equilibrium-constant}) were substituted
into $f$. Thus, the elastic field was eliminated. It turns out that
the resulting Euler-Lagrange equation for $P_{\rm r}$ is possible to
rewrite in the form
\begin{eqnarray}
\label{eq:el1d}
g\frac{d^2p(s)}{ds^2}=2a_1p(s)+4a_{11}p^3(s)+6a_{111}p^5(s)~,\label{eqn_general_wall}
\end{eqnarray}
where $p(s)$ stands for $P_{\rm r}(s)$, and where the coefficients
$g, a_1, a_{11}$ and $a_{111}$, different for each domain type,
depend only on the material tensors. The boundary values are
$-P_{\rm r}(\infty)=P_{\rm r}(-\infty)=p_{\infty}$.
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton3.eps}}
\caption{ Profile of the "reversed" polarization component $P_{\rm r
}$ (Eqn.\,\ref{eqn_wall_profile}) for a domain wall with the shape
factor $A=1$ (solid) and $A=3$ (dashed). Both profiles have the same
derivative for $s/2\xi=0$ (dotted) and therefore also the same thickness
(indicated by vertical lines) according to the definition in
Eqn.\,(\ref{eqn_analytical-wall-thickness}).
}\label{fig_clanek_profile}
\end{figure}
Solution of the Euler-Lagrange equation (\ref{eq:el1d}) is well
known.\cite{art_zhirnov_1958, art_houchmandzadeh_lajzerowic_1991,
art_hudak, art_cao_cross_prb_1991, art_ishibashi_dvorak_1976} We
shall follow the procedure of
Ref.\,\onlinecite{art_hlinka_marton_intgrferro}. Integrating Eqn.
(\ref{eq:el1d}) one can obtain the equation:
\begin{eqnarray}
\frac{g}{2}\left(\frac{\partial p}{\partial s}\right)^2=f_{\rm EL}(p)
\end{eqnarray}
where (see Ref.\,\onlinecite{art_hlinka_marton_prb_2006})
\begin{eqnarray}
f_{EL}(p)=a_1 p(s)^2+a_{11} p(s)^4+a_{111} p(s)^6~.
\end{eqnarray}
The function $f_{\rm EL}(p)$ is a double-well "Euler-Lagrange"
potential with two minima $\pm p_{\infty}$, where
\begin{eqnarray}
p_{\infty}^2=\frac{-a_{11}+\sqrt{a_{11}^2-3a_{111}a_{1}}}{3a_{111}}.\label{eqn_19}
\end{eqnarray}
The differential Eqn.\,(\ref{eqn_general_wall}) has the analytical solution
\begin{eqnarray}
p(s)=p_{\infty}\frac{{\sinh}({s}/{\xi^\prime})}{\sqrt{A+{\sinh}^2({s}/{\xi^\prime})}}~,\label{eqn_wall_profile}
\end{eqnarray}
where
\begin{eqnarray}
A=\frac{3a_{111}p_\infty^2+a_{11}}{2a_{111}p_\infty^2+a_{11}}~.
\end{eqnarray}
and
\begin{eqnarray}
\xi^\prime=\frac{\xi}{\sqrt{A}}~.
\end{eqnarray}
Quantity $A$ determines deviation of the profile (\ref{eqn_wall_profile})
from the $tanh$ profile, which occurs
for the $4^{th}$-order potential (i.e., $a_{111}=0,\ A=1$).
The domain wall thickness (Fig.\,\ref{fig_clanek_profile}) is defined as
\begin{eqnarray}
2\xi=p_\infty\sqrt{\frac{2g}{U}}~.\label{eqn_analytical-wall-thickness}
\end{eqnarray}
with $U$ being the energy barrier between the domain states:
\begin{eqnarray}
U=f_{EL}(0)-f_{EL}(p_\infty)=2a_{111}p_\infty^6+a_{11}p_\infty^4~.\label{eqn_energy_on_pinfty}
\end{eqnarray}
The surface energy density of the domain wall is
\begin{eqnarray}
\Sigma&=&\int_{-\infty}^{\infty}\left(f_{EL}(s)-f_{EL}(p_\infty)\right)ds\nonumber\\
&=&\frac{4}{3}p_\infty \sqrt{2gU}\left[A^{5/2}I(A)\right]~,\label{correction_factor_sigma}
\end{eqnarray}
where
\begin{eqnarray}
I(A)=\frac{3}{4}\int_{-\infty}^{\infty}\frac{{\rm
cosh}^2(h)dh}{(A+{\rm cosh}^2(h)-1)^3}~.
\end{eqnarray}
The domain wall characteristics depend on the coefficients $g,\
a_1,\ a_{11},\ a_{111}$ through Eqs. (\ref{eqn_19},
\ref{eqn_wall_profile}, \ref{eqn_analytical-wall-thickness},
\ref{correction_factor_sigma}). The explicit expressions for these
coefficients are summarized for various domain walls in Table\,II.
The expressions are simplified using the notation inspired by
Ref.\,\onlinecite{art_cao_cross_prb_1991}. For all phases we are
using:
\begin{eqnarray}
a_{1}^{\rm
r}&=&\alpha_{1}-\left[\frac{1}{3}\frac{\hat{q}_{11}^2}{\hat{C}_{11}}+\frac{1}{6}\frac{\hat{q}_{22}^2}{\hat{C}_{22}}-\frac{(q_{11}+q_{12})q_{12}^\prime}{2C_{11}^\prime}\right]P_0^2\nonumber
\end{eqnarray}
\begin{eqnarray}
a_{11}^{\rm
r}&=&\frac{\alpha_{11}^{(e)}}{2}+\frac{\alpha_{12}^{(e)}}{4}-\frac{q_{12}^{\prime
2}}{2C_{11}^\prime}\nonumber\\
a_{12}^{\rm
rs}&=&3\alpha_{11}^{(e)}-\frac{\alpha_{12}^{(e)}}{2}-\frac{q_{11}^\prime
q_{12}^\prime}{C_{11}^\prime}-\frac{\hat{q}_{22}^2}{2\hat{C}_{22}}\nonumber
\end{eqnarray}
\begin{eqnarray}
a_{111}^\prime&=&\frac{1}{4}\left(\alpha_{111}+\alpha_{112}\right)\nonumber\\
a_{112}^\prime&=&\frac{1}{4}\left(15\alpha_{111}-\alpha_{112}\right)~,\nonumber\label{eqn_t90elcoefficients}
\end{eqnarray}
for tetragonal and orthorhombic phases we are abbreviating
\begin{eqnarray}
C_{11}^\prime&=&\frac{C_{11}+C_{12}+2C_{44}}{2}\nonumber\\
C_{12}^\prime&=&\frac{C_{11}+C_{12}-2C_{44}}{2}\nonumber\\
C_{66}^\prime&=&\frac{C_{11}-C_{12}}{2}\nonumber
\end{eqnarray}
\begin{eqnarray}
q_{11}^\prime&=&\frac{q_{11}+q_{12}+q_{44}}{2}\nonumber\\
q_{12}^\prime&=&\frac{q_{11}+q_{12}-q_{44}}{2}\nonumber\\
q_{66}^\prime&=&q_{11}-q_{12}~,\nonumber\nonumber\label{eqn_transformace45}
\end{eqnarray}
and for the rhombohedral phase
\begin{eqnarray}
C_{33}^\prime&=&\frac{C_{11}+C_{12}+2C_{44}}{2}\nonumber\\
q_{11}^\prime&=&\frac{q_{11}+2q_{12}+2q_{44}}{3}\nonumber\\
q_{13}^\prime&=&\frac{q_{11}+2q_{12}-q_{44}}{3}~.\label{eqn_transformation_rhombo}
\end{eqnarray}
The expressions for the coefficients of the T180 and T90 domain
walls are equivalent to the previously published
expressions.\cite{art_cao_cross_prb_1991,
art_hlinka_marton_prb_2006} Derivations for O60, O120, and
R180\{$\bar{2}$11\} walls lead to complicated formulas, and
therefore only numerical results for $g, a_1, a_{11}$, and $a_{111}$
coefficients are presented here.
\begin{table*}
\caption{ Parameters characterizing various mechanically compatible
and neutral domain wall species of BaTiO$_3$-like ferroelectrics
within the SPP treatment described in the Section IV. Results for
O60, O120, and R180\{$\bar{2}$11\} walls as well as a few other
parameters are omitted because the corresponding analytical
expressions through the GLD model parameters and spontaneous values
of polarization and strain are too complicated.
}\label{tab_analytical_expressions}
\begin{tabular}{lccccc}
\hline\hline
Domain wall&$p_\infty$&$g$&$a_{1}$&$a_{11}$&$a_{111}$\\
\hline
\multicolumn{6}{c}{Tetragonal phase}\\
T180\{001\} &$P_0$ &$G_{44}$ &$\alpha_1-e_\parallel q_{11}-e_\perp q_{12}+\frac{C_{12}}{C_{11}}(e_\parallel+e_\perp)q_{12}$ &$\alpha_{11}^{(e)}-\frac{1}{2}\frac{q_{12}^2}{C_{11}}$ &$\alpha_{111}$\\
T180\{011\} &$P_0$ &$G_{44}$ &$\alpha_1-e_\parallel q_{11}+\frac{C_{12}}{C_{11}^\prime}e_\parallel q_{12}-2\frac{C_{44}}{C_{11}^\prime}e_\perp q_{12}$ &$\alpha_{11}^{(e)}-\frac{1}{2}\frac{q_{12}^2}{C_{11}^\prime}$ &$\alpha_{111}$\\
T90 &$\frac{P_0}{\sqrt{2}}$ &$\frac{G_{11}-G_{12}}{2}$ &$\alpha_1^{\rm r}+\alpha_{12}^{\rm rs}\frac{P_0^2}{2}+\alpha_{112}^\prime\frac{P_0^4}{4}$ &$\alpha_{11}^{\rm r}+\alpha_{112}^\prime\frac{P_0^2}{2}$ &$\alpha_{111}^\prime$\\
\\
\hline
\multicolumn{6}{c}{Orthorhombic phase}\\
O180\{1$\bar{1}$0\} &$P_0$ &$\frac{G_{11}-G_{12}}{2}$ &$\alpha_1-e_{\rm b}(q_{11}^\prime-q_{12}^\prime)-e_{\rm c} q_{12}-e_{\rm a}(q_{11}^\prime+q_{12}^\prime)$ &$\frac{\alpha_{11}^{(e)}}{2}+\frac{\alpha_{12}^{(e)}}{4}-\frac{q_{12}^{\prime 2}}{2 C_{11}^\prime}$ &$\frac{1}{4}\left(\alpha_{111}+\alpha_{112}\right)$\\
&$ $ &$ $ &$+\frac{q_{12}^{\prime 2}}{C_{11}^\prime}P_0^2$ &$ $ &$ $\\
O180\{001\} &$P_0$ &$G_{44}$ &$\alpha_1-e_{\rm a}(q_{11}+q_{12})-e_{\rm b} q_{44}+2e_{\rm a} q_{12}\frac{C_{12}}{C_{11}}$ &$\frac{\alpha_{11}^{(e)}}{2}+\frac{\alpha_{12}^{(e)}}{4}-\frac{q_{12}^2}{2 C_{11}}$ &$\frac{1}{4}\left(\alpha_{111}+\alpha_{112}\right)$\\
O90 &$\frac{P_0}{\sqrt{2}}$ &$G_{44}$ &$\alpha_1^{\rm r}+\alpha_{12}^{\rm(e)}\frac{P_0^2}{2}+\alpha_{112}\frac{P_0^4}{4}$ &$\alpha_{11}^{(e)}+\frac{1}{2}\left[\alpha_{112}P_0^2-\frac{q_{12}^2}{C_{11}}\right]$ &$\alpha_{111}$\\
O120 &$\frac{\sqrt{3}P_0}{2}$&$\frac{G_{11}-G_{12}+4G_{44}}{6}$&***&***&$\frac{2\alpha_{111}+21\alpha_{112}+2\alpha_{123}}{108}$\\
\\
\hline
\multicolumn{6}{c}{Rhombohedral phase}\\
R180\{1$\bar{1}$0\} &$P_0$ &$\frac{G_{11}-G_{12}+G_{44}}{3}$ &$\alpha_1+\frac{1}{6 C_{33}^\prime}[C_{12}(e_{\rm a}(q_{11} + 2 q_{12} - 4 q_{44}) - 6 e_{\rm b} q_{44})$ &$\frac{\alpha_{11}^{(e)}+\alpha_{12}^{(e)}}{3}-\frac{q_{13}^{\prime 2}}{2C_{33}^\prime}$ &$\frac{3\alpha_{111}+6\alpha_{112}+\alpha_{123}}{27}$\\
&$ $ &$ $ &$-C_{11} (6 e_{\rm b} q_{44} + 3e_{\rm a} q_{11}^\prime)$ &$ $ &$ $\\
&$ $ &$ $ &$-2 C_{44} (3 e_{\rm a} (q_{11} + 2 q_{12}) + 6e_{\rm b} q_{11}^\prime)]$ &$ $ &$ $\\
R180\{$\bar{2}$11\} &$P_0$
&$\frac{G_{11}-G_{12}+G_{44}}{3}$ &*** &*** &$\frac{3\alpha_{111}+6\alpha_{112}+\alpha_{123}}{27}$\\
R109 &$\frac{P_0}{\sqrt{3}}$ &$G_{44}$ &$\alpha_{1}+\frac{\alpha_{12}^{(e)}P_0^2}{3}+\frac{\alpha_{112}P_0^4}{9}+\frac{2q_{12}^2 P_0^2}{3C_{11}}-\frac{q_{44}^2 P_0^2}{6C_{44}}$ &$\frac{\alpha_{11}^{(e)}}{2}+\frac{\alpha_{12}^{(e)}}{4}-\frac{q_{12}^2}{2C_{11}}$ &$\frac{\alpha_{111}+\alpha_{112}}{4}$\\
&$ $ &$ $
&$-e_{\rm a}(q_{11}+2q_{12})-e_{\rm b} q_{44}$ &$+\frac{1}{12}\left(2\alpha_{112}+\alpha_{123}\right)P_0^2$ &$ $\\
R71 &$\frac{P_0}{\sqrt{3}}$ &$G_{44}$ &$\alpha_1+\frac{2\alpha_{12}^{(e)}P_0^2}{3}+\frac{2\alpha_{112}P_0^4}{9}+\frac{\alpha_{123}P_0^4}{9}$ &$\alpha_{11}^{(e)}+\frac{2\alpha_{112}P_0^2}{3}-\frac{q_{12}^2}{2C_{33}^\prime}$ &$\alpha_{111}$\\
&$ $ &$ $ &$-e_{\rm a}(q_{11}+2q_{12})+\frac{q_{12}^2 P_0^2}{3C_{33}^\prime}-\frac{q_{44}^2P_0^2}{3C_{44}}$ &$ $ &$ $\\
\hline\hline
\end{tabular}
\end{table*}
\section{\label{sec_results} QUANTITATIVE RESULTS}
\begin{table*}
\caption{Predicted values of thickness and planar energy density
of domain wall species illustrated in Fig.\,2 together with the
determining parameters appearing in the SPP treatment. Results are
evaluated from the BaTiO$_3$-specific GLD model at three selected
temperatures corresponding to the tetragonal, orthorhombic and
rhombohedral phase, respectively. Domain walls for which relaxing
of the $P_{\rm t}$ component results in a lower-energy CPP
solution (see in Section\,VI.) are denoted by $^\dag$. Numerical
values are in SI units ($2\xi$ in nm; $\Sigma$ in mJ/m$^{2}$; $U$
in MJ/m$^3$; $p_\infty$ in C/m$^2$; $g$ in
10$^{-11}$~kg\,m$^5$\,s$^{-2}$\,C$^{-2}$; $a_1$ in
10$^7$~kg\,m$^3$\,s$^{-2}$\,C$^{-2}$; $a_{11}$ in
10$^8$~kg\,m$^7$\,s$^{-2}$\,C$^{-4}$; $a_{111}$ in
10$^9$~kg\,m$^{11}$\,s$^{-2}$\,C$^{-6}$).}\label{tab_numerical_results}
\begin{tabular}{lrrrrrrrrr}
\hline\hline
Domain wall&~~~~~~~~~~$2\xi$&~~~~~~~~~~$\Sigma$&~~~~~~~~~~~~~~$U$&~~~~~~~~~~$A$&$~~~~~~~~~~p_\infty$&~~~~~~~~~~$g$&~~~~~~~~~~$a_{1}$&~~~~~~~~~~$a_{11}$&~~~~~~~~~~$a_{111}$\\
\hline
\multicolumn{10}{c}{Tetragonal phase (298\,K)}\\
T180\{001\} &0.63 &5.9 &6.41 &1.43 &0.265 &2.0 &-14.26 &1.69 &8.00\\
T180\{011\} &0.63 &5.9 &6.41 &1.43 &0.265 &2.0 &-14.26 &1.69 &8.00\\
T90 &3.59 &7.0 &1.45 &1.09 &0.188 &26.5 &-7.86 &9.53 &3.12\\
\\
\hline
\multicolumn{10}{c}{Orthorhombic phase (208\,K)}\\
O180\{1$\bar{1}$0\}$^\dag$ &2.66 &31.0 &8.20 &1.70 &0.331 &26.5 &-9.71 &-2.75 &4.36\\
O180\{001\} &0.70 &8.9 &8.95 &1.64 &0.331 &2.0 &-11.07 &-2.13 &4.36\\
O90 &0.72 &4.3 &4.26 &1.50 &0.234 &2.0 &-11.62 &-0.08 &12.97\\
O60 &3.62 &5.3 &1.09 &1.08 &0.166 &26.1 &-7.64 &12.14 &4.36\\
O120$^\dag$ &1.70 &13.7 &5.82 &1.47 &0.287 &10.2 &-10.82 &0.50 &4.92\\
\\
\hline
\multicolumn{10}{c}{Rhombohedral phase (118\,K)}\\
R180\{1$\bar{1}$0\}$^\dag$ &2.13 &36.0 &11.81 &1.83 &0.381 &18.3 &-9.53 &-3.64 &3.17\\
R180\{$\bar{2}$11\}$^\dag$ &2.13 &36.0 &11.81 &1.83 &0.381 &18.3 &-9.53 &-3.64 &3.17\\
R109$^\dag$ &0.70 &7.8 &7.81 &1.65 &0.311 &2.0 &-10.84 &-2.56 &5.60\\
R71 &0.74 &3.7 &3.52 &1.58 &0.220 &2.0 &-10.31 &-2.42 &17.94\\
\hline\hline
\end{tabular}
\end{table*}
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton4.eps}}
\caption{Dependence of correction factor in the expression for
domain wall energy density (25) as a function of the shape
coefficient $A$. Full points indicate values of $A$ for particular
domain walls considered in Table\,III.}\label{fig_integral}
\end{figure}
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton5.eps}}
\caption{Course of strain components along the domain wall normal
coordinate $s$ for the T90 domain wall. Indices refer to cubic axes
of the parent phase.}\label{fig_T90__e}
\end{figure}
\begin{figure*}
\centerline{
\includegraphics[width=17.6cm, clip=true]{FigMarton6.eps}}
\caption{ Temperature dependence of the thickness (a) and energy
density (b) for various mechanically compatible and neutral domain
walls of BaTiO$_3$ estimated within SPP approximation. The SPP
values for domain walls for which relaxing of the $P_{\rm t}$
component results in a lower-energy CPP solution (see in
Section\,VI.) are shown by dashed lines. Thickness of O180\{001\},
O90 has a very close temperature dependence. The results for T180
as well as for R180 walls are not distinguished by specific
orientations of the wall normal, since for both cases the angular
dependence of thickness and energy of SPP solutions on the domain
wall normal is negligible (see Tab.\,\ref{tab_numerical_results}).
Vertical lines mark phase-transition
temperatures.}\label{fig_sirka_energie}
\end{figure*}
Advantage of the GLD approach is that domain wall properties can
be obtained at any temperature. In
Table\,\ref{tab_numerical_results} we give numerical results for
domain wall parameters at one particular temperature for each of
the ferroelectric phases: at 298\,K, 208\,K, and 118\,K for the
tetragonal, orthorhombic, and rhombohedral phase, resp.
Corresponding numerical values of the spontaneous quantities
appearing in Table\,\ref{tab_boundary_conditions} are:
$P_0=0.265\,{\rm C\,m^{-2}}$,
$e_\parallel=Q_{11}P_0^2=7.77\times10^{-3}$ and
$e_\perp=Q_{12}P_0^2=-3.18\times10^{-3}$ (the tetragonal phase,
298\,K); $P_0=0.331\,{\rm C\,m^{-2}}$, $e_{\rm
a}=\frac{Q_{11}+Q_{12}}{2}P_0^2=3.58\times10^{-3}$, $e_{\rm
c}=Q_{12}P_0^2=-4.96\times10^{-3}$ and $e_{\rm
b}=\frac{Q_{44}}{4}P_0^2=0.79\times10^{-3}$ (the orthorhombic
phase, 208\,K); $P_0=0.381\,{\rm C\,m^{-2}}$, $e_{\rm
a}=\frac{Q_{11}+2Q_{12}}{3}P_0^2=0.97\times10^{-3}$ and $e_{\rm
b}=\frac{Q_{44}}{4}P_0^2=0.70\times10^{-3}$ (the rhombohedral
phase, 118\,K). The base vectors for the O60 domain wall (see
Eqn.\,\ref{eqn_O60rst_coord}) at 208\,K are
\begin{eqnarray}
{\bf r}_{\rm O60}&=&\left(0.707,0,0.707\right)\nonumber\\
{\bf s}_{\rm O60}&=&\left(0.701,0.130,-0.701\right)\nonumber\\
{\bf t}_{\rm
O60}&=&\left(-0.092,0.991,0.092\right)~.\label{eqn_O60rstnum_coord}
\end{eqnarray}
The right four columns of the Table\,\ref{tab_numerical_results}
contain corresponding numerical values of the domain wall
coefficients $g$, $a_{1}$, $a_{11}$, and $a_{111}$ derived from
GLD parameters and spontaneous order-parameter values using above
derived analytical expressions, mostly given explicitly in
Table\,\ref{tab_analytical_expressions}. The left part of the
Table\,\ref{tab_numerical_results} contains the key domain wall
properties such as wall thickness $2\xi$ and energy density
$\Sigma$.
Clearly, T90 and O60 domain walls are considerably broader then
others. T90 wall in the tetragonal phase is predicted to be
3.59\,nm thick at 298\,K (as already calculated in
Ref.\,\onlinecite{art_hlinka_marton_prb_2006}) and S-type O60
domain wall in the orthorhombic phase is predicted to be 3.62\,nm
thick at 208\,K. Since the wall thickness is greater than the
lattice spacing, the pinning of the walls is
weak\cite{art_ishibashi} and they can be easily moved. Moreover,
in case of O60 wall, the pinning is further suppressed due to
'incommensurate' character of Miller indices of the wall normal.
As follows from Eqn.\,(\ref{eqn_analytical-wall-thickness}),
domain wall thickness is determined by quantities $g$, $U$, and
$p_{\infty}$. By inspection of their values in
Table\,\ref{tab_numerical_results}, we see that the most important
factor is the coefficient $g$. Indeed, the domain walls R71, R109,
O90, O180\{001\}, T180 with $g=2\times10^{-11}~{\rm kg\,m^5
s^{-2}\,C^{-2}}$ are all very narrow (thickness below $1$\,nm),
and the thickness of various walls monotonically increases with
increasing value of $g$. Let us stress that coefficients $a_{1}$,
$a_{11}$, $a_{111}$, and $g$ depend on the direction of the domain
wall normal. For example, O180\{1$\bar{1}$0\} with
$g=26.5\times10^{-11}~{\rm kg\,m^5 s^{-2}\,C^{-2}}$ is almost four
times broader than O180\{001\} with $g=2\times10^{-11}~{\rm
kg\,m^5 s^{-2}\,C^{-2}}$.
In the absence of other constraints, the probability of appearance of domain
wall species should be determined by the surface energy density $\Sigma$.
Therefore, in the orthorhombic phase, the thinner O180$\{001\}$ wall is more
likely to occur than the 0180$\{1\bar{1}0\}$ one. Interestingly, the
O180$\{001\}$ wall has almost the same thickness as the O90 wall, while in the
tetragonal phase, it is the 90$^\circ$ wall which is much thicker than
180$^\circ$ wall (3.59\,nm compared to 0.63\,nm).
In general, the normal of the neutral 180$^\circ$-domain wall can take any
direction perpendicular to the spontaneous polarization of the adjacent domains.
Therefore also $\Sigma$ depends on the orientation of the wall normal (in the
$s-t$ plane). We have checked in the orthorhombic phase that the 0180$\{001\}$
and 0180$\{1\bar{1}0\}$ correspond to the extremes in the angular dependence of
$\Sigma$, which is monotonous between them. This strong angular dependence is
correlated with the anisotropy of the tensor $G_{ijkl}$. However, $g$ is
independent of the wall direction in the tetragonal and rhombohedral phase, and
also the variation of coefficients $a_{1}$, $a_{11}$, and $a_{111}$ is
insignificantly small, so that the effective directional dependence of
180$^\circ$ domain wall properties in these phases is negligible.
The values of the shape factor $A$ (see Table\,\ref{tab_numerical_results})
determines the deviation of the $P_{\rm r}(s)$ polarization profile of the
domain walls from the simple $tanh$ form. For all studied cases the values of
$A$ range between 1 and 2, where the correction factor $A^{5/2}I(A)$ appearing
in the Eqn.\,(\ref{correction_factor_sigma}) is almost linear function of $A$,
as it can be seen in Fig.\,\ref{fig_integral}. It means that the shape
deviations are much smaller than those shown by broken line in
Fig.\,\ref{fig_clanek_profile}. Unfortunately, in the case of broad walls T90
and O60, which are good candidates for study of the structure of the wall
central part, $A$ is almost one.
Eqn.\,(\ref{eqn_mechanical-equilibrium-constant}) can be also used to evaluate
local strain variation in the wall. In Fig.\,\ref{fig_T90__e} the profiles of
strain tensor components for T90 domain wall are shown for illustration.
Obviously, $e_{33}, e_{23}$, and $e_{13}$ strain components are strictly
constant and equal to their boundary values as it follows from mechanical
compatibility conditions. The $e_{11}$ and $e_{22}$ components and polarization
$P_{\rm r}$ vary between their spontaneous values. Let us stress that the
're-entrant' shear component $e_{12}$ (it has the same value in both adjacent
domains) approaches the non-zero value of about 5$\times 10^{-4}$ in the middle
of the domain wall.
The temperature dependence of the thickness and surface energy density of the 12
studied domain wall species is plotted in Fig.\,\ref{fig_sirka_energie}a and
Fig.\,\ref{fig_sirka_energie}b, respectively. Although some properties do vary
considerably, e.g. thickness of T90 wall in the vicinity of the
paraelectric-ferroelectric phase transition, the sequence of the thickness
values as well as the surface energy-density values of different domain wall
types remain conserved within each phase.
The temperature dependence of domain wall thickness follows the
trend given by Eqn.\,(\ref{eqn_analytical-wall-thickness}). It
increases with increasing temperature due to the dependence on
$p_\infty$ (see dependence of $U$ on $p_\infty$ in
Eqn.\,(\ref{eqn_energy_on_pinfty})). Such behavior is well known
also from the experimental observations.\cite{art_andrews_1986,
art_robert_1996, art_huang_jiang_hu_liu_JPCM_1996}
\section{\label{sec_cpp} CURVED POLARIZATION PATH SOLUTIONS}
\begin{figure} \centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton7.eps}}
\caption{Equipotential contours of the Euler-Lagrange potentials
showing ELPS associated with the $P_{\rm s}$=$P_{\rm
s}(\pm\infty)$ plane for a set of BaTiO$_3$ domain walls
considered in Table\,III. Polarization scale given for the bottom
left inset (in Cm$^{-2}$) is equally valid for all shown ELPS's.
Energetically most favorable domain wall solutions with
trajectories restricted to the $P_{\rm s}$=$P_{\rm s}(\pm\infty)$
plane are indicated with bold lines. The solutions corresponding
to a linear segment are denoted in the text as SPP walls, as
opposed to the CPP solutions (see in Section\,VI.), which have a
curved order-parameter trajectory.}\label{fig_rt_pot}
\end{figure}
So far we have investigated domain walls within the SPP
approximation, i.e. components $P_{\rm s}$ and $P_{\rm t}$, which
are the same in both domain states, were kept constant inside the
whole wall. This is quite usual assumption made for a
ferroelectric domain wall. Nevertheless, the full variational
problem, where all three components of $\bf P$ could vary along
$\bf s$ coordinate, leads in general to a lower energy solution
corresponding to a curved polarization path (CPP) in the 3D
primary order-parameter space. Appearance of the non-zero
're-entrant' components within a ferroelectric domain wall was
considered e.g. in works of
Refs.\,\onlinecite{art_huang_jiang_hu_liu_JPCM_1996,
art_hlinka_marton_prb_2006, art_lee_2009,
art_meyer_vanderbilt_2002}. For 180$^\circ$ walls, the
polarization profile associated with SPP is sometimes denoted as
the Ising-type wall. In contrast, the CPP solutions with non-zero
$P_{\rm s}$, $P_{\rm t}$ are often considered as N\'eel and
Bloch-like,\cite{art_lee_2009} even though, in contrast with
magnetism, the modulus of $\bf P$ is far from being conserved
along the wall normal $\bf s$.
We have previously considered non-constant $P_{\rm s}$ component of polarization
in the 90$^\circ$ wall with explicit treatment of electrostatic interaction and
realized that the deviations from SPP approximation are quite
negligible.\cite{art_hlinka_marton_prb_2006} In general, non-constant $P_{\rm
s}$ would lead to non-vanishing $\nabla \cdot \bf {\rm P}$ and finite local
charge density, which in a perfect dielectric causes a severe energy penalty.
The same situation is expected for all domain wall species. However, there is no
such penalty for non-constant $P_{\rm t}$-solutions. It was previously
argued\cite{art_huang_jiang_hu_liu_JPCM_1996} that the Bloch-like (with a
considerable magnitude of $P_{\rm t}$ at the domain wall center) solutions could
occur in orthorhombic BaTiO$_3$. Therefore, it is interesting to systematically
check for existence of such solutions using our model.
In order to study such Bloch-like solutions, we have calculated
Euler-Lagrange potential in the order-parameter plane $P_{\rm
s}=P_{\rm s}(\pm\infty)$ by integrating Euler-Lagrange equations
(Eqn.\,\ref{eqn_mechanical-equilibrium} and
\ref{eqn_mechanical-equilibrium-constant}) for all domain wall
species from Table\,\ref{tab_numerical_results} similarly as e.g.
in the Refs.\,\onlinecite{art_cao_cross_prb_1991,
art_huang_jiang_hu_liu_JPCM_1996}. Resulting 2D Euler-Lagrange
potential surfaces (ELPS) are displayed in Fig.\,\ref{fig_rt_pot}.
In each ELPS, the bold lines indicate numerically obtained
domain-wall solution with the lowest energy. The spatial step was
chosen as 0.1\,nm, $P_{\rm r}$ and $P_{\rm t}$ were fixed to
boundary conditions in sufficient distance from domain wall
(6\,nm) and initial conditions for $P_{\rm t}$ were chosen so that
the polarization path bypasses the energy maximum of the ELPS, and
the system was relaxed to the equilibrium.
Among the twelve treated wall species, there are six cases where only the SPP
solutions with $P_{\rm t}$=const exist: T180\{001\}, T180\{011\}, T90,
O180\{1$\bar{1}$0\}, O90, and R71. These solutions are clearly "Ising-like". In
all these cases, the ($P_{\rm r}=0$, $P_{\rm t}=0$) point is the only saddle
point of the ELPS. In the other six cases -- O180\{001\}, O60, O120,
R180\{1$\bar{1}$0\}, R180\{$\bar{2}$11\}, and R109 -- the ELPS has a maximum at
the ($P_{\rm r}=0$, $P_{\rm t}=0$), and the lowest energy solutions correspond
to curved polarization paths. This suggests that the previously discussed SPP
description may not necessarily be the proper approximation for these walls.
Nevertheless, in the case of O60 and O120 walls the deviations from the SPP
model are marginal, and only the remaining four solutions exhibit strong
Bloch-like behavior. Moreover, the energy differences between SPP and CPP
solutions were found to be almost negligible, except for R180, where the CPP
energy is by about 10\% lower in the entire temperature range of stability of
the rhombohedral phase. Therefore, it is quite possible that in the case of
O180\{001\}, R180\{1$\bar{1}$0\}, R180\{$\bar{2}$11\}, and R109 walls both
Bloch-like and Ising-like solutions may be realized.
The deviations from SPP in the case of the almost Ising-like O60 and O120 walls
are associated with the fact that ELPS is not symmetric with respect to $P_{\rm
t}=P_{\rm t}(\pm\infty)$ mirror plane. In these cases not only the polarization
path and wall energies, but also domain wall thicknesses of the SPP and CPP
counterpart solutions are very similar. This is demonstrated in
Fig.\,\ref{fig_p_profiles_os}, which shows polarization profiles of both SPP and
CPP solutions for the O60 wall.
Much more pronounced difference between domain wall profiles of SPP and CPP
solutions are found in case of O180\{001\}, R180\{1$\bar{1}$0\},
R180\{$\bar{2}$11\}, and R109 Bloch-like walls where the CPP trajectories bypass
the ($P_{\rm r}=0$, $P_{\rm t}=0$) maximum near the additional minima, which
originate from 'intermediate' domain states either of the same phase or even of
the different ferroelectric phases. In these cases, the inadequacy of SPP
approximation is obvious. For example, the CPP of R180\{$\bar{2}$11\} wall seems
to pass through a additional minimum corresponding to an 'orthorhombic'
polarization state (consult corresponding inset in the Fig.\,\ref{fig_plochy}).
As expected, the profile of such CPP solution deviates strongly from $tanh$
shape and even definition of the wall thickness would be problematic (see
Fig.\,\ref{fig_p_profiles_r180}).
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton8.eps}}
\caption{Predicted profiles of polarization components for O60
domain wall in BaTiO$_3$ at $T= 208\,$K. Full line stands for the
$P_{\rm r}$ component of polarization vector, broken line for the
$P_{\rm t}$ one. Analytically obtained SPP solution (a) practically
does not differ from the numerically obtained CPP
solution (b) with an unconstrained $P_{\rm t}$ component in
this case.}\label{fig_p_profiles_os}
\end{figure}
\begin{figure}
\centerline{
\includegraphics[width=8.6cm, clip=true]{FigMarton9.eps}}
\caption{Profiles of polarization components in R180\{1$\bar{1}$0\}
domain wall showing the SPP stationary trajectory (a) as well as the
lower energy Bloch-like solution (b), both for the model parameters
corresponding to $T= 118\,$K. Full line stands for the $P_{\rm r}$
component of polarization vector, broken line for the $P_{\rm t}$
one. The bottom panel demonstrates that Bloch solution may
considerable modify the domain domain wall
profile.}\label{fig_p_profiles_r180}
\end{figure}
\section{\label{sec_conclusion}CONCLUSION}
The work reports detailed study of mechanically compatible and electrically
neutral domain walls in BaTiO$_3$. The investigation was done within the
framework of the GLD model. Using the SPP approximation it was possible to
compare properties of various kinds of domain wall species from the same
perspective.
The phenomenological nature of the GLD model allowed to predict the temperature
dependence of the domain wall characteristics in the whole temperature range of
ferroelectric phases. Its continuous nature gave us even the opportunity to
deal conveniently with the non-crystallographic S-type domain wall, which has a
general orientation with respect to the crystal lattice, and which is therefore
difficult to cope with in discrete models relying on periodic boundary
conditions.
The S-wall in the orthorhombic, as well as the 90$^\circ$ wall in tetragonal
phase were both found to be about 4\,nm thick and consequently are expected to
be mobile, i.e. they could be easily driven by external fields, and they may
thus significantly contribute to the dielectric or piezoelectric response of the
material.
For several temperatures, we have numerically investigated domain walls allowing
for more complicated CPP solutions with non-constant $P_{\rm t}$. We have
identified solutions, which could be considered as analogues of Bloch walls
known from magnetism. Interestingly, in contrast with
Ref.\,\onlinecite{art_huang_jiang_hu_liu_JPCM_1996}, our model predicts the
Ising-type profile of the O180\{1$\bar{1}$0\} wall. At the same time the
Bloch-like structure of the O180\{001\} wall is predicted.
We believe that this kind of somewhat exotic walls actually represent important
generic examples of ferroelectric domain species, which should be anticipated in
all ferroelectrics with several equivalent domain states distinguished
simultaneously by the orientation of the spontaneous polarization and strain.
They should be certainly taken into account in investigations of domain-wall
phenomena in ferroelectric perovskites. At the same time, the energy differences
between the Bloch-like and Ising-like solutions are rather subtle here.
Therefore, in spite of the fairly good agreement for 180$^\circ$ and 90$^\circ$
domain walls between ab-initio calculations and predictions of this model in the
tetragonal phase,\cite{art_hlinka_marton_prb_2006} the preference for the
calculated Bloch-like trajectories may not necessarily reproduced for the domain
walls encountered in real BaTiO$_3$ crystal, since there is obviously a
considerable uncertainty in the adopted material-specific GLD parameters. In
addition, predictions for the domain walls with very small thickness must be
considered with a particular caution since the description of the sharp domain
wall profiles obviously touches the limits of the applicability of the
continuous model.
In conclusion, we have derived a number of qualitative and quantitative
predictions for mechanically compatible neutral domain walls of tetragonal,
orthorhombic and rhombohedral BaTiO$_3$ on the basis of the previously proposed
material-specific GLD model. We believe that the insight into the domain wall
properties mediated by the provided analytical and numerical analysis could be
helpful for understanding of domain wall phenomena in BaTiO$_3$ as well as in
some other intensively investigated members of the ferroelectric perovskite
family with same sort of macroscopic ferroelectric phases, for example in
KNbO$_3$, BiFeO$_3$, PbTiO$_3$ or even PZT and perovskite relaxor-related
materials.
\begin{acknowledgments}
Authors are grateful to Prof. V. Janovec for helpful comments and
critical reading of the manuscript. The work has been supported by
the Czech Science Foundation (Projects Nos. P204/10/0616 and
202/09/0430).
\end{acknowledgments}
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"redpajama_set_name": "RedPajamaArXiv"
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The logo design and promotional content for a non-profit organization that organizes concerts to bring more extensive music education to the St. Lucie County public school system.
Brochure was created to showcase the schedule and sponsorship information for BlueBird Educational Production's 2014/15 season.
Poster designs that were created to promote both a music festival in Fort Pierce, and music concerts at the Sunrise Theatre, Fort Pierce, Florida.
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"redpajama_set_name": "RedPajamaC4"
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SOL UPDATE – FLORIDA APPELLATE COURT INDICATES EXPANSION OF FIVE-YEAR LIMITATIONS PERIOD
Home Resources SOL UPDATE – FLORIDA APPELLATE COURT INDICATES EXPANSION OF FIVE-YEAR LIMITATIONS PERIOD
The Fifth DCA recently affirmed entry of a final judgment in favor of the bank finding the mortgage foreclosure action was not barred by the statute of limitations even though the bank filed its complaint more than five years after the borrower defaulted on the loan. Velden v. Nationstar Mortg., LLC.[i] In Velden, the bank filed a foreclosure complaint in July 2014 based on the borrower's failure to pay the February 2009 payment and all subsequent payments. The borrower moved for an involuntary dismissal based on the five-year statute of limitations.[ii] The lower court tried the matter and entered a final judgment of foreclosure in favor of the bank for the full amount of the unpaid note, plus interest from January 2009. The borrower appealed the judgment asserting the lower court erred by not dismissing the complaint based on the five-year statute of limitations. In the alternative, the borrower argued the court erred in awarding the bank amounts which accrued beyond the five-year limitations period.
The Fifth DCA affirmed the entry of judgment based on the fact that the bank alleged and proved the borrower remained in default after missing the February 2009 payment. The Court explained:
"Because the Bank alleged and proved missed payments within the five years prior to the filing of its complaint, its action was not barred by the statute of limitations." Velden, quoting itself in Klebanoff v. Bank of N. Y. Mellon[iii].
However, the Fifth DCA agreed with the borrower that the judgment should not include "amounts which accrued beyond the five-year limitations period." The Court cited several Florida courts which reached the same conclusion.[iv] The Fifth DCA remanded the matter to the trial court with instructions "to exclude [from the judgment]…damages for any defaults that occurred more than five years prior to the filing date of the current lawsuit."
However, there is indication that the law has not settled in this area and there is some judicial support for the position that the Florida Statute of Limitations does not exclude a future claim based on default of the subsequent month of that plead in the dismissed case. Judge Lambert offered an insightful concurring opinion wherein he agreed entry of judgment was proper and he conceded the Court properly followed binding precedent when it limited the award of damages, but he noted:
"[I]f I were writing on a clean slate, I would not exclude these sums from the judgment and would affirm the final judgment of foreclosure for the entire balance owed on the thirty-year note at issue."
Judge Lambert reasoned that the initial foreclosure lawsuit was dismissed without prejudice which placed the parties "back in their respective pre-acceleration positions." He also noted:
"that when the right to accelerate the debt for non-payment is optional with the
holder of the note, the statute of limitations does not run until the note is due."
Judge Lambert concluded that forbearance of the right to accelerate does "not constitute a waiver or defense against future collection of all sums due and owing under the note" and the bank "should not be deemed to have waived or forfeited its right to have included in the final judgment of foreclosure those monies owed for non-payments on the note that are more than five years from the filing of the lawsuit based on the statute of limitations defense."[v]
Judge Lambert's concurring opinion is consistent with the concurring opinion in the Florida Supreme Court case of Bollettieri v. Bank of New York.[vi] Although Judge Lambert provided insightful and well-reasoned logic in his concurring opinion, the Fifth DCA's opinion in Velden makes it clear that a bank seeking to foreclose and collect the full amount of the note should continue to file their complaint within five years of the initial default. As the issue evolves, it is reasonable to presume the Florida Supreme Court will eventually have the opportunity to consider adopting the concurring opinions into precedential law.
[i] 5D16-3628, 2018 Fla. App. LEXIS 359, 1-2 (5th DCA Jan. 12, 2018).
[ii] § 95.11(2)(c), Fla. Stat. 2017.
[iii] Klebanoff v. Bank of N. Y. Mellon, 228 So. 3d 167,168-69 (Fla. 5th DCA 2017).
[iv] U.S. Bank, N.A. v. Diamond, 228 So. 3d 177, 178 (Fla. 5th DCA 2017); U.S. Bank National Association v. Bartram, 140 So. 3d 1007 (Fla. 5th DCA 2014); Kaan v. Wells Fargo Bank, 981 F. Supp. 2d 1271, 1274 (S. D. Fla. 2013); Greene v. Bursey, 733 So. 2d 1111 (Fla. 4th DCA 1999); Cent. Home Tr. Co. of Elizabeth v. Lippincott, 392 So. 2d 931 (Fla. 5th DCA 1991).
[v] Velden v. Nationstar Mortg., LLC, No. 5D16-3628, 2018 Fla. App. LEXIS 359, at *6 (5th DCA Jan. 12, 2018).
[vi] See November e-Blast.
SERVICERS' BOARDING PROCEDURES | Key PointsSOL UPDATE | Key Points
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Rails.application.config.session_store :cookie_store, key: '_react-commentsbox-rails_session'
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"redpajama_set_name": "RedPajamaGithub"
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Narcos and Vikings, the slot machine projects announced by Netent at ICE 2018
Posted on February 8, 2018 by Admin in Gambling
Netent announced on the sidelines of the London ICE conference this year's releases of the Vikings and Narcos slot machines, both from the two TV series of the same name. After having transcribed in slots of blockbusters such as Aliens ™ or Planet of the Apes ™, Netent revealed to want to focus on TV series, aware that some of them are more popular than most movies. However, Netent does not give up the films since it has planned to release in 2018 the Jumanji ™ slot machine.
The Vikings series dear to Netent
Netent is a Swedish game developer; to take out a slot machine on the Vikings series is something that was important to the company, especially to its CEO Per Eriksson. He told the London ICE conference that he was delighted with this particular project because "coming from Scandinavia, this project is special ".
Vikings is an Irish-Canadian series depicting the story of a group of Vikings led by the legendary warrior Ragnar Lothbrok, who was the first to attack the French and English sides to plunder Western riches. His name has allowed the Vikings to be feared all over Europe for centuries.
Released since 2013, it has millions of fans around the world and will be one of Netent's innovations for the current 2018. The firm announced its next release this week during ICE 2018, as well as that of another very awaited slot machine: Narcos.
Narcos is about to become a slot machine
Just before announcing the Vikings Slot Machine, Netent unveiled its plan to make a slot on the Narcos series, Netflix's top-rated series. Narcos is an event series about the character of Pablo Escobar, the richest drug trafficker in history.
Narcos traces the history of the Cartel of Medellin and Pablo Escobar for seasons 1 and 2, then the Cartel of Cali for season 3. Narcos is with only three seasons one of the most popular series of all time and its online slot machine is already eagerly awaited by many players.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
Quantum key distribution (QKD) \cite{Bennett1984Quantum,Ekert1991Quantum} is the most successful application in quantum information science, whose security was proved at the end of the last century \cite{Mayers2001Unconditional,Lo1999Unconditional,Shor2000Simple}. Since then, there has been a tremendous interest in developing this quantum technology for real-life applications, starting from the first 32-cm demonstration in the early 1990s \cite{bennett1992experimental} to the recent satellite QKD over 1200 km \cite{liao2017satellite}. In these implementations, photons are used as information carriers, owing to their fast transmission speed and robustness against decoherence from the environment. Also, optical quantum communication can be easily integrated with the current telecommunication network infrastructure.
Now, the transmission loss of photons has become a major obstacle in practical implementations. The quantum channel transmission efficiency is characterized by the transmittance $\eta$, defined as the probability of a photon being successfully transmitted through the channel and being detected. For most of the current implemented schemes, such as the well-known Bennet-Brassard 1984 (BB84) protocol \cite{Bennett1984Quantum}, single-photon sources\footnote{In practice, single-photon sources are often replaced with weak coherent state sources or heralded single-photon sources. Nevertheless, only the single-photon components are used for secure key distribution.} are employed for key information encoding. Since the photon carries the quantum information, when it is lost in the channel, no secure key can be distributed. Thus, the transmittance $\eta$ becomes a natural upper bound of the key generation rate. A more strict derivation shows a linear key-rate bound with respect to the transmittance \cite{Curty2004Entanglement,takeoka2014fundamental,Pirandola2017Fundamental}, $R\le O(\eta)$. Since the transmittance $\eta$ decays exponentially with the communication distance in the fiber-based network, this linear key-rate bound severely limits the key generation rate.
The following two approaches to overcome this rate limit have been considered: quantum repeaters \cite{EntSwap1993,Briegel1998Repeater,azuma2015all} and trusted relays. Unfortunately, using quantum repeater schemes with current technology is infeasible because they require high-quality quantum memory and complicated local entanglement distillation operations. The trusted-relay approach, however, relies on the assumption that the quantum relays between two users are trustworthy, which is difficult to ensure or verify practically; this severely undermines the primary goal of QKD, i.e., security. In 2012, the measurement-device-independent quantum key distribution (MDI-QKD) scheme was proposed to close all the detection loopholes \cite{Lo2012Measurement}, which enhances the security of a practical QKD system. Nevertheless, the key rate of the MDI-QKD scheme is still bounded by $O(\eta)$. Therefore, the linear key-rate bound \cite{takeoka2014fundamental,Pirandola2017Fundamental} was widely believed to hold for practical QKD systems without relays.
Significant efforts have been devoted to improve the key rate by proposing different schemes. Recently, Lucamarini\textit{~et~al.} proposed a novel phase-encoding QKD protocol, called twin-field quantum key distribution (TF-QKD) \cite{Lucamarini2018TF}, which shows the possibility to overcome the key-rate limit and make a quadratic improvement over phase-encoding MDI-QKD \cite{Tamaki2012PhaseMDI}. In both schemes, single-photon detection is used, whereas coincident detection is required in other MDI-QKD schemes \cite{Lo2012Measurement,Ma2012Alternative}. From technical point of view, the single-photon detection is the key reason for the quadratic improvement. Unfortunately, a rigorous security proof is still missing at the moment. In fact, as shown later, the widely used photon number channel model \cite{Ma2008PhD} used in the security proof of MDI-QKD is proven to be invalid for this kind of setting.
Following the TF-QKD scheme \cite{Lucamarini2018TF}, we investigate phase-encoding MDI-QKD schemes with single detection \cite{Tamaki2012PhaseMDI} and propose a phase-matching quantum key distribution (PM-QKD) scheme that can surpass the linear key-rate bound, inspired by the relative-phase-encoding Bennett-1992 \cite{Bennett1992Quantum}, phase-encoding MDI-QKD \cite{Tamaki2012PhaseMDI,Ma2012Alternative}, and passive differential-phase-shift QKD \cite{Guan2015RRPDS}. The two communication parties prepare two coherent states independently, encode the key information onto the common phase, and match phases via interference detection at an untrusted measurement site. Details are given in Sec. \ref{Sc:protocol}. By developing an optical-mode-based security proof, we show that the key rate of the proposed scheme scales with the square root of the transmittance, $R=O(\sqrt{\eta})$, in Sec. \ref{Sc:security} with technical details given in Appendix \ref{Sc:SecureProof}. Also, the proposed phase-matching scheme falls into the MDI framework, which is immune to all possible detection attacks. Our security proof can be directly applied to TF-QKD. In Sec. \ref{Sc:practical}, we deal with related practical issues and develop a phase postcompensation technique to ease the experimental requirements. In Sec. \ref{Sc:simulation}, we simulate the key-rate performance of PM-QKD and compare it to former QKD protocols, with all the practical factors taken into account. Finally, in Sec. \ref{Sc:outlook}, we discuss possible future work directions.
\section{PM-QKD protocol}\label{Sc:protocol}
In PM-QKD, the two communication parties, Alice and Bob, generate coherent state pulses independently. For a $d$-phase PM-QKD protocol, Alice and Bob encode their key information $\kappa_a, \kappa_b \in \{0, 1, \dots, d-1\}$, into the phases of the coherent states, respectively, and send them to an untrusted measurement site that could be controlled by Eve, as shown in Fig.~\ref{fig:PM}(a). Eve is expected to perform interference detection. Define a successful detection as the case where one and only one of the two detectors clicks, denoted by $L$ click and $R$ click. This interference measurement would match the phases of Alice and Bob's signals. Conditioned on Eve's announcement, Alice and Bob's key information is correlated.
In this work, we mainly focus on PM-QKD with $d=2$ and phase randomization. That is, Alice and Bob add extra random phases on their coherent state pulses before sending these pulses to Eve. After Eve's announcement, Alice and Bob announce the extra random phases and postselect the signals based on the random phases. This PM-QKD scheme is detailed below and shown in Fig.~\ref{fig:PM}(b). For simplicity, by using the name ``PM-QKD'' in the text below, we refer to the case of $d=2$ plus phase randomization.
\begin{itemize}[label={}]
\item
\textbf{State preparation}: Alice randomly generates a key bit $\kappa_a$ and a random phase $\phi_a\in[0,2\pi)$ and then prepares the coherent state $\ket{\sqrt{\mu_a}e^{i(\phi_a+\pi\kappa_a)}}_{A}$. Similarly, Bob generates $\kappa_b$ and $\phi_b\in[0,2\pi)$ and then prepares $\ket{\sqrt{\mu_b}e^{i(\phi_b+\pi\kappa_b)}}_{B}$.
\item
\textbf{Measurement}: Alice and Bob send their optical pulses, $A$ and $B$, to an untrusted party, Eve, who is expected to perform an interference measurement and record the detector ($L$ or $R$) that clicks.
\item
\textbf{Announcement}: Eve announces her detection results. Then, Alice and Bob announce the random phases $\phi_a$ and $\phi_b$, respectively.
\item
\textbf{Sifting}: Alice and Bob repeat the above steps many times. When Eve announces a successful detection, (a click from exactly one of the detectors $L$ or $R$), Alice and Bob keep $\kappa_a$ and $\kappa_b$ as their raw key bits. Bob flips his key bit $\kappa_b$ if Eve's announcement was an $R$ click. Then, Alice and Bob keep their raw key bit only if $|\phi_a-\phi_b| = 0$ or $\pi$. Bob flips his key bit $\kappa_b$ if $|\phi_a-\phi_b| = \pi$.
\item
\textbf{Parameter estimation}: For all the raw data that they have retained, Alice and Bob analyze the gains $Q_{\mu}$ and quantum bit error rates $E^Z_{\mu}$. They then estimate $E^{X}_{\mu}$ using Eq.~\eqref{eq:Emu}.
\item
\textbf{Key distillation}: Alice and Bob perform error correction and privacy amplification on the sifted key bits to generate a private key.
\item
Notations: Denote a coherent state in mode $A$ to be $\ket{\sqrt{\mu}e^{i\phi}}_A$, where $\mu$ is the intensity and $\phi$ is the phase; $\mu_a = \mu_b = \mu/2$; Alice's (Bob's) key bit $\kappa_{a(b)}\in\{0,1\}$; total gain $Q_\mu$; phase error rate $E_\mu^X$; and bit error rate $E_\mu^Z$.
\end{itemize}
\begin{figure}[htbp]
\centering
\resizebox{8cm}{!}{\includegraphics{ProGPEsecurity3.pdf}}
\caption{
(a) Schematic diagram of PM-QKD. Alice generates a coherent state, $\ket{\sqrt{\mu_a}e^{2\pi i\kappa_a/d}}$, where $\kappa_a\in \{0, 1, \dots, d-1\}$. Similarly, Bob generates $\ket{\sqrt{\mu_b}e^{2\pi i\kappa_b/d}}$. Alice and Bob send the two coherent states which interfere at an untrusted measurement site. (b) Schematic diagram of PM-QKD with $d=2$ plus phase randomization. Alice prepares $\ket{\sqrt{\mu_a}e^{i(\phi_a+\pi\kappa_a)}}$ and Bob prepares $\ket{\sqrt{\mu_b}e^{i(\phi_b+\pi\kappa_b)}}$. The two coherent states interfere at an untrusted measurement site. If the phase difference $|(\phi_a+\pi\kappa_a)- (\phi_b+\pi\kappa_b)|$ is $0$, detector $L$ clicks; if the phase difference is $\pi$, detector $R$ clicks. After Eve announces her measurement result, Alice and Bob publicly announce $\phi_a$ and $\phi_b$. (c) Equivalent scenario for the postselected signals with $\phi_a = \phi_b$. A trusted party (Charlie) prepares $\ket{\Psi}_{C}$, splits it and sends it to both Alice and Bob. Without loss of generality, we consider the case where Alice and Bob both modulate this by the same phase $0$ or $\pi$ to create the systems $A$ and $B$. If $\ket{\Psi}_{C}$ only contains odd- or even-photon number components, we can see that $\ket{\Psi_0} = \ket{\Psi_\pi}$.
}
\label{fig:PM}
\end{figure}
The above implementation of the PM-QKD protocol clearly resembles the phase-encoding MDI-QKD protocol \cite{Ma2012Alternative,Tamaki2012PhaseMDI}, where the key bits are encoded in the relative phase of two coherent pulses (the reference and signal pulses).
However, in the PM-QKD protocol, the reference pulse can be regarded as being shared by Alice and Bob. Therefore, they no longer need to send the reference pulse, and the key becomes the global phase of the coherent signal pulses. Another significant difference between PM-QKD and the former phase-encoding MDI-QKD scheme is that no basis switching is required. In this respect, it resembles the Bennett-1992 \cite{Bennett1992Quantum} and passive DPS \cite{Guan2015RRPDS} QKD protocols. Note that, a similar proposal named ``MDI-B92'' protocol has been proposed by Ferenczi in 2013 \cite{Ferenczi2013}. With the decoy-state method, the quantum part of PM-QKD would be similar to that of the TF-QKD protocol without basis sifting.
\section{Security of PM-QKD} \label{Sc:security}
To provide an intuitive understanding of the manner in which PM-QKD works, we demonstrate its security by considering an equivalent scenario shown in Fig.~\ref{fig:PM}(c). Here, a trusted party (Charlie) prepares a pure state $\ket{\Psi}_{C}$, splits it using a $50$--$50$ beam splitter, and sends it to Alice and Bob separately. Alice and Bob encode their key information $\kappa_a$ and $\kappa_b$ into systems $A$ and $B$ by modulating the phases, and then they send these to Eve who is supposed to tell whether $|\kappa_a-\kappa_b|=0$ or 1. Thus, the four possible output states that could be sent to Eve can be expressed as $\ket{\Psi_{0, 0}}$, $\ket{\Psi_{0,\pi}}$, $\ket{\Psi_{\pi, 0}}$, and $\ket{\Psi_{\pi, \pi}}$.
Without the loss of generality, we consider the following case in which both encoded key bits are the same, $\kappa_a = \kappa_b$. Eve attempts to learn the key bit $\kappa_{a(b)}$ from the state sent to her, which is either $\ket{\Psi_{0, 0}}$ or $\ket{\Psi_{\pi, \pi}}$. Here, the phase, controlled by $\kappa_a$ and $\kappa_b$ and modulated into $A$ and $B$, has become the ``global phase'' of the combined system $A$ and $B$. If $\ket{\Psi}_C$ is a Fock state $\ket{k}_C$ with $k$ photons, then $\ket{\Psi_{0,0}} = \ket{\Psi_{\pi,\pi}}$, since the global phases of Fock states are meaningless. In this case, Eve cannot tell in principle whether the modulated phases are $0$ or $\pi$ and can only learn that $\kappa_a = \kappa_b$.
In our PM-QKD protocol, both Alice and Bob transmit weak coherent pulses, $\ket{\sqrt{\mu_{a}}e^{i(\phi_a+\pi\kappa_a)}}_{A}$ and $\ket{\sqrt{\mu_{b}}e^{i(\phi_b+\pi\kappa_b)}}_{B}$ to Eve. The phase sifting condition $\phi_a = \phi_b =\phi$ is equivalent to imagining that Charlie employs a source state of $\ket{\Psi}_{C} = \ket{\sqrt{\mu} e^{i\phi}}_{C}$ in Fig.~\ref{fig:PM}(b). For a phase-randomized state $\ket{\sqrt{\mu} e^{i\phi}}_{C}$, it is equivalent for Charlie to prepare a Fock state $\ket{\Psi}_{C} = \ket{k}_{C}$ with a probability of $P(k) = e^{-\mu} \mu^k/k!$. Thus, the PM-QKD protocol is secure if Eve cannot learn the phase $\phi$. However, in the real PM-QKD protocol, the phase $\phi$ will eventually be announced during the sifting process. When this happens, Charlie's source $\ket{\Psi}_C$ can no longer be regarded as combinations of different photon-number states $\ket{k}_C$. The key challenge of the security proof of PM-QKD lies in the fact that the quantum source cannot be regarded as a mixture of photon number states, after Alice and Bob announce the phases, $\phi_a$ and $\phi_b$. That is, the photon number channel model \cite{Ma2008PhD} and the ``tagging'' method used in the security proof by Gottesman et al.~\cite{gottesman04} (we will refer it as GLLP security proof) can no longer be applied. In Appendix~\ref{sc:attack}, a beam-splitting attack is proposed to show that the GLLP formula is incorrect after the phase announcement. Therefore, one cannot simply reduce a randomized-phase coherent state protocol to a single-photon-based protocol.
Our security proof of PM-QKD is based on analyzing the distillable entanglement of its equivalent entanglement-based protocol. Following the Shor-Preskill security argument \cite{Shor2000Simple}, the key rate of PM-QKD protocol (for the sifted signals) is given by
\begin{equation} \label{eq:keyrate}
\begin{aligned}
r_{PM} &\ge 1 - H(E_{\mu}^Z) - H(E_{\mu}^{X}), \\
\end{aligned}
\end{equation}
where $E^Z_\mu$ is the quantum bit error rate (QBER) that can be directly estimated in the experiment; $E^X_\mu$ is the phase error rate, which reflects the information leakage; and $H(x)=-x\log_2x-(1-x)\log_2(1-x)$ is the binary Shannon entropy function. We demonstrate in Appendix \ref{Sc:SecureProof} that $E^X_\mu$ can be bounded by
\begin{equation} \label{eq:Emu}
\begin{aligned}
E_{\mu}^X &\le e^Z_0q_0 +\sum_{k=0}^\infty e_{2k+1}^Z q_{2k+1} + (1 - q_0 - \sum_{k=0}^\infty q_{2k+1}). \\
\end{aligned}
\end{equation}
Here, $q_k$ is the estimated ratio of the ``$k$-photon signal'' to the full detected signal:
\begin{equation} \label{eq:qk}
\begin{aligned}
q_k &= \frac{(e^{-\mu} \mu^k/k!)Y_k }{Q_\mu}, \\
\end{aligned}
\end{equation}
where $Q_\mu$ is the total gain of the pulses, and $Y_k$ and $e^Z_k$ are the yield and bit error rate, respectively, if Charlie's light source is a $k$-photon number state. Alice and Bob can estimate the yield and bit error rate via the decoy-state method \cite{Hwang2003Decoy,Lo2005Decoy,Wang2005Decoy}. Note that the parameters $Y_k$ and $ e^Z_k$ can still be used to characterize Eve's behavior even though the source is not actually a combination of photon-number states.
Unlike most of the existing security analysis of discrete-variable QKD, our analysis is not single-qubit based. For a long time, the sources in QKD implementations, such as weak coherent sources and spontaneous parametric down-conversion sources, have been fabricated as an approximation of single qubit, following the BB84 protocol \cite{Bennett1984Quantum}. Here, we show the security of PM-QKD with a coherent light source by directly applying the Lo-Chau entanglement distillation argument \cite{Lo1999Unconditional} on analyzing the optical modes. This technique could be helpful for both a new QKD scheme design and security analysis.
In the equivalent scenario considered above, shown in Fig.~\ref{fig:PM}(b), a trusted party Charlie is introduced. We need to emphasize that the virtual Charlie will be removed in the real implementation in Sec. \ref{Sc:practical}. If Charlie does exist, Eve may inject some probes after Charlie's outputs, and then she measures them at the output of Alice and Bob to learn their operations. This is the main problem of detection-device-independent QKD \cite{lim2014detector,gonzalez2015quantum}. In the PM-QKD protocol shown in Fig.~\ref{fig:PM}(a), Alice and Bob can simply isolate their light source and modulators in an optical circulator to prevent such Trojan-horse-like attacks. Hence, the PM-QKD scheme, like other MDI-QKD schemes, is secure against Trojan-horse-like attacks.
\section{Practical implementation}\label{Sc:practical}
Now we address a few practical issues. In the protocol shown in Fig.~\ref{fig:PM}, Alice and Bob only retain their signals when their announced phases, $\phi_a$ and $\phi_b$ are either exactly the same or with a $\pi$ difference. However, since the announced phases are continuous, the successful sifting probability tends to zero. Moreover, we assume that Alice's and Bob's laser sources are perfectly locked, such that their phase references meet, but it is very challenging in practice to achieve such phase locking.
To address these practical issues, we employ a phase postcompensation method \cite{Ma2012Alternative}, where Alice and Bob first divide the phase interval $[0,2\pi)$ into $M$ slices $\{\Delta_j\}$ for $0\le j \le M-1$, where $\Delta_j = [2\pi j/M, 2\pi (j+1)/M)$. Instead of comparing the exact phases, Alice and Bob only compare the slice indexes. This makes the phase-sifting step practical, but introduces an intrinsic misalignment error. Also, Alice and Bob do not perform the phase-sifting immediately in each round, and instead, they do it in data postprocessing. In the parameter estimation step, they perform the following procedures, as shown in Fig.~\ref{fig:phasesel}.
\begin{enumerate}
\item
For each bit, Alice announces the phase slice index $j_a$ and randomly samples a certain amount of key bits and announces them for QBER testing.
\item
In the phase postcompensation method, given an offset compensation $j_d\in \{0,1,...,M/2 -1\}$, Bob sifts the sampled bits with the phase sifting condition $|j_b + j_d - j_a| \mod M = 0$ or $M/2$. For the case of $M/2$, Bob flips the key bit $\kappa_b$. After sifting, Bob calculates the QBER $E^Z$ with Alice's sampling key bits. Bob tries all possible $j_d\in\{0,1,\cdots,M-1\}$, and figures out the proper $j_d$ to minimize the sampling QBER. Using the phase sifting condition with the proper $j_d$, Bob sifts (and flips if needed) the unsampled bits and announces the locations to Alice. Alice sifts her key bits accordingly.
\item
Alice and Bob analyze the overall gain $Q_{\mu_i}$ and QBER $E^Z_{\mu_i}$ for different values of intensities $\mu_a =\mu_b =\mu_i/2$. They estimate the phase error rate $E^{(X)}_{\mu}$ by Eq.~\eqref{eq:Emu}.
\end{enumerate}
Here, in the phase postcompensation step, Bob does not need to fix a $j_d$ for the whole experiment. Bob can group the raw data into data blocks, and then he is able to adjust the offset $j_d$ for different data blocks. Alternatively, Bob can also adjust $j_d$ in real time based on a prediction via data-fitting samples nearby. A detailed description of the operations and security arguments is presented in Appendix \ref{Sc:SecurePost}. We emphasize that the fluctuation of phases will only introduce additional bit errors, but not affect the security.
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{phasesel.pdf}
\caption{Phase postcompensation. Without loss of generality, here we consider the case $\kappa_a = \kappa_b$. Denote the phase references of Alice and Bob as $\phi_{a0}$ and $\phi_{b0}$, and hence the reference deviation is $\phi_0=\phi_{b0}-\phi_{a0} \mod 2\pi$. Bob can figure out the proper phase compensation offset $j_d$ by minimizing the QBER from random sampling as follows. Bob sets up a $j_d$, sifts the bits by $|j_b + j_d - j_a| = 0$, and evaluates the sample QBER. He tries all possible $j_d\in\{0,1,\cdots,M-1\}$ and figures out the proper $j_d$ to minimize the sample QBER. Then, he announces the sifted locations of unsampled bits to Alice. As shown in the figure, we set $M=12$, and the reference deviation $\phi_0= 70^\circ$; hence Bob can set $j_d =2$ to compensate the effect of $\phi_0$.
} \label{fig:phasesel}
\end{figure}
An alternative method to phase postcompensation is that, Alice and Bob can generate strong pulses independent of quantum signals for phase calibration. The two pulses interfere at the measurement site so that Alice and Bob can identify the phase fluctuations between two channels. Note that the strong pulses can be set in an optical mode slightly deviated from the quantum signals to reduce cross-talks. According to the phase difference measured from the strong calibration pulses, Alice and Bob can estimate the offset $j_d$ accurately. This is a feasible way to replace the phase postcompensation method. The main difference of this phase calibration method from the usual phase-locking method \cite{santarelli1994heterodyne} is that, no active feedback is required. Alice and Bob learn the phase difference only for phase sifting in data postprocessing.
Note that similar ideas of phase postcompensation and phase calibration have already been adopted in some continuous-variable QKD protocols, such as the Gaussian-modulated coherent-state protocol \cite{Qi2007experimental,Qi2015generating} and the self-referenced protocol \cite{Soh2015self}.
Another important issue is how Alice and Bob set random phases $\phi_a, \phi_b$. In practice, it would be experimentally challenging to continuously set an accurate phase to a coherent state. As discussed above, they only need to set the slice indexes $j_a, j_b$ instead of exactly modulating phases. There are two methods to achieve this.
\begin{itemize}
\item
Alice and Bob first generate strong laser pulses with (unknown) randomized phases, either by turning on/off the lasers or active phase randomization. They then split the pulses and apply homodyne detection on one beam to measure the phase $\phi_a, \phi_b$ accurately enough to determine the slice indexes $j_a, j_b$. They use the rest beam for further quantum encoding.
\item
Alice and Bob first generate pulses with stable phases. Then, they actively randomize the phase using phase modulators. Alice and Bob can record the corresponding random numbers as for slice indexes $j_a, j_b$. Note that the phase randomization can be discrete, which has been proven to be secure and efficient with a few discrete phases \cite{zhu2015discrete}.
\end{itemize}
In order to optimize the estimated phase slice shift $j_d$ for minimizing the bit error rate, the phase compensation or calibration should be resettled with respect to phase drift. In practice, there are two major factors which may cause phase drift. One is the laser linewidth $\Delta \nu$, which causes a dispersion effect on the output pulse. The phase varies randomly with respect to the coherent time $T_{coh} \approx (\Delta \nu)^{-1}$. To alleviate the dispersion, a CW laser source with a long coherence time should be employed. The other major factor for phase drift is the variation of optical path length $\Delta L$. In a recent work of TF-QKD \cite{Lucamarini2018TF}, Lucamarini\text{~et~al.} experimentally tested the phase drift in a MDI setting. The results show that the phase drift follows a Gaussian distribution with zero mean and a standard deviation of about $6.0$ rad ms$^{-1}$ for a total distance of $550$ km. To enhance the performance of PM-QKD protocol, former works on phase stabilization of optical fibers \cite{Droste2013optical,Carvacho2015postselection,Lipka2017optical} can be employed.
With all the practical factors taken into account, the final key-rate formula can be expressed as
\begin{equation} \label{eq:keyRate}
\begin{aligned}
R_{PM} &\ge \dfrac{2}{M}Q_\mu [ - f H(E_{\mu}) + 1 - H(E_{\mu}^X)], \\
\end{aligned}
\end{equation}
where the phase error rate $E_{\mu}^X$ is given by Eq.~\eqref{eq:Emu}, $2/M$ is the sifting factor and $f$ is the error correction efficiency.
\section{Simulation results} \label{Sc:simulation}
We simulate the performance of PM-QKD with the parameters given in Fig.~\ref{fig:simulation}(b), assuming a lossy channel that is symmetrical for Alice and Bob. The dark count rate $p_d$ is from Ref. \cite{Tang2014MDI200}, and the other parameters are set to be typical values. The simulation formulas for $Q_\mu, E^Z_\mu$ and $E^X_\mu$ of PM-QKD are given in Eqs.~\eqref{eq:Qmu}, ~\eqref{eq:Emusimu}, and ~\eqref{eq:EmuXzoom}, respectively, in Appendix \ref{Sc:GainQBER}. The simulation formulas for BB84 and MDI-QKD are listed in Appendix \ref{Sc:Others}.
The simulation results are shown in Fig.~\ref{fig:simulation}(a). From the figure, one can see that PM-QKD is able to exceed the linear key-rate bound when $l> 250$ km with practical settings such as dark counts, misalignment errors, and sifting factors. Compared with MDI-QKD, PM-QKD can achieve a longer transmission distance of $l=450$ km and the key rate is increased by approximatedly $4\sim6$ orders of magnitude when $l>300$ km. Moreover, if we set up a cutoff line of key rate $R = 10^{-8}$ as real-life consideration, then the longest practical transmission distance of PM-QKD is over $400$ km, whereas the ones of BB84 and MDI-QKD are all lower than $250$ km.
Several QKD schemes are compared in Fig.~\ref{fig:simulation}(c). The comparison shows that the PM-QKD scheme outperforms the existing protocols in the following aspects. First, the PM-QKD scheme has a quadratic improvement of key rate, $O(\sqrt{\eta})$. In the security aspect, PM-QKD enjoys the measurement-device-independent nature that is immune to all detection attacks. In the practical aspect, it removes the requirement of the basis switch, which can simplify the experiment apparatus and reduce the randomness consumption.
Here, we would like to clarify that the linear key-rate bound by Pirandola et al.~\cite{Pirandola2017Fundamental} is derived for point-to-point QKD protocols. In the PM-QKD or other MDI-QKD schemes, there is an untrusted relay held by Eve. The quadratic improvement in PM-QKD seems unsurprising if we regard the untrusted middle node as a quantum repeater.
\begin{figure*}
\centering
\includegraphics[width=16cm]{simulationComb2.pdf}
\caption{(a) Simulation of our PM-QKD protocol. For the considered simulation parameters, the key rate of PM-QKD surpasses that of the conventional BB84 protocol when $l>120$ km and it exceeds the linear key-rate bound by Pirandola et al.~\cite{Pirandola2017Fundamental} when $l>250$ km. In addition, our protocol is also able to achieve a long transmission distance of $l=418$ km. (b) Parameters used for simulation. (c) Comparison of different QKD protocols: PM-QKD; TF-QKD \cite{Lucamarini2018TF}; MDI-QKD \cite{Lo2012Measurement}; passive differential phase-shift (DPS) QKD \cite{Guan2015RRPDS}; Bennett-1992 (B92) QKD \cite{Bennett1992Quantum}. Key rate: dependence of the key rate on the channel transmittance $\eta$; Detection loophole: whether the protocol is immune to all detection loopholes; Qubit based: whether the security analysis is based on the single-qubit case; Basis switch: whether the source (measurement device) must prepare (measure) states in complementary bases; Phase locking: whether the protocol requires a fixed phase reference frame for the two users. *Our security proof and performance analysis apply to TF-QKD if the basis information $X$, $Y$ is ignored.}
\label{fig:simulation}
\end{figure*}
Note that the security of recently proposed TF-QKD protocol \cite{Lucamarini2018TF} can be reduced to the security of PM-QKD protocol if the information for the two bases $X$, $Y$ is ignored in TF-QKD It remains an open problem whether we can reach a higher key rate by taking advantage of the basis information together.
\begin{comment}
\begin{figure*}
\begin{tcolorbox}[colback=white,colframe=white, equal height group=nobefaf,width=78mm,nobeforeafter,arc=0mm, left=0mm, right=0mm, top=0mm, bottom=0mm, boxrule=0pt,halign=flush center]
\tcbox[left=0mm, right=0mm, top=0mm, bottom=0mm, boxsep=0mm,boxrule=0pt, arc=0mm,colback=white, colframe=white]{
\centering
\includegraphics[width=8cm]{simulationPy.pdf}
}
(a)
\end{tcolorbox}
\begin{tcolorbox}[colback=white,colframe=white, equal height group=nobefaf,width=95mm,nobeforeafter,arc=0mm, left=0mm, right=0mm, top=0mm, bottom=0mm, boxrule=0.3pt,halign=flush center]
\vskip 3mm
\tcbox[left=0mm, right=0mm, top=0mm, bottom=0mm, boxsep=0mm,boxrule=0pt, arc=0mm,colback=white, colframe=white, fontupper = \small]{
\begin{tabular}{cccccc}
\hline
Parameters & Values \\
\hline
Dark count rate $p_d$ & $7.2\times 10^{-8}$ \\
Error correction efficiency $f$ & $1.15$ \\
Detector efficiency $\eta_d$ & $14.5\%$ \\
Number of phase slices $M$ & $16$ \\
Misalignment error $e_d$ & $1.5\%$ \\
\hline
\end{tabular}
}
(b)
\vskip 3mm
\tcbox[left=0mm, right=0mm, top=0mm, bottom=0mm, boxsep=0mm,boxrule=0pt, arc=0mm,colback=white, colframe=white, fontupper=\small]{
\centering
\begin{tabular}{cccccc}
\hline
{Protocol} & PM & TF & MDI & passive DPS & B92\\
\hline
Key rate & $O(\sqrt \eta)$ & * & $O(\eta)$ & $O(\eta)$ & $O(\eta^2)$ \\
Detection loophole & No & * & No & Yes & Yes \\
Qubit based & No & Yes & Yes & Yes & Yes \\
Basis switch & No & Yes & Yes & No & No \\
Phase locking & No & Yes & No & Yes & No \\
\hline
\end{tabular}
}
(c)
\end{tcolorbox}
\caption{(a) Simulation of our PM-QKD protocol. For the considered simulation parameters, the key rate of PM-QKD surpasses that of the conventional BB84 protocol when $l>120$ km and it exceeds the TGW bound when $l>250$ km. In addition, our protocol is also able to achieve a long transmission distance of $l=450$ km. (b) Parameters used for simulation. (c) Comparison of different QKD protocols: Phase-matching (PM) QKD; twin-field (TF) QKD \cite{Lucamarini2018TF}; measurement-device-independent (MDI) QKD \cite{Lo2012Measurement}; passive differential phase-shift (DPS) QKD \cite{Guan2015RRPDS}; Bennett-1992 (B92) QKD \cite{Bennett1992Quantum}. Key rate: dependence of the key rate on the channel transmittance $\eta$; Detection loophole: whether the protocol is immune to all detection loopholes; Single-photon-based: whether the security analysis is based on the single-photon case; Requires basis switch: whether the source (measurement device) must prepare (measure) states in complementary bases; Phase-locking: whether the protocol requires a fixed phase reference frame for the two users. *The security and key rate formula of TF-QKD presented in \cite{Lucamarini2018TF} have not been verified. }
\label{fig:simulation}
\end{figure*}
\end{comment}
\section{Outlook} \label{Sc:outlook}
\begin{comment}
We have proposed a new QKD scheme, named PM-QKD, where the key information is encoded into the global phase of two coherent states. We developed a new security proof technique based on analysing the optical modes, by which we show that the key rate of PM-QKD protocol scales with the square root of the transmittance, $O(\eta)$. Besides, the protocol is measurement-device-independent, i.e., it is immune to all possible detection attacks. Moreover, we developed a phase postcompensation technique to make the protocol feasible with current technology. The simulation result shows that PM-QKD protocol is able to surpass the linear key rate bound \cite{Pirandola2017Fundamental} with all the practical issues taken into account.
\end{comment}
There are a few interesting directions on PM-QKD. First, it is interesting to work out the security of the general $d$-phase PM-QKD protocol shown in Fig.~\ref{fig:PM}(a) with and without phase randomization. Meanwhile, in the above discussions, PM-QKD in Fig.~\ref{fig:PM}(b) is treated as a single-basis scheme with phase randomization. In a dual viewpoint, we can regard the different global phases $\phi_a, \phi_b$ as different bases and naturally treat the phase-sifting step as basis-sifting. This is interesting, since it shows the advantage of QKD using multi-nonorthogonal bases. Moreover, this multibases view may help us to generalize the phase-matching scheme to, for example, a polarization-based one.
Second, the phase-sifting factor $2/M$ is very small, which undermines the advantage of PM-QKD for near-distance (i.e., $l<120$ km) communication. One possible solution is to apply a biased phase randomization; i.e., the $\phi_{a(b)}$ is not uniformly randomized in $[0, 2\pi)$.
Third, we bound the total phase error $E^X_\mu$ by pessimistically considering the phase errors $e^X_k$ for even photon number components to be $1$ in Eq.~\eqref{eqn:EX Fock}, as shown in Appendix \ref{Sc:decoy}. There may be a scope to improve the bound of total phase error $E^X_\mu$. For example, the two-photon error $e_2^Z$ and $e_2^X$ can be better estimated with more decoy states, which leads to a tighter bounds of $E^X_\mu$.
Finally, since we can overcome the key-rate linear bound, it is interesting to investigate a repeaterless secret key capacity bound for QKD, for example, whether it is possible to push the key rate to $R = O(\eta^{1/3})$ or $O(\eta^{1/4})$ without repeaters. Note that the key-rate bound has been derived for the single-repeater case \cite{Pirandola2016Capacities}, $-\log(1-\sqrt{\eta})$, which is close to $\sqrt{\eta}$ when $\eta$ is small. So far, our key rate, Eq.~\eqref{eq:keyRate}, is still far away from this bound. It is an interesting direction to improve the PM-QKD protocol to approach this bound.
\begin{comment}
There are a few interesting directions on PM-QKD. Firstly, the continuous phase-randomization in PM-QKD protocol above can be replaced with discrete phase-randomization to simplify the practical implementation. Secondly, the sifting factor $2/M$ in the Eq.~\eqref{eq:keyRate} is considerably small, which compromise the advantage of PM-QKD in the near distance regime (i.e., $l<120$ km). A possible solution is to randomize the phase with bias. Thirdly, here to bound the total phase error $E^X_\mu$, we pessimistically set the phase error $e^X_k$ of even photon numbers as $1$, as shown in Appendix \ref{Sc:decoy}. There may be a room improve the bound of total phase error $E^X_\mu$. Finally, since we can overcome the key rate linear bound, it is interesting to investigate a repeater-less secret key capacity bound for QKD; for example, whether it is possible to push the key rate to $R = O(\eta^{1/3})$ or $O(\eta^{1/4})$ without repeaters.
\end{comment}
\acknowledgments
This work was supported by the National Natural Science Foundation of China Grant No.~11674193 and the National Key R\&D Program of China Grants No. 2017YFA0303900 and No. 2017YFA0304004. We thank T.~Chen, S.~Pirandola, and F.~Xu for enlightening discussion, especially T.~Chen for providing the alternative methods for compensating the phase reference difference.
\begin{appendix}
\section{Security Proof of PM-QKD}\label{Sc:SecureProof}
In this section, we provide a security proof for the PM-QKD protocol with $d=2$ via entanglement distillation \cite{Bennett1996BDSW}. Note that the existing security proofs of discrete-variable QKD assume qubit (or qudit) states transmitted through the channel. Here, we develop a new security proof by exploring continuous optical modes directly.
The organization of the proof is presented as follows. First, we briefly review the main results of the Lo-Chau \cite{Lo1999Unconditional} and Shor-Preskill \cite{Shor2000Simple} security proofs in Appendix~\ref{Sc:SecureEDP}. Then, we provide a virtual entanglement-based protocol (Protocol I) in Appendix~\ref{Sc:SecureScenario} and present a key result as Lemma \ref{Lem:parity}. In Appendix~\ref{Sc:SecureCoherent}, we employ a few equivalency arguments, and eventually, we prove the security of PM-QKD in Appendix~\ref{Sc:SecureAnnounce}. In Appendix~\ref{Sc:decoy}, we employ the decoy-state method to give tight bounds on phase error rates. In Appendix~\ref{Sc:SecurePost}, we solve the phase-reference issue with the phase postcompensation technique.
Here, we introduce some definitions and notations for later discussions. For an optical mode $A$, whose creation operator is $a^\dag$, its Hilbert space is denoted as $\mathcal{H}^A$. Let $\mathcal{D}(\mathcal{H}^A)$ denote the space of density operators acting on $\mathcal{H}^A$ and $\mathcal{L}(\mathcal{H}^A)$ denote the space of linear operators acting on $\mathcal{H}^A$. A Fock state $\ket{k}_A$ with $k$ photons in mode $A$ is defined as
\begin{equation} \label{eq:defFock}
\ket{k}_A \equiv \dfrac{(a^\dag)^k}{\sqrt{k!}} \ket{0}_A,
\end{equation}
where $\ket{0}_A$ is the vacuum state. A coherent state $\ket{\alpha}_A$ is defined as
\begin{equation}
\begin{aligned}
\ket{\alpha}_A &\equiv e^{-\frac{1}{2}|\alpha|^2} \sum_{k=0}^{\infty} \dfrac{\alpha^k}{\sqrt{k!}} \ket{k}_A \\
&= e^{-\frac{1}{2}|\alpha|^2} \sum_{k=0}^{\infty} \dfrac{(\alpha a^\dag)^k}{k!} \ket{0}_A \\
&= e^{-\frac{1}{2}|\alpha|^2} e^{\alpha a^\dag} \ket{0}_A.
\end{aligned}
\end{equation}
The photon number of $\ket{\alpha}_A$ follows a Poisson distribution,
\begin{equation}
P(k) = e^{-\mu} \dfrac{\mu^k}{k!},
\end{equation}
where $\mu = |\alpha|^2$ is the mean photon number or the light intensity.
Define the odd subspace $\mathcal{H}^A_{odd}\subseteq\mathcal{H}^A$ which is spanned by the odd Fock states $\{\ket{k}_A\}$, where all the photon numbers $k$ are odd. Similarly, define the even subspace $\mathcal{H}^A_{even}\subseteq\mathcal{H}^A$, where the photon numbers are even. Name a state $\rho\in \mathcal{D}(\mathcal{H}^A_{odd})$ to be the odd state and a state $\rho\in \mathcal{D}(\mathcal{H}^A_{even})$ to be even state. Name a state $\rho\in \mathcal{D}(\mathcal{H}^A_{odd})$ or $\rho\in \mathcal{D}(\mathcal{H}^A_{even})$ to be parity state.
Denote
\begin{equation} \label{eqn:coherent state parity}
\begin{aligned}
\ket{\alpha_{odd}}_A &\equiv \dfrac{1}{2\sqrt{c_{odd}}} (\ket{\alpha}_A - \ket{-\alpha}_A) \\
&= \dfrac{1}{\sqrt{c_{odd}}} e^{-\frac{1}{2} |\alpha|^2 } \sum_{k=0}^{\infty} \dfrac{(\alpha)^{2k+1}}{\sqrt{(2k+1)!}}\ket{2k+1}_A, \\
\ket{\alpha_{even}}_A &\equiv \dfrac{1}{2\sqrt{c_{even}}} (\ket{\alpha}_A + \ket{-\alpha}_A) \\
&= \dfrac{1}{\sqrt{c_{even}}} e^{-\frac{1}{2} |\alpha|^2 } \sum_{k=0}^{\infty} \dfrac{(\alpha)^{2k}}{\sqrt{(2k)!}}\ket{2k}_A, \\
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
\label{eqn: c normal}
c_{odd} &= e^{-\mu} \sum_{k=0}^{\infty} \dfrac{\mu^{2k+1}}{(2k+1)!} = e^{-\mu} \sinh \mu, \\
c_{even} &= e^{-\mu} \sum_{k=0}^{\infty} \dfrac{\mu^{2k}}{(2k)!} = e^{-\mu} \cosh \mu, \\
\end{aligned}
\end{equation}
are the normalization factors with $c_{odd} + c_{even} =1$, and $\mu = |\alpha|^2$ is the light intensity. It is not hard to see that $\ket{\alpha_{odd}}_A \in\mathcal{H}^A_{odd}$ and $\ket{\alpha_{even}}_A\in\mathcal{H}^A_{even}$.
Denote the photon number measurement $\{M_k\}_k$ as
\begin{equation}
M_k \equiv \ket{k}_A\bra{k}.
\end{equation}
Denote the parity measurement $\{M_{odd}, M_{even}\}$ as
\begin{equation} \label{eqn:Mpar}
\begin{aligned}
M_{odd} &\equiv \sum_{k=0}^{\infty} \ket{2k+1}_A\bra{2k+1}, \\
\quad M_{even} &\equiv \sum_{k=0}^{\infty} \ket{2k}_A\bra{2k}. \\
\end{aligned}
\end{equation}
For a beam splitter (BS), we express the input optical modes as $A, B$, with creation operators $a^\dag, b^\dag$, respectively, and the output optical modes as $C, D$, with creation operators $c^\dag, d^\dag$, respectively. The BS transforms modes $A$ and $B$ to $C$ and $D$ according to
\begin{equation}
\begin{pmatrix}
c^\dag \\
d^\dag
\end{pmatrix}
=
\dfrac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
\begin{pmatrix}
a^\dag \\
b^\dag
\end{pmatrix}.
\end{equation}
For a qubit system $A^\prime$, the Hilbert space is denoted by $\mathcal{H^{A^\prime}}$. The Pauli operators on $\mathcal{H^{A^\prime}}$ are denoted as $X_{A^\prime}, Y_{A^\prime}$ and $Z_{A^\prime}$. The eigenstates of $X_{A^\prime}, Y_{A^\prime}$ and $Z_{A^\prime}$ are denoted by $\{\ket{\pm}_{A^\prime}\}; \{\ket{\pm i}_{A^\prime}\}$ and $\{\ket{0}_{A^\prime},\ket{1}_{A^\prime}\}$, respectively. The $X$-basis measurement is denoted by $M_X: \{\ket{+}_{A^\prime}\bra{+}, \ket{-}_{A^\prime}\bra{-} \}$. The $Z$-basis measurement is denoted by $M_Z: \{\ket{0}_{A^\prime}\bra{0}, \ket{1}_{A^\prime}\bra{1} \}$.
A control-phase gate, $C_\pi$, from a qubit $A^\prime$ to an optical mode $A$, is defined as
\begin{equation} \label{eq:Cpi}
\begin{aligned}
C_{\pi} \equiv \ket{0}_{A^\prime}\bra{0}\otimes U_A(0) + \ket{1}_{A^\prime}\bra{1}\otimes U_A(\pi), \\
\end{aligned}
\end{equation}
where $U_A(\phi)\equiv e^{i\phi a^\dag a}$ is a $\phi$-phase shifter operation on the mode $A$.
\begin{definition} \label{def:equiv}
Two QKD protocols are \emph{equivalent} if the following criteria are satisfied:
\begin{enumerate}
\item
The quantum states transmitted in the channel are the same.
\item
All announced classical information is the same.
\item
Alice and Bob perform the same measurement on the same quantum states to obtain the raw key bits.
\item
Alice and Bob use the same postprocessing to extract secure key bits.
\end{enumerate}
\end{definition}
Obviously, equivalent QKD protocols will lead to identical key rates.
\subsection{Security proof via entanglement distillation}\label{Sc:SecureEDP}
Here, we briefly review the security proof based on entanglement distillation \cite{Lo1999Unconditional,Shor2000Simple}. Suppose that in QKD, Alice generates an $l$-bit key string $S$, and Bob generates an estimate of the key string $S'$. Denote the space of $S$ as $\mathcal{S}$, whose dimension is $2^l$. An adversary Eve attempts to learn about $S$ from the information leakage.
After QKD, Alice and Bob should share the \emph{same} key \emph{privately}. A key $S$ is called ``correct'', if $S'= S$ for any strategy of Eve, and is called ``$\epsilon_{cor}$-correct'', if
\begin{equation} \label{eqn:Defcorrect}
\begin{aligned}
Pr[S'\neq S]\leq \epsilon_{cor}.
\end{aligned}
\end{equation}
To define a private key, consider the quantum state $\rho_{AE}$ that describes the correlation between Alice's classical key $S$ and Eve's system $E$ (for any attacks). A key $S$ is called ``$\epsilon_{sec}$-private'' from $E$ if \cite{Ben2005composable,Renner2005universally}
\begin{equation}
\min_{\sigma_E} \dfrac{1}{2} || \rho_{AE} - \omega_A \otimes \sigma_E ||_1 \leq \epsilon_{sec},
\end{equation}
where $\omega_A = (|\mathcal{S}|)^{-1} \sum_{S=0}^{|\mathcal{S}|-1} \ket{S}_A\bra{S}$ is the equally mixed key state over all possible keys in space $\mathcal{S}$ and $||\cdot||_1$ is the trace norm. A QKD protocol is called ``secure'' if the generated key is both correct and private. It is called ``$\epsilon$-secure'' if the generated key is $\epsilon_{cor}$-correct and $\epsilon_{sec}$-private with $\epsilon=\epsilon_{cor} + \epsilon_{sec}$.
Here, we consider the case where Alice and Bob share an $m$-pair-of-qubits gigantic state $\rho_{A^\prime B^\prime}^{(m)}$. If we can show that
\begin{equation}
F(\rho_{A^\prime B^\prime}^{(m)}, \ket{\Phi^+}_{A^\prime B^\prime}^{(m)}) \geq \sqrt{1 - \epsilon_l^2}
\end{equation}
where $\ket{\Phi^+}_{A^\prime B^\prime}^{(m)}$ is the state of $m$ perfect EPR pairs $\ket{\Phi^+} = (\ket{00}+\ket{11})/\sqrt{2}$, and $0\le\epsilon_l\le1$, then it can be shown that it is $\epsilon_l$-private and $\epsilon_l$-correct, and then it is $2\epsilon_l$-secure. That is, to show the security of a QKD protocol, we only need to show that the state for key extraction $\rho_{A^\prime B^\prime}^{(m)}$ is close to the perfect EPR pairs $\ket{\Phi^+}_{A^\prime B^\prime}^{(m)}$.
Here is the intuition of the Lo-Chau security proof \cite{Lo1999Unconditional}. Suppose Alice and Bob share an $n$-pair-of-qubit gigantic state $\rho_{A^\prime B^\prime}^{(n)}$ at the beginning. If they perform an efficient entanglement distillation protocol (EDP) to distill $m$ EPR pairs, then they will be able to share nearly $m$ bit correct and private keys. Now the task becomes how to find the right EDP.
Bennett, DiVincenzo, Smolin, and Wootters (BDSW) show that \cite{Bennett1996BDSW}, if an $n$-pair-of-qubit state $\rho_{A^\prime B^\prime}^{(n)}$ can be written as a classical mixture of Bell state products,
\begin{equation} \label{eqn:classical Bell}
\begin{aligned}
&\rho_{A^\prime B^\prime}^{(n)} = \sum_{b_1, b_2, ...,b_n} p_{b_1, b_2,...,b_n} \ket{b_1, b_2,...,b_n}\bra{b_1, b_2,...,b_n}, \\
&\ket{b_1, b_2,...b_n} = \bigotimes_{i=1}^{n} \ket{\Phi^{(i)}_{b_i}}, \\
\end{aligned}
\end{equation}
where $\ket{\Phi^{(i)}_{b_i}}$ is one of the four Bell states labeled by $b_i \in \{0,1,2,3\}$ on the $i$ th qubit, then one can distill entanglement by employing EDPs. In the one-way hashing method \cite{Bennett1996BDSW}, a specific type of one-way EDP, Alice measures a series of commuting operators based on some random-hashing matrix on her $n$ qubits, and she sends the results to Bob. Bob measures the same operators on his $n$ qubits and infers the locations and types of the errors from the difference in the measurement results. After that, Alice and Bob correct the errors and obtain $m$ ($m\leq n$, almost perfect) EPR pairs.
In general, of course, the initial state $\rho_{A^\prime B^\prime}^{(n)}$ can deviate from Bell-diagonal states and become highly entangled between different pairs. In the Lo-Chau security proof \cite{Lo1999Unconditional}, it has been shown that, for the one-way hashing EDP introduced above, the error syndrome and EDP performance of such $\rho_{A^\prime B^\prime}^{(n)}$ is the same as that of the state after dephasing between pairs,
\begin{equation}
\begin{aligned}
\rho_{A^\prime B^\prime, dep}^{(n)} &\equiv W \rho_{A^\prime B^\prime}^{(n)} W, \\
W & = \sum_{b_1, b_2, ...,b_n} \ket{b_1, b_2,...,b_n}\bra{b_1, b_2,...,b_n}.
\end{aligned}
\end{equation}
where $\ket{b_1, b_2,...,b_n}$ is defined in Eq.~\eqref{eqn:classical Bell}. Therefore, one can reduce the EDP for general $\rho_{A^\prime B^\prime}^{(n)}$ (i.e., the case with coherent attacks) to the case in Eq.~\eqref{eqn:classical Bell}.
In the Shor-Preskill security proof \cite{Shor2000Simple}, the one-way EDP protocol is reduced to a ``prepare-and-measure'' QKD protocol, by employing the CSS code\cite{CSS1996qec}. Later, other techniques of decoupling $X$-error correction and $Z$-error correction are introduced for this reduction. For a one-way EDP protocol based on the CSS code, the distillation rate of EPR pairs, $r \equiv \lim_{n\rightarrow\infty}m/n$, is given by
\begin{equation} \label{eqn:Shor-Preskill key rate}
r = 1 - H(E^Z) - H(E^X),
\end{equation}
where $E^Z$ and $E^X$ are the $Z$-error rate and $X$-error rate, respectively, and $H(x)=-x\log_2x-(1-x)\log_2(1-x)$ is the binary Shannon entropy function. The error rates can defined as measurement results on $\rho^{(n)}_{A^\prime B^\prime}$,
\begin{equation}
\begin{aligned}
\label{eqn: EX EZ}
E^Z &\equiv Tr[ \bigoplus_{j=1}^{n} \dfrac{1}{2}(1- Z^{(j)}_{A^\prime} \otimes Z^{(j)}_{B^\prime}) \rho^{(n)}_{A^\prime B^\prime}], \\
E^X &\equiv Tr[ \bigoplus_{j=1}^{n} \dfrac{1}{2}(1- X^{(j)}_{A^\prime} \otimes X^{(j)}_{B^\prime}) \rho^{(n)}_{A^\prime B^\prime}], \\
\end{aligned}
\end{equation}
where $X^{(j)}_{A^\prime(B^\prime)}$ is the Pauli $X$ operator and $Z^{(j)}_{A^\prime(B^\prime)}$ is the Pauli $Z$ operator, on the $j$ th pair-of-qubits system $A^\prime (B^\prime)$. Since the bit value is measured in the $Z$-basis and the phase value is measured in the $X$-basis, we also call the $Z$-basis error the ``bit error'' and the $X$-basis error the ``phase error''.
Following the Shor-Preskill security proof, distillable entanglement of the one-way EDP protocol is the key rate for some prepare-and-measure QKD protocols such as BB84 \cite{Bennett1984Quantum}. In the BB84 protocol, we can estimate $E^Z, E^X$ by randomly measuring the qubits in the $X$- and $Z$-basis. In the PM-QKD protocol, on the other hand, Alice and Bob can only measure in the $Z$-basis. In the following sections, we will introduce entanglement-based PM-QKD protocols and discuss how to infer the value of $E^X$.
\subsection{Entanglement-based PM-QKD protocol} \label{Sc:SecureScenario}
We first introduce an entanglement-based PM-QKD protocol, called Protocol I, as shown in Fig.~\ref{fig:Pro1}.
\begin{figure*}[htbp]
\centering
\includegraphics[width=16cm]{Pro1.pdf}
\caption{(a) Schematic diagram of Protocol I. (b) Equivalent view of Protocol I, where the $X$-basis measurement on $A^\prime$ or $B^\prime$ is realized by a Hadamard gate followed by the $Z$-basis measurement.} \label{fig:Pro1}
\end{figure*}
\textbf{\uline{Protocol I}}
\begin{enumerate}
\item \label{StepPrep}
State preparation: A trusted party, Charlie, picks a state $\rho$ on optical mode $C$, splits $\rho$ into two pulses, $A$ and $B$; and sends them to Alice and Bob, respectively. Alice and Bob initialize their qubits in $\ket{+i}$. Alice applies the control gate $C_{\pi}$, defined in Eq.~\eqref{eq:Cpi}, to qubit $A^\prime$ and optical pulse $A$. Similarly, Bob applies $C_{\pi}$ to $B^\prime$ and $B$.
\item \label{StepMeasure}
Measurement: The two optical pulses $A$ and $B$ are sent to an untrusted party, Eve, who is supposed to perform interference measurement and record which detector ($L$ or $R$) clicks.
\item \label{StepAnnounce}
Announcement: Eve announces the detection result, $L/R$ click or failure, for each round.
\item \label{StepSifting}
Sifting: When Eve announces an $L/R$ click, Alice and Bob keep the qubits of systems $A^\prime$ and $B^\prime$. In addition, Bob applies a Pauli $Y$-gate to his qubit if Eve's announcement is $R$ click.
\item \label{StepParaEst}
Parameter estimation: After many rounds of the above steps, Alice and Bob end up with a joint $2n$-qubit state, denoted by $\rho_{A^\prime B^\prime}^{(n)}$. They then perform random sampling on the remaining $\rho_{A^\prime B^\prime}^{(n)}$ to estimate $E^Z$ and infer $E^X$ by Eq.~\eqref{eqn:EX par}.
\item\label{StepExtraction}
Key distillation: Alice and Bob apply a standard EDP when the error rates are below a certain threshold. The distillation ratio $r$ is given by Eq.~\eqref{eqn:Shor-Preskill key rate}. Once Alice and Bob obtain $nr$ (almost) pure EPR pairs, they both perform local $Z$ measurements on the qubits to generate private keys.
\end{enumerate}
Our first observation is that, if the state $\rho$ prepared by Charlie is a parity state, namely, $\rho\in\mathcal{D}(\mathcal{H}^C_{odd})$ or $\rho\in\mathcal{D}(\mathcal{H}^C_{even})$, then the $X$-error rate and $Z$-error rate are correlated. Here, we denote the $Z$-error rate and $X$-error rate for an odd state $\rho_{odd}$ as $e_{odd}^Z$ and $e_{odd}^X$, respectively, and similarly, denote for an even state $\rho_{even}$ as $e_{even}^Z$ and $e_{even}^X$.
\begin{lemma}\label{Lem:parity}
In Protocol I, for $\rho\in\mathcal{D}(\mathcal{H}^C_{odd})$, $e_{odd}^X = e_{odd}^Z$; and for $\rho\in\mathcal{D}(\mathcal{H}^C_{even})$, $e_{even}^X = 1- e_{even}^Z$.
\end{lemma}
\begin{proof}
First consider the case when $\rho$ is a Fock state $\ket{k}_C$, defined in Eq.~\eqref{eq:defFock}. After passing through the BS, as shown in Fig.~\ref{fig:Pro1}, the state on modes $A$ and $B$ becomes
\begin{equation}
\dfrac{1}{\sqrt{2^k k!}} (a^\dag + b^\dag)^k \ket{00}_{AB}.
\end{equation}
The joint state on system $A^\prime, B^\prime, A, B$ before the $C_\pi$ operations is
\begin{equation}
\begin{aligned}
& \dfrac{1}{\sqrt{2^k k!}} \ket{+i+i}_{A^\prime B^\prime} (a^\dag + b^\dag)^k \ket{00}_{AB} \\
& = \dfrac{1}{\sqrt{2^k k!}}\dfrac{1}{2} [(\ket{00} - \ket{11}) + i (\ket{01} + \ket{10}) ]_{A^\prime B^\prime} (a^\dag + b^\dag)^k \ket{00}_{AB}.
\end{aligned}
\end{equation}
After the $C_\pi$ operations, this state becomes
\begin{widetext}
\begin{equation} \label{eqn:rho0 fock}
\begin{aligned}
\ket{\Psi^{(k)}_0} = &
\begin{cases}
&\dfrac{1}{2\sqrt{2^k k!}} [ (\ket{00} + \ket{11} )_{A^\prime B^\prime} (a^\dagger + b^\dagger)^k \ket{00}_{A B} + i(\ket{01} - \ket{10} )_{A^\prime B^\prime} (a^\dagger - b^\dagger)^k \ket{00}_{AB} ], \\
& \text{if } k \text{ is odd}, \\
&\dfrac{1}{2\sqrt{2^k k!}} [ (\ket{00} - \ket{11} )_{A^\prime B^\prime} (a^\dagger + b^\dagger)^k \ket{00}_{A B} + i(\ket{01} + \ket{10} )_{A^\prime B^\prime} (a^\dagger - b^\dagger)^k \ket{00}_{AB} ], \\
& \text{if } k \text{ is even}.
\end{cases} \\
\end{aligned}
\end{equation}
\end{widetext}
As shown in Fig.~\ref{fig:Pro1}(b), the $X$-basis measurement on $A^\prime$ or $B^\prime$ is realized by a Hadamard gate followed by the $Z$-basis measurement. Denote the state after local Hadamard gates as $ \ket{\Psi^{(k)}_{HH}} \equiv (H^{A^\prime}\otimes H^{B^\prime}) \ket{\Psi^{(k)}_0}$; then,
\begin{widetext}
\begin{equation} \label{eqn:rhoHH fock}
\begin{aligned}
\ket{\Psi^{(k)}_{HH}} = &
\begin{cases}
&\dfrac{1}{2\sqrt{2^k k!}} [ (\ket{00} + \ket{11} )_{A^\prime B^\prime} (a^\dagger + b^\dagger)^k \ket{00}_{A B} + i(\ket{01} - \ket{10} )_{A^\prime B^\prime} (a^\dagger - b^\dagger)^k \ket{00}_{AB} ], \\
& \text{if } k \text{ is odd}, \\
&\dfrac{1}{2\sqrt{2^k k!}} [ (\ket{01} + \ket{10} )_{A^\prime B^\prime} (a^\dagger + b^\dagger)^k \ket{00}_{A B} + i(\ket{00} - \ket{11} )_{A^\prime B^\prime} (a^\dagger - b^\dagger)^k \ket{00}_{AB} ], \\
& \text{if } k \text{ is even}.
\end{cases} \\
\end{aligned}
\end{equation}
\end{widetext}
In other words, the $X$-error rate $e^X_k$ can be understood as the error rate by performing the $Z$-basis measurement on the state of $\ket{\Psi^{(k)}_{HH}}$. The relation between the $X$- and $Z$-error rates can be obtained by comparing Eqs.~\eqref{eqn:rho0 fock} and \eqref{eqn:rhoHH fock}. For the odd-photon-number case, since $\ket{\Psi^{(2k+1)}_{HH}} = \ket{\Psi^{(2k+1)}_0}$, we have $e^X_{2k+1} = e^Z_{2k+1}$. For the even-photon-number case, since $\ket{\Psi^{(2k)}_{HH}} = I^{A^\prime}\otimes Y^{B^\prime}\ket{\Psi^{(2k)}_0}$, we have $e^X_{2k} = 1 - e^Z_{2k}$.
Now, let us consider the case of pure parity states. For an odd state $\ket{\psi_{odd}}_C = \sum c_{2k+1} \ket{{2k+1}}_C $, where $\sum_k |c_{2k+1}|^2 = 1$, since the BS and $C_\pi$ are unitary operations, the state after these operations can be written as
\begin{equation}
\label{eq:pure odd psi}
\ket{\Psi_0^{(odd)}} = \sum_{k} c_{2k+1} \ket{\Psi_0^{(2k+1)}}.
\end{equation}
After local Hadamard gates, the state becomes $ \ket{\Psi^{(odd)}_{HH}} \equiv (H^{A^\prime}\otimes H^{B^\prime}) \ket{\Psi^{(odd)}_0} $. From Eqs.~\eqref{eqn:rho0 fock}, \eqref{eqn:rhoHH fock}, \eqref{eq:pure odd psi}, we can see that $\ket{\Psi^{(odd)}_{HH}} = \ket{\Psi^{(odd)}_0}$, and hence $e^Z_{odd} = e^X_{odd}$. With the same argument, we have $e^Z_{even} = 1 - e^X_{even}$.
For general parity states, we can regard them as mixtures of pure parity states,
\begin{equation}
\begin{aligned}
\rho_{odd} &= \sum_{i} p_i \ket{\psi_{odd}^{(i)}}\bra{\psi_{odd}^{(i)}}, \\
\rho_{even} &= \sum_{i} p_i \ket{\psi_{even}^{(i)}}\bra{\psi_{even}^{(i)}}.
\end{aligned}
\end{equation}
This is equivalent to Charlie sending out $\ket{\psi_{odd(even)}^{(i)}}$ with probability $p_i$. For each pure state component, we have $e^X_{odd(i)} = e^Z_{odd(i)}$ and $e^X_{even(i)} = 1- e^Z_{even(i)}$. Thus, the relations hold for all (mixed) parity states.
\end{proof}
In general, Charlie might not use a parity state. Consider the case that Charlie performs the parity measurement $\{M_{odd},M_{even}\}$, defined in Eq.~\eqref{eqn:Mpar}, before sending to Alice and Bob. Denote the measurement outcome probabilities for the odd and even parity to be $p_{odd}$ and $p_{even}$, respectively. This is equivalent to Charlie preparing an odd state $\rho_{odd}$ and an even state $\rho_{even}$ with probabilities $p_{odd}$ and $p_{even}$, respectively. Then, the state can be written as
\begin{equation}
\rho = p_{odd} \rho_{odd} + p_{even} \rho_{even},
\end{equation}
that is, $\rho\in \mathcal{D}(\mathcal{H}_{odd}^C \oplus \mathcal{H}_{even}^C)$.
Suppose Charlie announces the parity information publicly to Alice and Bob.
Then, they can label the sifted qubits with ``odd" and ``even". Denote $q_{odd}$ and $q_{even}$, with $q_{odd}+q_{even}=1$, to be the fractions of odd- and even-labeled states in the sifted $n$-pairs of qubit states $A^\prime, B^\prime$, respectively. Then, according to Lemma \ref{Lem:parity}, the total $X$-error rate can be calculated by
\begin{equation} \label{eqn:EX par}
\begin{aligned}
E^X &= q_{odd} e^X_{odd} + q_{even} e^X_{even} \\
&= q_{odd} e^Z_{odd} + q_{even} (1 - e^Z_{even}). \\
\end{aligned}
\end{equation}
Here, the parameters $q_{odd}$, $q_{even}$, $e^Z_{odd}$, and $e^Z_{even}$ can be evaluated according to Charlie's parity announcement. When the parity information is missing, Alice and Bob need to estimate these parameters, which will be described later in Appendix \ref{Sc:decoy}.
\subsection{Coherent state protocol and equivalent process} \label{Sc:SecureCoherent}
From Protocol I to the PM-QKD protocol, there are a few practical issues that need to be addressed. First, the trusted party Charlie and the BS should be removed. In order to do this, we consider a special case where Charlie prepares a coherent state $\ket{\sqrt{\mu}}_C$ as the photon source. In addition, Charlie adds a random phase $\phi\in\{0,\pi\}$ with equal probabilities to the coherent state, so the density matrix can be written as
\begin{equation} \label{eqn:coherent state parity addition}
\begin{aligned}
\rho(\alpha, -\alpha) &= \dfrac{1}{2}(\ket{\alpha}\bra{\alpha} + \ket{-\alpha}\bra{-\alpha}) \\
&= c_{odd}\ket{\alpha_{odd}}\bra{\alpha_{odd}} + c_{even}\ket{\alpha_{even}}\bra{\alpha_{even}}, \\
\end{aligned}
\end{equation}
where $\alpha=\sqrt{\mu}$, and $\ket{\alpha_{odd}}$, $\ket{\alpha_{even}}$, $c_{odd}$ and $c_{even}$ are defined in Eqs.~\eqref{eqn:coherent state parity} and Eq.~\eqref{eqn: c normal}. Clearly, one can see that $\rho(\alpha, -\alpha) \in \mathcal{D}(\mathcal{H}_{odd}^C \oplus \mathcal{H}_{even}^C)$.
\begin{figure*}[htbp]
\centering
\includegraphics[width=16cm]{Pro2.pdf}
\caption{(a) A specific realization of Protocol I, where Charlie prepares $\ket{\sqrt{\mu}}$ and $\ket{\sqrt{-\mu}}$ with equal probabilities. (b) Schematic diagram of Protocol II, where Alice and Bob prepare $\ket{\sqrt{\mu/2}}$ and add the same random phase $\phi\in\{0,\pi\}$.} \label{fig:Pro2}
\end{figure*}
Now, Alice and Bob want to prepare the state given in Eq.~\eqref{eqn:coherent state parity addition} without Charlie's assistance. They first locally prepare two coherent states $\ket{\sqrt{\mu/2}}_{A(B)}$ and add the same random phase $\phi\in\{0,\pi\}$ to their states, as shown in Fig.~\ref{fig:Pro2}(b). Then, the state becomes
\begin{widetext}
\begin{equation}
\label{eqn:two coherent state parity addition}
\begin{aligned}
\rho_{AB}(\sqrt{\mu},-\sqrt{\mu}) = \dfrac{1}{2}(\ket{\sqrt{\mu/2}}_A\bra{\sqrt{\mu/2}} \otimes \ket{\sqrt{\mu/2}}_B\bra{\sqrt{\mu/2}}
+ \ket{\sqrt{-\mu/2}}_A\bra{\sqrt{-\mu/2}} \otimes \ket{\sqrt{-\mu/2}}_B\bra{\sqrt{-\mu/2}}).
\end{aligned}
\end{equation}
\end{widetext}
Apparently, $\rho_{AB}(\sqrt{\mu},-\sqrt{\mu})$ is the state after $\rho(\alpha, -\alpha)$ defined in Eq.~\eqref{eqn:coherent state parity addition} going through the BS. Thus, this new protocol (Protocol II) is equivalent to Protocol I with the input state $\rho(\sqrt{\mu},-\sqrt{\mu})$.
Protocol II runs as follows, as shown in Fig.~\ref{fig:Pro2}(b). Here, $\mu_a = \mu_b = \mu/2$.
\textbf{\uline{Protocol II}}
\begin{enumerate}
\item
State preparation:
Alice and Bob prepare coherent states $\ket{\sqrt{\mu_a}}$ and $\ket{\sqrt{\mu_b}}$ on optical modes $A$ and $B$, respectively. They initialize the qubits $A^\prime$ and $B^\prime$ in $\ket{+i}$. They add the same random phase $\phi\in\{0,\pi\}$ on the optical modes $A$ and $B$. Alice applies the control gate $C_{\pi}$, defined in Eq.~\eqref{eq:Cpi}, to qubit $A^\prime$ and optical pulse $A$. Similarly, Bob applies $C_{\pi}$ to $B^\prime$ and $B$.
\item
Measurement: The two optical pulses $A$ and $B$ are sent to an untrusted party, Eve, who is supposed to perform interference measurement and record which detector ($L$ or $R$) clicks.
\item
Announcement: Eve announces the detection result, $L/R$ click or failure, for each round.
\item
Sifting: When Eve announces an $L/R$ click, Alice and Bob keep the qubits of systems $A^\prime$ and $B^\prime$. In addition, Bob applies a Pauli $Y$-gate to his qubit if Eve's announcement is an $R$ click.
\item
Parameter estimation: After many rounds of the above steps, Alice and Bob end up with a joint $2n$-qubit state, denoted by $\rho_{A^\prime B^\prime}^{(n)}$. They then perform random sampling on the remaining $\rho_{A^\prime B^\prime}^{(n)}$ to estimate $E^Z$ and infer $E^X$ by Eq.~\eqref{eqn:EX par}.
\item
Key distillation: Alice and Bob apply a standard EDP when the error rates are below a certain threshold. The distillation ratio $r$ is given by Eq.~\eqref{eqn:Shor-Preskill key rate}. Once Alice and Bob obtain $nr$ (almost) pure EPR pairs, they both perform local $Z$ measurements on the qubits to generate private keys.
\end{enumerate}
From Protocol II to the PM-QKD protocol, there are still some missing links.
\begin{enumerate}
\item
Alice and Bob need to add the same random phase $\phi\in\{0,\pi\}$ to their states, which would cost them a private bit.
\item
They use Eq.~\eqref{eqn:EX par} to infer $E^X$, which needs information on the parameters of $ q_{odd}, q_{even}, e^Z_{odd}$ and $e^Z_{even}$.
\item
The phase references of Alice and Bob's coherent states are locked. That is, the phases of the initial coherent states, $\ket{\sqrt{\mu_a}}$ and $\ket{\sqrt{\mu_b}}$, are the same, and hence remote phase locking is required.
\end{enumerate}
We shall remove the simultaneous phase randomization requirement in Appendix \ref{Sc:SecureAnnounce}, bound $q_{odd}, q_{even}, e^Z_{odd}$ and $e^Z_{even}$ in Appendix \ref{Sc:decoy}, and remove the phase-locking requirement in Appendix \ref{Sc:SecurePost}.
\subsection{Security with phase announcement}\label{Sc:SecureAnnounce}
To remove the simultaneous phase randomization requirement in Protocol II, a straightforward idea is that Alice and Bob add random phases $\phi_{a(b)}\in\{0,\pi\}$ independently (with equal probability $1/2$); during the sifting step, they announce the phases $\phi_a, \phi_b$ and postselect the bits only for $\phi_a=\phi_b$, as shown in Fig.~\ref{fig:Pro3}. All the other steps remain unchanged. Here, there are two modifications that need to be analyzed, the phase announcement of $\phi_a, \phi_b$ and postselection of $\phi_a=\phi_b$.
We deal with the phase announcement of $\phi_a, \phi_b$ first. Consider Protocol IIa, as shown in Fig.~\ref{fig:Pro3}, where Alice and Bob announce the random phase $\phi$ in Protocol II during postprocessing.
\begin{figure*}[htbp]
\centering
\includegraphics[width=18cm]{Pro3-eps-converted-to.pdf}
\caption{The schematic diagram of protocols II, IIa, and III. In Protocol II, Alice and Bob add the same phase $\phi\in\{0,\pi\}$ on their coherent state $\sqrt{\mu/2}$. In Protocol IIa, Alice and Bob announce the phase $\phi$ after Eve's announcement. In Protocol III, Alice and Bob add phases $\phi_a$ and $\phi_b$, independently. They announce $\phi_a$ and $\phi_b$ after Eve's announcement.}
\label{fig:Pro3}
\end{figure*}
\textbf{\uline{Protocol IIa}}
\begin{enumerate}
\item
State preparation:
Same as Protocol II.
\item
Measurement: Same as Protocol II.
\item
Announcement: Same as Protocol II.
\item
Sifting: The first part is the same as Protocol II. After that, Alice and Bob announce the random phase $\phi$.
\item
Parameter estimation: Same as Protocol II.
\item
Key distillation: Same as Protocol II.
\end{enumerate}
Note that the only difference between Protocol II and IIa is that Alice and Bob announce their phase $\phi$ after Eve's announcement. Since the classical information announced during postprocessing is different, according to Definition~\ref{def:equiv}, Protocol II and IIa are not equivalent. In fact, from some specific attacks (see Appendix~\ref{sc:attack}), one can see that the security of the two protocols with and without phase announcement can be very different.
In Protocol II, as shown in Fig.~\ref{fig:Pro3}, because of phase randomization, one can always assume that Alice and Bob (or Eve) perform the total parity measurement $\{M_{odd}^t, M_{even}^t\}$, which is defined as, similar to Eq.~\eqref{eqn:Mpar},
\begin{equation}
\begin{aligned}
\label{eqn:MeasureTotalParity}
M_{odd}^t &\equiv \sum_{k_1 + k_2 \text{ is odd}} \ket{k_1 k_2}_{AB}\bra{k_1 k_2}, \\
M_{even}^t &\equiv \sum_{k_1 + k_2 \text{ is even}} \ket{k_1 k_2}_{AB}\bra{k_1 k_2}. \\
\end{aligned}
\end{equation}
Here, $\{\ket{k_1 k_2}_{AB}\}$ are the Fock states on mode $A,B$, with $k_1$ photons on $A$ and $k_2$ photons on $B$. That is, in Protocol II, the photon source can be regarded as a mixture of odd states and even states. The parity-state channel model is similar to the photon number channel used in the security proof of the decoy-state method \cite{Lo2005Decoy,Ma2008PhD}.
In Protocol IIa, as shown in Fig.~\ref{fig:Pro3}, on the other hand, no such parity measurement is allowed because it does not commute with the phase announcement. That is, Eve can distinguish whether or not Alice and Bob perform $\{M_{odd}^t, M_{even}^t\}$ after the announcement of phases $\phi_a$ and $\phi_b$. In other words, the quantum signals sent by Alice and Bob can no longer be regarded as a mixture of $\rho_{odd}$ and $\rho_{even}$ in Protocol IIa. To analyze the security of Protocol IIa, we notice the following observation.
\begin{observation}\label{obs:same rho}
The joint states $\rho_{A^\prime B^\prime}^{(n)}$ obtained in protocols I, II, and IIa are the same after sifting in Step \ref{StepSifting}.
\end{observation}
Note that qubit systems $A^\prime$ and $B^\prime$ are local and never sent to Eve. Before the sifting step, these qubits are these qubits are identical and sampled independently (i.i.d.) and the same for protocols I, II, and IIa. The state $\rho_{A^\prime B^\prime}^{(n)}$ is postselected by Eve's announcement. Of course, Eve can manipulate $\rho_{A^\prime B^\prime}^{(n)}$ by different measurement or announcement strategies. We emphasize that \emph{Eve announces before the phase announcement in Protocol IIa}. Then, her strategy cannot depend on the phase announcement. Therefore, in all three protocols, the state $\rho_{A^\prime B^\prime}^{(n)}$ remains the same. This is crucial to our security analysis. From Observation \ref{obs:same rho}, we can have the following Corollary.
\begin{corollary}\label{Cor:same EX}
Both the $X$- and $Z$-error patterns of the joint states $\rho_{A^\prime B^\prime}^{(n)}$ in protocol II and IIa are the same. Then, the $X$-error rates $E^X$ of protocols II and IIa are the same, as given by Eq.~\eqref{eqn:EX par}.
\end{corollary}
Now, we deal with the postselection of $\phi_a=\phi_b$. Regardless of the values of $\phi_a, \phi_b$, because of control-phase gates $C_\pi$, the state $\rho_{AB}$ sent to Eve is
\begin{equation}
\rho_{AB} = \rho_A(\sqrt{\mu/2}, -\sqrt{\mu/2}) \otimes \rho_B(\sqrt{\mu/2}, -\sqrt{\mu/2}),
\end{equation}
where $\rho(\sqrt{\mu/2}, -\sqrt{\mu/2})$ is defined in Eq.~\eqref{eqn:coherent state parity addition}.
The state $\rho_{AB}$ is independent of $\phi_a, \phi_b$; hence Eve's attack cannot depend on $\phi_a, \phi_b$, and the sifted qubits are also independent of the value of $\phi_a (=\phi_b)$.
Furthermore, we notice that discarded qubits with $|\phi_a-\phi_b|=\pi$ can also be used for entanglement distillation. Here, adding a phase $\pi$ to system $B$ is equivalent to performing a Pauli $Y$-gate to system $B^\prime$ for a parity-state source in protocols I, II and IIa. Thus, for the qubits with $|\phi_a-\phi_b|=\pi$, Bob performs $Y$-gate on qubit $B^\prime$.
Therefore, if Alice and Bob randomize their phases $\phi_a, \phi_b\in \{0,\pi\}$ independently and perform phase-sifting operations after Eve's announcement, shown in Fig.~\ref{fig:Pro3}, the modified protocol is equivalent to Protocol IIa. We call this Protocol III, which runs as follows, as shown in Fig.~\ref{fig:Pro3}. Here, $\mu_a = \mu_b = \mu/2$.
\textbf{\uline{Protocol III}}
\begin{enumerate}
\item
State preparation:
Alice and Bob prepare coherent states $\ket{\sqrt{\mu_a}}$ and $\ket{\sqrt{\mu_b}}$ on optical modes $A, B$, separately. They initial their qubits $A^\prime, B^\prime$ in $\ket{+i}$. They independently add random phases $\phi_a, \phi_b \in \{0,\pi\}$ on optical modes $A, B$. Alice applies the control gate $C_{\pi}$, defined in Eq.~\eqref{eq:Cpi}, to qubit $A^\prime$ and optical pulse $A$. Similarly, Bob applies $C_{\pi}$ to $B^\prime$ and $B$.
\item
Measurement: The two optical pulses, $A$ and $B$, are sent to an untrusted party, Eve, who is supposed to perform an interference measurement and record which detector ($L$ or $R$) clicks.
\item
Announcement: Eve announces the detection result, $L$/$R$ click or failure, for each round.
\item
Sifting: When Eve announces an $L/R$ click, Alice and Bob keep the qubits of systems $A^\prime$ and $B^\prime$. Bob applies a Pauli $Y$-gate to his qubit if Eve's announcement is $R$ click. After Eve's announcement, Alice and Bob announce their encoded phase $\phi_a, \phi_b$. Bob applies a Pauli $Y$-gate to his qubit if $|\phi_a-\phi_b|=\pi$.
\item
Parameter estimation: After many rounds of the above steps, Alice and Bob end up with a joint $2n$-qubit state, denoted by $\rho_{A^\prime B^\prime}^{(n)}$. They then perform random sampling on the remaining $\rho_{A^\prime B^\prime}^{(n)}$ to estimate $E^Z$ and infer $E^X$ by Eq.~\eqref{eqn:EX par}.
\item
Key distillation: Alice and Bob apply a standard EDP when the error rates are below a certain threshold. The distillation ratio $r$ is given by Eq.~\eqref{eqn:Shor-Preskill key rate}. Once Alice and Bob obtain $nr$ (almost) pure EPR pairs, they both perform local $Z$ measurements on the qubits to generate private keys.
\end{enumerate}
\subsection{Decoy-state method and phase randomization} \label{Sc:decoy}
Here, we introduce a method to estimate $q_{odd}$, $q_{even}$, $e_{odd}^Z$ and $e_{even}^Z$. Without loss of generality, we mainly discuss the decoy-state method in Protocol I. Similar arguments can be applied to protocols II, IIa, and III.
Recall that in Protocol I, if Charlie prepares coherent state $\ket{\sqrt{\mu}}$ and adds a random phase $\{0,\pi\}$ on it, it is equivalent to preparing odd- and even-parity states with probabilities $p_{odd}^\mu = c_{odd}$ and $p_{even}^\mu = c_{even}$, respectively, as defined in Eq.~\eqref{eqn: c normal}. Define the yield $Y_{odd}^\mu$ ($Y_{even}^\mu$) as the probability of successful detection conditional on the odd-parity (even-parity) state. The fraction of odd- and even-parity states in the final detected signal is given by
\begin{equation}
\begin{aligned}
\label{eqn:q parity}
q_{odd}^\mu &= p_{odd}^\mu \dfrac{Y_{odd}^\mu}{Q_\mu}, \\
q_{even}^\mu &= p_{even}^\mu \dfrac{Y_{even}^\mu}{Q_\mu},
\end{aligned}
\end{equation}
where $Q_\mu$ is the total gain of the signals. For signals with intensity $\mu$, we have
\begin{equation}
\begin{aligned}
\label{eqn:decoy parity}
Q^{\mu} &= p_{odd}^{\mu} Y^{\mu}_{odd} + p_{even}^{\mu} Y^{\mu}_{even}, \\
E^{Z,\mu} Q^{\mu} &= e_{odd}^{Z,\mu}p_{odd}^{\mu} Y^{\mu}_{odd} + e_{even}^{Z,\mu}p_{even}^{\mu} Y^{\mu}_{even}, \\
\end{aligned}
\end{equation}
where $E^Z_\mu$, $e^{Z,\mu}_{add}$ ($e^{Z,\mu}_{even}$) are the quantum bit error rate (QBER) and the $Z$-error rate of odd-state (even-state) signals with total a intensity of $\mu$, respectively.
If we directly estimate $q_{odd}^\mu$, $q_{even}^\mu$, $e_{odd}^{Z,\mu}$ and $e_{even}^{Z,\mu}$ from Eq.~\eqref{eqn:q parity}, Eq.~\eqref{eqn:decoy parity}, and the constriant that
\begin{equation}
q_{odd}^\mu, q_{even}^\mu, e_{odd}^{Z,\mu}, e_{even}^{Z,\mu}, Y_{odd}^\mu, Y_{even}^\mu \in [0,1],
\end{equation}
then the estimation is too loose to bound the $X$-error $E^X$ in Eq.~\eqref{eqn:EX par}.
Now, we introduce a more efficient method to estimate $q_{odd}$, $q_{even}$, $e_{odd}^Z$ and $e_{even}^Z$. Essentially, we employ the idea of the decoy-state method \cite{Lo2005Decoy}. That is, in Protocol I, Charlie adjusts the intensity $\mu$ of his prepared coherent lights. After Eve's announcement, Charlie announces the value of $\mu$.
Furthermore, Charlie randomizes the phase $\phi$ on state $\ket{\sqrt{\mu}e^{i\phi}}$ continuously from $[0,2\pi)$. In this case, the state prepared by Charlie can be written as
\begin{equation} \label{eqn:phase random}
\dfrac{1}{2\pi}\int_0^{2\pi} d\phi \ket{\sqrt{\mu} e^{i\phi}} \bra{\sqrt{\mu} e^{i\phi}} = \sum_{k=0}^{\infty} P(k) \ket{k}\bra{k}.
\end{equation}
That is, in Protocol I, if Charlie prepares $\ket{\sqrt{\mu} e^{i\phi}}$ with random phase $\phi$, this is equivalent to preparing the Fock states $\{\ket{k}\}$ with probability $P(k)$. Obviously, Fock states $\{\ket{k}\}$ are parity states. Then, by directly applying Lemma \ref{Lem:parity}, we can estimate the $X$-error by
\begin{equation} \label{eqn:EX Fock}
\begin{aligned}
E^X &= \sum_{k=0}^{\infty} q_{2k+1} e^Z_{2k+1} + \sum_{k=0}^{\infty} q_{2k} (1 - e^Z_{2k}), \\
\end{aligned}
\end{equation}
where $q_0$ is the detection caused by the vacuum signal (i.e.~dark counts) and $e_0^Z = e_0=1/2$ is the vacuum $Z$-error rate.
The source components are Fock states $\{\ket{k}\}$, whose yields $\{Y_k\}$ and $Z$-error rates $\{e^Z_k\}$ are independent of $\mu$. The fractions $q_k^\mu$ of the ``$k$-photon component'' in the final detected signals are given by
\begin{equation}
\label{eqn:q Fock}
q^{\mu}_k = P^\mu(k) \dfrac{Y_k}{Q_\mu}.
\end{equation}
The overall gain and QBER are given by
\begin{equation}
\begin{aligned}
\label{eqn:decoy Fock}
Q_\mu &= \sum_{k=0}^{\infty} P^\mu(k) Y_k, \\
E^Z_\mu Q_\mu &= \sum_{k=0}^{\infty} e_{k}^{Z} P^\mu(k) Y_k. \\
\end{aligned}
\end{equation}
The main idea of the decoy-state method is that Alice and Bob can obtain a set of linear equations in the form of Eqs.~\eqref{eqn:q Fock} and \eqref{eqn:decoy Fock} by using a few values of $\mu$. When an infinite amount of decoy states is used, Alice and Bob can estimate all the parameters $Y_{k}$ and $e_{k}^Z$ accurately, with which they can estimate $q_k^\mu$, $e^Z_k$ and the upper bound of the $X$-error by Eq.~\eqref{eqn:EX Fock}.
To apply the decoy-state method to protocols II and III, we modify the phase randomization requirements accordingly. In the decoy-state version of Protocol III, the phases $\phi_a, \phi_b$ should be randomized independently in $[0, 2\pi)$ rather than $\{0, \pi\}$. Also, there will be an additional phase-sifting condition, $|\phi_a-\phi_b|=0$ or $\pi$. Such random-phase announcement and postselection would not affect the security, with the following reasons, similar to the argument of the equivalence between protocols II and III. In particular, Observation \ref{obs:same rho} in Appendix~\ref{Sc:SecureAnnounce} still holds.
First, we want to argue that the random-phase postselection of $|\phi_a-\phi_b|=0$ or $\pi$ would not affect the security. In fact, any phase postselection would not be affected by Eve's announcement. Note that the random phases $\phi_a, \phi_b$ are determined by Alice and Bob locally. Thus, the sifted qubit pairs, $\rho_{A^\prime B^\prime}^{(n)}$, would be the same in the two cases: 1) Alice and Bob independently randomize the phases, $\phi_a, \phi_b$, and employ this phase postselection $|\phi_a-\phi_b|=0$ or $\pi$; 2) Charlie randomizes the same phases, $\phi_a=\phi_b$, to $A$ and $B$.
Second, we want to argue that the random-phase announcement would not affect the security. Eve announces the detection events before Alice and Bob's random-phase announcement. Thus, her hacking strategy cannot depend on the random phases. Note that with the postselection of $|\phi_a-\phi_b|=0$ or $\pi$, the announced phase $\phi_a$ (or $\phi_b$) can be regarded as the phase reference in Protocol III.
Third, we apply infinite decoy states to estimate $q_k^\mu$ and $e^Z_k$ accurately. In practical implementations, it is interesting to explore if finite decoy states, such as the widely applied vacuum and weak decoy states, are enough to make a valid estimation.
Now, we can modify Protocol III with continuous phase randomization $\phi_a, \phi_b$ and the decoy-state method, namely Protocol IV, as shown in Fig.~\ref{fig:Pro4}(a).
\textbf{\uline{Protocol IV}}
\begin{enumerate}
\item
State preparation:
Alice and Bob randomly select $\mu_a,\mu_b$ from a given set $\{\mu_0, \mu_1,...\}/2$. They prepare coherent state $\ket{\sqrt{\mu_a}}$ and $\ket{\sqrt{\mu_b}}$ on optical modes $A, B$ separately. They initial their qubits $A^\prime, B^\prime$ in $\ket{+i}$ and independently add random phase $\phi_a, \phi_b\in [0,2\pi)$ on the optical modes $A, B$. Alice applies the control gate $C_{\pi}$, defined in Eq.~\eqref{eq:Cpi}, to qubit $A^\prime$ and optical pulse $A$. Similarly, Bob applies $C_{\pi}$ to $B^\prime$ and $B$.
\item
Measurement: The two optical pulses $A$ and $B$, are sent to an untrusted party, Eve, who is supposed to perform interference measurement and record which detector ($L$ or $R$) clicks.
\item
Announcement: Eve announces the detection result, $L$/$R$ click or failure, for each round.
\item
Sifting: When Eve announces an $L/R$ click, Alice and Bob keep the qubits of systems $A^\prime$ and $B^\prime$. Bob applies a Pauli $Y$-gate to his qubit if Eve's announcement is $R$ click. After Eve's announcement, Alice and Bob announce their encoded intensities $\mu_a, \mu_b$ and phases $\phi_a, \phi_b$. They keep the signal if $\mu_a = \mu_b$ and $|\phi_a-\phi_b|=0$ (or $\pi$). Bob applies a Pauli $Y$-gate to his qubit if $|\phi_a-\phi_b|=\pi$.
\item
Parameter estimation: After many rounds of the above steps, Alice and Bob end up with a joint $2n$-qubit state, denoted by $\rho_{A^\prime B^\prime}^{(n)}$. They then perform random sampling on the remaining $\rho_{A^\prime B^\prime}^{(n)}$ to estimate $E^Z$ and infer $E^X$ by Eq.~\eqref{eqn:EX Fock}.
\item
Key distillation: Alice and Bob apply a standard EDP when the error rates are below a certain threshold. The distillation ratio $r$ is given by Eq.~\eqref{eqn:Shor-Preskill key rate}. Once Alice and Bob obtain $nr$ (almost) pure EPR pairs, they both perform local $Z$ measurements on the qubits to generate private keys.
\end{enumerate}
Finally, we need to reduce the entanglement-based protocol to a prepare-and-measure protocol. Following the Shor-Preskill argument, we move the key measurement in Step \ref{StepExtraction} before the EDP, the parameter estimation and the $C_{\pi}$ gates. That is, they measure the systems $A^\prime$ and $B^\prime$ at the beginning. Therefore, the entanglement-based protocol (Protocol IV) becomes the PM-QKD protocol, as shown in Fig.~\ref{fig:Pro4}(b).
\begin{figure*}[htbp]
\centering
\includegraphics[width=16cm]{Pro4.pdf}
\caption{(a) Schematic diagram of Protocol IV, where the decoy-state method is applied. The random phase $\phi_a, \phi_b\in [0, 2\pi)$. (b) Schematic diagram of PM-QKD protocol.}
\label{fig:Pro4}
\end{figure*}
\textbf{\uline{PM-QKD protocol}}
\begin{enumerate}
\item
State preparation:
Alice randomly generates a key bit $\kappa_a\in\{0,1\}$, a random phase $\phi_a\in[0,2\pi)$, and a random intensity $\mu_a\in\{\mu/2, \mu_1/2, \mu_2/2, \cdots\}$. She prepares a coherent-state optical pulse $\ket{\sqrt{\mu_a}e^{i(\phi_a + \pi\kappa_a)}}_A$. Similarly, Bob generates a key bit $\kappa_b$ and prepares $\ket{\sqrt{\mu_b}e^{i(\phi_b+\pi\kappa_b)}}_B$.
\item
Measurement: The two optical pulses $A$ and $B$ are sent to an untrusted party, Eve, who is supposed to perform interference measurement and record which detector ($L$ or $R$) clicks.
\item
Announcement: Eve announces the detection result, $L$/$R$ click or failure, for each round.
\item
Sifting: When Eve announces an $L$/$R$ click, Alice and Bob keep the key bits, $\kappa_a$ and $\kappa_b$. Bob flips his key bit $\kappa_b$ if Eve's announcement is $R$ click. After Eve's announcement, Alice and Bob announce their encoded intensities and phases $\mu_a, \phi_a; \mu_b, \phi_b$. They maintain the signal if $\mu_a = \mu_b$ and $|\phi_a - \phi_b| = 0$ or $\pi$. Bob flips his key bit $\kappa_b$ if $|\phi_a-\phi_b|=\pi$.
\item
Parameter estimation: After many rounds of the above steps, Alice and Bob end up with joint $n$ pairs of the raw key. They then perform random sampling on this raw key. With Eqs.~\eqref{eqn:q Fock} and \eqref{eqn:decoy Fock}, they estimate $q_{k}$ and $e^Z_{k}$.
\item
Key distillation: Alice and Bob apply classical communication for error correction and privacy amplification. The key distillation ratio $r$ is given by Eq.~\eqref{eqn:Shor-Preskill key rate}.
\end{enumerate}
Note that the PM-QKD protocol mentioned above is the exact PM-QKD protocol mentioned in the main text with the decoy-state method contained.
Here, in Protocol IV and the PM-QKD protocol, there are some unrealistic assumptions remaining.
\begin{enumerate}
\item
The phase of Alice and Bob's coherent state is locked; that is, the phase reference of Alice's and Bob's coherent state is the same.
\item
The phase-sifting condition, $\phi_a = \phi_b$ can be satisfied with an acceptable probability.
\end{enumerate}
We will discuss how to remove these two requirements in Appendix~\ref{Sc:SecurePost}, and, hence, make the protocol practical.
\subsection{Phase PostCompensation}\label{Sc:SecurePost}
Here, we modify the PM-QKD protocol to remove the requirement of phase locking between Alice and Bob, and also relax the postselection condition of $|\phi_a-\phi_b|=0$ or $\pi$. The method is shown in Fig.~\ref{fig:phasesel}.
First, we relax the postselection condition of $|\phi_a-\phi_b|=0$ or $\pi$ by dividing the phase interval $[0,2\pi)$ into $M$ slices $\{\Delta_j\}$ for $0\le j \le M-1$, where $\Delta_j = [2\pi j/M, 2\pi (j+1)/M)$. We change the phase postselection condition of $\phi_a = \phi_b$ to slice postselection $j_a=j_b$. Here $j_a, j_b$ are defined as
\begin{equation} \label{eq:jajb}
\begin{aligned}
j_a &= [\frac{\phi_a}{2\pi} M] \mod M, \\
j_b &= [\frac{\phi_b}{2\pi} M] \mod M,
\end{aligned}
\end{equation}
where $[\cdot]$ is the round function to output the nearest integer. That is, Alice announces the $j_a$ th slice $\Delta_{j_a}$, which her random phase $\phi_a$ falls into. Similarly, Bob announces $j_b$. The sifting condition is $|j_b - j_a| = 0$ or $M/2$. This technique is first used in the phase-encoding MDI-QKD \cite{Ma2012Alternative}. Announcing the phase slices $j_a$ and $j_b$ rather than the exact phase $\phi_a$ and $\phi_b$ will only leak less information to Eve. Thus, all the security-proof results from previous subsections apply here. Applying Eq.~\eqref{eqn:Shor-Preskill key rate}, we also need to add a phase-sifting factor $2/M$. Therefore, the final key rate is given by
\begin{equation} \label{eq:keyRateapp}
\begin{aligned}
R_{PM} &\ge \frac{2}{M} Q_\mu [ 1 - f H(E_{\mu}^Z) - H(E_{\mu}^X) ], \\
\end{aligned}
\end{equation}
where $Q_\mu$ is the total gain of the pulses, $E_{\mu}^Z$ is the overall QBER, the phase error rate $E_{\mu}^X$ is given by Eq.~\eqref{eqn:EX Fock}, and $f$ is the error correction efficiency. We need to point out that, when $M$ is too small, the misalignment caused by phase slices $\Delta_{j_a}$ and $\Delta_{j_b}$ is big and results in a large QBER $E_{\mu}^Z$.
Second, we remove the requirement of phase locking. In the phase postselection step, either Alice or Bob announces that the random phase is sufficient. Without loss of generality, we assume that Alice announces the phase and Bob performs the sifting of $|j_b - j_a| = 0$ or $M/2$. With less information announced, the key rate still holds.
Suppose that the difference between Alice's and Bob's phase references, denoted by $\phi_0\in[0,2\pi)$, is fixed but unknown. In sifting, Bob needs to figure out the value of $\phi_0$. Define an offset, where $0\le j_0\le M-1$, as
\begin{equation} \label{eq:j0}
\begin{aligned}
j_0 \equiv [\frac{\phi_0}{2\pi} M] \mod M,
\end{aligned}
\end{equation}
During sifting, Bob can estimate the offset $j_0$ for each pulse. The estimation accuracy would not affect the security. In fact, in the security proofs, we assume that Eve knows the phase references ahead. Suppose Bob compensates the offset by $j_d$, and he has the freedom to choose $j_d$ from $\{0,1,\cdots,M-1\}$. In a practical scenario, normally the phase references (and, hence, the offset $j_0$) change slowly with time. Then, Bob can figure out the proper phase compensation offset $j_d$ by minimizing the QBER from random sampling, as shown in Fig.~\ref{fig:phasesel}.
For the case where the phase reference difference $\phi_0$ varies, we can treat the changing caused by Eve. Such deviation will introduce bit errors, but not help Eve learn key information, since the variation of $\phi_0$ is independent of the key $\kappa_a, \kappa_b$. Note that in the security proof, we assume that Eve knows the phase references accurately.
With all the modifications above, we propose the following practical version of PM-QKD protocol.
\textbf{\uline{PM-QKD protocol with phase postcompensation}}
\begin{enumerate}
\item
State preparation: Alice randomly generates a key bit $\kappa_a\in\{0,1\}$, a random phase $\phi_a \in[0,2\pi)$, and a random intensity $\mu_a\in\{\mu/2, \mu_1/2, \mu_2/2, \cdots\}$. She prepares a coherent-state optical pulse $\ket{\sqrt{\mu_a}e^{i(\phi_a + \pi\kappa_a)}}_A$. Similarly, Bob generates a key bit $\kappa_b$ and prepares $\ket{\sqrt{\mu_b}e^{i(\phi_b + \pi\kappa_b)}}_B$.
\item
Measurement: The optical pulses, systems $A$ and $B$, are sent to an untrusted party, Eve, who is supposed to perform interference measure and record which detector ($L$ or $R$) clicks.
\item
Announcement: Eve announces the detection results, $L$/$R$ click or failure, for each round.
\item
Sifting: When Eve announces an $L$/$R$ click, Alice and Bob keep the key bits, $\kappa_a$ and $\kappa_b$. Bob flips his key bit $\kappa_b$ if Eve's announcement is $R$ click. After Eve's announcement, Alice and Bob announce their encoded intensities $\mu_a$ and $\mu_b$, respectively. They keep the bits if $\mu_a = \mu_b$.
\item
Parameter estimation:
Alice and Bob run the above procedures many times and then run the following procedures.
\begin{enumerate}
\item
For each bit, Alice announces the phase slice index $j_a$, given in Eq.~\eqref{eq:jajb}, and she randomly samples a certain amount of key bits and announces them for QBER testing.
\item
In the phase postcompensation method, given an offset compensation $j_d\in \{0,1,...,M/2 -1\}$, Bob sifts the sampled bits with the phase postselection condition $|j_b + j_d - j_a| \mod M = 0$ or $M/2$. For the case of $M/2$, Bob flips the key bit $\kappa_b$. After sifting, Bob calculates the QBER $E^Z$ with Alice's sampling key bits. Bob tries all possible $j_d\in\{0,1,\cdots,M-1\}$, and figures out the proper $j_d$ to minimize the sampling QBER. Using the phase postselection condition with the proper $j_d$, Bob sifts (and flips if needed) the unsampled bits and announces the locations to Alice. Alice sifts her key bits accordingly.
\item
Alice and Bob analyze the overall gain $Q_{\mu_i}$ and QBER $E^Z_{\mu_i}$ for different values of intensities $\mu_a =\mu_b =\mu_i/2$. They estimate the phase error rate $E^{(X)}_{\mu}$ by Eq.~\eqref{eqn:EX Fock}.
\end{enumerate}
\item
Key distillation: Alice and Bob apply error correction and privacy amplification. The key rate $r$ is given by Eq.~\eqref{eq:keyRateapp}.
\end{enumerate}
Here, in the phase postcompensation, Bob does not need to fix a $j_d$ for the whole experiment. Instead, he can adjust $j_d$ in real time. Given the total number of slices $M$, define the phase-stable time, $\tau_M$, to be the time period during which the phase fluctuation is smaller than $\pi/M$. Roughly speaking, Bob adjusts $j_d$ for every $\tau_M$. In practice, Bob would randomly sample some detections for tests. Here, in order to obtain a good estimation of $j_d$, Bob should sample enough detections for the offset test within $\tau_M$. Then, the phase postcompensation method would put a requirement on the detection rate.
Now, we consider a practical example. Considering the slice number $M=16$, the tolerable fluctuation should be about $\pi/M=0.196$ rad. The phase of a continuous-wave (CW) laser pulse usually fluctuates randomly, and its behavior can be modeled as a random walk. In an experimental work of testing the phase drift for a 36.5-km Mach-Zehnder interferometer \cite{minavr2008phase}, within a time duration of $0.2$ ms, the mean phase fluctuation is about $0.15$ rad. In the data presented in recent TF-QKD \cite{Lucamarini2018TF}, via the total transmission distance around $200$ km, the phase drift rate is about $3$ rad ms$^{-1}$. For a time period of $0.05$ ms, the mean phase fluctuation is about $0.15$ rad, which is less than $\pi/M$. Then, in this case, one can set $\tau_M=0.05$ ms for testing. Considering a GHz QKD system, there are $5\times10^4$ signals sent out within $\tau_M$. Using the same parameters for a longer transmission distance of $100$ km from Alice to Eve (same distance for Bob) using standard telecom fiber (0.2 dB/km), the transmission loss is $\eta = 20\,dB = 10^{-2}$, and there will be $500$ data left for post-processing, which is sufficient for sampling tests. From here, one can see that our phase postcompensation method is feasible with current technology. For a longer transmission distance, the phase postcompensation method may not be sufficient. One can use the phase calibration method as an alternative, as introduced in the main text.
\section{Model and Simulation} \label{Sc:SimuPara}
Here, we show the details for the simulations of different QKD schemes. We mainly follow the simulation model used in the literature \cite{Ma2005Practical,Ma2008PhD}. We calculate the detection probabilities of PM-QKD in Appendix~\ref{Sc:DetProb}, and then evaluate the yields, gains, and error rates in Appendix~\ref{Sc:GainQBER}. In simulations, we also compare the performance of PM-QKD with the decoy-state BB84 \cite{Bennett1984Quantum}, MDI-QKD \cite{Lo2012Measurement}, and the linear key-rate bound \cite{Pirandola2017Fundamental}. We list the used formulas and simulation parameters in Appendix~\ref{Sc:Others}.
\subsection{Detection probabilities} \label{Sc:DetProb}
The channel is assumed to be a pure loss one and symmetric for Alice and Bob with transmittance $\eta$ (with detector efficiency $\eta_d$ taken into account). We suppose that the lights from Alice and Bob faithfully interfere and are measured with dark-count rate $p_d$. Without loss of generality, we only consider the case where $\kappa_a = \kappa_b = 0$, $j_a = j_b = 0$.
In PM-QKD, the global phases $\phi_a, \phi_b$ are divided into $M$ slices. Without phase locking, the phase references for Alice and Bob can differ. Therefore, even if the announced slices meet, $|j_a - j_b| = 0$, there may be a considerable difference $\phi_\delta \equiv \phi_b-\phi_a$ between the two global phases $\phi_a, \phi_b$.
First, for a fixed $\phi_\delta$, we calculate the detection probability in the single-photon case. After Alice and Bob's encoding, the state becomes
\begin{equation}
(e^{i\phi_a}a^\dagger + e^{i\phi_b}b^\dagger)\ket{00}_{A0,B0} = (a^\dagger + e^{i\phi_\delta}b^\dagger)\ket{0}.
\end{equation}
Here, the transmittance $\eta$ includes channel losses and detection efficiencies, which transfer $a^\dagger,b^\dagger$ according to
\begin{equation}
\begin{aligned}
a^\dagger &\rightarrow \sqrt{\eta}a^\dagger + \sqrt{1-\eta}s^\dagger, \\
b^\dagger &\rightarrow \sqrt{\eta}b^\dagger + \sqrt{1-\eta}t^\dagger,
\end{aligned}
\end{equation}
where $s^\dagger, t^\dagger$ are the modes coupled to the environment. Before Eve's interference, the state is
\begin{equation}
(\sqrt{\eta}(a^\dagger + e^{i\phi_\delta}b^\dagger) + \sqrt{1 - \eta}(s^\dagger + e^{i\phi_\delta}t^\dagger))\ket{0},
\end{equation}
and after Eve's interference, the state becomes
\begin{equation}
\{\sqrt{\dfrac{\eta}{2}}[ (1+e^{i\phi_\delta})l^\dagger + (1-e^{i\phi_\delta})r^\dagger ] + \sqrt{1 - \eta}(s^\dagger + e^{i\phi_\delta}t^\dagger) \}\ket{0},
\end{equation}
where $l^\dagger$ and $r^\dagger$ are the creation operators for the modes to $L$- and $R$-detectors, respectively. Then, the detection probabilities are given by
\begin{equation} \label{eq:p1}
\begin{aligned}
p_0^{(1)} &= 1 - \eta, \\
p_l^{(1)} &= \eta \cos^2(\dfrac{\phi_\delta}{2}), \\
p_r^{(1)} &= \eta \sin^2(\dfrac{\phi_\delta}{2}), \\
p_{lr}^{(1)} &= 0,
\end{aligned}
\end{equation}
where $p_0^{(1)}$, $p_l^{(1)}$, $p_r^{(1)}$, and $p_{lr}^{(1)}$ are, respectively, the probabilities for no click, $L$ click, $R$ click, and double click for the single-photon case.
Then, we consider the $k$-photon case, for which we regard as $k$ identical and independent click events of the single-photon case; the click probabilities are given by
\begin{equation}\label{eq:pk}
\begin{aligned}
p_0^{(k)} &= (p_0^{(1)})^k,\\
p_l^{(k)} &= (p_0^{(1)} + p_l^{(1)})^k - (p_0^{(1)})^k, \\
p_r^{(k)} &= (p_0^{(1)} + p_r^{(1)})^k - (p_0^{(1)})^k, \\
p_{lr}^{(k)} &= 1 - p_0^{(k)} - p_l^{(k)} - p_r^{(k)} \\
&= 1 + (p_0^{(1)})^k -(p_0^{(1)} + p_l^{(1)})^k - (p_0^{(1)} + p_r^{(1)})^k,
\end{aligned}
\end{equation}
where $p_0^{(k)}$, $p_l^{(k)}$, $p_r^{(k)}$, and $p_{lr}^{(k)}$ are, respectively, the probabilities for no-click, $L$ click, $R$ click, and double click for the $k$-photon case.
Now, we take into account the effects caused by the detector dark counts $p_d$. Since the dark counts are independent of the photon-click events, we can draw a table for all the cases, as shown in Table \ref{tab:darkprob}. The final click probabilities for the $k$-photon case are given by
\begin{equation}\label{eq:Pk}
\begin{aligned}
P_0^{(k)} &= (1-p_d)^2 p_0^{(k)}, \\
P_L^{(k)} &= (1-p_d)^2 p_l^{(k)} + p_d(1-p_d)(p_0^{(k)}+p_l^{(k)}) \\
&= p_d(1-p_d)p_0^{(k)}+(1-p_d) p_l^{(k)}, \\
P_R^{(k)} &= (1-p_d)^2 p_r^{(k)} + p_d(1-p_d)(p_0^{(k)}+p_r^{(k)}) \\
&= p_d(1-p_d)p_0^{(k)}+(1-p_d) p_r^{(k)}, \\
P_{LR}^{(k)} &= (1-p_d)^2 p_{lr}^{(k)} + p_d(1-p_d)(p_l^{(k)} + p_r^{(k)} + 2 p_{lr}^{(k)}) + p_d^2 \\
&= (1-p_d^2) p_{lr}^{(k)} + p_d(1-p_d)(p_l^{(k)} + p_r^{(k)}) + p_d^2.
\end{aligned}
\end{equation}
\begin{table*}[htbp]
\begin{tabular}{cc|cccc}
\hline
\multicolumn{2}{c|}{\textbf{Dark count condition}} & \multicolumn{4}{c}{\textbf{Contributions to the overall click probabilities}} \\
\hline
$L$-dark count & $R$-dark count & No-click $P^{(k)}_0$ & $L$ click $P^{(k)}_L$ & $R$ click $P^{(k)}_R$ & Double-click $P^{(k)}_{LR}$ \\
\hline
$(1-p_d)$ & $(1-p_d)$ & $p^{(k)}_0$ & $p^{(k)}_l$ & $p^{(k)}_r$ & $p^{(k)}_{lr}$ \\
$(1-p_d)$ & $p_d$ & 0 & 0 & $p^{(k)}_0 + p^{(k)}_r$ & $p^{(k)}_l + p^{(k)}_{lr}$ \\
$p_d$ & $(1-p_d)$ & 0 & $p^{(k)}_0 + p^{(k)}_l$ & 0 & $p^{(k)}_r + p^{(k)}_{lr}$ \\
$p_d$ & $p_d$ & 0 & 0 & 0 & 1 \\
\hline
\end{tabular}
\caption{Probability table of clicks with dark counts present.} \label{tab:darkprob}
\end{table*}
Similarly, we can derive the formulas for detection probabilities with coherent state inputs. The states sent out by Alice and Bob are $\ket{\sqrt{\mu_a} e^{i\phi_a}}$ and $\ket{\sqrt{\mu_b} e^{i\phi_b}}$, respectively, where $\mu_a=\mu_b=\mu/2$. After channel and detection losses, the states become $\ket{\sqrt{\eta\mu_a} e^{i\phi_a}}$ and $\ket{\sqrt{\eta\mu_b} e^{i\phi_b}}$. By going through the BS, the states become
\begin{equation}
\begin{aligned}
\ket{\alpha_L} & = \ket{\dfrac{\sqrt{\eta\mu}}{2} (e^{i\phi_a} + e^{i\phi_b})}= \ket{\dfrac{\sqrt{\eta\mu}}{2} e^{i\phi_a}(1 + e^{i\phi_\delta}) }, \\
\ket{\alpha_R} & = \ket{\dfrac{\sqrt{\eta\mu}}{2} (e^{i\phi_a} - e^{i\phi_b})}= \ket{\dfrac{\sqrt{\eta\mu}}{2} e^{i\phi_a}(1 - e^{i\phi_\delta}) }. \\
\end{aligned}
\end{equation}
Then, the detection click probabilities are
\begin{equation} \label{eq:Pmu}
\begin{aligned}
P_\mu(\bar{L}) &= (1 - p_d)\exp{(-|\alpha_L|^2)} \\
&= (1 - p_d) \exp{(\, -\eta\mu\, cos^2(\dfrac{\phi_\delta}{2}) \,)}, \\
P_\mu(L)&= 1-P_\mu(\bar{L}), \\
P_\mu(\bar{R}) &= ( 1 - p_d )\exp{(-|\alpha_R|^2)} \\
&= (1 - p_d) \exp{(\, -\eta\mu\, sin^2(\dfrac{\phi_\delta}{2}) \,)}, \\
P_\mu(R) &= 1-P_\mu(\bar{R}),
\end{aligned}
\end{equation}
where $P_\mu(L)$ and $P_\mu(\bar{L})$ are the probabilities of the $L$ click and no $L$ click, respectively, and $P_\mu(R)$ and $P_\mu(\bar{R})$ are for the $R$-detector. These probabilities are similar to Eq.~\eqref{eq:Pk}. The difference is that the probabilities in Eq.~\eqref{eq:Pk} are mutually exclusive, while in Eq.~\eqref{eq:Pmu}, the probabilities $P_\mu(L)$ and $P_\mu(R)$ are independent.
All the above probability formulas are functions of the phase difference $\phi_\delta$. In the simulation, one needs to integrate $\phi_\delta$ over its probability distribution $f_{\phi_\delta}(\phi)$. Recall that the phase reference deviation between Alice and Bob is $\phi_0$. Here, with the phase postcompensation method introduced in Appendix~\ref{Sc:SecurePost}, we assume that $\phi_0$ is uniformly distributed in $[-\pi /M,\pi /M)$, denoted by
\begin{equation}
\phi_0 \sim U[-\pi /M,\pi /M).
\end{equation}
In this case, $\phi_a$ and $\phi_b$ are uniformly distributed in $[0,2\pi/M)$ and $[\phi_0, 2\pi/M + \phi_0)$, respectively,
\begin{equation}
\begin{aligned}
\phi_a &\sim U[0, 2\pi/M), \\
\phi_b &\sim U[\phi_0, 2\pi/M + \phi_0).
\end{aligned}
\end{equation}
For a fixed $\phi_0$, the probability distribution of the phase difference $\phi_\delta$ is given by
\begin{equation} \label{eq:fdelta}
f_{\phi_\delta}^{(\phi_0)}(\phi) =
\begin{cases}
(\frac{M}{2\pi})^2 [\phi + (\frac{2\pi}{M} - \phi_0)] \quad &\phi\in [\phi_0 - \frac{2\pi}{M}, \phi_0) ,\\
(\frac{M}{2\pi})^2 [-\phi + (\frac{2\pi}{M} + \phi_0)] \quad &\phi\in [\phi_0 , \phi_0 + \frac{2\pi}{M}) ,\\
0 & \textit{otherwise}.
\end{cases}
\end{equation}
Furthermore, in the simulation, one needs to take another integration of $\phi_0$ over $[-\pi /M,\pi /M)$ to get the yields, gain, and error rates.
\subsection{Yields, gain, and error rates} \label{Sc:GainQBER}
We first analyze the yield $\{Y_k\}$ for $k\ge0$ and the total gain $Q_\mu$ according to the probability formulas given in Appendix~\ref{Sc:DetProb}. Note that, in PM-QKD, both no-click and double-click events are regarded as failure detection. The yield $Y_k$ of the $k$-photon state is given by Eqs.~\eqref{eq:p1}, \eqref{eq:pk}, and \eqref{eq:Pk},
\begin{widetext}
\begin{equation} \label{eq:Yk}
\begin{aligned}
Y_k &= P_L^{(k)} + P_R^{(k)} \\
&= (1-p_d) (p_l^{(k)} + p_r^{(k)}) + 2p_d(1-p_d) p_0^{(k)} \\
&= (1-p_d) [ (1-\eta\, \sin^2(\dfrac{\phi_\delta}{2}))^k + (1-\eta\, \cos^2(\dfrac{\phi_\delta}{2}))^k ] - 2(1-p_d)^2 (1-\eta)^k \\
&\approx ( 1 - p_d) ( 1 + (1 - \eta)^k ) - 2( 1 - p_d )^2 ( 1 - \eta )^k \\
&= (1-p_d) [ 1 - (1 - 2p_d)(1 -\eta)^k ] \\
&\approx 1 - (1 - 2p_d)(1 -\eta)^k. \\
\end{aligned}
\end{equation}
\end{widetext}
Here, in the first approximation, we take the first order of a small $\phi_\delta$ and ignore $\sin^2(\dfrac{\phi_\delta}{2})=0$. In the second approximation, we omit the higher-order term $p_d(1-(1-2p_d)(1-\eta)^k)=O(p_d^2+p_d\eta)$, since normally $p_d$ is small.
The total gain $Q_\mu$ is given by Eq.~\eqref{eq:Pmu},
\begin{widetext}
\begin{equation} \label{eq:Qmu}
\begin{aligned}
Q_\mu &= P(L)P(\bar{R}) + P(\bar{L})P(R) \\
&= (1 - p_d) \exp{(\, -\eta\mu\, \sin^2(\dfrac{\phi_\delta}{2}) \,)}[1 - (1 - p_d) \exp{(\, -\eta\mu\, \cos^2(\dfrac{\phi_\delta}{2}) \,)} ] \\&\quad\quad + (1 - p_d) \exp{(\, -\eta\mu\, \cos^2(\dfrac{\phi_\delta}{2}) \,)}[1 - (1 - p_d) \exp{(\, -\eta\mu\, \sin^2(\dfrac{\phi_\delta}{2}) \,)} ] \\
&\approx (1-p_d)[1 - (1-p_d)\exp{(-\eta\mu_b)}] + p_d(1-p_d)\exp{(-\eta\mu_b)} \\
&= (1-p_d) [ 1 - (1 - 2p_d)e^{-\eta\mu} ] \\
&\approx 1 -e^{-\eta\mu} +2p_de^{-\eta\mu},
\end{aligned}
\end{equation}
\end{widetext}
where the two approximations are the same as the ones used in Eq.~\eqref{eq:Yk}. Since $\phi_\delta$ does not affect the yields $Y_k$ and gain $Q_\mu$ much with the first-order approximation, the average yields $Y_k$ and gain $Q_\mu$ over $\phi_\delta\in[-\pi /M,\pi /M)$ can be regarded as the ones when $\phi_\delta=0$. Also, the results of Eqs.~\eqref{eq:Yk} and \eqref{eq:Qmu} are consistent with the ones presented in the regular QKD model \cite{Ma2005Practical}, as shown in Eqs.~\eqref{eq:YemuBB84} and \eqref{eq:QEmuBB84}.
Then, we calculate the bit error rate for the $k$-photon signal $e^Z_k$ and the QBER $E^Z_\mu$ with coherent states input $\mu_a = \mu_b = \mu/2$. The bit error rate $e^Z_k(\phi_\delta)$ is given by
\begin{equation} \label{eq:ek}
e^Z_k(\phi_\delta) = \dfrac{P_R^{(k)}}{P_L^{(k)} + P_{R}^{(k)}},
\end{equation}
where $P_L^{(k)}$ and $P_{R}^{(k)}$ are given by Eqs.~\eqref{eq:p1}, \eqref{eq:pk}, and \eqref{eq:Pk}.
The average bit error rate of $e^Z_k$ over $\phi_\delta$ is given by the following integral,
\begin{equation} \label{eqn:eZkint}
e^Z_k =\dfrac{M}{2\pi} \int^{\pi/M}_{-\pi/M} d\phi_0 \int^{3\pi/M}_{-3\pi/M} d\phi \; f_{\phi_\delta}^{(\phi_0)} (\phi) e^Z_k(\phi), \\
\end{equation}
where $f_{\phi_\delta}^{(\phi_0)} (\phi)$ is given by Eq.~\eqref{eq:fdelta}. For the case $k=1$, we explicitly calculate the error rate $e^Z_1$, given by Eqs.~\eqref{eq:p1}, \eqref{eq:pk}, \eqref{eq:Pk}, and \eqref{eq:ek},
\begin{equation}
\begin{aligned} \label{eqn:eZ1}
e^Z_1(\phi_\delta) &= \dfrac{(1-p_d)^2 p_l^{(1)} + p_d(1-p_d)(p_0^{(1)}+p_l^{(1)})}{(1-p_d)^2 \eta + p_d(1-p_d)(2-\eta)} \\
&= \dfrac{\sin^2(\phi_\delta/2) \eta + p_d(1-\eta)}{\eta+2p_d(1-\eta)}, \\
\end{aligned}
\end{equation}
and integrate Eq.~\eqref{eqn:eZ1} with Eq.~\eqref{eqn:eZkint},
\begin{equation} \label{eq:e1}
\begin{aligned}
e^Z_1 &=\dfrac{M}{2\pi} \int^{\pi/M}_{-\pi/M} d\phi_0 \int^{3\pi/M}_{-3\pi/M} d\phi\; f_{\phi_\delta}^{(\phi_0)} (\phi) e^Z_1(\phi) \\
&= \dfrac{ e_{\delta} \eta + p_d(1-\eta)}{\eta+2p_d(1-\eta)}, \\
\end{aligned}
\end{equation}
\vskip 8mm
where
\begin{equation} \label{eq:edelta}
\begin{aligned}
e_{\delta} = \frac{\pi }{M}-\frac{M^2 }{\pi ^2}\sin ^3\left(\frac{\pi }{M}\right)
\end{aligned}
\end{equation}
can be regarded as the misalignment error rate. Equation~\eqref{eq:e1} can be understood that, if a signal causes a click, the error rate is $e_{\delta}$; and if a dark count causes a click, the error rate is $e_0=1/2$. Since the double-click events are discarded, when both a signal and a dark count cause clicks, the error rate is $e_{\delta}$. Then, we can approximate $e^Z_k$ with the same spirit, for the contributions in $Y_k$, given in Eq.~\eqref{eq:Yk},
\begin{equation} \label{eq:ekapp}
\begin{aligned}
e^Z_k &\approx \frac{p_d(1 -\eta)^k+e_\delta[1 -(1 -\eta)^k]}{Y_k}. \\
\end{aligned}
\end{equation}
Here, note that in Eq.~\eqref{eq:ekapp}, we ignore the double clicks caused by multiphoton signals, which would further reduce the misalignment error and, hence, the value of $e^Z_k$. The QBER $E^Z_\mu(\phi_\delta)$ for a given $\phi_\delta$ is given by substituting Eq.~\eqref{eq:Pmu},
\begin{widetext}
\begin{equation}\label{eq:EZ}
\begin{aligned}
E^Z_\mu(\phi_\delta) &= \dfrac{P(\bar{L})P(R)}{P(\bar{L})P(R) + P(L)P(\bar{R})} \\
&= \frac{1}{Q_\mu}{(1 - p_d) \exp{(\, -\eta\mu\, \cos^2(\dfrac{\phi_\delta}{2}))}\{1-(1 - p_d) \exp[-\eta\mu \sin^2(\frac{\phi_\delta}{2})]\}} \\
&= \frac{1}{Q_\mu} (1-p_d) \{\exp{[\, -\eta\mu\, \cos^2(\dfrac{\phi_\delta}{2})]}-(1 - p_d) e^{-\eta\mu}\} \\
&\approx \frac{e^{-\eta\mu}}{Q_\mu} [p_d + \eta\mu\sin^2(\frac{\phi_\delta}{2})] \\
\end{aligned}
\end{equation}
\end{widetext}
similar to Eq.~\eqref{eq:e1},
\begin{equation} \label{eq:Emusimu}
\begin{aligned}
E^Z_\mu &= \dfrac{M}{2\pi}\int^{\pi/M}_{-\pi/M} d\phi_0 \int^{3\pi/M}_{-3\pi/M} d\phi\; f_{\phi_\delta}^{(\phi_0)} (\phi) E^Z_\mu(\phi_\delta) \\
&\approx \frac{(p_d + \eta\mu e_\delta)e^{-\eta\mu}}{Q_\mu},
\end{aligned}
\end{equation}
where $e_\delta$ is given in Eq.~\eqref{eq:edelta}. The results of Eqs.~\eqref{eq:Emusimu} and \eqref{eq:e1} are consistent with the one presented in the regular QKD model \cite{Ma2005Practical}, as shown in Eqs.~\eqref{eq:YemuBB84} and \eqref{eq:QEmuBB84}, with a slight difference. The difference is caused by how the double-click events are processed. In BB84, Alice and Bob need to randomly assign a bit to double clicks, while in PM-QKD, double clicks are discarded.
Now, we can evaluate the key rate with the above model. Let us restate the key-rate formula, Eq.~\eqref{eq:keyRateapp},
\begin{equation}
R_{PM} \ge \frac{2}{M} Q_\mu [ 1 - f H(E_{\mu}^Z) - H(E_{\mu}^X) ],
\end{equation}
where the phase error rate is given by Eq.~\eqref{eqn:EX Fock},
\begin{equation} \label{eq:EmuX}
\begin{aligned}
E_{\mu}^X &= \sum_{k=0}^{\infty} q_{2k+1} e^Z_{2k+1} + \sum_{k=0}^{\infty} q_{2k} (1 - e^Z_{2k}), \\
&\le q_0 e^Z_0 +\sum_{k=0}^\infty e_{2k+1}^Z q_{2k+1} + (1 - q_0 - q_{odd}).
\end{aligned}
\end{equation}
In simulation, $Q_\mu$ and $E_{\mu}^Z$ are given by Eqs.~\eqref{eq:Qmu} and \eqref{eq:Emusimu}. The fractions, $\{q_k\}$ with $k\ge0$, of different photon components $k$ contributing to the valid detections ($L/R$ clicks) are given by
\begin{equation} \label{eq:q}
\begin{aligned}
q_k = \dfrac{P(k)Y_k}{Q_\mu} = e^{-\mu}\dfrac{\mu^k}{k!}\dfrac{Y_k}{Q_\mu}.
\end{aligned}
\end{equation}
For the odd-photon-number component, we have
\begin{equation} \label{eq:qevenodd}
\begin{aligned}
q_{odd} &= \sum_{k=0}^{\infty} q_{2k+1} \\
&= \frac{1}{Q_\mu}\sum_{k=0}^{\infty} \frac{Y_{2k+1} \mu^{2k+1} e^{-\mu}}{(2k+1)!} \\
&= \frac{e^{-\mu}}{Q_\mu}\sum_{k=0}^{\infty} \{ \dfrac{\mu^{2k+1}}{(2k+1)!} - (1-2p_d)\dfrac{[(1-\eta)\mu]^{2k+1}}{(2k+1)!} \} \\
&= \frac{e^{-\mu}}{Q_\mu}\{\sinh(\mu) - (1-2p_d)\sinh[(1-\eta)\mu] \}.
\end{aligned}
\end{equation}
To simplify the simulation, we explicitly calculate the $e^Z_k$ and $q_k$ for $0\le k\le 5$. Further zooming into Eq.~\eqref{eq:EmuX}, we have
\begin{equation} \label{eq:EmuXzoom}
\begin{aligned}
E_{\mu}^X &= \sum_{k=0}^{\infty} q_{2k+1} e^Z_{2k+1} + \sum_{k=0}^{\infty} q_{2k} (1 - e^Z_{2k}), \\
&\le q_0 e^Z_0 +(q_1 e^Z_1 + q_3 e^Z_3 + q_5 e^Z_5) + (1 - q_0 - q_1 - q_3 -q_5),
\end{aligned}
\end{equation}
where $e_k^Z$ and $q_k$ are given by Eqs.~\eqref{eq:ek} and \eqref{eq:q}, respectively.
Here, we attach the MALTAB code of PM-QKD key rate for reference.
\twocolumngrid
\begin{lstlisting}[language={Matlab}]
function [RPM] = PMKey(eta,mu,pd,M,Ef)
Y0 = 2*pd;
Y1 = 1 - (1 - 2*pd).*(1 - eta);
Y3 = 1 - (1 - 2*pd).*(1 - eta)^3;
Y5 = 1 - (1 - 2*pd).*(1 - eta)^5;
Qmu = 1 - (1 - 2*pd).*exp(-mu.*eta);
q0 = Y0.*exp(-mu)./Qmu;
q1 = Y1.*mu.*exp(-mu)./Qmu;
q3 = Y3.*(mu.^3).*exp(-mu)./(factorial(3)*Qmu);
q5 = Y5.*(mu.^5).*exp(-mu)./(factorial(5)*Qmu);
e0 = 0.5;
edelta = pi/M - (M/pi)^2 * (sin(pi/M))^3;
e1Z = ( pd*(1-eta) + edelta*(1 - (1-eta) ) )/Y1;
e3Z = ( pd*(1-eta)^3 + edelta*(1 - (1-eta)^3) )/Y3;
e5Z = ( pd*(1-eta)^5 + edelta*(1 - (1-eta)^5) )/Y5;
EZPM = ( pd + eta*mu*edelta ).*exp(-eta*mu)./Qmu;
EX = q0.*e0 + q1.*e1Z + q3.*e3Z + q5.*e5Z + (1 - q0 - q1 - q3 - q5);
EX = min(EX,0.5);
rPM = -Ef.*h(EZPM) + 1 - h(EX);
RPM = (2/M).*Qmu.*rPM;
RPM = max(RPM,0);
end
function [entropy] = h(p)
entropy = -p.*log2(p)-(1-p).*log2(1-p);
entropy(p<=0 | p>=1) = 0;
end
\end{lstlisting}
\subsection{Simulation formulas for BB84 and MDI-QKD protocol and simulation parameters} \label{Sc:Others}
We compare our derived key rate with that of the prepare-and-measure BB84 \cite{Bennett1984Quantum} protocol, whose key rate is given by the GLLP-decoy method \cite{gottesman04}.
The key rate of the decoy-state BB84 protocol is given by \cite{Lo2005Decoy}
\begin{equation}
R_{BB84} = \dfrac{1}{2} Q_\mu\{- f H(E_{\mu}^Z) + q_1[1 - H(e^X_1)] \}, \\
\end{equation}
where $1/2$ is the basis sifting factor.
In the simulation, the yield and error rates of the $k$-photon component are given by \cite{Ma2008PhD}
\begin{equation} \label{eq:YemuBB84}
\begin{aligned}
Y_k &= 1-(1-Y_0)(1-\eta)^{k}, \\
e_k &= e_d + \frac{(e_0-e_d)Y_0}{Y_k}, \\
\end{aligned}
\end{equation}
where $e_d$ is the intrinsic misalignment error rate caused by a phase reference mismatch. The gain and QBER are given by
\begin{equation} \label{eq:QEmuBB84}
\begin{aligned}
Q_\mu &= \sum_{k=0}^{\infty} \frac{\mu^k e^{-\mu}}{k!} Y_k, \\
&= 1-(1-Y_0)e^{-\eta\mu}, \\
E_\mu &= \sum_{k=0}^{\infty} \frac{\mu^k e^{-\mu}}{k!} e_kY_k, \\
&= e_d + \frac{(e_0-e_d)Y_0}{Q_\mu}, \\
\end{aligned}
\end{equation}
where $Y_0=2p_d$ and $e_0 = 1/2$.
The key rate of MDI-QKD is given by \cite{Lo2012Measurement}
\begin{equation}
\begin{aligned}
R_{MDI} = \dfrac{1}{2}\{ Q_{11} [1 - H(e_{11})] -f Q_{rect} H(E_{rect}) \},
\end{aligned}
\end{equation}
where $Q_{11} = \mu_a\mu_b e^{-\mu_a-\mu_b}Y_{11}$ and $1/2$ is the basis sifting factor. We take this formula from Eq.~(B27) in Ref.~\cite{Ma2012Alternative}. In simulation, the gain and error rates are given by
\begin{widetext}
\begin{equation}
\begin{aligned}
Y_{11} & = (1-p_d)^2 [ \dfrac{\eta_a\eta_b}{2} + (2\eta_a + 2\eta_b -3\eta_a\eta_b)p_d + 4(1-\eta_a)(1-\eta_b)p_d^2 ], \\
e_{11} & = e_0Y_{11} - (e_0- e_d)(1-p_d^2)\dfrac{\eta_a\eta_b}{2}, \\
Q_{rect} & = Q_{rect}^{(C)} + Q_{rect}^{(E)}, \\
Q_{rect}^{(C)} & = 2(1-p_d)^2 e^{-\mu^\prime/2} [1 - (1-p_d)e^{-\eta_a\mu_a/2}][1 - (1-p_d)e^{-\eta_b\mu_b/2}], \\
Q_{rect}^{(E)} & = 2p_d(1-p_d)^2 e^{-\mu^\prime/2}[I_0(2x) - (1-p_d)e^{-\mu^\prime/2}]; \\
E_{rect} Q_{rect} & = e_d Q_{rect}^{(C)} + (1 - e_d) Q_{rect}^{(E)}, \\
\end{aligned}
\end{equation}
\end{widetext}
Here,
\begin{equation} \label{eqn:MDI mu x}
\begin{aligned}
\mu^\prime & = \eta_a \mu_a + \eta_b \mu_b, \\
x & = \frac12\sqrt{\eta_a \mu_a \eta_b \mu_b}, \\
\end{aligned}
\end{equation}
where $\mu^\prime$ denotes the average number of photons reaching Eve's beam splitter, and $\mu_a = \mu_b = \mu/2, \eta_a = \eta_b = \eta$. We take these formulas from Eqs.~(A9),~(A11),~(B7),~and (B28)-(B31) in Ref.~\cite{Ma2012Alternative}.
In 2014, Takeoka\textit{~et~al.} derived an upper bound of the key rate of the point-to-point-type QKD protocols \cite{takeoka2014fundamental},
\begin{equation}
R_{TGW} = -\log_2(\dfrac{1-\eta}{1+\eta}).
\end{equation}
Later, Pirandola\textit{~et~al.} established a tight upper bound \cite{Pirandola2017Fundamental},
\begin{equation}\label{eq:plob}
R_{PLOB} = - \log_2(1-\eta),
\end{equation}
which is the linear key-rate bound used in the main text. Note that Eq.~\eqref{eq:plob} is the secret key capacity of the pure loss channel, as it coincides with the lower bound previously known \cite{pirandola2009direct}.
\begin{comment}
We list the parameters used in the simulation in Table.~\ref{tab:simupara}. The dark count rate $p_d$ is from the work \cite{Tang2014MDI200}, and the other parameters are set to be typical values.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{cccccc}
\hline
Parameters & Values \\
\hline
Dark count rate $p_d$ & $8\times 10^{-8}$ \\
Error correction efficiency $f$ & $1.15$ \\
Detector efficiency $\eta_d$ & $14.5\%$ \\
Number of phase slices $M$ & $16$ \\
Misalignment error $e_d$ & $1.5\%$ \\
\hline
\end{tabular}
\end{center}
\caption{List of simulation parameters.} \label{tab:simupara}
\end{table}
\end{comment}
\section{Beam-splitting Attack: Invalidation of the photon number channel model} \label{sc:attack}
Here, we argue that the tagging technique does not work for PM-QKD, by showing that the lower bound of the key rate from naively employing the tagging technique can be higher than an upper bound derived from a specific attack, in some parameter regime. The failure of the tagging technique stems from the failure of the photon number channel model used in the security proof \cite{Ma2008PhD}. In other words, the photon number channel model does not hold for the case when the random phases are announced during postprocessing.
\subsection{Key rate with or without the photon number channel}
Assuming the existence of the photon number channel model in PM-QKD, we can apply the tagging method and obtain a key-rate formula following the GLLP analysis \cite{gottesman04,Lo2005Decoy},
\begin{equation}\label{eq:keygllp}
r_{GLLP}=q_1[1-H(e^X_1)]- f H(E_{\mu}^Z),
\end{equation}
where $q_1$ is the single-photon detection ratio, $e^X_1$ and $E_{\mu}^Z$ are the single-photon phase error rate and total QBER, and $f$ is the error correction efficiency.
As for comparison, our PM-QKD key-rate formula is given by Eq.~\eqref{eqn:Shor-Preskill key rate},
\begin{equation}
r_{PM} = 1 - H(E^Z_\mu) - H(E^X_\mu),
\end{equation}
where $E_{\mu}^X$ is bounded by Eq.~\eqref{eq:EmuXzoom}
\begin{equation}
\begin{aligned}
E_{\mu}^X &= \sum_{k=0}^{\infty} q_{2k+1} e^Z_{2k+1} + \sum_{k=0}^{\infty} q_{2k} (1 - e^Z_{2k}), \\
&\le q_0 e^Z_0 +(q_1 e^Z_1 + q_3 e^Z_3 + q_5 e^Z_5) + (1 - q_0 - q_1 - q_3 -q_5).
\end{aligned}
\end{equation}
\subsection{Beam-splitting attack}
Now, we consider a beam-splitting attack (BS-attack), where Eve sets beam splitters with transmittance $\eta$ on both Alice and Bob's side to simulate a lossy channel. She intercepts the pulses on $A$ and $B$, regardless of the keys and bases of the pulses, and stores the reflected lights on her quantum memories, modes $A0$ and $B0$. Eve interferes the two transmitted pulses, modes $A1$ and $B1$, and announces the results. After Alice's and Bob's phase announcements, Eve performs an unambiguous state discrimination (USD) \cite{jaeger1995optimal} on the states on modes $A0$ and $B0$ separately to guess the key information. Since the states on $A0(B0)$ and $A1(B1)$ are uncorrelated because of the beam splitting of coherent states, the BS-attack will not introduce any error or other detectable effects. Apparently, this BS-attack is an individual attack. We will calculate Eve's successful probability for guessing the key bits unambiguously, and calculate the mutual information $I(\kappa_{a(b)}:E)$, from which we can derive an upper bound on the secure key rate.
In PM-QKD, Alice and Bob prepare coherent states $\{\ket{\sqrt{\mu_a}e^{i(\phi_a+\kappa_a)}}_A$ and $\ket{\sqrt{\mu_b}e^{i(\phi_b+\kappa_b)}}_B\}$, where $\mu_a = \mu_b = \mu/2$, $\phi_a,\phi_b\in[0,2\pi)$ are the random phases, and $\kappa_a,\kappa_b\in\{0,\pi\}$ are the key information. The states on system $A0,B0$ reflected by Eve's beam splitters are given by
\begin{equation}
\begin{aligned}
\ket{\sqrt{(1-\eta)\mu_a} e^{i(\phi_a+\kappa_a)}}_{A0}, \\
\ket{\sqrt{(1-\eta)\mu_b} e^{i(\phi_b+\kappa_b)}}_{B0}.
\end{aligned}
\end{equation}
The two remaining pulses on the system $A1,B1$, used for interference are given by
\begin{equation}
\begin{aligned}
\ket{\sqrt{\eta\mu_a} e^{i (\phi_a+\kappa_a)}}_{A1}, \\
\ket{\sqrt{\eta\mu_b} e^{i (\phi_b+\kappa_b)}}_{B1}.
\end{aligned}
\end{equation}
Suppose Eve interferes systems $A1$ and $B1$; then the total gain $Q_\mu$ is given by
\begin{equation}
\label{eqn:gain}
Q_\mu = 1 - e^{-\eta\mu}.
\end{equation}
Here, we consider the signal with $\phi_a = \phi_b$, and Eve holds perfect single-photon detectors.
Without loss of generality, we consider Eve's attack on system $A0$.
If Eve cannot get the sifted phase information $\phi_a$, then no matter the value of key phases $\kappa_a$, the state on $A0$ from Eve's perspective is
\begin{widetext}
\begin{equation}
\dfrac{1}{2\pi}\int_{0}^{2\pi} d\phi_a \ket{\sqrt{(1-\eta)\mu_a}e^{i(\phi_a+\kappa_a)}}_{A0}\bra{\sqrt{(1-\eta)\mu_a}e^{i(\phi_a+\kappa_a)}} = \sum_{k=0}^{\infty} P^{(1-\eta)\mu_a}(k) \ket{k}_{A0}\bra{k},
\end{equation}
\end{widetext}
where
\begin{equation}
\begin{aligned}
P^{(1-\eta)\mu_a}(k) = \frac{[(1-\eta)\mu_a]^k}{k!}e^{-(1-\eta)\mu_a} .
\end{aligned}
\end{equation}
In this case, Eve cannot obtain any information about $\kappa_a$. While in PM-QKD, the phase $\phi_a$ will be announced. In this case, Eve can first rotate the states on system $A0$ by $-\phi_a$. Now, the state becomes
\begin{equation}
\begin{aligned}
\dfrac{1}{2}&(\ket{\sqrt{(1-\eta)\mu_a}}_{A0}\bra{\sqrt{(1-\eta)\mu_a}} \\
&+ \ket{-\sqrt{(1-\eta)\mu_a}}_{A0}\bra{-\sqrt{(1-\eta)\mu_a}}).
\end{aligned}
\end{equation}
Then, Eve only needs to determine whether the phase $\kappa_a$ is $0$ or $\pi$ to obtain the key information. As shown in Ref.~\cite{jaeger1995optimal}, if two pure states $\ket{p}$ and $\ket{q}$ are prepared with the same \textit{a priori} $1/2$, the maximum probability of \emph{unambiguous} discrimination is
\begin{equation}
P_{des} = 1 - |\braket{p|q}|.
\end{equation}
Therefore, the probability of successfully distinguishing the states $\ket{\sqrt{(1-\eta)\mu_a}}$ and $\ket{-\sqrt{(1-\eta)\mu_a}}$ is given by
\begin{equation}
P_{suc} = 1 - |\braket{\sqrt{(1-\eta)\mu_a}|-\sqrt{(1-\eta)\mu_a}}| = 1 - e^{-(1-\eta)\mu}.
\end{equation}
Note that Eve already know the value $\kappa_a - \kappa_b$ with interference results on $A1, B1$. Eve only needs to learn either of $\kappa_a$ or $\kappa_b$. Thus, Eve's successful unambiguous measurement probability is
\begin{equation}
P_{BS} = 1- (1-P_{suc})^2 = 1 - e^{-2(1-\eta)\mu}.
\end{equation}
If the light intensity $\mu$ is large enough, $P_{BS} = 1$, then Eve can learn the key information with a high probability.
With unambiguous state discrimination, the mutual information of Eve's measurement result (denoted by variable $E$) and Alice's and Bob's key $\kappa_{a(b)}$ is
\begin{equation}
I(\kappa_{a(b)}:E) = P_{BS} =1 - e^{-2(1-\eta)\mu}.
\end{equation}
Suppose that there is no extra error induced by other factors; then the BS-attack provides an upper bound on the key rate,
\begin{equation} \label{eqn: rBS}
\begin{aligned}
r_{BS} & = [1-I(\kappa_{a(b)}:E)] \\
& = e^{-2(1-\eta)\mu} \\
& = \frac{e^{-2\mu}}{(1-Q_\mu)^2}.
\end{aligned}
\end{equation}
Here, the third equality is based on Eq.~\eqref{eqn:gain}.
\subsection{Comparison}
Now we make a simulation to compare the key rate in the following cases: with the ``tagging'' method, by our security proof, and the upper bound under BS-attack.
In the discussion, we neglect the misalignment error and dark counts.
First, under BS-attack, the yield is given by
\begin{equation}
Y_k = 1 - (1-\eta)^k,
\end{equation}
and the fraction of the $k$-photon component is
\begin{equation}
q_k = (e^{-\mu}\dfrac{\mu^k}{k!})\dfrac{Y_k}{Q_\mu} = [1 - (1-\eta)^k]\dfrac{e^{-\mu}\mu^k}{k! Q_\mu}.
\end{equation}
Note that $q_0=0$.
Second, in the BS-attack scenario, Eve's operations will not introduce any error; hence,
\begin{equation}
\begin{aligned}
e^Z_k &= 0, \quad\forall k \ge 0, \\
E^Z_\mu &= 0. \\
\end{aligned}
\end{equation}
Then Eqs.~\eqref{eq:keygllp}, ~\eqref{eqn:Shor-Preskill key rate}, and ~\eqref{eq:EmuXzoom} can be simplified to
\begin{equation}
\begin{aligned} \label{eqn: runderBS}
r_{GLLP} &= q_1 \\
&= \eta\mu e^{-\mu}, \\
r_{PM} &= [1 - H(E^X)] \\
&\ge [1 - H(1 - q_1 - q_3 - q_5)].
\end{aligned}
\end{equation}
We compare the key-rate formula $r_{GLLP}, r_{PM}$ in Eq.~\eqref{eqn: runderBS} with the key-rate upper bound by BS-attack $r_{BS}$ in Eq.~\eqref{eqn: rBS}. We set the total light intensity to be a typical value $\mu=0.5$, and we adjust $\eta$ to compare the key-rate performance. As shown in Fig.~\ref{fig:BSattackMu05}, the GLLP ``tagging'' formula for the ``single-photon'' component cannot hold under the BS-attack for a transmittance $\eta<0.6$.
If we fix the transmittance $\eta=0.2$ and adjust $\mu$, Fig.~\ref{fig:BSattackEta02} shows that the GLLP tagging formula cannot hold under the BS-attack.
\begin{figure}[hbtp]
\centering
\resizebox{8cm}{!}{\includegraphics{BSattackMu05.pdf}}
\caption{Key-rate comparison with fixed light intensity $\mu=0.5$. Here we can see that the GLLP tagging formula for the single-photon component cannot hold under the BS-attack when the transmittance $\eta<0.6$.} \label{fig:BSattackMu05}
\end{figure}
\begin{figure}[hbtp]
\centering
\resizebox{8cm}{!}{\includegraphics{BSattackEta02.pdf}}
\caption{Key-rate comparison with fixed transmittance $\eta=0.2$. The GLLP tagging formula for the single-photon component cannot hold under the BS-attack.} \label{fig:BSattackEta02}
\end{figure}
The BS-attack results serve to invalidate the tagging method even the single-photon component cannot exist, since the announcement of phase $\phi_a,\phi_b$ will leak more information of the key bits $\kappa_a $ and $\kappa_b$.
Recently, Wang\textit{~et~al.} proposed an eavesdropping strategy to the TF-QKD protocol \cite{Wang2018Effective}. The attack can also be performed against the PM-QKD protocol. Under such an attack,
Eve can learn all the key bits. The key rate provided by the GLLP formula, Eq.~\eqref{eq:keygllp}, is 0.5 (for all the clicked signals) while that given by our security proof is strictly 0. This also shows the invalidation of the GLLP key-rate formula, and our security proof is still valid under such an attack.
\section{Comparison with other phase-encoding MDI-QKD}
Here, we compare three related phase-encoding MDI-QKD protocols, including the previous phase-encoding MDI-QKD (Scheme I in Ref.~\cite{Tamaki2012PhaseMDI}), the recently proposed TF-QKD \cite{Lucamarini2018TF}, and our protocol (PM-QKD). For the simplicity of the statements, we assume that the phase reference of Alice and Bob is locked in all the protocols.
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{tfandpm.pdf}
\caption{(a) Schematic diagram of phase-encoding MDI-QKD (Scheme I in Ref.~\cite{Tamaki2012PhaseMDI}). (b) Schematic diagram of TF-QKD, which is the decoy-state version of the scheme in (a). (c) Schematic diagram of $d$-phase PM-QKD.}
\label{fig:mdivstfvspm}
\end{figure}
\subsection{Phase-encoding MDI-QKD} \label{AppSub:PEMDI}
In the phase encoding MDI-QKD (Scheme I in Ref.~\cite{Tamaki2012PhaseMDI}), Alice generates two random bits $\kappa_a, \beta_a$ as the key and the random choice of the $X$ or $Y$ basis, respectively, and then generates a coherent pulse $\ket{\sqrt{\mu/2} e^{i(\pi\kappa_a + \pi\beta_a/2)} }_A$. Similarly, Bob generates $\ket{\sqrt{\mu/2} e^{i(\pi\kappa_b + \pi\beta_b/2)} }_B$. They send their coherent pulses to Eve, who is supposed to perform an interference measurement and announce the detection results [Fig.~\ref{fig:mdivstfvspm}(a)]. After Eve's announcement, Alice and Bob announce the basis information $\beta_a, \beta_b$ and perform basis sifting.
Note that there is only single-photon detection in this protocol. Therefore, the number of clicked signals scales with $O(\sqrt{\eta})$, where $\eta$ is the total transmittance between Alice and Bob. However, there is a considerable source flaw caused by using a coherent state as an approximation of an ideal single photon state, resulting in a final key rate of $O(\eta)$.
\subsection{TF-QKD}
To avoid the performance deterioration caused by the source flaw in the phase-encoding MDI-QKD protocol above, a natural idea is to apply the decoy-state method \cite{Lo2005Decoy}, that is, to perform phase randomization, and to estimate the fraction of $q_1$ and the error rate $e_1$ of the single-photon component.
Recently, Lucamarini\textit{~et~al.} modified the phase-encoding QKD protocol, namely TF-QKD \cite{Lucamarini2018TF}. As shown in Fig.~\ref{fig:mdivstfvspm}(b), Alice generates two random bits $\kappa_a, \beta_a$ as the key, chooses randomly the $X$ or $Y$ basis, and modulates another random phase $\phi_a$ for the phase randomization in the decoy-state method. She generates a coherent pulse $\ket{\sqrt{\mu/2} e^{i(\pi\kappa_a + \pi\beta_a/2 + \phi_a)} }_A$. Similarly, Bob generates a coherent pulse $\ket{\sqrt{\mu/2} e^{i(\pi\kappa_b + \pi\beta_b/2 + \phi_b)} }_B$. They send their coherent pulses to Eve, who is supposed to perform the interference measurement and announce the detection results. After Eve's announcement, Alice and Bob announce $\beta_a, \beta_b; \phi_a, \phi_b$ and perform basis sifting and phase sifting.
\subsection{$d$-phase phase-matching QKD}
In the $d$-phase PM-QKD, Alice first generates a random integer $\kappa_a\in\{0, 1,..., d-1\}$ and then prepares a coherent pulse $\ket{\sqrt{\mu/2} \exp{(i\kappa_a \dfrac{2\pi}{d})}}_A$. Similarly, Bob generates $\kappa_b\in\{0,1,..., d-1\}$ and prepares a similar state $\ket{\sqrt{\mu/2} \exp{(i\kappa_b \dfrac{2\pi}{d}})}_B$. They send their coherent pulses to Eve, who is supposed to perform an interference measurement and announce the detection results. If Eve announces detector $L/R$ clicks, Alice and Bob keep the numbers $\kappa_a, \kappa_b$ and which detector clicks.
After many rounds of the steps above, Alice and Bob randomly select some of the data, announce them, and calculate the $L/R$ detection probability for each case. By the estimated probabilities, they calculate the key rate and extract private keys.
As we can see, after Eve's announcement, Alice and Bob will share some mutual information. While Eve cannot fully learn the variable $\kappa_a$, the information-secure key can be generated between Alice and Bob.
The PM-QKD protocol in the main text and Appendix \ref{Sc:SecureProof} corresponds to the case of $d=2$, which is combined with a random phase announcement and the decoy-state method. In Appendix \ref{Sc:SecureProof}, we completed the proof for the PM-QKD of $d=2$ with a random phase announcement. The security proof for this generalized case is left for future work.
\subsection{Comparison of different protocols}
In all three protocols, phase-encoding MDI-QKD \cite{Tamaki2012PhaseMDI}, TF-QKD \cite{Lucamarini2018TF}, and PM-QKD, single-detection clicks on the untrusted node are used as successful measurement events, whose rate scales with $O(\sqrt{\eta})$. Technically, this is the precondition of the key rate having square-root scaling\cite{Lucamarini2018TF}.
Nevertheless, in the phase-encoding MDI scheme, without phase randomization, the single-photon source can be approximated by two weak coherent states, which decreases significantly the key rate of the protocol to $O(\eta)$. In TF-QKD, a single-qubit view is taken and a higher key rate is targeted. Unfortunately, the phase announcement invalidates the existence of the photon number channel model \cite{Ma2008PhD}, and, hence, no enhancement can be claimed by directly applying the decoy-state method onto the phase-encoding MDI-QKD scheme.
Here, in the phase-matching QKD scheme, we switch from the qubit-based view to the optical-mode-based view. As shown above, our proposed phase-matching QKD follows the phase-encoding MDI-QKD scheme and the TF-QKD scheme, by modifying the encoding and basis choice. The name ``phase-matching (MDI-)QKD'' follows ``phase-encoding MDI-QKD''. We removed ``MDI'' (which is not the key point to our work) to make the name concise.
The PM-QKD protocol in the main text and Appendix \ref{Sc:SecureProof} is the one with $d=2$ and random phase announcement. Though the current practical implementations of phase-encoding MDI-QKD / TF-QKD and PM-QKD do resemble each other, the security scenarios are quite different. The TF-QKD can be taken as an extension of the BB84 protocol, which utilizes the single-photon source and is a discrete-variable QKD protocol. In the future, many discrete-variable QKD design techniques, such as a six-state protocol and reference-frame-independent protocol, can be employed in this framework. On the other hand, in PM-QKD, we focus more on the optical modes rather than single-photon states (qubits). It would be interesting to see whether the security proof techniques developed in continuous-variable QKD can be applied to PM-QKD.
\subsection{Recent related works on TF-QKD}
There are some recent works based on TF-QKD. Tamaki\textit{~et~al.} provided a security proof of the TF-QKD protocol \cite{tamaki2018information}. They modified the TF-QKD protocol by introducing a ``test mode'', where Alice and Bob do not announce the phase information and photon-number-channel model holds. The original TF-QKD protocol is called ``code mode''. Following the original security proof of phase-encoding MDI-QKD \cite{Tamaki2012PhaseMDI}, they estimated the phase error $E_X$ by considering the imbalance of different bases. With a fair sampling argument on the test mode and the code mode, the basis imbalance in the code mode can be estimated by the one in the test mode. A square-root-scaling key rate has been derived, but it is significantly lower than ours. Note that, there are still two bases in Tamaki\textit{~et~al.}'s protocol and security proof, which follows the qubit-based view of BB84. The two-basis requirement in the security proof implies that it cannot be applied to our PM-QKD protocol directly, which also highlights the difference between the phase-encoding MDI-QKD / TF-QKD and PM-QKD.
Wang\textit{~et~al.} proposed a ``sending or not sending'' TF-QKD protocol \cite{Wang2018Sending}, which aims to utilize the $Z$-basis of single-photon for key generation, following the viewpoint of TF-QKD. However, the definition of the $Z$-basis encoding seems confusing. In the original TF-QKD protocol, the definitions of the $X$ or $Y$ bases refer to the ancillary bits rather than the real single photon. Also, the $Z$-basis encoding on the real photon does not correspond to a $Z$-basis encoding on the ancillary bits. This protocol needs further studies.
\end{appendix}
\normalem
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"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How to place a changeable html element on ther same line? This is a javascript beginner question. I just want to put some text on the same line (as rendered in html), but where I can change the second element. Something like
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style="display: inline"
Function to solve: <div id="eq" style="display: inline" >equation</div>
A: Change <div id="eq">equation</div> to <span id="eq">equation</span>
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}\label{sec:intro}
A $t$-\emph{design} $\mathcal{D}=(\mathcal{P},\mathcal{B})$ with parameters $(v,k,\lambda)$ is an incidence structure consisting of a set $\mathcal{P}$ of $v$ \emph{points}, and a set $\mathcal{B}$ of $k$-element subsets of $\mathcal{P}$, called \emph{blocks}, such that every $t$-element subset of points lies in exactly $\lambda$ blocks. The design $\mathcal{D}$ is \emph{non-trivial} if $t < k < v-t$, and is \emph{symmetric} if $|\mathcal{B}| = v$. By \cite[Theorem 1.1]{a:Camina94}, if $\mathcal{D}$ is symmetric and non-trivial, then $t\leq 2$, see also \cite[Theorem 1.27]{b:Hugh-design}. Thus we study non-trivial symmetric $2$-designs with parameters $(v,k,\lambda)$ which we simply call non-trivial \emph{symmetric $(v,k,\lambda)$ designs}.
A \emph{flag} of $\mathcal{D}$ is an incident pair $(\alpha,B)$, where $\alpha$ and $B$ are a point and a block of $\mathcal{D}$, respectively. An \emph{automorphism} of a symmetric design $\mathcal{D}$ is a permutation of the points permuting the blocks and preserving the incidence relation. An automorphism group $G$ of $\mathcal{D}$ is called \emph{flag-transitive} if it is transitive on the set of flags of $\mathcal{D}$. If $G$ leaves invariant a non-trivial partition of $\mathcal{P}$, then $G$ is said to be \emph{point-imprimitive}, otherwise, $G$ is called \emph{point-primitive}. We here adopt the standard notation as in \cite{b:Atlas,b:Wilson} for finite simple groups of Lie type, for example, we use $PSL_{n}(q)$, $PSp_{n}(q)$, $PSU_{n}(q)$, $P\Omega_{2n+1}(q)$ and $P\Omega_{2n}^{\pm}(q)$ to denote the finite classical simple groups.
Symmetric and alternating groups on $n$ letters are denoted by $S_{n}$ and $A_{n}$, respectively. Further notation and definitions in both design theory and group theory are standard and can be found, for example in \cite{b:Dixon,b:Hugh-design,b:lander}. We also use the software \textsf{GAP} \cite{GAP4} for computational arguments.
Flag-transitive incidence structures have been of most interest. In 1961, Higman and McLaughlin \cite{a:HigMcL61} proved that a flag-transitive automorphism group of a linear space must act primitively on its points set, and then Buekenhout, Delandtsheer and Doyen \cite{a:buekenhout-1988} studied this action in details and proved that a linear space admitting a flag-transitive automorphism group (which is in fact point-primitive) is either of affine, or almost simple type.
Thereafter, a deep result \cite{a:buekenhout-1990}, namely the classification of flag-transitive finite linear spaces relying on the Classification of Finite Simple Groups (CFSG) was announced.
Although, flag-transitive symmetric designs are not necessarily point-primitive, Regueiro \cite{a:reg-reduction} proved that a flag-transitive and point-primitive automorphism group of such designs for $\lambda\leq 4$ is of affine or almost simple type, and so using CFSG, she determined all flag-transitive and point-primitive biplanes ($\lambda=2$). In conclusion, she gave a classification of flag-transitive biplanes except for the $1$-dimensional affine case \cite{t:Regueiro}. Tian and Zhou \cite{a:Zhou-lam100} proved that a flag-transitive and point-primitive automorphism group of a symmetric design with $\lambda\leq 100$ must be of affine or almost simple type. Generally, Zieschang \cite{a:zieschang-1988} proved in 1988 that if a flag-transitive automorphism group of a $2$-design with $\gcd(r,\lambda)=1$ is a point-primitive group of affine or almost simple type, and this result has been generalised by Zhuo and Zhan \cite{a:Zhou-lamcond18} for $\lambda \geq \gcd(r, \lambda)^2$. In this paper, we study flag-transitive automorphism groups of symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$, and we show that such an automorphism group is not necessarily point-primitive:
\begin{theorem}\label{thm:main}
Let $\mathcal{D}=(\mathcal{P},\mathcal{B})$ be a non-trivial symmetric $(v, k, \lambda)$ design with $\lambda\geq 1$, and let $G$ be a flag-transitive automorphism group of $\mathcal{D}$. If $\lambda$ divides $k$ and $k\geq \lambda^2$, then one of the following holds:
\begin{enumerate}[{\rm (a)}]
\item $G$ is point-primitive of affine or almost simple type;
\item $G$ is point-imprimitive and $v=\lambda^{2}(\lambda+2)$ and $k=\lambda(\lambda+1)$, for some positive integer $\lambda$. In particular, if $G$ has $d$ classes of imprimitivity of size $c$, then there is a constant $l$ such that, for each block $B$ and each class $\Delta$, the size $|B \cap \Delta|$ is either $0$, or $l$, and $(c,d,l)=(\lambda^2,\lambda+2,\lambda)$ or $(\lambda+2,\lambda^{2},2)$.
\end{enumerate}
\end{theorem}
We highlight here that if $\lambda$ divides $k$, then $\gcd(k,\lambda)^2=\lambda^2>\lambda$ which does not satisfy the conditions which have been studied in \cite{a:Zhou-lamcond18,a:zieschang-1988}. Moreover, in Section \ref{sec:example}, we provide some examples to show that both possibilities in Theorem \ref{thm:main} can actually occur.
In order to prove Theorem \ref{thm:main}(a), we apply O'Nan-Scott Theorem \cite{a:LPS-Onan-Scott} and discuss possible types of primitive groups in Section \ref{sec:prim}. We further note that our proof for part (a) relies on CFSG. To prove part (b), we use an important result by Praeger and Zhou \cite[Theorem 1.1]{a:Praeger-imprimitive} on characterisation of imprimitive flag-transitive symmetric designs.
\subsection{Examples and comments on Theorem~\ref{thm:main}}\label{sec:example}
Here, we give some examples of symmetric $(v,k,\lambda)$ designs admitting flag-transitive automorphism groups, where $\lambda$ divides $k$ and $k\geq \lambda^2$. In Table \ref{tbl:examples}, we list some small examples of such designs with $\lambda\leq 3$. To our knowledge the design in line 2 is the only point-primitive example of symmetric designs with $v\leq 2500$ satisfying the conditions of Theorem \ref{thm:main} and this motivates the authors to investigate symmetric designs admitting symplectic automorphism groups \cite{a:ADO-PSp4}. More examples of symmetric designs admitting flag-transitive and point-imprimitive automorphism groups can be found in \cite{a:Praeger-imprimitive} and references therein.\smallskip
\noindent \textbf{Line 1.} Hussain \cite{a:hussain1945} showed that there are exactly three symmetric $(16, 6, 2)$ designs, and Regueiro proved that exactly two of such designs are flag-transitive and point-imprimitive \cite[p. 139]{a:reg-reduction}.\smallskip
\noindent \textbf{Line 2.} The symmetric design in this line arises from the study of primitive permutation groups with small degrees. This design belongs to a class of symmetric designs with parameters $(3^m(3^m+1)/2,3^{m-1}(3^m-1)/2,3^{m-1}(3^{m-1}-1)/2)$, for some positive integer $m>1$, see \cite{a:Braic-2500-nopower,a:Dempwolff2001}.
If $m=2$, then we obtain the symmetric $(45,12,3)$ design admitting $PSp_4(3)$ or $PSp_4(3): 2$ as flag-transitive automorphism group of rank $3$, see \cite{a:Braic-2500-nopower}.\smallskip
\noindent \textbf{Lines 3-4.} Mathon and Spence \cite{a:mathon-1996} constructed $2616$ pairwise non-isomorphic symmetric $(45,12,3)$ designs with non-trivial automorphism groups. Praeger \cite{a:praeger-45point} proved that there are exactly two flag-transitive symmetric $(45, 12, 3)$ designs, exactly one of which admits a point-imprimitive group, and this example satisfies Line 4, but not Line 3.\smallskip
\begin{table}
\centering
\small
\caption{Some symmetric designs satisfying the conditions in Theorem \ref{thm:main}} \label{tbl:examples}
\begin{tabular}{lllllllllllp{3cm}}
\hline
Line & $v$ & $k$ &$\lambda$ & $c$ & $d$ & $l$ & Case & Examples & Reference & Comments \\
\hline
%
$1$ & $16$ & $6$ & $2$ & $4$ & $4$ & $2$ & (b) & $2$ & \cite{a:hussain1945}, \cite{a:reg-reduction}& imprimitive\\
$2$ & $45$ & $12$ & $3$ & - & - & - & (a) & $1$ & \cite{a:Braic-2500-nopower}& primitive\\
$3$ & $45$ & $12$ & $3$ & $5$ & $9$ & $2$ & (b) & None & \cite{a:praeger-45point} & imprimitive\\
$4$ & $45$ & $12$ & $3$ & $9$ & $5$ & $3$ & (b) & 1 & \cite{a:praeger-45point} & imprimitive\\
\hline
\end{tabular}
\end{table}
\section{Preliminaries}\label{sec:pre}
In this section, we state some useful facts in both design theory and group theory.
\begin{lemma}\label{lem:six}{\rm \cite[Lemma 2.1]{a:ABD-PSL2}}
Let $\mathcal{D}$ be a symmetric $(v,k,\lambda)$ design, and let $G$ be a flag-transitive automorphism group of $\mathcal{D}$. If $\alpha$ is a point in $\mathcal{P}$ and $H:=G_{\alpha}$, then
\begin{enumerate}[\rm (a)]
\item $k(k-1)=\lambda(v-1)$;
\item $k\mid |H|$ and $\lambda v<k^2$.
\end{enumerate}
\end{lemma}
\begin{lemma}{\rm \cite[Corollary 4.3]{a:AB-Large-15}}\label{lem:bound}
Let $T$ be a finite simple classical group of dimension $n$ over a finite field $\mathbb{F}_q$ of size $q$. Then
\begin{enumerate}[{\rm (a)}]
\item If $T=PSL_n(q) $ with $n\geq 2$, then $ |T|>q^{n^2-2} $;
\item If $T=PSU_n(q) $ with $n\geq 3$, then $ |T|>(1-q^{-1})q^{n^2-2} $;
\item If $T=PSp_n(q) $ with $n\geq 4$, then $|T|>q^{\frac{1}{2}n(n+1)}/(2\alpha)$, where $\alpha=\gcd(2, q-1)$;
\item If $T=P\Omega ^{\epsilon}_n(q) $ with $n\geq 7$, then $ |T|> q^{\frac{1}{2}n(n-1)}/(4\beta)$, where $\beta=\gcd(2, n)$.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{lem:out}
Let $T$ be a non-abelian finite simple group satisfying
\begin{align}\label{eq:out}
|T|<8\cdot |\mathrm{Out}(T)|^{3}.
\end{align}
Then $T$ is isomorphic to $A_{5}$ or $A_{6}$.
\end{lemma}
\begin{proof}
If $T$ is a sporadic simple group or an alternating group $A_{n}$ with $n\geq 7$, then $|\mathrm{Out}(T)|\in\{ 1,2\}$, and so by~\eqref{eq:out}, we must have $|T|<64$, which is a contradiction. Note that the alternating groups $A_{5}$ and $A_{6}$ satisfy \eqref{eq:out} as claimed. Therefore, we only need to consider the case where $T$ is a finite simple group of Lie type. In what follows, we discuss each case separately.
Let $T=PSL_{n}(q)$ with $q=p^{a}$ and $n\geq 2$. If $n=2$, then $q\geq 4$ and $|\mathrm{Out}(T)|=a\cdot \gcd(2,q-1)$, and so by Lemmas~\ref{lem:bound}(a) and \eqref{eq:out}, we have that $q^{2}<|PSL_{2}(q)|<8a^{3}\cdot \gcd(2,q-1)^{3}\leq 64a^{3}$. Thus, $q^{2}<64a^{3}$. This inequality holds only for $(p,a)\in\{(2,1),(2,2), (2,3),(2,4), (2,5),(2,6),(2,7), (3,1), (3,2),(3,3),(5,1),(7,1)\}$. Note in this case that $q\geq 4$, and hence by \eqref{eq:out}, we conclude that $T$ is either $PSL_{2}(4)\cong PSL_{2}(5)\cong A_{5}$, or $PSL_{2}(9)\cong A_{6}$, as claimed. If $n=3$, then by Lemma~\ref{lem:bound}(a), we have that $q^{7}<64a^{3}\cdot \gcd(3,q-1)^{3}< 64a^{3}q^{3}$, and so $q^{4}<64a^{3}$. If $q$ would be odd, then we would have $3^{4a}<64a^{3}$, which is impossible. If $q=2^{a}$, then $2^{a}<64a^{3}$ would hold only for $a=1,2$. Therefore, $T$ is isomorphic to $PSL_{3}(2)$ or $PSL_{3}(4)$. These simple groups do not satisfy \eqref{eq:out}. If $n\geq 4$, then \eqref{eq:out} implies that $ q^{11}<64a^{3}$, but this inequality has no possible solution.
Let $T=PSU_{n}(q)$ with $q=p^{a}$ and $n\geq 3$. By Lemma~\ref{lem:bound}(b), we have that $|T|> (1-q^{-1})q^{n^2-2}$, and so \eqref{eq:out} follows that $(1-q^{-1})q^{n^2-2}<64a^{3}\cdot \gcd(n, q+1)^{3}$. If $n=3$, then $(1-q^{-1})q^7<64a^{3}\cdot \gcd(n,q+1)^{3}$, and so $ q^6<27\cdot 64 a^3 $. This inequality holds only for $(p,a)\in\{(2,1), (2,2),(3,1) \}$. Note that $PSU_{3}(2)$ is not simple. Therefore, $T$ is isomorphic to $PSU_{3}(3)$ or $PSU_{3}(4)$. These simple groups do not satisfy \eqref{eq:out}. If $n\geq 4$, then since $(q+1)^3<4\cdot q^3(q-1)$, we would have $q^{n^2-3}<64a^{3}\cdot \gcd(n, q+1)^{3}/(q-1)<4\cdot 64a^{3}(q+1)^{3}/4(q-1)<4\cdot 64 a^{3}q^{3}$, and so $q^{n^2-6}<4\cdot 64 a^{3}$, and hence $q^{10}<4\cdot 64a^{3}$, which is impossible.
Let $T=PSp_{n}(q)$ with $q=p^{a}$ and $n\geq 4$. By Lemma~\ref{lem:bound}(c), we observe that $|T|> q^{\frac{1}{2}n(n+1)}/2\gcd(2,q-1)\geq q^{\frac{1}{2}n(n+1)}/4$. By \eqref{eq:out}, we have that $q^{10}\leq q^{\frac{1}{2}n(n+1)}<4\cdot 64a^{3}$, and so $q^{10}<4\cdot 64a^{3}$, which is impossible.
Let $T=P\Omega_n(q)$ with $q=p^{a}$ odd and $n\geq 7$. Then we conclude by Lemma~\ref{lem:bound}(d) that $|T|>q^{\frac{1}{2}n(n-1)}/8$. Since $|\mathrm{Out}(T)|=2a$ and $n\geq 7$, it follows from \eqref{eq:out} that $q^{21}<8^3a^{3}$, which is impossible.
Let $T=P\Omega^{\epsilon}_n(q)$ with $q=p^{a}$ and $n\geq 8$ and $\epsilon=\pm$. It follows from Lemma~\ref{lem:bound}(d) that $|T|>q^{\frac{1}{2}n(n-1)}/8$. Note that $|\mathrm{Out}(T)|\leq 6a\cdot \gcd(4,q^{\frac{n}{2}}-\epsilon)\leq 24 a$. Then \eqref{eq:out} implies that $q^{28}<8^2\cdot 24^3a^{3}$, which is impossible.
Let $T$ be one of the finite exceptional groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$, $^2\!F_4(q)$ ($q=2^{2m+1}$), $^3\!D_4(q)$ and $^2\!E_6(q)$. Then $|T|>q^{20} $, and so \eqref{eq:out} implies that $q^{20}<8\cdot 2^3\cdot 3^3a^{3}$, which is impossible. If $T=G_2(q) $ with $q=p^{a} \neq 2$. Then by \eqref{eq:out}, we have that $q^{12}< q^{6}(q^2-1)(q^6-1)<8\cdot 2^3a^{3}$, and so $q^{12}<8\cdot 2^3a^{3} $, which is impossible. Similarly, if $T$ is one of the groups $^{2}\!B_2(q) $ with $q=2^{2m+1}$ and $^{2}\!G_2(q) $ with $q=3^{2m+1}$, then $|T|>q^{4}$, and so \eqref{eq:out} implies that $q^{4}<8a^3$, which is impossible.
\end{proof}
\section{Point-primitive designs}\label{sec:prim}
In what follows, we assume that $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is a non-trivial symmetric $(v, k, \lambda)$ design admitting a flag-transitive and point-primitive automorphism group $G$. Let also $\lambda$ divide $k$ and $k\geq \lambda^2$ and set $t:=k/\lambda$. Notice that $\lambda <k$, and so $t \geq 2$. We moreover observe by Lemma~\ref{lem:six}(a) that
\begin{align}
k&=\dfrac{v+t-1}{t};\label{eq:k} \\
\lambda &=\dfrac{v+t-1}{t^2}. \label{eq:lam}
\end{align}
Since also $G$ is a primitive permutation group on $\mathcal{P}$, then by O'Nan-Scott Theorem \cite{a:LPS-Onan-Scott}, $G$ is of one of the following types:
\begin{enumerate}[{\rm (a)}]
\item Affine;
\item Almost simple;
\item Simple diagonal;
\item Product;
\item Twisted wreath product.
\end{enumerate}
\subsection{Product and twisted wreath product type}
In this section, we assume that $G$ is a primitive group of product type on $\mathcal{P}$, that is to say, $G\leq H \wr S_{\ell}$, where $H$ is of almost simple or diagonal type on the set $\Gamma$ of size $m:=|\Gamma |\geq 5$ and $\ell\geq 2$. In this case, $\mathcal{P} = \Gamma^{\ell}$.
\begin{lemma}\label{lem:sub-prod}
Let $G$ be a flag-transitive point-primitive automorphism group of product type. Then $k$ divides $\lambda \ell(m-1)$.
\end{lemma}
\begin{proof}
See the proof of \rm{Lemma~4} in \cite{a:reg-reduction}.
\end{proof}
\begin{proposition}\label{prop:prod}
If $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is a non-trivial symmetric $(v, k, \lambda)$ design admitting a flag-transitive and point-primitive automorphism group $G$, where $\lambda$ divides $k$ and $k\geq \lambda^2$, then $G$ is not of product type.
\end{proposition}
\begin{proof}
Assume the contrary. Suppose that $G$ is of product type. Then $v=m^{\ell}$. Note by Lemma~\ref{lem:sub-prod} that $k$ divides $\lambda\ell(m-1)$, and so $t=k/\lambda$ divides $\ell(m-1)$. We also note by Lemma \ref{lem:six}(b) that $\lambda v <k^{2}$. Then $v<\lambda t^2$, and since $\lambda\leq t$, we have that $v < t^3$. Recall that $t$ divides $\ell(m-1)$. Hence
\begin{align}\label{eq:prod}
m^\ell<\ell^3(m-1)^3.
\end{align}
Then $m^{\ell}<\ell^{3} m^{3}$, or equivalently, $m^{\ell-3}<\ell^{3}$. Since $m\geq 5$, it follows that $5^{\ell-3}<\ell^{3}$, and this is true for $2\leq \ell\leq 6$. If $\ell=6$, then since $m^{6-3}<6^3$, we conclude that $m=5$, but $(m,\ell)=(5,6)$ does not satisfy \eqref{eq:prod}. Therefore, $2\leq \ell\leq 5$.
Suppose first that $\ell=5$. Then by~\eqref{eq:prod}, we have that $m^{5}<5^{3}(m-1)^{3}$, and so $5\leq m\leq 9$. It follows from \eqref{eq:k} that $t$ divides $m^{5}-1$. For each $5\leq m\leq 9$, we can obtain divisors $t$ of $m^{5}-1$. Note by \eqref{eq:lam} that $t^{2}$ must divide $m^{5}-t+1$. This is true only for $m=7$ when $t=2$ or $6$ for which $(v,k,\lambda)=(16807, 8404, 4202)$ or $(16807, 2802, 467)$, respectively. Since $\lambda^{2}\leq k$, these parameters can be ruled out.
Suppose that $\ell=4$. Then by~\eqref{eq:prod}, we have that $m^{5}<4^{3}(m-1)^{3}$, and so $5\leq m\leq 9$. By the same argument as in the case where $\ell=5$, by \eqref{eq:k} and \eqref{eq:lam}, we obtain possible parameters $(m,t,v,k,\lambda)$ as in Table~\ref{tbl:l4}. Note by Lemma~\ref{lem:sub-prod} that $k$ must divide $4\lambda(m-1)$, and this is not true, for all parameters in Table~\ref{tbl:l4}.
\begin{table}[h]
\centering
\small
\caption{Possible values for $(m,t,v,k,\lambda)$ when $\ell=4$.}\label{tbl:l4}
\begin{tabular}{lllll}
\hline
$m$ & $t$ & $v$ & $k$ & $ \lambda $ \\
\hline
%
13 & 51 & 28561 & 561 & 11\\
31 & 555 & 923521 & 1665 & 3\\
47 & 345 & 4879681 & 14145 & 41\\
57 & 416 & 10556001 & 25376 & 61\\
\hline
\end{tabular}
\end{table}
Suppose now that $\ell=3$. We again apply Lemma~\ref{lem:sub-prod} and conclude that $t$ divides $3(m-1)$. Then there exists a positive integer $x$ such that $3(m-1)=tx$, and so $m=(tx+3)/3$. By \eqref{eq:lam}, we have that
\begin{align*}
\lambda=\frac{m^2+t-1}{t^2}=\frac{t^2x^3+9tx^2+27x+27}{27t}.
\end{align*}
Then $27\lambda t=t^2x^3+9tx^2+27x+27$. Therefore, $t$ must divide $27x+27$, and so $ty=27x+27$, for some positive integer $y$. Thus,
\begin{align}\label{eq:lam3}
\lambda=\frac{t(ty-27)^3+9\cdot 27(ty-27)^2+27^{3}y}{27^{4}},
\end{align}
for some positive integers $t$ and $y$. Since $\lambda^2\leq k $, we have that $\lambda\leq t$, and so
\begin{align}\label{eq:l3}
t(ty-27)^3+9\cdot 27(ty-27)^2+27^{3}y\leq 27^{4}t.
\end{align}
If $y\geq 32$, then $t(ty-27)^3+9\cdot 27(ty-27)^2+27^{3}y\geq t(32t-27)^3+9\cdot 27(32t-27)^2+32\cdot 27^{3}>27^{4}t$, for $t\geq 2$. Thus $1\leq y\leq 31$, and so by \eqref{eq:l3}, we conclude that $2\leq t\leq 107$. For each such $y$ and $t$, by straightforward calculation, we observe that $\lambda$ as in \eqref{eq:lam3} is not a positive integer.
Suppose finally that $\ell=2$. Recall by Lemma~\ref{lem:sub-prod} that $t$ divides $2(m-1)$. Then $2(m-1)=tx$ for some positive integer $x$, and so $m=(tx+2)/2$. It follows from \eqref{eq:lam} that $\lambda =(tx^2+4x+4)/4t$, or equivalently, $4t\lambda=tx^2+4x+4$. This shows that $t$ divides $4x+4$, and so $ty=4x+4$, for some positive integer $y$. Therefore, $4^{3}\lambda=(ty-4)^2+16y$. Since $\lambda^2\leq k $, we have that $\lambda\leq t$, and so $(ty-4)^2+16y\leq4^3 t$. If $y\geq 6$, then $(6t-4)^2+6\cdot 16\leq4^3 t$, which has no possible solution for $t$. Thus $1\leq y\leq 5$. Since also $(t-4)^2+16\leq4^3 t$, we conclude that $2\leq t\leq 71$, and so \eqref{eq:k} and \eqref{eq:lam} imply that
\begin{align*}
k= \frac{t(t^2y^2-8ty+16y+16)}{64} \text{ and } \lambda=\frac{(ty-4)^2+16y}{64},
\end{align*}
where $2\leq t\leq 71$ and $1\leq y\leq 5$. For these values of $t$ and $y$, considering the fact that $m\geq 5$, $k\geq \lambda^{2}$ and $\lambda$ divides $k$, we obtain $(v,k,\lambda)= (121, 25, 5)$ or $(441, 56, 7)$ respectively when $(t,y)=(5,4)$ or $(8,3)$. These possibilities can be ruled out by \cite{a:Braic-2500-nopower} or \cite[Theorem~1.1]{a:Zhou-lam100}.
\end{proof}
\begin{proposition}\label{prop:tw}
If $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is a non-trivial symmetric $(v, k, \lambda)$ design admitting a flag-transitive and point-primitive automorphism group $G$, where $\lambda$ divides $k$ and $k\geq \lambda^2$, then $G$ is not of twisted wreath product type.
\end{proposition}
\begin{proof}
If $G$ would be of twisted wreath product type, then by \cite[Remark 2(ii)]{a:LPS-Onan-Scott}, it would be contained in the wreath product $H \wr S_m$ with $H = T \times T$ of simple diagonal type, and so $G$ would act on $\mathcal{P}$ by product action, and this contradicts Proposition~\ref{prop:prod}.
\end{proof}
\subsection{Simple diagonal type}\label{sec:sd}
In this section, we suppose that $G$ is a primitive group of diagonal type. Let $M=\mathrm{Soc}(G)=T_1\times\ldots\times T_m $, where $T_i\cong T$ is a non-abelian finite simple group, for $i=1,\ldots, m$. Then $G$ may be viewed as a subgroup of $M\cdot(\mathrm{Out}(T)\times S_m)$. Here, $G_\alpha$ is isomorphic to a subgroup of $\mathrm{Aut}(T ) \times S_m$ and $M_{\alpha} \cong T$ is a diagonal subgroup of $M$, and so $|\mathcal{P}|=|T|^{m-1}$.
\begin{lemma}\label{lem:sub-sd}
Let $G$ be a flag-transitive point-primitive automorphism group of simple diagonal type with socle $T^m$. Then $k$ divides $\lambda m_1 h$, where $m_1\leq m$ and $h$ divides $|T|$.
\end{lemma}
\begin{proof}
See the proof of \rm{Proposition~3.1} in \cite{a:Zhou-lam100}.
\end{proof}
\begin{proposition}\label{prop:diag}
If $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is a non-trivial symmetric $(v, k, \lambda)$ design admitting a flag-transitive and point-primitive automorphism group $G$, where $\lambda$ divides $k$ and $k\geq \lambda^2$, then $G$ is not of simple diagonal type.
\end{proposition}
\begin{proof}
Suppose by contradiction that $G$ is a primitive group of simple diagonal type. Then $v=|T|^{m-1}$, and so by Lemma \ref{lem:six}(b), $\lambda v < k^2$. This implies that $\lambda |T|^{m-1} < k^2=\lambda^2t^2 $. Since $\lambda\leq k^2$, we must have $\lambda\leq t$, and hence
\begin{align}\label{eq:diag}
|T|^{m-1}< t^3.
\end{align}
Note by Lemma~\ref{lem:sub-sd} that $k$ divides $\lambda m_1h$ and $m_1h\leq m|T|$. Then $t$ divides $m_1h$, and so $t\leq m|T|$. We now apply \eqref{eq:diag} and conclude that
$|T|^{m-1} < m^3|T|^3$. Therefore, $|T|^{m-4}<m^3$. Since $|T|\geq 60$, we must have $m< 6$. If $m=5$, then $|T|<5^{3}$, and this follows that $T\cong A_{5}$. Note that $k$ divides $\lambda(v-1)=\lambda(|T|^{m-1}-1)$. Then $t$ divides $|T|^{m-1}-1=60^{4}-1=13\cdot 59\cdot 61\cdot 277$. Since $t\leq m|T|=300$ and $t\geq 2$, it follows that $t\in\{13, 59, 61, 277\}$. For each such $t$, we have that $\lambda\leq t$ and $k=t\lambda$, and so we easily observe that these parameters does not satisfy Lemma~\ref{lem:six}(a). Therefore $m\in \{2,3,4\}$. Note that $G_{\alpha}$ is isomorphic to a subgroup of $\mathrm{Aut}(T ) \times S_m$. Then by Lemma~\ref{lem:six}(b), the parameter $k$ divides $|G_{\alpha}|$, and so $k$ divides $(m!)\cdot |T |\cdot |\mathrm{Out}(T )|$. On the other hand, Lemma~\ref{lem:six}(a) implies that $k$ divides $\lambda (|T|^{m-1}-1)$, and so $t$ divides $|T|^{m-1}-1$ implying that $\gcd(t, |T|)=1$. Since $k$ divides $(m!)\cdot |T |\cdot |\mathrm{Out}(T )|$ and $t$ is a divisor of $k$, we conclude that $t$ divides $(m!)\cdot |\mathrm{Out}(T)| $. Recall by \eqref{eq:diag} that $|T |^{m-1}< t^3$. Therefore,
\begin{equation}\label{eq:diag-2}
|T |^{m-1}<(m!)^{3}\cdot |\mathrm{Out}(T)|^3,
\end{equation}
where $m\in \{2,3,4\}$.
If $m = 2$, then $|T |< 8\cdot |\mathrm{Out}(T )|^3$. If $m = 3$, then $|T |^2< 6^3|\mathrm{Out}(T )|^3$, and so
$|T |< 6^{\frac{3}{2}}|\mathrm{Out}(T )|$. If $m = 4$, then $|T |^3 < 24^3|\mathrm{Out}(T )|^3$, and $|T | < 24|\mathrm{Out}(T )|$. Thus
for $m\leq 4$, we always have
\begin{equation*}
|T | < 8\cdot |\mathrm{Out}(T )|^3,
\end{equation*}
where $T$ is a non-abelian finite simple group. We now apply Lemma~\ref{lem:out} and conclude that $T$ is isomorphic to $A_5$ or $A_6$. If $m=2$, then since $t$ divides $|T|^{m-1}-1=|T|-1$, we have that $t$ divides $59$ or $359$ when $T$ is isomorphic to $A_5$ or $A_6$, respectively. Thus $(v, k, \lambda)= (60, 59\lambda, \lambda)$ or $(v, k, \lambda)=(360, 359\lambda, \lambda)$. Since $\lambda>1$, in each case , we conclude that $k>v$, which is a contradiction. For $m=3,4$, since $|\mathrm{Out}(A_5)|=2$ and $|\mathrm{Out}(A_6)|=4$, it follows from \eqref{eq:diag-2} that $|T|< 48$ or $|T|<96 $ when $T$ is isomorphic to $A_5$ or $A_6$, respectively, which is a contradiction.
\end{proof}
\section{Proof of the main result}
In this section, we prove Theorem~\ref{thm:main}. Suppose that $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is a non-trivial symmetric $(v, k, \lambda)$ design with $\lambda$ divides $k$ and $k\geq \lambda^2$. Suppose also that $G$ is a flag-transitive automorphism group of $\mathcal{D}$.
\begin{proof}[Proof of Theorem \ref{thm:main}]
If $G$ is point-primitive, then by O'Nan-Scott Theorem \cite{a:LPS-Onan-Scott} and Propositions~\ref{prop:prod}, \ref{prop:tw} and \ref{prop:diag}, we conclude that $G$ is of affine or almost simple type. Suppose now that $G$ is point-imprimitive. Then $G$ leaves invariant a non-trivial partition $\mathcal{C}$ of $\mathcal{P}$ with $d$ classes of size $c$. By \cite[Theorem~1.1]{a:Praeger-imprimitive}, there is a constant $l$ such that, for each $B\in \mathcal{B}$ and $\Delta \in \mathcal{C}$, $|B \cap\Delta|\in \{0,l\}$ and one of the following holds:
\begin{enumerate}[{\rm (a)}]
\item $k \leq \lambda(\lambda -3)/2$;
\item $(v, k, \lambda) = (\lambda^2(\lambda +2), \lambda(\lambda + 1), \lambda)$ with $(c, d, l) = (\lambda^2,\lambda +2,\lambda)$ or $(\lambda +2,\lambda^2, 2)$;
\item $(v, k,\lambda, c, d, l) = ( \dfrac{(\lambda+2)(\lambda^2-2\lambda+2)}{4} , \dfrac{\lambda^2}{2} , \lambda, \dfrac{\lambda+2}{2} , \dfrac{\lambda^2-2\lambda+2}{2} , 2)$, and either $\lambda \equiv 0 \mod 4$, or $\lambda = 2u^2$, where $u$ is odd, $u\geq 3$, and $2(u^2-1)$ is a square;
\item $(v, k, \lambda, c, d, l) = (\dfrac{(\lambda + 6)(\lambda^2+4\lambda-1)}{4} , \dfrac{\lambda (\lambda+5)}{2} , \lambda, \lambda+6, \dfrac{\lambda^2+4\lambda-1}{4},3)$, where $\lambda\equiv 1$ or $3$ $\mod 6$.
\end{enumerate}
We easily observe that the cases (a) and (c) can be ruled out as $k\geq \lambda^{2}$. If case (d) occurs, then $\lambda(\lambda+5)/2=k\geq \lambda^{2}$ implying that $\lambda\leq 5$. Since $\lambda \equiv 1 \text{ or } 3 \mod{6}$, it follows that $\lambda=3$ for which $(v,k,\lambda,c,d,l)=(45,12,3,9,5,3)$ which satisfies the condition in Theorem~\ref{thm:main}(b). Therefore, the case (b) can occur as claimed.
\end{proof}
\bibliographystyle{amcjoucc}
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Can we learn a second language like we learned our first?
By Robert William McCaul
11 February 2016 - 09:11
An anxiety-free environment can help learners process comprehensible input. Photo ©
martinak15, licensed under CC BY 2.0 and adapted from the original (link no longer available).
Robert William McCaul, winner (with Marek Kiczkowiak) of the TeachingEnglish blog award, examines the influential ideas of linguist Stephen Krashen, and the implications they have for the language classroom.
If you've ever doubted whether you're a good language learner, then bear in mind that you've already learned one language very well indeed – your first. But this raises an interesting question: can adults learn a second language in the same way they learned their first as children? And if so, what are the implications for the classroom?
Stephen Krashen and the acquisition of languages
Perhaps no-one has looked at the question more closely than the linguist Stephen Krashen, who has introduced some of the most influential concepts to the study of second-language acquisition.
In his input hypothesis, first proposed in an article published in 1977, and expanded upon in later years, he makes the distinction between learning: the conscious, traditional grammar-based process in the classroom; and acquisition: essentially how we, as children, pick up our first language. He says that our mistake is trying to teach languages in the same way we teach science, history and mathematics. Instead, he believes that learners should acquire second languages in the same way children learn their first.
Krashen sums up the idea in a famous documentary on the subject called A child's guide to learning languages, produced by BBC Horizon in 1983. In the documentary, he says that acquisition is 'where the action is'. In other words, in every successful example of language-learning – an infant mastering a first language, an adult learner of English scoring a band 9 on the IELTS test – the reason for their success is that they have 'acquired' rather than 'learned' the language.
So, how do children and proficient adult learners perform the seemingly magical trick of mastering a language, and what can teachers learn from this? Krashen offers the following ideas:
1. We acquire languages when we can understand messages
Learners need to be exposed to what Krashen calls 'comprehensible input' – that is, exposure to interesting and understandable listening and reading material. In Krashen's view, we acquire languages when we understand messages. He stipulates that the emphasis should be on meaningful interactions and not on form. When parents speak to their children, for example, the emphasis is on meaning rather than the correct use of grammar. If the child says, 'Daddy fish water!', the parent is likely to respond, 'Yes, you're right, there's a fish in the river', rather than by correcting the child's grammar. The theory here is that exposure to sufficient quantities of comprehensible input always results in acquisition.
2. Getting the right level is crucial
Krashen makes the important point that comprehensible input needs to be at the right level for the learner, namely just higher than the learner's own. He calls this theoretical level 'i + 1'. A good practical example of this in action are graded readers. These are books that are specially created for learners of foreign languages at various levels, such as A2, B1, C2, etc, on the common European framework (CEFR).
3. The silent period
Children don't start speaking their mother tongue straight away. Until they utter their first words, they are acquiring language, even if they are not using it. The miraculous first words and sentences that quickly follow are the result of this acquisition. Adult learners, both inside and outside the classroom, need this silent period, too. Teachers shouldn't be afraid when their students don't participate in debates in class – perhaps they are simply acquiring the language. Moreover, putting pressure on the learner to speak before they are ready will result in anxiety.
4. Anxiety is the student's arch enemy
This brings me to one of Krashen's most famous insights, namely the affective filter. This means that the rate of acquisition decreases if we are under stress, or if we experience anxiety. Luckily, most children have a virtually stress-free language-learning environment at home with their mothers and fathers. But for learners of a second language, the classroom can be a cause of anxiety, greatly affecting the way they receive and process comprehensible input.
By contrast, a house party with lots of international guests is a great place to practise languages, as everybody is relaxed and having a good time. Such an environment offers the language learner plenty of comprehensible input, but (hopefully) none of the anxiety. The lesson here for teachers is that they can create a similar environment by turning the classroom into a sort of house party where people feel comfortable and relaxed.
5. The monitor hypothesis
According to Krashen, conscious language-learning cannot be the source of spontaneous speech, it can only monitor output, i.e., production in speech or writing. In other words, when learners freely formulate an utterance in the target language, they can only draw upon their repertoire of acquired language to check whether it is grammatically correct. This reduces errors as the learner can apply consciously learned rules to an utterance before producing it, or after production through self-correction. As many people place a high value on accuracy, especially in formal situations, the existence of the 'monitor' could be seen as a reason for retaining a grammar focus in a given lesson.
One way to apply this in the classroom would be to have learners notice grammatical features in listening and reading texts using a guided discovery approach. For example, if the learners were given a listening task to do on the biography of a famous person who is still alive, the teacher could hand out the transcript and get the students to underline all of the examples of the present perfect tense. This might be followed by a short discussion, led by the teacher, as to why the tense is being used in this particular situation, followed by some concept-checking questions to ensure students understand how to use the target language. However, Krashen is clear that the main focus of classroom activity should be on giving learners as much comprehensible input as possible. Teachers should base their lessons on meaningful interactions with plenty of graded listening and reading input.
6. The natural order hypothesis
The grammar and vocabulary of a language are acquired in the same general order, irrespective of who the learner is, which language they are acquiring and the order of the grammar syllabus. You can teach students reported speech, such as in the sentence, 'she mentioned that she had been at the shop that morning', but learners won't acquire it unless they are ready to. Certain elements of grammar are 'late-acquired', such as the third person '-s', and others are 'early-acquired'. This explains why my little niece continues to say things like 'Daddy go to work every day', even when she has already mastered more complex grammatical structures such as a conditional sentence like, 'I would do it if I had time'. Evidence for this 'natural sequence' of language acquisition can be found in the morpheme studies by Dulay and Burt. This casts doubt on the teaching of many points of grammar too early, that is, before students are ready to acquire them, such as the future perfect tense at intermediate level.
The advantages children have over adult learners
Before looking at the classroom implications of Krashen's insights, we should remind ourselves of some of the advantages that children learning their first language have over adults learning a second language. One of the principle advantages is that children are exposed to copious amounts of comprehensible input at just the right level, and there is no pressure on them to speak until they are ready to do so. Children can also take their time and wait until they feel confident before attempting to speak. Moreover, they often have lower expectations of themselves and this helps to ensure that their anxiety levels are low, which, in turn, increases their rate of acquisition.
One of the most surprising things is that when children acquire a language, the language acquisition itself is not their objective. Rather, it is a by-product of the achievement of some other purpose, such as making friends in a school playground. Moreover, they pick up the elements of their first language in its natural order. They are not 'force-fed' grammar too early before their language acquisition devices are ready for it. Instead, they acquire the language first and then consider its structure after acquisition has already taken place. Finally, they learn the elements of a language in the natural order.
The practical implications of Krashen's ideas for the classroom
From Krashen's theories, and having looked at the advantages that children have over adults when it comes to learning languages, we can draw certain conclusions about what conditions make for a successful learning environment. First, class time should be taken up with as much comprehensible input as possible. Second, classes should be stress-free environments where students are encouraged to relax and acquire the language by having fun with it.
One particularly important implication of Krashen's findings is that students, particularly at lower levels, should have lower demands made on them to speak, and materials and teacher talking time should be modified for each student's level. Furthermore, grammar instruction should be done on a need-to-know basis, and only with older learners. Last, but perhaps most important, lessons should not be based on grammar points, but rather on the exchange of meaning.
Listen to Robert William McCaul and Marek Kiczkowiak discuss Krashen's theories of language acquisition on their award-winning podcast.
Teachers, visit our TeachingEnglish website for more lesson plans and activities, and find out how you can become a TeachingEnglish blogger.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: How to get event/Occasion name from date picker? I want to know that it is possible to get event name from datepicker like if I select 25 december and get christmas as event name etc.
A: Yes it is possible to create such functionality. You will have to create NSDictionary and Map date with event name. Whenever user selects any date then look it for that date in NSDictionary and get event name for that date.
This is one possible way. There might be other ways but please note that there is no predefined control to achieve this functionality.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Tire World provides Replace Radiator services to Smyrna, TN, Murfreesboro, TN, LaVergne TN, and other surrounding areas.
Why Should You Have Replace Radiator Services Performed at Tire World?
We proudly service the Replace Radiator needs of customers in Smyrna, TN, Murfreesboro, TN, LaVergne TN, and surrounding areas.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Q: use the word "abandoned" in a sentence following a verb I want to say "we go on with life with feeling abandoned" is it correct?
does it sound like you're feeling abandoned?
Or does it sound like our feeling is abandoned?
A: You actually don't need the second "with", here. "feeling abandoned" is actually a phrase which acts as an adverb (modifying the verb "go on"), the same way that a word like "unhappily" might:
We go on with life unhappily.
So you can actually replace "unhappily" with "feeling abandoned" directly. However, since the adverbial phrase in this case includes another verb ("feeling"), it is generally separated from the rest of the sentence using a comma (to make it clearer which is the main sentence and which words are part of the adverbial phrase), so we get:
We go on with life, feeling abandoned.
To answer your other question, yes, this way of phrasing things implies that "we" are "feeling abandoned". Here, "feeling" is obviously a verb (we "feel"). If you actually wanted to say that (somebody's) feelings have been abandoned you would need to use the noun form ("feelings"), not the verb form ("feeling"), and would generally want another verb to connect them: "feelings having been abandoned".
A: Feeling is a present participle modifying we and taking the complement abandoned. A participle is a kind of adjective: a verb playing the part of an adjective. Just as with any other adjective, we don't introduce it with a preposition.
Here are some ways that you could rearrange the sentence:
Feeling abandoned, we go on with life.
We, feeling abandoned, go on with life.
These change the emphasis, but in all of them, feeling abandoned modifies we. It tells how "we" feel when we go on with life.
Here are some sentences with similar structure.
Claudia came to the costume party dressed as a caterpillar.
Malcolm walked home from school carrying his cello.
Ted returned from college 2 inches taller than when he left.
In all of these, the marked phrase functions as an adjective that modifies the subject of the sentence. It does not get introduced by a preposition. A preposition normally introduces some form of noun.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
Collisions between atomic nuclei at intermediate energies are often used to infer the bulk properties of nuclear matter.
Central among the bulk properties is the nuclear equation of state (EOS), which pertains to the state of stationary equilibrium of the matter and contains no information regarding the pace at which the equilibrium is reached.
Collective flow observables were successfully exploited to infer the EOS.
On the other hand, bulk properties that are tied to equilibration rate include transport coefficients such as shear viscosity and heat conduction.
In particular, shear viscosity is tied to momentum transport in a medium and, among reaction observables, it is natural to link it to stopping observables that quantify dissipation of momentum.
Knowledge of the shear viscosity is important for understanding the evolution of supernovae, the stability of rotating neutron stars, and the formation of black holes.
Besides its immediate practical importance, there have been conjectures regarding a fundamental quantum lower limit on the ratio of shear viscosity to entropy density ($\eta / s$) in a wide range of media \cite{danielewicz_dissipative_1985,kovtun_viscosity_2005, schafer_fluid_2014}. Among other situations, the limit is thought to be approached in the quark-gluon plasma and accessed in ultrarelativistic heavy ion collisions \cite{lacey_has_2007}.
The question remains as to whether freeing quark degrees of freedom is needed to approach such a limit in collisions.
We will make a quantitative assessment of how close nuclear matter, as seen in these lower-energy collisions, is to this ``perfect liquid'' limit.
In the present work, we use stopping, i.e.\ the degradation of the projectile longitudinal momentum due to interaction with the target, to constrain the elastic part of the in-medium nucleon-nucleon cross section, \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{}, in a \textsc{BUU} transport model.
These constraints are not without ambiguity. Different observables might lead to different conclusions. Therefore, we consider different stopping observables for different systems at different energies. What's more, different strategies for modifying the cross section in the nuclear medium can lead to the same degree of stopping, unsurprisingly.
Consequently, we inspect whether shear viscosity is actually correlated with the stopping observables in collisions.
We calculate the viscosity in a manner consistent with the Boltzmann equation used to describe the collisions and find a strong correlation between the predicted stopping observables and the magnitude of the predicted shear viscosity coefficients.
The correlation suggests a robustness in the conclusions on the viscosity, even when cross sections are not easy to pin down unambiguously based on the data alone.
\section{Boltzmann-Uehling-Uhlenbeck equation}\label{sec:buu}
To model central nuclear reactions and predict observables, we use a set of Boltzmann-Uehling-Uhlenbeck (\textsc{BUU}) equations, one for each species X, describing the time evolution of
a Wigner quasi-probability distribution in phase space,
$f_\mathrm{X} \equiv f_\mathrm{X}(\vec{r},\vec{p},t)$:
\begin{equation}\label{eqn:buu}
\frac{\partial f_\mathrm{X}}{\partial t}
+ \frac{\partial \epsilon_{\vec{p}}}{\partial \vec{p}} \frac{\partial f_\mathrm{X}}{\partial \vec{r}}
- \frac{\partial \epsilon_{\vec{p}}}{\partial \vec{r}} \frac{\partial f_\mathrm{X}}{\partial \vec{p}}
= I_{\mathrm{X}, \textrm{elastic}} + I_{\mathrm{X}, \textrm{inelastic}} \,.
\end{equation}
A prototype equation for the above is the Vlasov equation (single-particle Liouville equation), with vanishing r.h.s., describing the single-particle evolution of a phase space density in a mean field.
In the above, $\dfrac{\partial \epsilon_{\vec{p}}}{\partial \vec{p}}$ is the single-particle velocity, and $\dfrac{\partial \epsilon_{\vec{p}}}{\partial \vec{r}}$ is the force due to the mean field.
The r.h.s.\ of Eq.~\ref{eqn:buu} takes into account the effects of elastic and inelastic collisions. The elastic contribution can be expressed as
\begin{equation}\label{eq:buu-elastic}
I_{\mathrm{X}, \textrm{elastic}} = \sum_Y \frac{g_\mathrm{X}}{(2\pi\hbar)^3} \int \mathrm{d}\vec{p}_Y \,\mathrm{d}\Omega \,\, v_{XY} \,\, \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}
\left( \tilde{f}_{\mathrm{X}}^{} \tilde{f}_\mathrm{Y}^{} f_\mathrm{X}' f_\mathrm{Y}' - \tilde{f}_\mathrm{X}' \tilde{f}_\mathrm{Y}' f_\mathrm{X}^{} f_\mathrm{Y}^{} \right) \, .
\end{equation}
The first term accounts for particles with momenta $\vec{p}\phantom{'}'_\mathrm{X}$ and $\vec{p}\phantom{'}'_\mathrm{Y}$ colliding and acquiring the final momenta $\vec{p}_\mathrm{X}$ and $\vec{p}_\mathrm{Y}$, thus increasing the occupancy $f_\mathrm{X}$ (gain).
The second term describes, correspondingly, a decrease in the occupancy $f_\mathrm{X}$ in a reverse process (loss).
Here, for nucleons, $\tilde{f}_\mathrm{X} \equiv 1-f_\mathrm{X}$ represents the Pauli principle blocking scattering into the final state $\vec{p}_\mathrm{X}$.
The rate of scattering is governed by the elastic NN cross section $\dfrac{\mathrm{d} \sigma}{\mathrm{d} \Omega}$ (here, a function of relative momentum and the scattering angle $\theta$; $\Omega \equiv (\theta, \varphi)$).
It is this cross section of which modifications by in-medium effects are explored in Section~\ref{sec:cs-reductions}.
The second, inelastic term on the r.h.s.\ of Eq.~\ref{eqn:buu} represents interactions that create or annihilate particles of the given species.
There has been some work in producing medium modifications to inelastic processes \cite{li_isospin_2017}.
However, in the following sections, we only consider modification of elastic cross sections. Therefore, we must limit drawing conclusions to regimes of reaction dynamics where inelastic processes do not significantly affect the dynamics. Once beam energies are high enough, for example, pions are produced early in the collision. This affects the stopping, so until inelastic cross sections are also addressed, we restrict ourselves to lower energies. Formation and breakup of nuclear clusters is an inelastic process too, but we restrict this process to low densities, so that it plays a role only after the dynamics significant to stopping have taken place.
An implementation of a time-dependent solution to the Boltzmann equation set by Danielewicz and collaborators \cite{danielewicz_production_1991, danielewicz_blast_1992, pan_sideward_1993,danielewicz_determination_2000, danielewicz_hadronic_2002, barker_dissipation_2014}, often termed pBUU, is used to describe nuclear collisions.
In this implementation, the Wigner distributions are represented by a large number of test particles.
These particles move along classical trajectories under the influence of the mean field and then encounter binary collisions on a statistical basis with other test particles that are close to them in position space.
With an increase of test particle number, the simulation converges on a better sampled, stable solution \cite{bertsch_guide_1988}.
The single-particle energies $\epsilon_{\vec{p}}$ in Eq.~\ref{eq:buu-elastic} are derived from an assumed energy functional $\mathcal{E}\{f\}$ \cite{danielewicz_determination_2000} that accounts for modifications of the particle energies from free-space values $\epsilon^0_{\vec{p}}$ due to the average effect of interactions with particles in the medium.
The mean-field potential is $U=\epsilon_p - \epsilon_p^0$.
Unless otherwise indicated, we employ in the calculations an energy functional that yields a soft equation of state (EOS) and momentum-dependent $U$.
In the literature, the abbreviation ``SM'' is attributed to such functionals.
\subsection{Impact parameter selection}
Throughout this work, we will be comparing our simulation results to experimental data. In experiment, a range of impact parameters is selected for analysis. Most often, it is uncertain what precisely the distribution of those impact parameters is. In any single transport simulation, the initial state is prepared with one specific
impact parameter. To save computation time, an effective impact parameter, $b_\mathrm{eff}$, is commonly chosen that represents the median in probability for the impact parameter range. For a range bounded by $b_\mathrm{min}$ and $b_\mathrm{max}$, the effective impact parameter $b_\mathrm{eff}$ is normally taken from
\begin{equation}
\begin{split}
\pi b_\mathrm{eff}^2 &= \frac{\pi b_\mathrm{min}^2 + \pi b_\mathrm{max}^2}{2} \,, \\ \\
b_\mathrm{eff} &= \sqrt{\frac{1}{2} \left( b_\mathrm{min}^2 + b_\mathrm{max}^2 \right)} \,.
\end{split}
\end{equation}
In studies of the central collisions, often experimental ranges effectively start at $b_\mathrm{min}=0$, so $b_\mathrm{eff} = b_\mathrm{max}/\sqrt{2}$. We have tested in several cases that such a single parameter can indeed adequately represent the range, in that results from one parameter agree to a satisfactory degree with those from combining calculations from impact parameters spanning the range.
\section{Shear viscosity}
An elementary setting for introducing the concept of viscosity is that of laminar shear in a macroscopic system.
Consider two plates, with a medium between them, moving in antiparallel directions, in the steady state.
The layer adjacent to one plate induces a shear stress, $\tau$, on the layer below it, causing that layer to have a velocity $v(y-\mathrm{d}y) < v(y)$.
That layer induces a shear stress on the layer under it, and so on.
In the linear response approximation, these velocities can be related using the equation $\tau = \eta (\partial v / \partial y) $, where $\hat{y}$ is perpendicular to the plates.
Here, $\eta$ is the coefficient of shear viscosity, which is a measure of the efficiency of the momentum transfer in the medium.
In the nuclear context, many investigations concentrated on characteristics of giant resonances in order to infer the viscosity of nuclear matter (\cite{mondal_experimental_2017}, see references in \cite{auerbach_eta/s_2009}).
This relies on the validity of a hydrodynamic description down to zero temperature where the nucleon mean free path diverges.
We find that hydrodynamics fails to describe energetic reactions where the mean free path, while short, is not short enough for a hydrodynamic description to hold, requiring the use of transport theory to extrapolate to equilibrium or near-equilibrium situations.
Several groups have investigated the aforementioned $\eta / s$ ratio for different models utilized in nuclear collisions at intermediate energies, such as statistical multifragmentation \cite{pal_shear_2010} and quantum molecular dynamics (\textsc{QMD}) \cite{zhou_shear_2012}.
However, the latter investigations did not link viscosity to specific observables and did not aim at generality of the results beyond the specifics of the models.
Zhou \textit{et al.} noticed a correlation between shear viscosity and the strength of elliptic flow \cite{zhou_correlation_2014}. However, they did not validate that their model was accurately predicting viscosity-related observables by comparing to experimental data; therefore, their result is helpful for gaining a qualitative understanding of a theoretical relationship, but it is less reliable for learning about absolute bulk properties. Finally, the relaxation-time approaches \cite{xu_shear_2013-1, guo_isovector_2017} are suitable for order-of-magnitude estimates, but not for quantitative assessments.
The shear viscosity coefficient $\eta$, derived from the Boltzmann equation in Ref.~\cite{danielewicz_transport_1984} (see also \cite{shi_nuclear_2003}), is
\begin{equation}\label{eq:buu-viscosity}
\eta = \frac{5 T}{9} \frac{\left( \int \mathrm{d}\vec{p}_1 f_1 p_1^2 \right)^2} %
{\int \mathrm{d}\vec{p}_1 \,\mathrm{d}\vec{p}_2 \,\mathrm{d}\Omega f_1^{} f_2^{} \tilde{f}_1' \tilde{f}_2'%
v_{12} \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} q_{12}^4 \sin^2 \theta } \, .
\end{equation}
Here, the elastic scattering cross section, $\mathrm{d}\sigma/\mathrm{d}\Omega$, is scaled with a factor $q^4 \sin^2 \theta$, which emphasizes large relative momenta $q$, where $q_{12} = \abs{\vec{p}_1 - \vec{p}_2}/2\,$, and wide scattering angles $\theta$. Thus, in kinetic transport, viscosity is tied to $\dfrac{d\sigma}{d\Omega} q^4 \sin^2 \theta$, sometimes called the ``transport cross section'' --- the differential particle-particle cross section scaled with a weight that increases with relative momentum and scattering angle.
To learn about the shear viscosity, we will adjust the NN cross sections to match the stopping data, and we will draw conclusions about the viscosity using Eq.~\ref{eq:buu-viscosity}.
\section{The NN cross section in the nuclear medium}\label{sec:cs-reductions}
Looking ahead, comparisons to data clearly demonstrate that using the bare nucleon-nucleon cross section in the \textsc{BUU} equation (\ref{eqn:buu}) overestimates the amount of stopping found in central collisions at intermediate energy.
There are several different perspectives on the $\sigma_\mathrm{NN}$ in the medium.
Many groups follow the assumption that cross sections (CS) should scale with the nucleon effective mass \cite{pandharipande_nuclear_1992,persram_elliptic_2002,li_nucleon-nucleon_2005}.
This would require the nuclear transition matrix to stay the same in the medium as in vacuum, which is a perturbative approximation that does not hold for nuclear interactions. Therefore, there are questions about the validity of this assumption.
Further, the cross section should also be affected by the isospin asymmetry of the surrounding medium and several other factors not emphasized in the scaling.
In confronting the microscopic theory with the scaling \cite{sammarruca_microscopic_2014}, Sammaruca concluded that \underline{no} simple phenomenological ansatz following effective mass scaling is valid.
Some authors simply take \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} as a fraction, e.g.\ half, of the free cross section \cite{cai_yanhuang_semiclassical_1989,gaitanos_stopping_2004,zhou_thermodynamic_2013, basrak_aspects_2016}.
The deficiency of this assumption is that the free NN cross sections are not recovered when the matter becomes sufficiently dilute.
Following the transition matrix approach, one can derive, in the quasi-particle limit, both the mean field and in-medium cross section, making the development of the Boltzmann equation more self-consistent, in principle, changing both sides of Eq.~\ref{eqn:buu} \cite{alm_critical_1994, gaitanos_nuclear_2005}.
With this, though, the collision modification is only due to the mean field and statistics, and collisions do not affect each other.
In the early phenomenological parametrization of \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} in the literature, the cross section was assumed to change linearly with density \cite{westfall_mass_1993}. Eventually, with rise in density, this results in negative values. Another phenomenological approach was later adopted, where the cross section was assumed to reduce to a geometric unitary limit at high density \cite{danielewicz_hadronic_2002}.
In this paper, three scenarios for intermediate cross sections are discussed and then explored.
We resort to those scenarios because the free NN cross sections are found to be too large to describe data.
In the \textsc{BUU} simulations, the application of the \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} results in a reduced probability for NN collisions, compared to $\sigma_\textsc{NN}^\textrm{free}$.
For practical use in the simulations, each reduced cross section can be presented as a reduction factor multiplied by the free cross section.
That reduction factor gets sampled statistically in the simulations.
Pauli blocking of the final state is incorporated separately from this reduction factor.
\subsection{Tempered cross sections}
The Tempered cross section reduction scheme \cite{danielewicz_hadronic_2002} is arrived at by considering unitarity.
For a particle moving through a medium of number density $n$, the scattering partners are distributed at a relative distance of $\approx n^{-1/3}$. For two-body collisions to be independent from each other, the cross sections should be no larger than a value of the order of the distance squared, or $n^{-2/3}$,
\begin{equation}\label{eq:tempered-sigma0}
\ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} \lesssim \sigma_0 \equiv \nu n^{-2/3} \, ,
\end{equation}
where $\nu$ is of the order of 1. As the medium becomes more dilute, though, the cross sections should reach their free-space limit.
We parameterize the gradual change between the free and unitary limits with the formula
\begin{equation}\label{eqn:tempered}
\ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} = \sigma_0 \tanh \left( \frac{\sigma^\text{free}_\textsc{NN}}{\sigma_0} \right),
\end{equation}
where $\sigma_0$ is defined with the r.h.s.\ of Eq.~\ref{eq:tempered-sigma0} (in principle, other smooth interpolating functions could be used). As energies in an NN subsystem increase, the free NN cross sections become increasingly anisotropic, peaking in the forward and backward directions.
Those peaks are tied to higher angular momentum values. When particles are more tightly packed in a medium, these cross section contributions should be more suppressed than contributions from lower angular momenta.
For the anisotropic cross sections, we adopt a modification of Eq.~\ref{eqn:tempered}:
\begin{equation}\label{eqn:temper-omega}
\left( \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} \right)^\textrm{med}
= \frac{\sigma_0}{4\pi}
\tanh \left[ \frac{4\pi}{\sigma_0}
\left( \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} \right)^\textrm{free}
\right] \,.
\end{equation}
Here, for the purposes of the cross section, particles are treated as distinguishable, even when they belong to the same species.
The equation above accomplishes the goal of preferentially suppressing the forward and backward peaks, or contributions from high angular-momenta, relative to those from lower momenta.
With Eq.~\ref{eqn:temper-omega}, the cross sections become low and isotropic in the high-density limit, with the absolute limit on differential cross section of $\sigma_0 / 4\pi$. We stress here again that we treat the particles as distinguishable in the determination of cross section.
\subsection{Rostock cross sections}
Some early microscopic calculations of in-medium cross sections were carried out at the University of Rostock \cite{my_alm_critical_1994}, within a thermodynamic T-matrix approach.
In these calculations, the cross sections were modified to account for Pauli blocking in intermediate states and due to single-particle energy shifts \cite{alm_-medium_1995}.
Moreover, the results were derived assuming that the total momentum of the particles was zero in the frame of the local nuclear matter, in order to simplify the calculations.
We coarsely capture the essence of the results \cite{alm_critical_1994} with the following parametrization of the cross section reduction:
\begin{equation}
\ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} = \sigma^\textrm{free}_\textsc{NN} \exp \left( -0.6 \frac{\rho/\rho_0}{1+\left[T_\text{c.m.} / (150\,\text{MeV})\right]^2} \right),
\end{equation}
where $T_\text{c.m.}$ is the total kinetic energy of the two interacting particles, in the frame where the local medium is at rest.
\subsection{Fuchs cross sections}
Fuchs \textit{et al.} \cite{fuchs_off-shell_2001} underscored that in the \textsc{BUU} equation (\ref{eqn:buu}), the in-medium mean fields that the particles are subject to on the l.h.s.\ of the equation should be derived consistently with the in-medium NN cross sections $\sigma$ used in the collision integral on the r.h.s.
As the basis for these simultaneous alterations, they employed the in-medium Dirac-Brueckner T-matrix \cite{fuchs_off-shell_2001}.
Like the Rostock one, this cross section was derived for two particles with total momentum equal to zero in the rest frame of the local medium.
The cross section reduction of Fuchs \textit{et al.} is parameterized here with
\begin{equation}
\sigma_\textrm{nn}^\textrm{med} = \sigma_\text{nn}^\textrm{free}
\exp \left( -1.7 \frac{\rho/\rho_0}{1+\left[T_\text{c.m.}/ (12\,\text{MeV})\right]^{3/2}} \right)
\end{equation}
\begin{equation}
\sigma_\text{np}^\textrm{med} = \sigma_\text{np}^\textrm{free}
\exp \left( -1.4 \frac{\rho/\rho_0}{1+[T_\text{c.m.} / (33\,\text{MeV})]} \right).
\end{equation}
The different cross section reductions above will be used to compare predictions from pBUU to stopping data, to see if stopping is sensitive to the absolute or to the transport cross section similarly to the shear viscosity. If the latter dependence were monotonic, then we could tune the NN cross section to reproduce experimentally observed stopping and use that tuned cross section to calculate the shear viscosity self-consistently.
\section{Stopping observables}
Different observables have been used in the literature to characterize nuclear stopping. The observables tend to be optimized for a specific energy range where measurements are carried out. We use several of these observables to enable larger energy range coverage and better discern the robustness of our conclusions.
\subsection{Linear momentum transfer}
In a mass-asymmetric collision of a light projectile colliding with a heavy target, one can assess the momentum that is transferred to the target, and thus have a measure of the stopping power --- that is, a reflection of how much the projectile decelerates when it interacts with the target, provided the target survives in some form.
As in the schematic in Fig.~\ref{fig:lmt-schematic}, one finds the laboratory-frame velocity of the largest fragment emitted from the collision, assuming that this fragment stands out.
Because of the high mass, that fragment is assumed to originate from the target (the ``target-like fragment'').
The higher its velocity, the more momentum was transferred from the projectile.
This corresponds to a higher degree of stopping.
To compare the observable across reaction systems, this fragment velocity is divided by that of the center of mass.
Since the velocities involved are non-relativistic, they can be used to infer the scaled linear momentum transfer (\textsc{LMT}). This observable was originally used to distinguish between direct and compound fission reactions in heavy nuclei \cite{nicholson_direct-interaction_1959,sikkeland_momentum_1962}, then used more generally in nucleus-nucleus collisions \cite{viola_linear_1982}.
The technique relies on a clear determination of the target-like fragment, and as the beam energy increases, there are more violent collisions and the largest fragment produced becomes lighter.
In consequence, the practical energy range for this observable is from around the Coulomb barrier to around 150$\,$MeV/nucleon or so.
Above this range, any fragment that could be tied to the target is difficult to distinguish from other fragments of similar intermediate mass.
The observable \textsc{LMT} is defined \cite{colin_splintering_1998} as
\begin{equation}
\textsc{LMT} = \left\langle \frac{v_\parallel}{v_\text{c.m.}} \right\rangle \, ,
\end{equation}
where $v_\parallel$ is the velocity of the target-like residue in the beam direction, $v_\mathrm{c.m.}$ is the velocity of the reaction center of mass. A higher \textsc{LMT} corresponds to a higher degree of stopping.
\begin{figure}
\begin{center}
\input{limot-schematic.pdf_tex}
\end{center}
\caption{
Schematic of asymmetric collision.
Projectile transfers momentum to the target.
To assess linear momentum transfer, the longitudinal velocity of the target-like fragment is compared to the velocity of the center-of-mass of the collision.}\label{fig:lmt-schematic}
\end{figure}
Experimental \cite{colin_splintering_1998} and theoretical results for \textsc{LMT} in collisions of a $^{40}$Ar projectile with Cu, Ag, and Au targets are shown in Figs.~\ref{fig:lmt-arcu}, \ref{fig:lmt-arag}, and \ref{fig:lmt-arau}, respectively.
At low energies, \textsc{LMT}$\simeq 1$, indicating formation of a compound system and complete stopping. As beam energy increases, transparency sets in and \textsc{LMT} decreases.
In the experiment, it appears that targets were used with their natural isotopic content.
To determine $V_\parallel$ in the equation above, a filter on just the heaviest fragments was used, with the assumption that these heaviest fragments provided a good average estimate of the longitudinal momentum of the target remnant.
\begin{figure}
\begin{center}
\input{limot-arcu.pdf_tex}
\caption{Linear momentum transfer for $^{40}$Ar+$^{}$Cu.
Lines represent the theoretical results incorporating different in-medium NN cross sections.
The ``Tempered'' reductions are marked with their adjustable parameter $\nu$.
Symbols represent experimental data \protect\cite{my_colin_splintering_1998}.}\label{fig:lmt-arcu}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\input{limot-arag.pdf_tex}
\caption{Linear momentum transfer for $^{40}$Ar+Ag. Lines represent the theoretical results incorporating different in-medium NN cross sections. The ``Tempered''
reductions are marked with their adjustable parameter
$\nu$. Symbols represent experimental data \protect\cite{my_colin_splintering_1998}.}\label{fig:lmt-arag}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\input{limot-arau.pdf_tex}
\caption{Linear momentum transfer for $^{40}$Ar+Au. Lines represent the theoretical results incorporating different in-medium NN cross sections. The ``Tempered''
reductions are marked with their adjustable parameter
$\nu$. Symbols represent experimental data \protect\cite{my_colin_splintering_1998}.}\label{fig:lmt-arau}
\end{center}
\end{figure}
In the \textsc{BUU} calculations, we use the specific isotopes $^{63}$Cu, $^{107}$Ag, and $^{197}$Au for the targets. In central collisions at this energy, the one- or two-neutron differences in the target content should not impact \textsc{LMT} enough to matter. Within our simulation, the target remnant is explicitly tracked throughout the collision, and its velocity is directly calculated from the constituent particles.
In particular, nucleons that initially belonged to the target and continue to be bound are considered to be part of the target remnant.
To be considered bound, the particle energy must be at least $6\,$MeV below the continuum in the local frame. For charged particles, the energy excludes the Coulomb contribution, i.e.\ the continuum is counted from the top of the local Coulomb barrier.
The various \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} schemes described in Section~\ref{sec:cs-reductions} are tested in \textsc{BUU} calculations, with the corresponding results shown with lines in Figs.~\ref{fig:lmt-arcu}--\ref{fig:lmt-arau} alongside the experimental data.
The Rostock and Fuchs reductions, as well as the case with no reduction (``free''), are labeled with their names, while the Tempered CS is marked by its tunable parameter $\nu$.
It is clear from the \textsc{LMT} figures that the free cross section overestimates the stopping in all three reaction systems, and that the Tempered CS with $\nu=0.2$ underestimates it.
The Rostock and Fuchs reductions produce \textsc{LMT} values that are very close to each other in all cases, with Fuchs resulting in $\sim7\%$ higher values than Rostock in the 65$\,$MeV/nucleon region.
Generally, use of the free cross section results in a coarsely linear dependence of \textsc{LMT} on beam energy in all three systems at about 27$\,$MeV and higher, while the reductions all exhibit a positive concavity with energy in the $^{40}$Ar$\,+\,$Cu and $^{40}$Ar$\,+\,$Ag cases, which more closely resembles the data.
In the case of $^{40}$Ar$\,+\,$Au, all calculated lines show a roughly linear dependence on beam energy, while the experimental data shows an even larger concavity compared to the lighter systems.
Judging by eye, the cross section that best fits the $^{40}$Ar$\,+\,$Cu and $^{40}$Ar$\,+\,$Ag data is the Tempered one with $\nu=0.4$ or 0.6.
In the $^{40}$Ar$\,+\,$Au reaction, the cross section that best fits the data seems to be the Tempered CS with $\nu=0.8$.
\subsection{Rapidity variance ratio}
If particles in the hot, dense region of a nuclear collision undergo many collisions (because the mean free path becomes comparable to the typical interparticle distance), the region tends to equilibrate, and particles will lose memory of which direction they were originally traveling in.
With this, more stopping will occur and emission from this specific region will tend towards isotropy in the reaction center of mass.
The FOPI Collaboration provides a practical measure of this isotropy with the observable \textit{varxz}, defined as \cite{reisdorf_systematics_2007}
\begin{equation}
\textit{varxz} = \frac{\Delta y_x}{\Delta y_z},
\end{equation}
where $\Delta y_x$ is the variance of particle rapidity along a randomly chosen direction that is transverse to the beam and $\Delta y_z$ is the variance of the standard particle longitudinal rapidity.
Fig.~\ref{fig:varxz-nocomp-au} shows the experimental results from \textsc{FOPI} \cite{reisdorf_systematics_2010} as well as the p\textsc{BUU} transport simulation results with the various \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} reduction schemes used, for \coll{Au}{Au}, looking at the distribution of protons. The experiment was carried out using the heavy ion accelerator \textsc{SIS} at \textsc{GSI}/Darmstadt, and charged particles were detected with a good coverage of angles throughout the $4\pi$ region, using the \textsc{FOPI} detector and a set of other detectors that provided particle tracking, energy loss determinations, time of flight determination, and charged particle identification. The beam energies spanned the range from 0.09 to 1.5$\,$GeV/nucleon, and the impact parameter selection was limited to $b_{\textrm{red}} \equiv b/b_{\textrm{max}} \lesssim 0.15$.
The most startling finding in Fig.~\ref{fig:varxz-nocomp-au} is that simulations with free cross sections yield \textit{varxz} clearly in excess of 1, in a wide energy range of 0.09 -- 0.6 GeV/nucleon, while \textit{varxz} seems to always stay below 1 in measurements.
This clearly eliminates the possibility of cross sections staying the same in the medium as in free space.
Thus, just like LMT, the \textit{varxz} comparison points to a reduction of the cross sections in the medium.
In the simulations with free cross sections, the matter exhibits a strong hydrodynamic behavior, splashing to the sides in central collisions \cite{danielewicz_effects_1995}, yielding $\textit{varxz}>1$.
On the other hand, in the measurements, even the isotropy is never reached, with the medium always staying partially transparent.
Both in the calculations with free cross sections and in the measurements, \textit{varxz} eventually drops as energy increases.
This can be attributed to two factors: typical momenta in the center of mass become large compared to the Fermi momentum, and cross sections become increasingly isotropic with the increase in energy.
Additionally, as energy increases, inelastic processes give rise to $\Delta$ resonances and pions.
The higher the energy, the more important those inelastic processes become.
While adjusting the in-medium cross sections, we can arrive at similar \textit{varxz} values as in experiment at energies below 0.8 GeV/nucleon.
At higher bombarding energies, the theoretical results with different cross sections begin to merge and exceed experimental \textit{varxz}.
This is because, in the simulations, we adjust only elastic cross sections and leave inelastic intact, and the balance in the importance shifts to the latter cross sections as energy increases.
When examining the p\textsc{BUU} simulations with the tempered \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} at $\nu=0.6$ (a reasonable parameter), at energies from 90 to $1500\,$MeV/nucleon, the ratio of peak $\Delta$ production and absorption rates, which are the primary inelastic processes, to peak elastic collision rates varies from 0 to 0.8. Assuming that the inelastic collisions start significantly affecting the reaction dynamics when the ratio is about 0.2 or 0.3, then we should look at beam energies of less than $~600\,$MeV/nucleon in deciding on in-medium cross sections.
Given this caveat, it seems the \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} that best fits the data below 600$\,$MeV/nucleon is either $\nu=0.8$ or Rostock.
\begin{figure}
\begin{center}
\input{varxz-nocomp-au.pdf_tex}
\caption{Stopping observable \textit{varxz} for protons in \coll{Au}{Au} collisions at different beam energies at $b_\text{red}<0.15$.
Lines show the effects of different in-medium reductions of the NN cross section.
The ``Tempered'' cross-section reductions are marked with their tunable parameter $\nu$.
Symbols are experimental data from the \textsc{FOPI} Collaboration. \protect\cite{my_reisdorf_systematics_2010}.}\label{fig:varxz-nocomp-au}
\end{center}
\end{figure}
Another caveat concerning conclusions on in-medium cross sections concerns another type of inelastic process, namely cluster production and breakup.
The clusters get more copiously produced in colder regions of the matter and the production predominantly takes place at subnormal densities.
In the pBUU simulations, we have the option of activating the production of $A = 2$ and $A = 3$ fragments.
The production is limited in the simulations to the densities $\rho \lesssim 0.6 \rho_0$, but the production and breakup rates are calculated based on processes taking place in the vacuum without in-medium modifications. As the processes of cluster production and breakup can compete with two-body collisions, there may be concerns about the ability to make conclusions about the medium modifications of the two-body processes.
Most often, due to concerns about the impact of any double-counting of interactions, we carry out the calculations of stopping with the cluster production switched off. However, for the sake of testing the validity of concerns tied to the cluster production and absorption, we also carried out calculations of Au+Au reactions with the cluster production activated.
The peak region in \textit{varxz} vs.\ energy, where the comparison between data and theory may be most telling, seems to be generally best described with the Rostock cross section or the tempered cross section with $\nu = 0.8$, regardless of whether the calculation includes cluster production or not.
\begin{figure}
\begin{center}
\input{varxz-comp-au-species.pdf_tex}
\caption{Analogous to Fig.~\ref{fig:varxz-nocomp-au}, except that $A = 2, 3$ composite particle formation is enabled in the simulations, so a comparison to results for different particle species in the experiment \cite{reisdorf_systematics_2010} becomes possible.}\label{fig:varxz-comp-au-species}
\end{center}
\end{figure}
\subsection{Isospin tracer}
Another observable that we use for assessing cross sections, the so-called isospin tracer, attempts to identify the relative yield of particles from the target and projectile in a given region of momentum space by examining isospin content.
This is done by studying collisions between nuclei with identical mass number but different charge number, interchanging the projectile and target roles, and then comparing the results to those from collisions of identical nuclei.
The method is described in more detail here:
\begin{quotation}
``The ($N/Z$)-tracer method is based on the following idea:
let us assume that we are observing the final number of
protons, $Z$ in a given cell of the momentum space. The
expected yield $Z^\text{Ru}$ measured for the Ru $+$ Ru reaction
is higher than $Z^\text{Zr}$ of the \coll{Zr}{Zr} reaction since Ru has
44 protons as opposed to 40 for Zr. Such measurements
using identical projectile and target deliver calibration values
$Z^\text{Ru}$ and $Z^\text{Zr}$ for each observed cell. In the case of a
mixed reaction, \coll{Ru}{Zr} or \coll{Zr}{Ru}, the measured proton
yield $Z$ takes values intermediate between the calibration
values ($Z^\text{Ru}$, $Z^\text{Zr}$). If, e.g., $Z$ is close to $Z^\text{Ru}$ in a \coll{Ru}{Zr}
reaction, means that the cell is populated predominantly
from nucleons of the Ru projectile while if it is close to $Z^\text{Zr}$
it is mostly populated from nucleons of the Zr target. In
this way it is possible to trace back the relative abundance
of target to projectile nucleons contributing to a given cell.'' \cite{rami_isospin_2000}
\end{quotation}
Within the method, one constructs the observable $R_Z$, defined as
\begin{equation}\label{eq:rz}
R_Z = \frac{2 \times Z - Z^\text{Zr} - Z^\text{Ru}}{Z^\text{Zr} - Z^\text{Ru}} \, ,
\end{equation}
which assesses relative abundances of the projectile-target nucleons. In this case, $Z$ represents proton yield in a reaction with different projectile and target for a given location in momentum space. Yields for other particles can also be used in velocity space \cite{rami_isospin_2000}.
With the above definition, one arrives at $R_Z = 1\: (-1)$ when a momentum cell gets populated by protons originating exclusively from
the Zr (Ru) nucleus, as long as the dynamics do not depend on the charge content.
The case of complete stopping would mean that the protons completely mix and, for any cell, half come from Zr and half from Ru. Thus, full
stopping is expected to yield $R_Z \equiv 0$.
\begin{figure}
\begin{center}
\input{rz-nc.pdf_tex}
\caption{Isospin tracer observable for central collisions of \collpt{96}{Zr}{96}{Ru} (bottom) and \collpt{96}{Ru}{96}{Zr} (top) at 400$\,$MeV/nucleon vs.\ scaled center-of-mass rapidity, such that $y_{z0}=-1$ is the initial projectile rapidity.
Data is from the FOPI collaboration \protect\cite{rami_isospin_2000}. A tendency of $R_z$ to stay closer to zero at finite rapidities indicates a higher degree of mixing, and thus stopping.}\label{fig:rz-nc}
\end{center}
\end{figure}
The experimental results \cite{rami_isospin_2000}, along with results from the \textsc{BUU} transport model, with the $Z$ in $R_Z$ representing proton yield, for collisions between $^{96}$Zr and one of its $A=96$ counterparts, $^{96}$Ru, are shown in Fig.~\ref{fig:rz-nc} plotted against rapidity, for beam energy of 400$\,$MeV/nucleon and $b_\text{red}<0.12$.
As rapidity (in the center of mass) gets more negative, the momentum cells are increasingly populated by protons from the target, according to $R_Z$ and the interpretation above.
This makes sense, as the target's particles are more likely to persist in the backward rapidity region for limited momentum transfers in interactions.
As the bins closer to midrapidity are examined, it is seen that those bins get populated by protons from both colliding nuclei, as $R_Z$ is close to zero there.
It is again clear from Fig.~\ref{fig:rz-nc} that use of the free \textsc{NN} cross section overestimates the stopping or mixing in this case, when assessed with the $R_Z$ observable.
It seems that the \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} best fitting the data here is either Rostock, Fuchs, or Tempered with $\nu \sim 0.8$.
The $R_Z$ observable is challenging for Monte Carlo calculations, due to statistical fluctuations that get emphasized in the subtraction of similar values, $Z^\text{Zr} - Z^\text{Ru}$, and further amplified in the division by the resulting small number in Eq.~\ref{eq:rz}.
\subsection{Summary of in-medium cross section analysis}
A summary of the stopping observables that were investigated in different systems and the optimal \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} for each is given in Table~\ref{tab:stopping}. Overall, there is not one \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{} that optimally reproduces the stopping across all observables, sizes, and energies.
It is clear, though, that the cross section is reduced. For the remaining investigations, we use Tempered with $\nu=0.6$ as it is the most representative of the range of conclusions.
\begin{table}
\begin{tabular}{l|c|c|c}
observable & reaction system & energies [MeV] & best cross section reduction \\
\hline
\textsc{LMT} & $^{40}$Ar$\,+\,$Cu & 17--115 & Tempered w/ $0.4 \lesssim \nu \lesssim 0.6$ \\
\textsc{LMT} & $^{40}$Ar$\,+\,$Ag & 17--115 & Tempered w/ $0.4 \lesssim \nu \lesssim 0.6$ \\
\textsc{LMT} & $^{40}$Ar$\,+\,$Au & 27--115 & Tempered w/ $\nu \simeq 0.8$ \\
\textit{varxz} & Au$\,+\,$Au & 90--1500 & Tempered w/ $\nu \simeq 0.8$ or Rostock \\
$R_z$ & $^{96}$Zr+$^{96}$Ru & 400 & Tempered w/ $\nu \simeq 0.8$, Rostock, or Fuchs \\
& (and inverse) & &
\end{tabular}
\caption{Summary of cross section determination results. No single cross section reduction is favored universally. The Tempered cross section with $\nu \sim 0.6$ is deemed to be the best compromise.}\label{tab:stopping}
\end{table}
As the last issue of potential concern in drawing conclusions from stopping observables, we discuss possible competition between cross sections and mean fields in shaping those observables.
\subsection{Mean-Field Sensitivity}\label{sec:sensitivity-eos}
Choices made regarding the energy functional can make the matter less or more compressible. For more incompressible matter, the mean field potentials become more quickly repulsive with increase in net density $\rho$. The mean field potentials can also depend on momentum $p$ in the local rest frame. The incompressibility of matter is commonly described in terms of the constant \cite{blaizot_microscopic_1995}
\begin{equation}
K = 9 \rho_0 \left. \frac{\partial^2 (E/A)}{{\partial \rho}^2} \right|_{\rho=\rho_0} \, ,
\end{equation}
where $E/A$ is the energy per nucleon in cold symmetric matter, and the derivative is evaluated at the normal density $\rho_0$.
The calculations so far were all done employing a relatively conventional functional yielding incompressibility $K = 210\:$MeV and effective mass at Fermi momentum at $\rho_0$ of $m^*/m = 0.7$, where the latter characterizes the momentum dependence of $U$.
Some uncertainty regarding the incompressibility and momentum dependence remains, though, and the FOPI Collaboration \cite{reisdorf_systematics_2010} found some sensitivity of the stopping to the decisions made on the mean-field interactions in the Isospin Quantum Molecular Dynamics (IQMD) model \cite{hartnack_modelling_1998}.
Here we test whether we can observe any similar sensitivity.
An excessive sensitivity would hamper the efforts to learn about the in-medium cross sections.
To test sensitivity to the mean field, we show in Fig.~\ref{fig:eos-effects} results obtained for stopping observables when using our standard mean field (soft, momentum-dependent, or ``SM''), corresponding to incompressibility $K=210\:$MeV, as well as a mean field with no momentum dependence, corresponding to incompressibility $K=380\:$MeV (hard, or ``H''). The two mean fields yield similar results for flow in semicentral collisions \cite{danielewicz_determination_2002}. However, the momentum-dependent mean field fails to explain flow at high impact parameters or high transverse momenta \cite{danielewicz_determination_2000}. While the H mean field is not realistic, its use allows us to assess the general sensitivity of the stopping observable to the choice of mean field.
\begin{figure}
\begin{center}
\input{limot-arag-eos.pdf_tex}\input{limot-arau-eos.pdf_tex}
(a) linear momentum transfer
\hspace*{\fill}
\input{varxz-eos-au.pdf_tex}\hspace{\fill}
\input{rz-eos.pdf_tex}\hspace*{\fill}
\hspace*{\fill} (b) \textit{varxz} \hspace{\fill} (c) isospin tracer \hspace*{\fill}
\caption{Mean-field sensitivity of stopping observables. S (H) refers to a soft (hard) compressibility, while M refers to the
inclusion of momentum-dependence in the mean field. \textsc{SM} and \textsc{H} are limiting combinations that give similar predictions for flow in semicentral collisions \cite{pan_sideward_1993}.
}\label{fig:eos-effects}
\end{center}
\end{figure}
Surprisingly, even when a quite extreme mean field like H is used, the stopping observables at high energies, \textit{varxz} and isospin tracer $R_Z$, change very little, suggesting relative robustness of conclusions there. We find some sensitivity to the mean field at lower energies, in the stopping observable LMT. Interestingly, no matter what mean field is used, the need for in-medium cross section modifications is apparent, in order to match the data in Fig.~\ref{fig:eos-effects}. The reduction in the momentum dependence in $U$, accompanied by an increase in the incompressibility to meet flow data from semicentral collisions, results in an enhanced stopping when judging that stopping with LMT. With this, to meet the data with reduced momentum dependence in $U$, one would need a stronger reduction in in-medium cross section. However, realistically, the uncertainty in $U$ and in incompressibility spans approximately a third of the range between SM and H. Given Figs.~\ref{fig:lmt-arau} and \ref{fig:eos-effects}, any deemed change in the reduction for in-medium cross section would be small compared to the ambivalence we already have.
\section{Stopping and the transport cross section}
In-medium cross sections are obviously not directly observable, and they are tied to the transport equation that relies on the concept of quasiparticles, which brings in a level of phenomenology.
So a question might be asked whether more robust conclusions may be drawn from the studies of stopping that extend beyond the cross sections.
To illustrate the precarious nature of the conclusions on cross sections, we show in Fig.~\ref{fig:collsvstime}a the collision number in \coll{Au}{Au} collisions at $400\,$MeV/nucleon obtained in simulations, with three different in-medium cross sections: free-space, Rostock, and tempered with $\nu=0.8$.
The stopping is similar in those collisions for the Rostock and $\nu=0.8$ cross sections, when quantified in terms of \textit{varxz}, and significantly reduced compared to free cross sections, as seen in Fig.~\ref{fig:varxz-nocomp-au}. Yet, in spite of the similar stopping in that figure, the collision count for the two cross sections is different by a factor of 2 in Fig.~\ref{fig:collsvstime}a. Apparently, the stopping does not directly correlate to the typical elementary cross section in a reaction, which in turn does not bode well for reaching physics conclusions from stopping.
Taking another perspective, the collision count includes some collisions that are hard, occurring at high relative velocity with large momentum transfer, and some that are soft, occurring at low relative velocity and low momentum transfer.
Those soft collisions contribute little to momentum transfer across the system as represented by observables such as LMT, \textit{varxz}, or $R_Z$.
Instead, one can consider that the most elementary macroscopic characteristics for a system, which are tied to cross sections, are transport coefficients.
The one tied directly to momentum transfer is the shear viscosity, and it involves the so-called transport cross sections.
The transport-type cross section may be recognized in Eq.~\ref{eq:buu-viscosity}. Here, the shear viscosity coefficient $\eta$ is dependent on the NN cross section, with the cross section's weight dependent on the relative momentum and scattering angle. In Fig.~\ref{fig:collsvstime}b, we show the weighted collision count, with each collision multiplied by its viscous weight factor $q^4 \sin^2 \theta$. The weighted collision count is similar for the Rostock and $\nu=0.8$ cross sections, consistent with \textit{varxz} values being similar for those two cross sections in Fig.~\ref{fig:varxz-nocomp-au}.
To provide more insight, in Fig.~\ref{fig:collsvstime} we plot the unweighted (top) and weighted (bottom) collision counts up to $100\,$fm$/c$ vs.\ \textit{varxz}, for a variety of \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{}. While the stopping correlates with the unweighted collision count, the correlation is fairly broad, with the count differing by up to a factor of 2 for some different plausible cross section reductions.
Here, one can see that the stopping poorly tests the overall number of particle-particle collisions. However, the correlation of the stopping is fairly tight with the collision count when the collisions are weighted with the viscous weight, as seen in Fig.~\ref{fig:collvstop}. The broader systematics further support the view that the stopping tests transport cross sections and more broadly the medium shear viscosity.
\begin{figure}
\input{collrates.pdf_tex}
\caption{Cumulative number of elastic NN collisions vs.\ elapsed time in central \coll{Au}{Au} collisions at $400\,$MeV/nucleon, for three different in-medium cross sections. Panels (a) and (b) show, respectively, net number of collisions and number of collisions weighted with the viscous weight.}\label{fig:collsvstime}
\end{figure}
\begin{figure}
\input{collvstop.pdf_tex}
\caption{Correlation between the number of nucleon-nucleon collisions that took place up until $100\,$fm$/c$ and the stopping observable $varxz$ for the central $400\,$MeV/nucleon collision. The top panel utilizes the unweighted collision number, and the bottom panel utilizes the number weighted with the viscous weight. The weighted collision number rises monotonically with the stopping, unlike the unweighted number.}\label{fig:collvstop}
\end{figure}
\subsection{Viscosity from \textsc{BUU}}\label{sec:viscosity}
As Eq.~\ref{eq:buu-viscosity} was derived using the same assumptions as the transport model used to constrain the \ensuremath{\sigma^\textrm{med}_\textsc{NN}}{}, that cross section can be inserted into this equation to find a viscosity coefficient $\eta$ that is hopefully of greater generality than even the transport model itself, given the correlation shown in, for example, Fig.~\ref{fig:collvstop}. The calculation is performed with the effective mass described in Section~\ref{sec:sensitivity-eos}, which tends to increase the viscosity somewhat, compared to using the free mass.
The results of viscosity calculations are displayed in Fig.~\ref{fig:viscos}.
At all densities and cross section reductions presented, the viscosity grows indiscriminately at low temperatures.
This occurs when the nucleon system becomes degenerate and collisions become strongly Pauli-suppressed, with the weighted rate in the denominator in Eq.~\ref{eq:buu-viscosity} tending towards zero.
As temperatures increase and Pauli effects diminish, the collisions become more frequent.
The viscosity goes through a minimum and at high temperatures, it behaves in a classical fashion, growing like $\sqrt{T}$.
Eventually, inelastic processes set in and calculation of viscosity using just elastic processes in Eq.~\ref{eq:buu-viscosity} will start overestimating the actual viscosity. For situations where consideration of only elastic cross sections is still justified, we demonstrated that the stopping data imply a significant in-medium cross section reduction, as compared to free, and thus we demonstrate an enhancement of the shear viscosity as compared to that calculated with free cross sections. For reference, shear viscosity calculated with free cross sections and velocities is also given in Fig.~\ref{fig:viscos}.
\begin{figure}
\begin{center}
\input{viscos-rho0_5.pdf_tex}
\end{center}
\caption{Shear viscosity for symmetric nuclear matter at different temperatures and densities, deduced from Boltzmann equation with in-medium cross sections.}\label{fig:viscos}
\end{figure}
We now use our newly determined viscosity to explore how close nuclear matter is to being the touted ``perfect fluid''.
\subsection{Lower quantum limit of ratio of viscosity to entropy density}
It has been found theoretically that certain strong coupling limits of gauge theories have a constant ratio of shear viscosity to entropy density regardless of the metric used \cite{danielewicz_dissipative_1985, kovtun_viscosity_2005},
\begin{equation}
\frac{\eta}{s} = \frac{\hbar}{4 \pi k_\mathrm{B}} \, .
\end{equation}
Moreover, it has been speculated that this value represents a lower limit for all relativistic, finite temperature quantum field theories with zero chemical potential, and for single-component nonrelativistic gases of particles with spin 0 or $1/2$ \cite{kovtun_viscosity_2005}.
We calculate this ratio at intermediate energies to find the proximity of nuclear matter at these energies to this conjectured lower limit.
To find the ratio $\eta/s$, we calculate $\eta$ and $s$ separately.
We simplify finding the entropy density by utilizing the model's ability to describe deuteron yields and use equilibrium conditions relating the ratio of the yield of deuterons and deuteron-like correlations to that of total charge, following the prescriptions of Bertsch and Cugnon \cite{bertsch_entropy_1981} as formulated in Ref.~\cite{csernai_entropy_1986}. We reproduce the formula here:
\begin{equation}
\sigma = S/A = 3.945 - \ln \left( N_\textrm{d-like} / Z \right) - \frac{1}{8} N_\textrm{d-like} / Z \, ,
\end{equation}
where $N_\textrm{d-like} = N_\textrm{d} + \frac{3}{2}(N_\textrm{t} + N_\textrm{h}) + 2 N_\alpha + \cdots$ and $Z = N_\textrm{p} + N_\textrm{d} + N_\textrm{t} + 2(N_\textrm{h} + N_\alpha) + \cdots$ \cite{bertsch_entropy_1983}.
The bulk of the entropy is produced in regions of hot, dense matter, during the compression and thermalization phase of the reaction. This is also where the stopping signals are generated. Therefore, the density and temperature in that specific space-time region should be used to determine the entropy per volume, $s = \sigma n$, as well as the temperature at which to find the viscosity. In the simulation, we choose a $2\,$fm-radius spherical region centered at the reaction center of mass, during the time of maximal density in that region. The temperature is found assuming that the momentum distribution of the nucleons approximates that of a degenerate relativistic Fermi gas.
We choose several representative reactions to find the characteristic temperatures and densities reached at intermediate energies. Listed here in order of decreasing $\eta/s$ and increasing maximal temperature, as represented by open circles in Fig.~\ref{fig:eta-s}, they are \collpt{197}{Au}{197}{Au} at beam energies of 100, 400, and $1000\,$MeV/nucleon, each with a reduced impact parameter $b_\textrm{red} = 1.5$.
Even though this work is at a much lower beam energy, the trend of the nuclear matter looks to match findings at \textsc{RHIC} energies, which use a Monte Carlo Bayesian framework \cite{bernhard_quantifying_2015} following initializations in the Glauber \cite{miller_glauber_2007} and KLN \cite{adil_eccentricity_2006} models.
Indeed, as the temperature approaches the critical temperature for nuclear matter, $\sim 170\,$MeV \cite{karsch_quark_2001}, $\eta/s$ is seen to approach the conjectured lower bound.
\begin{figure}
\centering
\input{eta-s.pdf_tex}
\caption{This work's estimate for $\eta/s$ values in selected reactions (open circles), alongside the estimates arrived at RHIC energies following the KLN model (square) \cite{adil_eccentricity_2006} and the Glauber model (triangle) \cite{miller_glauber_2007} for the initial conditions. The ratio is given in units of $\hbar/k_\mathrm{B}$. The conjectured lower quantum limit, $1/4\pi$ \cite{kovtun_viscosity_2005}, is shown in a dotted line.}\label{fig:eta-s}
\end{figure}
\section{Conclusion}
We investigated the viscosity of nuclear matter by adjusting the in-medium nucleon-nucleon cross section to fit nuclear stopping data in terms of several different stopping observables across a wide range of nuclear mass and beam energy.
We found that, for p\textsc{BUU}, an in-medium reduction in the \textsc{NN} cross section is necessary to match a variety of experimental data, and that this need for reduction is consistent across a range of reasonable choices of nuclear mean field.
Using this in-medium nucleon-nucleon cross section, we calculate shear viscosity $\eta$ in nuclear matter, at densities and temperatures representing those encountered in the collisions from which we draw the stopping observables, in a manner consistent with the way we simulate the collisions to match the measured observables.
We argue that the stopping observables better correlate with viscosity than with the details in the cross sections. In calculations of viscosity, the use of reduced cross sections, compared to free-space, increases the viscosity values.
We subsequently calculated the ratio of shear viscosity to entropy per unit volume, $\eta / s$, which is often mentioned in the literature.
We demonstrate that our values for the ratio trend towards that deduced in ultrarelativistic collisions as temperature increases, corresponding to changing beam energy. The calculated ratio is only a few times larger than the speculated absolute lower bound of the ratio.
To benefit from data on stopping at higher energies, where pion production starts to influence the stopping at a significant level, modifications of inelastic processes need to be explored, from which we refrain at present. We do not systematically incorporate the effect of inelastic processes on viscosity at high temperatures either. However, Fig.~\ref{fig:varxz-nocomp-au} suggests a reduction in the rates for inelastic processes in the medium, as compared to free-space extrapolations, and a corresponding increase in viscosity, just as in the case of elastic processes only.
\begin{acknowledgments}
This work was supported by the US National Science Foundation under Grants PHY-1068571 and PHY-1403906.
\end{acknowledgments}
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| 4,099
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11 Dec, 2020 17:16
HomeUSA News
Activists plead with Biden to 'ABOLISH the death penalty' after federal execution of Brandon Bernard
FILE PHOTO © Reuters / Bryan Woolston
Calls to "abolish the death penalty" trended on Twitter after the US government executed Brandon Bernard for his part in a 1999 double homicide, with activists pleading with presumed President-elect Joe Biden to end the policy.
Bernard's death on Thursday night garnered national attention because President Donald Trump's administration resumed the practice of federal capital punishment after a nearly two decade lapse earlier this year.
Following the execution, the hashtag #AbolishTheDeathPenalty began trending on Twitter in the US, uttered by some of the most prominent left-wing leaders, including Senator Bernie Sanders (D-Vermont) and firebrand Congresswoman Alexandria Ocasio-Cortez (D- NY).
"In a world of incredible violence, the state should not be involved in premeditated murder," tweeted Sanders. "Brandon Bernard should be alive today," wrote the account of the leftist Gravel Institute.
Brandon Bernard should be alive today. We must end all federal executions and abolish the death penalty. In a world of incredible violence, the state should not be involved in premeditated murder. https://t.co/TeppKF667T
— Bernie Sanders (@BernieSanders) December 11, 2020
Abolish the death penalty.
— Alexandria Ocasio-Cortez (@AOC) December 11, 2020
— Rep. Barbara Lee (@RepBarbaraLee) December 11, 2020
Brandon Bernard should be alive today. Abolish the death penalty.
— Gravel Institute (@GravelInstitute) December 11, 2020
New York's Congressman-elect Jamaal Bowman pleaded with Biden to push the issue when he enters the White House next month. He and others demanded Biden put an "end to federal executions," which reemerged under Trump – and some appeared to want Biden to abolish capital punishment altogether, which would require attempting to override a number of states' laws on the issue.
Brandon Bernard.Heartbroken and fatigued from mourning but we cannot stop raising our voices and organizing for justice.4/5 of the remaining federal executions this year will be Black men. @JoeBiden we need an end to federal executions. We must abolish the death penalty.
— Jamaal Bowman (@JamaalBowmanNY) December 11, 2020
I think Biden will abolish the federal death penalty. Unfortunately, I don't think he can make the states give theirs up.
— GenaG (@literati26) December 11, 2020
Another reason to hate Trump and another issue we must demand of Biden and Dems: abolish the death penalty. https://t.co/Hm1pnQr0i2
— Shelly Anand (@maanandshelly) December 11, 2020
Biden needs to abolish the death penalty day one https://t.co/4MBsIA4YOh
— Tyler (@dr_mousebrain8) December 11, 2020
The top Democrat himself, however, is yet to comment on the matter.
Last month, Biden's press secretary TJ Ducklo said that the former vice president "opposes the death penalty now and in the future."
Biden had already promised on his campaign website to restore the pre-Trump status quo. Not only did he say he would abolish it at the federal level, he said his administration would "incentivize states to follow the government's example."
Also on rt.com US executes death row inmate Brandon Bernard in 9th federal execution in 2020
While Biden is apparently not a fan of the death penalty anymore, he was once a champion of it.
In a grim coincidence, one of the three people executed in November by the Trump administration was originally tried in 1994 under the Crime Bill, infamously co-authored by the current presumed president-elect.
Earlier, in 1992, Biden bragged that the legislation in the works was so tough that it did "everything but hang people for jaywalking."
Death penaltyJoe BidenTwitter
Global mass extinctions follow regular 27-million-year cycle, may be linked to Earth's journey through Milky Way
Father & Mother no more? Sweden considers introduction of gender-neutral terms for parents to suit 'rainbow families'
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{"url":"http:\/\/crypto.stackexchange.com\/tags\/hash\/hot?filter=month","text":"# Tag Info\n\n4\n\nThe padding scheme isn't collision resistant. For any message $m$ where $|m| \\not\\equiv 0 \\pmod n$, there will always be a collision between $m$ and $m || 0$.\n\n4\n\nWord size is not a term originating in cryptography. Rather it is a term which came from the area of computer architecture. Here it specifies either how many bits are transferred over a bus at a time or the size of a register. When used in the description of cryptographic primitives (such as a compression function or a block cipher), it covers the size of ...\n\n3\n\nNo, there is no known way. It would actually be rather surprising if there were even a theoretical way; the SHA-256 and the SHA-512 compression functions are rather different (for one, one works with 32 bit words and the other works with 64 bit words); one wouldn't expect them to share any sort of relation.\n\n3\n\nFirst to explain you, why you get 512-bit outputs from a 256-bit curve: The output is basically a point (x-coordinate is enough) and a message-dependant value, with the x-coordinate being expressed as integer. You can verify the signature by checking for a specific relationship between the point and the message-dependant value and the public key point. In ...\n\n3\n\nSipHash is a MAC (aka Pseudo Random Function Family) with 64-bit output and 128-bit key, rather than a hash (aka random public member of a Pseudo Random Function Family). It is explicitly designed to be used with a secret random key. Quoting Jean-Philippe Aumasson and Daniel J. Bernstein's SipHash: a fast short-input PRF (in proceedings of Indocrypt 2012): ...\n\n3\n\nThe word size in hash functions means the size of the integral unit of operation for the internal transformations. For example: for SHA-512, you'll get some input, split it and then perform operations on 64-bit words (=unsigned integers, like modulo addition or shift) whereas for SHA-256 you'll use 32-bit operations (= operations on 32-bit-integers, like ...\n\n3\n\nThe construction you are proposing is called the \"envelope\" or \"sandwich\" MAC, it predates HMAC, and it is in fact secure\u2014provided the key and message are appropriately padded. That is, $$\\text{SHA256}(k \\parallel m \\parallel 1 \\parallel 0^{b - 1 - (|m| \\bmod b)} \\parallel k)$$ is secure, as long as $k$ is the underlying hash function's block length $b$ ...\n\n3\n\nWe learn the first 32 bits of the SHA1 hash of the secret (well, it's actually the SHA1 hash of the secret concatenated with some known stuff, but that amounts to the same thing). We can enumerate all $2^{56}$ possibilities for the secret. We expect that about $1\/2^{32}$ of them will produce a matching 32-bit value, so about $2^{24}$ candidates for the ...\n\n2\n\nIf you got $n$ blocks, then you compute the encryption of each block, and let's look at one bit at position $j$. Let's call this $c_{i,j}= E(m_i)[j]$. Now what you will get at position $i$ in your output is $(((c_{i,1} \\bar{\\vee} c_{i,2}) \\bar{\\vee}c_{i,3}) \\bar{\\vee} \\dots )\\bar{\\vee}c_{i,n}$. If we assume that all $c_{i,j}$ are evenly distributed in ...\n\n2\n\nIf you have an $m-$bit output decent hash function (such as SHA1) and you're hashing $k-$bit values, and your list $L$ of candidates is not tiny ($2^{56}$ which is the size of possible candidates, isn't) the function will behave like a random mapping on any substring of its' output. If your prefix consists of 8 4-bit characters, then you'll be filtering ...\n\n2\n\nWell, if you construct what you described you basically create a function $f: \\{0,1\\}^{2m} \\rightarrow \\{0,1\\}^m$. As you correctly pointed out these two strings give the same when xor'ed. So the messages $10101111$ and $00000101$ will result in the same xor and hence will get mapped to the same hash, resulting in a second preimage as you found two $x,x'$ ...\n\n1\n\nI'll take the (previous version of the) question as: how to implement a secure hash function $H3$ with 3 arguments, from one hash function $H$ with 1 argument, all with argument(s) in $\\{0,1\\}^*$. Note: For a concrete $H$ with the destination set $\\mathbb Z_p^*$ thought in the question, we could use SHA3-512 followed by a suitable function; for example, ...\n\n1\n\nA cryptographic hash function will produce an output with pseudorandom properties, therefore when expressed in hexadecimal, a list of hash values will have an almost equal number of each character. Pseudorandom data will not compress, as compression looks for patterns. If you had duplicates, compression could reduce the data size. If you want to compress ...\n\n1\n\nI don't think this problem is solvable as specified. With a small message space, and deterministic hashing (or encryption), a generic attack involves exhaustively searching all likely messages to find one that corresponds to the known hash \/ ciphertext. If all of the digits of the ID numbers were random, an exhaustive search would require about $10^{10} ... 1 I'm not aware of any case where somebody actually searched for such a collision. However it would certainly be possible as the same workload ($2^{64}$) was already accomplished a few years ago (2002) by this project, having brute-forced RC5-64. Now assume you'd use the full power of the bitcoin blockchain (300 Peta-Hashes \/ s = 600 Peta-Hashes \/s for ... 1 To build on tylo's answer, here's a practical internal collision attack on this construction, assuming that the block cipher$\\rm Enc$has a 128-bit block size (like AES, for example) or less: Pick an arbitrary initial block$m_0$, and calculate$c_0 = {\\rm Enc}_{m_0}(m_0)$. If$c_0$has less than$n\/2 = 64$bits set, pick a new$m_0$and repeat. (On ... 1 You could do something fairly simple, such as$UserSecret = Random()UserID = HMAC(ServerSecret, UserSecret)$Send the user the two values. When he reconnects, he sends the two values back. If re-calculating$UserID$with the user's$UserSecret$gives the same$UserID\\$ then that proves (to a high degree of certainty) that it's the same person that was ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2015-05-28 16:23:12","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\": 1, \"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\": 0.5110880136489868, \"perplexity\": 905.5084032864135}, \"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-2015-22\/segments\/1432207929422.8\/warc\/CC-MAIN-20150521113209-00339-ip-10-180-206-219.ec2.internal.warc.gz\"}"}
| null | null |
MIT announced today that it is massively doubling down on the future of computer science with the launch of a new college of computing. The university is committing $1 billion in resources to the new school, and the university received a $350 million donation from Stephen A. Schwarzman, who will be the naming donor. MIT said that its commitment is the largest of a university yet to the discipline.
For students, MIT is taking an even more bullish stance: that every graduate should encounter computer science and AI before graduation. The objective of the new school will be to ensure that all MIT students become familiar with the field regardless of their chosen profession. The school will be housed in a new building on MIT's Cambridge, Mass. campus.
Creating a separate school for computer science will change the Institute, and will position it more similarly to its peer rival Carnegie Mellon, which has had a separate School of Computer Science for some time.
The $350 million gift from Stephen A. Schwarzman for naming rights is in line with other massive recent gifts to MIT's close neighbor Harvard, which received $400 million from John A. Paulsen to name the School of Engineering and Applied Sciences and $350 million from Gerald Chan to name the School of Public Health.
Schwarzman, the co-founder, CEO, and chairman of alternative investment firm Blackstone, has been on a philanthropic binge recently. In addition to this MIT gift, he has donated $100 million to the main branch of the New York Public Library near Bryant Park, $150 million to Yale University, and created the Schwarzman Scholars program as a China-based competitor to the Rhodes Scholarship with a $100 million grant.
MIT said that the new college is expected to open in fall of next year, and its new building will be completed in 2022.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,362
|
Q: A REAL iTunes server on NAS I am certainly not the only one dealing with the issue (looking into questions here which deal with some aspects related to an iTunes Server or looking at the Apple Support Forum). So I ask:
What is a way of having a real (explanation of real follows) iTunes Server on a NAS?
Driving force of course is the desire to centralize my iTunes Library, photos, big data, that I do not want to carry around with my me. Solution: A Network Attached Storage (NAS). For all the other stuff except the iTunes library this solution is great. Of course I can put the library on it BUT:
*
*I will not be able to stream content to my iDevice directly (I could if my Mac is, but not directly)
*I cannot tell my library to stream some music to an AirPlay device from my iDevice (again, only if my Mac is on)
*In principle: Without any other device(s) running, my library can do nothing. It is not stand alone! To stream content requires additional devices running.
Any NAS that can store any of my data (ok that's every NAS :)) and serves as a stand alone iTunes library I consider a real iTunes Server. I would consider it a real real iTunes Server if I can remote it via Apples Remote App and stream music/videos via Apple appropriate apps (yes I kinda favor Apples solution whenever available).
I will give myself an answer for one possibility (an additional Mac) with pros and mainly cons. But I am very interested in other solutions (please as well do not hide disadvantages from your proposed solutions)!
A: One possibility: An additional Mac.
Pros:
*
*Could do everything asked for.
Cons:
*
*Expensive.
*Power consuming.
*To save some power, one would let it sleep. So everytime one wants to connect to the library, the Mac has to be waked up.
*Setup not as easy as NAS.
A: I have invested many many years into research and experiments on Entertainment Setups in Home Environments.
Gathering all my Knowledge, I will now try to answer your question, while explaining you, how you could do it and why.
Let's Start with Apple. Apple is limiting iTunes heavily because of their own interests. iTunes Match is one of them.
To Stream Content to your iDevice, you need a Mac, iTunes, an AppleID and Home Sharing activated.
Otherwise it won't work.
If you have all these things you can only do 1 thing : Stream Music. Nothing Less and nothing more. Simple, Stupid Streaming.
Streaming Content from your Library via Airplay to iDevices.
Well.. in theory iTunes could do it, but practically, it is not possible. Simple as that.
And you are therefore totally right, when saying : Without any other Device, your iTunes Library won't even work at all.
And even if it did with another Device/Machine, it wouldn't work either.
Now we talk NAS.
I have to admit, that the Synology DS have some good UIs for those things.
But if you want to do more than just sharing Music, you are pretty much stuck in the mud.
So how to go about it ?
At first, I would not recommend any of the Apple-Solutions. iTunes is a pain if you want to centralize it.
But it is indeed possible and here is my approach on how to do it :
I have a 10 TB Self-Build Macintosh NAS System. But you could use a FreeNAS System as well. Or even a Ubuntu/Fedora Machine.
(If you don't need iTunes)
I have 6x2TB, 1xTB for the System and stuff, 5x2 TB in a ZFS RAID-Z Pool.
I chose Subsonic for Streaming Music to computers, AudioTap (App for iDevices) for streaming Music to iDevices.
Even though AudioTap is not developed any more, you can absolutely use it. You can Edit Playlists, Give ratings, and so on.
I have a very clean and well Tagged Music Collection that is synced with Subsonic automatically and synced with iTunes Manually.
iTunes Sharing allows every Mac to simply connect to the Music, AudioTap is great for all the iDevices.
But you could also use Subsonic Clients as well.
But this is all Audio.
For Movies and TV-Shows I prefer Plex. And I have some Additional Apps/Clients as well.
Plex could also handle Music, but I prefer Subsonic for that.
Plex can be installed on every OS. Windows, Mac, Linux. Whatever you prefer.
Plex can stream to any device, and you can control it from every Device.
Plex automatically updates the Library if you change something. Plex can handle Playlists.
Both of the Solutions are free. Plex is Free and Subsonic is free (except for streaming to Devices). But you could
use Madsonic instead, which allows streaming without extra fees.
If you have time and Money, I would consider a Mac Mini (older model) in combination with attached Storage.
You can then install iTunes, Subsonic, Plex and many other Apps.
You could also install things like "AirServer", which extends you AirPlay needs.
In fact, the "Media Centralization" is very complicated, if you want to have a good Experience in the end.
I've gone through all of it and even now I am not satisfied with the solution. But there is nothing else.
And iTunes is in my view the worst.
Speaking of iPhoto. You could then also put an Instance of iPhoto onto your mac and Have some script for auto-adding pictures to the library.
I use Piwigo for my Photo-Collection, but I am pretty sure there is something similar to "Automatically add to iTunes" for iPhoto.
All the Other Big Data can be stored on the Server too.
So that is maybe not the answer you wanted to hear, but there is no good way that I am aware of when using iTunes as a Media Server.
A: There are in fact iTunes Servers on some NAS devices. All of these require iTunes to be running on a Mac or PC on the network as they don't have direct access. These will show up in iTunes as a network server. I have done some research as i already have the drives for a Netgear ReadyNAS and am waiting on delivery. The new range 100/300/500 series running OS6 all have the iTunes Server on them. I do not know at this stage whether the IOS remote will work with this Server, i certainly hope so, or what is the point. I will test it when received and get back to you. NAS devices are all DLNA enables so you can stream direct form any DLNA device, TV's etc.
A: SyncPhoto is working well for me as an iPhoto solution. It will keep all your iPhoto instances synced with each other. I agree with Plex and have been using it for years. Similar to XBMC but Plex has a central database server and it communicates to clients, iOS, OS X, Windows, Roku, ATV with some hacking, ....
A: There is a simpler solution. You need a client that can render the video/audio. The presumption is that you need an iTunes server to send the stream to the device.
I use FileBrowser to render the video or audio directly to the device without my Mac delivering the stream. FileBrowser gives you the ability to browse through the files on your NAS without iTunes.
There are probably other apps that will do the same functions. By the way, FileBrowser also will AirPlay a file without iTunes as well. I am not getting any money for this, but FileBrowser does a lot more than just media rendering.
A: The Outclass AppleTV server claims to support homesharing. I'm not sure if this is just secretly a Windows box running iTunes or where you even purchase one of the things, though.
A: here is the answer.
rysnas.com/
You can run a VM of windows for iTunes Home Sharing on your NAS box now.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,512
|
Thank you for contacting us about your interest in a 4+1 program!
A graduate admission counselor will follow up with you soon to answer any questions you have.
You can reach out to the Graduate Admission office any time at (805) 493-3325 or clugrad@callutheran.edu.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,032
|
Q: Scraping PDF data into Excel *absolute beginner* This is literally day 1 of python for me. I've coded in VBA, Java, and Swift in the past, but I am having a particularly hard time following guides online for coding a pdf scraper. Since I have no idea what I am doing, I keep running into a wall every time I want to test out some of the code I've found online.
Basic Info
*
*Windows 7 64bit
*python 3.6.0
*Spyder3
*I have many of the pdf related code packages (PyPDF2, pdfminer, pdfquery, pdfwrw, etc)
Goals
To create something in python that allows me to convert PDFs from a folder into an excel file (ideallY) OR a text file (from which I will use VBA to convert).
Issues
Every time I try some sample code from guides i've found online, I always run into syntax errors on the lines where I am calling the pdf that I want to test the code on. Some guide links and error examples below. Should I be putting my test.pdf into the same file as the .py file?
*
*How to scrape tables in thousands of PDF files?
*
*I got an invalid syntax error due to "for" on the last line
*PDFMiner guide (Link)
runfile('C:/Users/U587208/Desktop/pdffolder/pdfminer.py', wdir='C:/Users/U587208/Desktop/pdffolder')
File "C:/Users/U587208/Desktop/pdffolder/pdfminer.py", line 79
print pdf_to_csv('test.pdf', separator, threshold)
^
SyntaxError: invalid syntax
A: It seems that the tutorials you are following make use of python 2. There are usually few noticable differences, the the biggest is that in python 3, print became a funtion so
print()
I would recomment either changing you version of python or finding a tutorial for python 3. Hope this helps
A: Here
Pdfminer python 3.5 an example, how to extract informations from a PDF.
But it does not solve the problem with tables you want to export to Excel. Commercial products are probably better in doing that...
A: I am trying to do this exact same thing! I have been able to convert my pdf to text however the formatting is extremely random and messy and I need the tables to stay in tact to be able to write them into excel data sheets. I am now attempting to convert to XML to see if it will be easier to extract from. If I get anywhere on this I will let you know :)
btw, use python 2 if you're going to use pdfminer. Here's some help with pdfminer https://media.readthedocs.org/pdf/pdfminer-docs/latest/pdfminer-docs.pdf
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,372
|
President Trump's support for Luther Strange wasn't his fault, this time
President Trump outside in press gaggle Wednesday Sept 27. / [Screenshot from the White House via YouTube:https://youtu.be/FWV50D99VOE]
Roy Moore was appropriately the perfect Trump candidate but the President went for the establishment candidate for a reason.
by Jon Mark
September 30, 2017 at 7:10 PM September 30, 2017 at 7:10 PM
It was reported on Wednesday that President Trump deleted all of his tweets that were related to endorsing the Republican candidate for Alabama, Sen. Luther Strange. This was right after the president had tweeted about all of the chances he claimed that Strange had to win the primary race for former Senator Jeff Sessions open seat. Instead, Roy Moore - who is more of the type of candidate that fits with Trump - won the election. This resulted in a tweet that was just as congratulatory as if he had been supporting Moore the entire time.
Spoke to Roy Moore of Alabama last night for the first time. Sounds like a really great guy who ran a fantastic race. He will help to #MAGA!
Despite the deletions, the president also appeared to act like a "good sport" by giving Strange some credit for running a good race.
Congratulations to Roy Moore on his Republican Primary win in Alabama. Luther Strange started way back & ran a good race. Roy, WIN in Dec!
President Trump's misdirection to support establishment candidate
Prior to this, Donald Trump became part of a situation that confused many when both Moore and Strange were coddling for the president's support. They fought over which one of them was the true Trump candidate. Many felt that it would make more sense for the president to put his support behind Roy Moore, especially since Moore got the support of Breitbart CEO and Trump's former chief strategist at the White House, Steve Bannon.
CNN reported that as soon as it was announced that Moore had won the primary Tuesday evening, that the President was furious about the loss and slammed Senate Majority Leader Mitch McConnell for pushing him to support Strange.
According to the article: "Trump infuriated after backing Alabama loser," the President also vented on his political team who he felt had misled him. The article also pointed out that McConnell had consolidated GOP support from the establishment behind Strange.
Reasons for why Trump supported Strange
The article also reported that Trump was already having second thoughts about Strange saying that he felt he was "too low energy." Because of this, the President was also concerned that his support for an establishment candidate would make him look weak.
This would make the populist effort generated by Steve Bannon another win for the far right. It also gave the appearance that the president was wrong in his decision because to many, it only made sense for Trump to give his support for Moore, who is just as controversial as Trump.
It has been pointed out that the reason for Trump's decision, however, wasn't entirely because it was made at McConnell's request.
Luther Strange was also quite loyal to Trump, who seems to prioritize loyalty over everything else. The loss, no doubt, also made an impact with him as it happened on the same day that Republicans failed again to pass a bill that would repeal Obamacare, which Trump has promised to do but the GOP has failed to do, repeatedly.
Jon Mark
I absorb the "blast" of news stories every day so you don't have to. My topics of interest are with national security, foreign affairs, world news, humanitarian and activist causes, politics, science and finance.
Follow jon on Facebook Follow jon on Twitter
Read more on the same topic from Jon Mark:
Private migrant detention company sued for toddler's death US Secretary of State Mike Pompeo is wild card for Trump cabinet Trump's non-stop tantrum over stalled immigration policy continues
Blasting News recommends Von Miller knows what Tom Brady is all about in the playoffs: 'He's incredible' 10-year-old cancer survivor on Tom Brady: 'He's kind of been my hero all along' Julian Edelman now roots for Tom Brady, celebrates Bucs win over Eagles as he wins bet Rams cornerback Jalen Ramsey calls Tom Brady 'great leader', expects TB12 to 'bring it' Donald Trump doubles down with second attack on San Juan Mayor in Twitter rant Trump harshly accuses San Juan mayor of poor leadership
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,426
|
const resetCssLoader = require('../loaders/css/reset');
/**
* Webpack 插件 - 重置 `koot-css-loader` 计数器
*/
class ResetCssLoaderPlugin {
apply(compiler) {
compiler.hooks.done.tapAsync.bind(
compiler.hooks.done,
'ResetCssLoaderPlugin'
)((compilation, callback) => {
if (process.env.WEBPACK_BUILD_ENV === 'prod') resetCssLoader();
callback();
});
}
}
module.exports = ResetCssLoaderPlugin;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 315
|
{"url":"https:\/\/q4interview.com\/quantitative-aptitude\/decimal-fraction\/discussion\/a-students-grade-in-a-course-is-determined-by-6-quizzes-and\/2859\/11","text":"Take FREE!! Online Mettl Mock Test to Crack TechM and Other Companies Written Exams.\nGet Off-Campus Placement Jobs Info !!!\nTCS NQT Technical and Managerial Round Interview Questions\n\n# Quantitative Aptitude :: Decimal Fraction - Discussion\n\nHome > Quantitative Aptitude > Decimal Fraction > MCQs Questions Discussion\n\n11 \/ 70\n\nChoose the correct option.\n\n## A student's grade in a course is determined by 6 quizzes and one examination. If the examination counts thrice as much as each of the quizzes, what fraction of final grade is determined by the examination?\n\nA$$\\frac{1}{6}$$\n\nB$$\\frac{1}{5}$$\n\nC$$\\frac{1}{3}$$\n\nD$$\\frac{1}{4}$$\n\nShow Explanation\n\nSolution: Let the marks allotted for one quiz be 10 marks so acc. to the question...\n\nMarks allotted for the examination will be its three times i.e. 30 marks\n\nSo total marks of 6 quizzes and 1 examination will be 60+30=90\n\nFraction of examination marks out of total marks is: 30\/90=1\/3","date":"2021-06-17 18:31:05","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\": 0.6775199770927429, \"perplexity\": 10180.769275982058}, \"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-2021-25\/segments\/1623487630518.38\/warc\/CC-MAIN-20210617162149-20210617192149-00325.warc.gz\"}"}
| null | null |
PacketFlagon
============
The main source code for the central node PacketFlagon.is and the various shards
Install
---------------
* Clone this repo
* Create an API key by navigiating to your website and following the instructions or with ```curl https://packetflagon.is/api/1/?action=create -d "{'domain':'YOURSERVERNAME.TLD','contact':'youremail@YOURSERVERNAME.TLD'}"```
* Move libs/config.example.php to libs/config.php
* Edit various elements of libs/config.php _($FQDN, $ShardName & $PacketFlagonAPIKey at a minimum)_
* Ensure your web server is configured to respect the .htaccess and that mod_rewrite _(or equivilent)_ is enabled
* If running a standalone instance ensure you import DB_STRUCTURE.sql to your DB and set appropriate credentials in libs/config.php
Open Source Licenses
----------
| Code Path | License | Credit |
|-----------------------|---------------|--------|
|css/zsocial.css | MIT | http://zocial.smcllns.com - by Sam Collins (@smcllns) |
|css/bootstrap.css | MIT | Twitter Inc |
|css/fontawesome.css | MIT | @davegandy - http://fontawesome.io - @fontawesome |
|js/jquery.js | MIT | jQuery Foundation, Inc. |
|js/jquery.mousewheel | MIT | Brandon Aaron (http://brandonaaron.net) |
|js/jquery.touchSwipe | MIT/GPLv2 | Matt Bryson (www.skinkers.com) |
|js/modernizr | MIT & BSD | - |
|js/jPages.js | MIT | Luís Almeida |
|font/fontawesome | MIT | @davegandy - http://fontawesome.io - @fontawesome |
|font/zsocial | MIT | Sam Collins (@smcllns) |
| | | |
|All other code | BSD | Brass Horn Communications (@BrassHornComms) |
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 248
|
import os
import boto3
import clamav
from common import AV_DEFINITION_PATH
from common import AV_DEFINITION_S3_BUCKET
from common import AV_DEFINITION_S3_PREFIX
from common import CLAMAVLIB_PATH
from common import S3_ENDPOINT
from common import get_timestamp
def lambda_handler(event, context):
s3 = boto3.resource("s3", endpoint_url=S3_ENDPOINT)
s3_client = boto3.client("s3", endpoint_url=S3_ENDPOINT)
print("Script starting at %s\n" % (get_timestamp()))
to_download = clamav.update_defs_from_s3(
s3_client, AV_DEFINITION_S3_BUCKET, AV_DEFINITION_S3_PREFIX
)
for download in to_download.values():
s3_path = download["s3_path"]
local_path = download["local_path"]
print("Downloading definition file %s from s3://%s" % (local_path, s3_path))
s3.Bucket(AV_DEFINITION_S3_BUCKET).download_file(s3_path, local_path)
print("Downloading definition file %s complete!" % (local_path))
clamav.update_defs_from_freshclam(AV_DEFINITION_PATH, CLAMAVLIB_PATH)
# If main.cvd gets updated (very rare), we will need to force freshclam
# to download the compressed version to keep file sizes down.
# The existence of main.cud is the trigger to know this has happened.
if os.path.exists(os.path.join(AV_DEFINITION_PATH, "main.cud")):
os.remove(os.path.join(AV_DEFINITION_PATH, "main.cud"))
if os.path.exists(os.path.join(AV_DEFINITION_PATH, "main.cvd")):
os.remove(os.path.join(AV_DEFINITION_PATH, "main.cvd"))
clamav.update_defs_from_freshclam(AV_DEFINITION_PATH, CLAMAVLIB_PATH)
clamav.upload_defs_to_s3(
s3_client, AV_DEFINITION_S3_BUCKET, AV_DEFINITION_S3_PREFIX, AV_DEFINITION_PATH
)
print("Script finished at %s\n" % get_timestamp())
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,591
|
Twitter/Simone_Biles
Olympic Gymnast Simone Biles' Brother Tevin Pleads Not Guilty in NYE Triple Homicide in Ohio
Tevin Biles-Thomas entered a not guilty plea to triple murder charges during his arraignment where a bond has been set for his release. The mother of one of the victims expressed her dismay.
Tevin Biles-Thomas, the brother of Olympic gymnast Simone Biles has entered a not guilty plea to triple murder charges filed against him.
THE ARRAIGNMENT
Biles-Thomas appeared on video for his arraignment on Friday at the Cuyahoga County Court. His lawyer pleaded not guilty on his behalf as he sat silently wearing an orange jumpsuit. He is currently being held at Cuyahoga County Jail with bail set at $1 million. He was also ordered not to make any contact with any surviving victims.
"I don't agree with a million dollars bond, not for the murder of three people."
HIS ARREST
Biles-Thomas was arrested in Georgia on August 29 as the prime suspect in the murder of three people at a New Year's Eve party in Cleveland. The 24-year-old was identified as the shooter who attended the party uninvited and claimed the lives of Delvante Johnson, 19, Toshaun Banks, 21, and De Vaughn Gibson, 23.
A MOTHER'S PAIN
Johnson's mother, Brandie Johnson expressed her disappointment after learning of Biles-Thomas' hearing and the bond that was set for his release. "I don't agree with a million dollars bond, not for the murder of three people," she told Cleveland.com.
The grieving mom shed tears as she recounted what happened to her son. She also revealed that Biles' biological father was her cousin and that her son and Banks left daughters who were born in July. "We're just trying to find out how to get through this for their babies," she said tearfully.
"There is nothing I can say that will heal anyone's pain, but I do want to express my sincere condolences to everyone affected by this terrible tragedy."
HIS SISTER'S HEARTBREAK
Biles-Thomas' arrest was a source of pain for Biles who held a special place in her heart for her brother even though they didn't grow up together. Both were adopted by different relatives after being placed in foster care in their early years.
In 2014, Biles wished her brother good luck when he entered the Army. In 2017, she posted a photo of the two of them and mentioned how they looked like twins.
Biles was initially reclusive when she learned of her brother's charges. She simply tweeted, "eating my feelings don't talk to me."
Woman Divorces Husband of 36 Years after Their Only Son Finds Out He Has a Brother – Story of the Day
Simone Biles Made the First Move On Fiance Jonathan Owens in a Dating App - He Had No Clue Who She Was
Brother Inherits a Mansion, Sister Inherits Old Key and a Note – Story of the Day
A few days later, she opened up in another tweet, this time expressing her heartache for the victims and their families.
"My heart aches for everyone involved, especially for the victims and their families. There is nothing I can say that will heal anyone's pain, but I do want to express my sincere condolences to everyone affected by this terrible tragedy," she wrote.
Biles-Thomas is scheduled to appear in the next hearing on September 18 at 9 am.
Sons Who Leave Sick Mother Alone Find Out She Left All Inheritance to Her Tenants – Story of the Day
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Young Woman Adopts an Orphan Boy and Finds Out He Is Her Biological Brother – Story of the Day
Mel Gibson's Ex & Mom of His Kid Went Bankrupt after He Left — She Could Not Pay Debts & Had $10 in Cash
'I Can't Believe I Actually Found You!': Birth Mom Reunites with Son after 33 Years of Separation
Drew Carey 'Never Got the Chance' to Talk to Ex-fiancée Who Texted Him 3 Days before She Passed
Kurt Russell Became 'Pa' to Goldie Hawn's Kids Whose Father Considers Them 'Dead'
After Abandoning Her Daughter 24 Years Ago, Mother Shows up at Her Wedding – Story of the Day
Husband Is Not Allowed into Sick Wife's Ward, Few Days Later She Hears His Voice on Radio – Story of the Day
Richard Thomas Said Divorce from Mom of His 4 Kids 'Brought Him to His Knees' — Inside His Blended Family
Devoted Dad-Of-Three Finds Out He's Always Been Infertile, Decides to Take a DNA Test
5-Year-Old Brother Carries His Little Sister's Coffin to Say Goodbye to Her Forever
20 Years after Newborn Daughter's Death, Man Hears, 'Dad, I Finally Found You' – Story of the Day
Tim McGraw's Daughter Shows Weight Loss Progress after She Was Blasted for Being Overweight & Displaying Body
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James Van Der Beek Shares New Family Pics after He Helped Save Wife's Life & They Moved to Ranch
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,344
|
package org.apache.hadoop.yarn.server.nodemanager.recovery;
import static org.fusesource.leveldbjni.JniDBFactory.asString;
import static org.fusesource.leveldbjni.JniDBFactory.bytes;
import java.io.File;
import java.io.IOException;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Map.Entry;
import org.apache.commons.logging.Log;
import org.apache.commons.logging.LogFactory;
import org.apache.hadoop.conf.Configuration;
import org.apache.hadoop.fs.FileSystem;
import org.apache.hadoop.fs.Path;
import org.apache.hadoop.fs.permission.FsPermission;
import org.apache.hadoop.yarn.api.records.ApplicationId;
import org.apache.hadoop.yarn.conf.YarnConfiguration;
import org.apache.hadoop.yarn.proto.YarnProtos.LocalResourceProto;
import org.apache.hadoop.yarn.proto.YarnServerNodemanagerRecoveryProtos.DeletionServiceDeleteTaskProto;
import org.apache.hadoop.yarn.proto.YarnServerNodemanagerRecoveryProtos.LocalizedResourceProto;
import org.apache.hadoop.yarn.server.utils.LeveldbIterator;
import org.apache.hadoop.yarn.util.ConverterUtils;
import org.fusesource.leveldbjni.JniDBFactory;
import org.fusesource.leveldbjni.internal.NativeDB;
import org.iq80.leveldb.DB;
import org.iq80.leveldb.DBException;
import org.iq80.leveldb.Logger;
import org.iq80.leveldb.Options;
import org.iq80.leveldb.WriteBatch;
public class NMLeveldbStateStoreService extends NMStateStoreService {
public static final Log LOG =
LogFactory.getLog(NMLeveldbStateStoreService.class);
private static final String DB_NAME = "yarn-nm-state";
private static final String DB_SCHEMA_VERSION_KEY = "schema-version";
private static final String DB_SCHEMA_VERSION = "1.0";
private static final String DELETION_TASK_KEY_PREFIX =
"DeletionService/deltask_";
private static final String LOCALIZATION_KEY_PREFIX = "Localization/";
private static final String LOCALIZATION_PUBLIC_KEY_PREFIX =
LOCALIZATION_KEY_PREFIX + "public/";
private static final String LOCALIZATION_PRIVATE_KEY_PREFIX =
LOCALIZATION_KEY_PREFIX + "private/";
private static final String LOCALIZATION_STARTED_SUFFIX = "started/";
private static final String LOCALIZATION_COMPLETED_SUFFIX = "completed/";
private static final String LOCALIZATION_FILECACHE_SUFFIX = "filecache/";
private static final String LOCALIZATION_APPCACHE_SUFFIX = "appcache/";
private DB db;
public NMLeveldbStateStoreService() {
super(NMLeveldbStateStoreService.class.getName());
}
@Override
protected void startStorage() throws IOException {
}
@Override
protected void closeStorage() throws IOException {
if (db != null) {
db.close();
}
}
@Override
public RecoveredLocalizationState loadLocalizationState()
throws IOException {
RecoveredLocalizationState state = new RecoveredLocalizationState();
LeveldbIterator iter = null;
try {
iter = new LeveldbIterator(db);
iter.seek(bytes(LOCALIZATION_PUBLIC_KEY_PREFIX));
state.publicTrackerState = loadResourceTrackerState(iter,
LOCALIZATION_PUBLIC_KEY_PREFIX);
iter.seek(bytes(LOCALIZATION_PRIVATE_KEY_PREFIX));
while (iter.hasNext()) {
Entry<byte[],byte[]> entry = iter.peekNext();
String key = asString(entry.getKey());
if (!key.startsWith(LOCALIZATION_PRIVATE_KEY_PREFIX)) {
break;
}
int userEndPos = key.indexOf('/',
LOCALIZATION_PRIVATE_KEY_PREFIX.length());
if (userEndPos < 0) {
throw new IOException("Unable to determine user in resource key: "
+ key);
}
String user = key.substring(
LOCALIZATION_PRIVATE_KEY_PREFIX.length(), userEndPos);
state.userResources.put(user, loadUserLocalizedResources(iter,
key.substring(0, userEndPos+1)));
}
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
} finally {
if (iter != null) {
iter.close();
}
}
return state;
}
private LocalResourceTrackerState loadResourceTrackerState(
LeveldbIterator iter, String keyPrefix) throws IOException {
final String completedPrefix = keyPrefix + LOCALIZATION_COMPLETED_SUFFIX;
final String startedPrefix = keyPrefix + LOCALIZATION_STARTED_SUFFIX;
LocalResourceTrackerState state = new LocalResourceTrackerState();
while (iter.hasNext()) {
Entry<byte[],byte[]> entry = iter.peekNext();
String key = asString(entry.getKey());
if (!key.startsWith(keyPrefix)) {
break;
}
if (key.startsWith(completedPrefix)) {
state.localizedResources = loadCompletedResources(iter,
completedPrefix);
} else if (key.startsWith(startedPrefix)) {
state.inProgressResources = loadStartedResources(iter, startedPrefix);
} else {
throw new IOException("Unexpected key in resource tracker state: "
+ key);
}
}
return state;
}
private List<LocalizedResourceProto> loadCompletedResources(
LeveldbIterator iter, String keyPrefix) throws IOException {
List<LocalizedResourceProto> rsrcs =
new ArrayList<LocalizedResourceProto>();
while (iter.hasNext()) {
Entry<byte[],byte[]> entry = iter.peekNext();
String key = asString(entry.getKey());
if (!key.startsWith(keyPrefix)) {
break;
}
if (LOG.isDebugEnabled()) {
LOG.debug("Loading completed resource from " + key);
}
rsrcs.add(LocalizedResourceProto.parseFrom(entry.getValue()));
iter.next();
}
return rsrcs;
}
private Map<LocalResourceProto, Path> loadStartedResources(
LeveldbIterator iter, String keyPrefix) throws IOException {
Map<LocalResourceProto, Path> rsrcs =
new HashMap<LocalResourceProto, Path>();
while (iter.hasNext()) {
Entry<byte[],byte[]> entry = iter.peekNext();
String key = asString(entry.getKey());
if (!key.startsWith(keyPrefix)) {
break;
}
Path localPath = new Path(key.substring(keyPrefix.length()));
if (LOG.isDebugEnabled()) {
LOG.debug("Loading in-progress resource at " + localPath);
}
rsrcs.put(LocalResourceProto.parseFrom(entry.getValue()), localPath);
iter.next();
}
return rsrcs;
}
private RecoveredUserResources loadUserLocalizedResources(
LeveldbIterator iter, String keyPrefix) throws IOException {
RecoveredUserResources userResources = new RecoveredUserResources();
while (iter.hasNext()) {
Entry<byte[],byte[]> entry = iter.peekNext();
String key = asString(entry.getKey());
if (!key.startsWith(keyPrefix)) {
break;
}
if (key.startsWith(LOCALIZATION_FILECACHE_SUFFIX, keyPrefix.length())) {
userResources.privateTrackerState = loadResourceTrackerState(iter,
keyPrefix + LOCALIZATION_FILECACHE_SUFFIX);
} else if (key.startsWith(LOCALIZATION_APPCACHE_SUFFIX,
keyPrefix.length())) {
int appIdStartPos = keyPrefix.length() +
LOCALIZATION_APPCACHE_SUFFIX.length();
int appIdEndPos = key.indexOf('/', appIdStartPos);
if (appIdEndPos < 0) {
throw new IOException("Unable to determine appID in resource key: "
+ key);
}
ApplicationId appId = ConverterUtils.toApplicationId(
key.substring(appIdStartPos, appIdEndPos));
userResources.appTrackerStates.put(appId,
loadResourceTrackerState(iter, key.substring(0, appIdEndPos+1)));
} else {
throw new IOException("Unexpected user resource key " + key);
}
}
return userResources;
}
@Override
public void startResourceLocalization(String user, ApplicationId appId,
LocalResourceProto proto, Path localPath) throws IOException {
String key = getResourceStartedKey(user, appId, localPath.toString());
try {
db.put(bytes(key), proto.toByteArray());
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
}
}
@Override
public void finishResourceLocalization(String user, ApplicationId appId,
LocalizedResourceProto proto) throws IOException {
String localPath = proto.getLocalPath();
String startedKey = getResourceStartedKey(user, appId, localPath);
String completedKey = getResourceCompletedKey(user, appId, localPath);
if (LOG.isDebugEnabled()) {
LOG.debug("Storing localized resource to " + completedKey);
}
try {
WriteBatch batch = db.createWriteBatch();
try {
batch.delete(bytes(startedKey));
batch.put(bytes(completedKey), proto.toByteArray());
db.write(batch);
} finally {
batch.close();
}
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
}
}
@Override
public void removeLocalizedResource(String user, ApplicationId appId,
Path localPath) throws IOException {
String localPathStr = localPath.toString();
String startedKey = getResourceStartedKey(user, appId, localPathStr);
String completedKey = getResourceCompletedKey(user, appId, localPathStr);
if (LOG.isDebugEnabled()) {
LOG.debug("Removing local resource at " + localPathStr);
}
try {
WriteBatch batch = db.createWriteBatch();
try {
batch.delete(bytes(startedKey));
batch.delete(bytes(completedKey));
db.write(batch);
} finally {
batch.close();
}
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
}
}
private String getResourceStartedKey(String user, ApplicationId appId,
String localPath) {
return getResourceTrackerKeyPrefix(user, appId)
+ LOCALIZATION_STARTED_SUFFIX + localPath;
}
private String getResourceCompletedKey(String user, ApplicationId appId,
String localPath) {
return getResourceTrackerKeyPrefix(user, appId)
+ LOCALIZATION_COMPLETED_SUFFIX + localPath;
}
private String getResourceTrackerKeyPrefix(String user,
ApplicationId appId) {
if (user == null) {
return LOCALIZATION_PUBLIC_KEY_PREFIX;
}
if (appId == null) {
return LOCALIZATION_PRIVATE_KEY_PREFIX + user + "/"
+ LOCALIZATION_FILECACHE_SUFFIX;
}
return LOCALIZATION_PRIVATE_KEY_PREFIX + user + "/"
+ LOCALIZATION_APPCACHE_SUFFIX + appId + "/";
}
@Override
public RecoveredDeletionServiceState loadDeletionServiceState()
throws IOException {
RecoveredDeletionServiceState state = new RecoveredDeletionServiceState();
state.tasks = new ArrayList<DeletionServiceDeleteTaskProto>();
LeveldbIterator iter = null;
try {
iter = new LeveldbIterator(db);
iter.seek(bytes(DELETION_TASK_KEY_PREFIX));
while (iter.hasNext()) {
Entry<byte[], byte[]> entry = iter.next();
String key = asString(entry.getKey());
if (!key.startsWith(DELETION_TASK_KEY_PREFIX)) {
break;
}
state.tasks.add(
DeletionServiceDeleteTaskProto.parseFrom(entry.getValue()));
}
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
} finally {
if (iter != null) {
iter.close();
}
}
return state;
}
@Override
public void storeDeletionTask(int taskId,
DeletionServiceDeleteTaskProto taskProto) throws IOException {
String key = DELETION_TASK_KEY_PREFIX + taskId;
try {
db.put(bytes(key), taskProto.toByteArray());
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
}
}
@Override
public void removeDeletionTask(int taskId) throws IOException {
String key = DELETION_TASK_KEY_PREFIX + taskId;
try {
db.delete(bytes(key));
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
}
}
@Override
protected void initStorage(Configuration conf)
throws IOException {
Path storeRoot = createStorageDir(conf);
Options options = new Options();
options.createIfMissing(false);
options.logger(new LeveldbLogger());
LOG.info("Using state database at " + storeRoot + " for recovery");
File dbfile = new File(storeRoot.toString());
byte[] schemaVersionData = null;
try {
db = JniDBFactory.factory.open(dbfile, options);
try {
schemaVersionData = db.get(bytes(DB_SCHEMA_VERSION_KEY));
} catch (DBException e) {
throw new IOException(e.getMessage(), e);
}
} catch (NativeDB.DBException e) {
if (e.isNotFound() || e.getMessage().contains(" does not exist ")) {
LOG.info("Creating state database at " + dbfile);
options.createIfMissing(true);
try {
db = JniDBFactory.factory.open(dbfile, options);
schemaVersionData = bytes(DB_SCHEMA_VERSION);
db.put(bytes(DB_SCHEMA_VERSION_KEY), schemaVersionData);
} catch (DBException dbErr) {
throw new IOException(dbErr.getMessage(), dbErr);
}
} else {
throw e;
}
}
if (schemaVersionData != null) {
String schemaVersion = asString(schemaVersionData);
// only support exact schema matches for now
if (!DB_SCHEMA_VERSION.equals(schemaVersion)) {
throw new IOException("Incompatible state database schema, found "
+ schemaVersion + " expected " + DB_SCHEMA_VERSION);
}
} else {
throw new IOException("State database schema version not found");
}
}
private Path createStorageDir(Configuration conf) throws IOException {
final String storeUri = conf.get(YarnConfiguration.NM_RECOVERY_DIR);
if (storeUri == null) {
throw new IOException("No store location directory configured in " +
YarnConfiguration.NM_RECOVERY_DIR);
}
Path root = new Path(storeUri, DB_NAME);
FileSystem fs = FileSystem.getLocal(conf);
fs.mkdirs(root, new FsPermission((short)0700));
return root;
}
private static class LeveldbLogger implements Logger {
private static final Log LOG = LogFactory.getLog(LeveldbLogger.class);
@Override
public void log(String message) {
LOG.info(message);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,803
|
\section{Introduction}
Precisely obtaining the weak phase $\alpha$, $\beta$ and $\gamma$ in
the Cabibbo-Kobayashi-Maskawa (CKM) matrix is one of the central
issues in the current studies of $B$ decays. Besides global fits to
all the indirect measurements in the Standard Model
(SM)\cite{Charles:2004jd,Bona:2005vz} or measurements on the
time-dependent CP asymmetry in $B \rightarrow J / \psi K_S$, the phase
angles $\alpha$ and $\gamma$ of the unitarity triangle can also be
probed from hadronic charmless $B$ decays. In the charmless decay
modes $B \rightarrow P P$ with $P$ denoting a pseudo-scalar final
state, the weak phase $\gamma$ can be determined either with
theoretical inputs such as QCD factorization
\cite{Beneke:2001ev,Beneke:2003zv}, perturbation QCD
\cite{Keum:2000ph,Keum:2002ri,Keum:2002vi} and soft-collinear
effective theories \cite{Bauer:2004tj,Bauer:2004dg,Grossman:2005jb} etc, or through
model independent phenomenological methods based on flavor SU(3)
symmetry
\cite{zeppenfeld:1981ex,Savage:1989jx,Chau:1990ay,Gronau:1994rj,Gronau:1995hn,Gronau:1995ng,Gronau:2002gj,Chiang:2005kz}.
Within the flavor SU(3) symmetry, direct $B$ decay amplitudes are
described by a set of flavor topological diagrams. The leading
diagrams involve: a tree diagram $\mathcal{T}$, a color suppressed
tree diagram $\mathcal{C}$, a flavor octet (singlet) QCD penguin
diagram $\mathcal{P} ( \mathcal{S} )$ and a color allowed (color-suppressed)
electroweak penguin diagram $\mathcal{P}_{EW}$ ($\mathcal{P}_{EW}^C$)
etc. The hierarchical structure in the size of these diagrams
simplifies the analysis and makes it powerful in exploring
the hadronic $B$ decays. Recent global fits using the
diagrammatic method have already shown that the weak phase $\gamma$
can be determined with a reasonable precision and the obtained value
agrees well with the one from the global CKM fit
\cite{Wu:2000rb,Zhou:2000hg,He:2000ys,Ali:2004hb,Chiang:2004nm,Wu:2004xx,Wu:2005hi}.
However, the current data also exhibit some puzzling patterns which
needs further understanding. The unexpected large branching ratio of
$\pi^0 \pi^0$ and the relative suppression of $\pi^+ \pi^-$ possess a
big theoretical challenge and may require large nonfactorizable
contributions\cite{Buras:2003dj,Buras:2004ub,Buras:2004th}; the relative enhancement of $\pi^0 \bar{K}^0$ to $\pi^+
K^-$ may lead to an enhancement of electroweak penguin which could be a
signal of new physics ( see, eg.
\cite{Yoshikawa:2003hb,Mishima:2004um,Buras:2003dj,Buras:2004ub,Buras:2004th%
,Wu:2004xx,Wu:2005hi}).
The recently measured mixing-induced CP asymmetries of ($\omega,
\phi$, $\pi^0, \eta' ) K_S$ though not conclusive yet, suggest a
possibility that the weak phase $\beta$ obtained from $b \rightarrow
s$ penguin-dominant processes may deviate from the one
determined from $b \rightarrow c$ tree-dominated process $J / \psi K_S$
\cite{Buchalla:2005us,Beneke:2005pu,Kim:2005pe}.
The global fit to all the charmless $B$ decay modes connected by flavor SU(3)
symmetry is the most consistent way to explore the weak phases and the
involved hadronic decay amplitudes. However, to get more insight on the
potential inconsistencies in the theory and a better understanding of the
strong dynamics in hadronic $B$ decays it is usefully to divide the whole
decay modes into several subsets in which the relevant parameters can be
investigated individually. The comparison among the same quantities obtained
from different subsets will not only provide us important cross-checks but
also shed light on the origins of those puzzles and possible signals of new
physics beyond the SM.
For instance, in $\pi \pi$ system the three decays modes $\pi^+ \pi^-,
\text{$\pi^0 \pi^0 $ and $\pi^0 \pi^-$ }$ provide at most seven
independent observables including three branching ratios, two direct
CP asymmetries (the direct CP asymmetry for $\pi^0 \pi^-$ is predicted
to be vanishing in SM) and two mixing-induced CP asymmetries, enough
to determine the involved hadronic amplitudes $\mathcal{T}$,
$\mathcal{C}$, $\mathcal{P}$ and also the weak phase $\gamma$. In
$\pi \pi$ modes, the electroweak penguins are small and negligible.
The recent fits taking weak phase $\beta$ as input show a good
determination of all the amplitudes. The ratio of
$\mathcal{C}/\mathcal{T}$ is found to be large close to
0.8\cite{Chiang:2004nm,He:2004ck,Charng:2004ed,Wu:2004xx,Wu:2005hi,Kim:2005jp},
the weak phase $\gamma$ is determined up to a multi-fold ambiguity and
one of them agrees well with the SM global fit value $\sim 62^\circ$.
In $\pi K$ system, the available data involve four CP averaged
branching ratios, three direct CP asymmetries ( the direct CP asymmetry
in $B^- \rightarrow \pi^- \bar{K}^0$ is predicted to be nearly zero
when annihilation diagram is negligible). Plussing a mixing-induced CP
asymmetry in $B \rightarrow \pi^0 K_S$, there are eight data points in
total. The independent flavor diagrams include $\mathcal{T}$,
$\mathcal{C}$ and $\mathcal{P}$. The electro-weak penguin
$\mathcal{P}_{\tmop{EW}}$ is significant but can be related to tree
type diagrams in the SM \cite{Neubert:1998re,Gronau:2003kj}. Other
parameters in the CKM matrix elements can be chosen as angles $\gamma$
and $\beta$ or the Wolfenstein parameter $\rho$ and $\eta$. Thus the
shape of the whole unitarity triangle can be in principle determined
in $\pi K$ modes alone\cite{Imbeault:2003it}. The current data of
$\pi K$ are not enough to perform such an independent determination.
Taking the SM value of weak phase $\gamma$ and $\beta$ as inputs, one
can extract other hadronic amplitudes. The recent fits show a even
larger value of $\mathcal{C}/\mathcal{T} \sim 1.7$ and enhancement of
$\mathcal{P}_{\tmop{EW}}$\cite{Baek:2004rp,Wu:2005hi}.
In the present paper, we discuss the determination of $\gamma$ from an
other important subset, the $\eta^{(')} K$ modes. The advantages of
using $\eta^{(' )} K$ final states over the $\pi\pi$ and $\pi K$
states are as follows
\begin{itemizedot}
\item All the four $\eta^{(' )} K$ modes are penguin dominant with
appreciable tree-penguin interferences. Nonvanishing direct CP
asymmetries are expected in {\tmem{all}} the four decay modes, while
in $\pi \pi$($\pi K$) one of the direct CP asymmetry in
$\pi^-\pi^0$($\pi^-\bar{K}^0$) is predicted to be nearly zero.
\item The two neutral modes in $\eta^{(')}K$ will provides two
additional data points from mixing-induced CP asymmetries in
$\eta^{(' )} K_S$, while in $\pi K$ there is only one.
\item Most importantly, the flavor topological structure in
$\eta^{(')}K$ amplitudes allows a regrouping of penguin type
diagrams in such a way that the number of independent hadronic
amplitudes can be reduced to four complex parameters.
\item The electroweak penguin diagram $\mathcal{P}_{EW}$ can be included
in the reduced hadronic parameters. It is not necessary to assume
the SM relation between electroweak penguin and tree type diagram.
This is of particular importance as the current data imply the
possibility of new physics beyond the SM.
\end{itemizedot}
Thus in $\eta^{(' )} K$ modes there will be at most ten observables
available, enough to simultaneously determine all the involved
diagrammatic amplitudes, the weak phase $\gamma$ and $\beta$ which
determine the apex of the unitarity triangle. This method
distinguishes itself from the previous ones in that it makes use of the
$\eta^{(')} K$ modes {\tmem{alone}} while the previous methods focus on
constructing quadrangles connecting to $\pi K$ modes using SU(3)
symmetry\cite{Gronau:1995ng,Dighe:1995bm}.
This paper is organized as follows, in section II, we present details
of determining weak phase $\gamma$ from $\eta^{(' )} K$ modes. In
section III, the implications from the current data of $\eta' K$ is
discussed. We take typical values of hadronic parameters as inputs to
constrain $\gamma$ from $\eta' K$ modes. In section IV, the new
physics effects on the $\gamma$ determination is discussed. We finally
conclude in section V.
\section{Determining $\gamma$ from $\eta^{(')}K$}
We assume flavor SU(3) symmetry and take the following diagrammatic
decomposition for $B\to \eta^{(')}K$ decay amplitudes
\cite{Dighe:1995gq}.
\begin{eqnarray}
\bar{\mathcal{A}} ( \eta \bar{K}^0 ) & = & \frac{1}{\sqrt{3}} \left(
\mathcal{C}+\mathcal{P}_{\eta} \right) , \nonumber\\
\bar{\mathcal{A}} ( \eta' \bar{K}^0 ) & =& \frac{1}{\sqrt{6}} \left(
\mathcal{C}+\mathcal{P}_{\eta'} \right) , \nonumber\\
\bar{\mathcal{A}} ( \eta K^- ) & = & \frac{1}{\sqrt{3}} \left(
\mathcal{T}+\mathcal{C}+\mathcal{P}_{\eta} \right) , \nonumber\\
\bar{\mathcal{A}} ( \eta' K^- ) &= & \frac{1}{\sqrt{6}} \left(
\mathcal{T}+\mathcal{C}+\mathcal{P}_{\eta'} \right) ,
\end{eqnarray}
which corresponds to the flavor contents of $\eta = ( -s
\bar{s} + u \bar{u} + d \bar{d} \text{} ) / \sqrt{3}$ and $\eta' = ( 2
s \bar{s} + u \bar{u} + d \bar{d} \text{} ) / \sqrt{6}$ respectively.
This is in accordance with an $\eta_8 - \eta_0$ mixing angle of $\theta =
\arcsin ( -1 / 3 ) \simeq - 19.5^{\circ}$\cite{Dighe:1995gq}. Such a simple
mixing scheme is a good approximation in phenomenology and is extensively used
in the recent analyses of hadronic $B$ and $D$ decays
\cite{Dighe:1997hm,Fu:2003fy,Chiang:2002mr,Wu:2004ht}.
The two penguin type diagram are given by
\begin{align
\mathcal{P}_{\eta} &\equiv\mathcal{S}+ \frac{2}{3} \mathcal{P}_{\tmop{EW}},
\nonumber\\
\mathcal{P}_{\eta'} &\equiv\ 3\mathcal{P}+ 4\mathcal{S}- \frac{1}{3}
\mathcal{P}_{\tmop{EW}}.
\end{align}
In the above expressions we assume that the color-suppressed
electro-weak penguin $\mathcal{P}_{\tmop{EW}}^C$ and annihilation
diagram $\mathcal{A}$ are small and negligible. We shall also assume
the $t$-quark dominance in the penguin diagrams. With these
assumptions, all the decay amplitudes depend on four complex parameters
$\mathcal{C}, \mathcal{T}, \mathcal{P}_{\eta} \mbox{ and }
\mathcal{P}_{\eta'}$.
The two weak
phases $\gamma$ and $\beta$ enter the expressions from direct and
mixing-induced CP asymmetries as additional free parameters. Removing
a overall strong phase, there are 9 real free parameters to be
determined by 10 observables in $B \rightarrow \eta^{(' )} K$ modes
which include four CP averaged decay rates, four direct CP asymmetries
and two mixing-induced CP asymmetries in $\eta^{(' )} K_S$.
Although the expressions of $P_{\eta}$ and $P_{\eta'}$ depends
on $\eta-\eta'$ mixing scheme, the isospin symmetry guarantees
that neutral($\eta(\eta')\bar{K}^0$) and charged ($\eta(\eta')K^-$) modes
have the same coefficients for $\mathcal{S},\mathcal{P}$ and also
$\mathcal{P}_{EW}$, which allows the reduction to a single
penguin type parameter. Thus the number of free parameters is the same
for other mixing schemes
such as FKS and two-mixing angle schemes ( see, e.g.
\cite{Leutwyler:1998yr,Feldmann:1999uf}).
The CP averaged branching ratio is
defined through
\begin{eqnarray}
\tmop{Br} & \equiv & \frac{1}{2} \tau ( | \bar{\mathcal{A}} |^2 + |\mathcal{A}|^2
) ,
\end{eqnarray}
where the factor $\tau$ stands for the life time difference in $B$ mesons and is
normalized to $\tau = 1 ( \tau_+ / \tau_0 )$ for neutral(charged) modes with
$\tau_0(\tau_+)$ the life time for neutral (charged) $B$ mesons
and $\tau_+/\tau_0=1.086$.
The definition of direct CP asymmetry is
\begin{eqnarray}
a_{\tmop{cp}} & \equiv & \frac{| \bar{\mathcal{A}} |^2 - |\mathcal{A}|^2}{|
\bar{\mathcal{A}} |^2 + |\mathcal{A}|^2} .
\end{eqnarray}
The mixing-induced CP asymmetry is defined as
\begin{eqnarray}
a_{\tmop{cp}} ( t ) & \equiv & \frac{\tmop{Br} ( \bar{B}^0 ( t ) \rightarrow f )
- \tmop{Br} ( B^0 ( t ) \rightarrow f )}{\tmop{Br} ( \bar{B}^0 ( t )
\rightarrow f ) + \tmop{Br} ( B^0 ( t ) \rightarrow f )}
\nonumber\\
& = &S \sin(\Delta m_B t)- C\cos (\Delta m_B t) ,
\end{eqnarray}
where
\begin{eqnarray}
S = \frac{\tmop{Im} \lambda}{| \lambda |^2 + 1} & , & \mbox{ and } \lambda = -
e^{- 2 i \phi_d} \frac{\bar{\mathcal{A}}}{\mathcal{A}} ,
\end{eqnarray}
with $\phi_d$ the weak phase appearing in $B^0 - \bar{B}^0$ mixing and $\phi_d =
\beta$ in the SM. The coefficient $C$ is related to the direct CP asymmetry by
$C=-a_{cp}$. The latest data involving $B \rightarrow \eta^{(' )} K$ are
summarized in Tab.\ref{data} \cite{Aubert:2005iy,Abe:2004xp,hfag}
\begin{table}[htb]\begin{center}
\begin{ruledtabular}
\begin{tabular}{ccccc}
& CLEO & BaBar & Belle & WA\\
\hline
$\tmop{Br} ( \eta \bar{K}^0 )$ & $< 9.3$ & $< 2.5$ & $< 2.0$ & $< 2.0$\\
\hline
$\tmop{Br} ( \eta K^- )$ & $2.2^{+ 2.8}_{- 2.2}$ & $3.3 \pm 0.6 \pm 0.3$ &
$2.1 \pm 0.6 \pm 0.2$ & $2.6 \pm 0.5$\\
\hline
$\tmop{Br} ( \eta' \bar{K}^0 )$ & $89^{+ 18}_{- 16} \pm 9$ & $67.4 \pm 3.3
\pm 3.3$ & $68 \pm 10^{+ 9}_{- 8}$ & $68.6 \pm 4.2$\\
\hline
$\tmop{Br} ( \eta' K^- )$ & $80^{+ 10}_{- 9} \pm 7$ & $68.9 \pm 2 \pm 3.2$ &
$78 \pm 6 \pm 9$ & $70.8 \pm 3.4$\\
\hline
$a_{\tmop{cp}} ( \eta K^- )$ & & $- 0.2 \pm 0.15 \pm 0.01$ & $- 0.49 \pm
0.31 \pm 0.07$ & $- 0.25 \pm 0.14$\\
\hline
$a_{\tmop{cp}} ( \eta' \bar{K}^0 )$ & & & & $(0.04 \pm 0.08)$\\
\hline
$a_{\tmop{cp}} ( \eta' K^-$) & $0.03 \pm 0.12 \pm 0.02$ & $0.033 \pm 0.028
\pm 0.005$ & $- 0.015 \pm 0.007 \pm 0.009$ & $0.027 \pm 0.025$\\
\hline
$S'$ & & $0.30 \pm 0.14 \pm 0.02$ & $0.65 \pm 0.18 \pm 0.04$ & $0.43 \pm
0.11$\\
\end{tabular}
\end{ruledtabular}
\end{center}
\caption{The latest world average of branching ratios (
in unit of $10^{-6}$), direct CP violations as well as mixing-induced CP violations for $B \rightarrow
\eta K, \eta' K$ modes. The direct CP asymmetry of $\eta' \bar{K}^0$ comes
from time-dependent CP asymmetry measurements of $\eta' K_S$.
}
\label{data}
\end{table}
It is well known that the unusually large branching ratios of $B
\rightarrow \eta' K$ modes may require a enhancement of flavor singlet penguin
diagrams $\mathcal{S}$, which possess an other theoretical challenge and
is still under extensive theoretical study
( see, e.g.\cite{etaK}).
The flavour singlet contribution can be systematically calculated
in QCD factorization, the results favour a smaller value with
significant theoretical uncertainties\cite{Beneke:2002jn}.
However, for the
purpose of extracting weak phases one needs only the ratios of
decay rates between
neutral and charged modes in which the penguin amplitudes cancel
in a great extent, making the results in sensitive to $\mathcal{S}$.
We then define a ratio between neutral and charged decay rates as
\begin{eqnarray}
R^{(' )} & \equiv & \frac{\tau_+}{\tau_0} \frac{\tmop{Br} ( \eta^{(' )}
\bar{K}^0 )}{\tmop{Br} ( \eta^{(' )} K^- )} ,
\end{eqnarray}
The current data of $B \rightarrow \eta' K$ gives
\begin{eqnarray}
R' = 1.04 \pm 0.08.
\end{eqnarray}
The corresponding ratio in $\eta K$ modes gives $R<0.83$.
The ratio between tree and penguin type diagrams are
parameterized as
\begin{eqnarray}
\zeta^{(' )} e^{i \delta^{(' )}} = \frac{\mathcal{C}}{\mathcal{P}_{\eta^{('
)}}}e^{i\gamma} & , & \chi^{(' )} e^{i \omega^{(' )}} =
\frac{\mathcal{T}+\mathcal{C}}{\mathcal{P}_{\eta^{(' )}}}e^{i\gamma} ,
\end{eqnarray}
where $\zeta^{(' )}$ and $\chi^{(' )}$ are both real-valued.
$\delta^{(')}$ and $\omega^{(')}$ are purely strong phases as the weak
phase $\gamma$ has been extracted from the definitions. We further
define a ratio between color-suppressed and color-allowed tree
diagrams
\begin{eqnarray}
r e^{i \varphi} \equiv \frac{\zeta^{(' )} e^{i \delta^{(' )}}}{\chi^{(' )}
e^{i \omega^{(' )}}} & = & \frac{\mathcal{C}}{\mathcal{T}+\mathcal{C}} ,
\end{eqnarray}
with $r = | \zeta^{(' )} / \chi^{(' )} |$ and $\varphi = \delta^{(' )}
- \omega^{(' )}$, which are common to both $\eta K$ and $\eta' K$ modes.
All the parameters can be solved numerically from the above equations.
They can also be solved analytically to the leading order expansion of
$\zeta^{(' )}$ and $\chi^{(' )}$. Taking the $\eta' K$ modes as an
example, to the leading order of $\zeta'$ and $\omega'$, the ratio of
the decay rates is given by
\begin{eqnarray}
R' \equiv 1+\Delta R' & \simeq &1+ 2 \zeta' \left[ \cos \delta' - \frac{1}{r} \cos
( \delta' - \varphi ) \right] \cos \gamma . \label{R}
\end{eqnarray}
The two direct CP asymmetries are
\begin{align}
a'_0 \equiv a_{\tmop{cp}} ( \eta' \bar{K}^0 ) & \simeq 2 \zeta' \sin \delta'
\sin \gamma, \\
a_-' \equiv a_{\tmop{cp}} ( \eta' K^- ) & \simeq 2 \frac{\zeta'}{r} \sin (
\delta' - \varphi ) \sin \gamma . \label{acp}
\end{align}
The mixing-induced CP violation is found to be
\begin{eqnarray}
S'\equiv S(\eta'K_S) & \simeq & \text{$\sin 2 \beta$+} 2 \zeta' \cos 2 \beta \cos \delta' \sin
\gamma . \label{S}
\end{eqnarray}
In the above expressions, we use the primed quantities such as $R'$,
$a'_0$, $a'_-$ and $S'$ to denote ratio of decay rates, direct
and mixing-induced CP asymmetries respectively in $\eta' K$ modes. For
$B \rightarrow \eta K$ process, equations similar to
Eq.(\ref{R})-(\ref{S}) can be constructed with the substitution of
primed quantities to be unprimed ones i.e $R$, $a_0$, $a_-$ and $S$
etc. The Eqs.(\ref{R})-(\ref{S}) together with the ones for $\eta K$
modes provide eight equations which constrain the eight parameters,
$$\zeta', \ \delta', \ \zeta, \ \delta, \ r,\ \varphi, \ \gamma \
\mbox{and} \ \beta .$$
A simultaneous determination of
$\gamma$ and $\beta$ will allow a reconstruction of the unitarity
triangle from $\eta^{(' )} K$ modes alone.
Since great success has already been achieved in the measurement of
$\sin 2 \beta$ from $B \rightarrow J / \psi K_S$ in the two
$B$-factories and the value obtained agrees remarkably with the one
from global fits to all the indirect measurements such as neutral $B$
and $K$ meson mixing and semileptonic $B$ decays etc, throughout this
paper we shall take the value of \cite{Aubert:2002ic,Abe:2002bx}
\begin{align
\sin 2 \beta = 0.687 \pm 0.032
\end{align}
from $B \rightarrow J / \psi K_S$ as input and focus on the
determination of the less known weak phase $\gamma$ in $\eta^{(' )} K$
modes.
Following this strategy, the value of $r$ and $\varphi$ are
determined purely by direct CP asymmetries and $\beta$
\begin{eqnarray}
\tan \varphi & \simeq &
\frac{(a'_0 a_- - a_0 a'_-) \cos2\beta}
{a_-(S'-\sin2\beta)- a'_-( S-\sin2\beta)} ,
\end{eqnarray}
and
\begin{eqnarray}
r & \simeq &
\frac{a'_0}{a'_-}\left( \cos\varphi - \frac{S'-\sin2\beta}{a'_0 \cos2\beta}\sin\varphi
\right) .
\end{eqnarray}
Note that the determination of $\varphi$ requires CP asymmetry
measurements for both $\eta K$ and $\eta' K$ modes.
The solution to $\gamma$ in terms of of $r$, $\varphi$ and $\beta$ is
found straight forwardly
\begin{eqnarray}
\tan \gamma & \simeq & \frac{1}{r \Delta R'} \left[ ( r - \cos \varphi ) \frac{S'
- \text{$\sin 2 \beta$}}{\cos 2 \beta} - a'_0 \cdot \sin \varphi \right] .
\label{gamma}
\end{eqnarray}
Thus $\gamma$ is determined up to discrete ambiguities. The above
expression forms the base of the present paper. The weak phase
$\gamma$ does not depend on the ratio $\zeta'$ and $\chi'$. It only
depends on the ratio between tree type diagrams $r$ and $\varphi$. The accuracy of
$\gamma$ depends heavily on the CP violation measurements. It also
depends on the ratios of the decay rates $R'$. Note again that in this
method the weak phase $\gamma$ is determined within a closed subset of
$\eta^{(' )} K$. No measurements from other modes are needed.
In a typical case where $\varphi$ is small and $r< \cos\varphi$,
the second term in the right handed side of Eq.(\ref{gamma}) is
negligible, the sign of $\tan\gamma$ depends on the sign of
$(S'-\sin2\beta)/\Delta R'$. Thus a positive $\tan\gamma$ nontrivially requires
$R'>1$ since the current data prefer $S'-\sin2\beta<0$.
The value of $\tan\gamma$ will be enhanced if $r$ or $\Delta R'$
is very small.
The main source of the uncertainties comes from the SU(3) breaking
between $\eta K$ and $\eta' K$ decay amplitudes. At present there is
no robust estimates for SU(3) breaking effects. In the naive
factorization approach the SU(3) breaking arises from two difference
pieces in $B\to \eta^{(')} K$ amplitudes. One is proportional to the
form factors
$F^{B\to\eta}_0(m^2_B-m^2_\eta)/F^{B\to\eta'}_0(m^2_B-m^2_{\eta'})\approx1.16$
with $F^{B\to \eta(\eta')}$ the form factor of $B\to \eta^{(')}$
transition. The other one is proportional to the decay constants
$f^u_\eta/f^u_{\eta'}\approx 1.22$. This is gives an estimate that
the SU(3) breaking effect is up to $\sim 20\%$.
It needs to be emphasised that $P_{\eta}$ and $P_{\eta'}$ are treated
as two independent parameters not related by SU(3) symmetry. Note that
the SU(3) symmetry could be broken in a more complicated way in the
strong phase\cite{Wu:2002nz} and radiative corrections may give
contributions not proportional to the decay constants
\cite{Beneke:2002jn}.
The accuracy of $r$, $\varphi$ and $\delta^{(' )}$ lies on the
precision of $a_{\tmop{cp}}$s to be measured from $\eta^{(' )} K$
modes. The branching ratios for $\eta' K$ are known to be large ( a
few $\times 10^{- 5}$), while the $\eta K$ modes
are expected to be an order of magnitude smaller due to it's flavor
structure \cite{Lipkin:1991us}. However, in the $\eta K$ modes the tree-penguin
interferences could be stronger and the direct CP
asymmetries could be more significant. With the increasing
statistics in the two $B$-factories, the precision of $a_{\tmop{cp}}
( \eta' K$) will be improved. Higher precision measurements can be
achieved in the future super-$B$ factories \cite{Hewett:2004tv}.
\section{Implications from the latest data}
The weak phase $\gamma$ obtained from $\eta^{(' )} K$ modes can be
compared with the one from other methods. The difference, if exists
will shed light on the nonstandard contributions or possible new
physics. At present, the data of the direct and mixing-induced CP
asymmetries for $\eta \bar{K}^0$ are not yet available, one can not
have a practical estimate of $\gamma $ from Eq.(\ref{gamma}).
However, $r$ and $\varphi$ can be extracted from other modes or
calculated theoretically. Taking $r$ and $\varphi$ as inputs, one can
infer the value of $\gamma$ from $\eta' K$ modes using the current
data and compare it with the SM fit value. For illustrations, we
consider two typical sets for the value of $r$ and $\varphi$
\begin{itemize}
\item[a)] The values of $r$ and $\varphi$ are extracted from global $\pi \pi$
and $\pi K$ fit based on flavor SU(3) symmetry. All the recent fits prefer a large
$\mathcal{C}$
\cite{Chiang:2003pm,Wu:2004xx,Wu:2005hi,He:2004ck,Charng:2004ed,Kim:2005jp}.
From an up to date fit in Ref.\cite{Wu:2005hi}, one finds the following values
\begin{eqnarray}
r = 0.56 \pm 0.05 & , & \varphi = (-33.2\pm 6.3)^\circ . \label{big-r}
\end{eqnarray}
The large $r$ is driven by the observed large branching ratio of
$\pi^0\pi^0$. The value of $r$ obtained in the $\pi\pi$ and $\pi K$
fits can be directly used in $\eta' K$ as the leading SU(3) breaking
effects cancel in the ratio between $\mathcal{C}$ and $\mathcal{T}$.
\item[b)] The values of $r$ and $\varphi$ are taken from QCD
factorization calculations \cite{Beneke:2003zv,Beneke:2005vv}, which
prefers smaller values with considerable uncertainties. In numerical
estimations we take the following typical values from the latest QCD
factorization estimate \cite{Beneke:2005vv}
\begin{eqnarray} r =
0.20\pm0.14 & , & \varphi =-(12\pm18)^\circ . \label{small-r}
\end{eqnarray}
\end{itemize}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{gammaS.eps}
\end{center}
\caption{Weak phase $\gamma$(in degree) as function of $S'$. The solid,
dashed and dotted bands corresponds to $R'=0.96,1.04$, and $1.12$
respectively. The cross indicates the current experimental
measurements with the horizontal bar representing the data of
$S'$ and the vertical one representing the favored range of $\gamma$
from global SM fit. The values of $r$ and $\varphi$ are taken from
Eq.(\ref{big-r}) with uncertainties taken into
account. }
\label{Fig-1}
\end{figure}
In Fig.\ref{Fig-1}. we plot $\gamma$ as a function of $S'$, taking
Eq.(\ref{big-r}) as inputs for three different values of $R'$=0.96,
1.04 and 1.12 respectively, corresponding to the $1\sigma$ allowed
range.
The figure shows a strong dependence of $\gamma$ on both $S'$ and
$R'$. For $R'<1$, $\gamma$ grows up with $S'$ increasing
and is always larger than the best fitted value from global CKM fit.
For $R'>1$ , it moves down to the opposite direction and reaches $\sim
60^\circ$ for $S'\simeq 0.5$. For $R'=1$, $\tan\gamma$ becomes
infinity which fix $\gamma$ at $\sim 90^\circ$. The current data of $R'$
can not definitely tell us if $R'$ is greater or smaller than unity.
To have a robust conclusion, higher precision data are urgently
needed.
From Fig.\ref{Fig-1}, one finds a overall consistency with the global
SM fit. For $S'$ and $R'$ varying in the $1\sigma$ range,
the value of $\gamma$ is found to be
\begin{equation}
45^{\circ} \lesssim \gamma \lesssim 110^{\circ} .
\end{equation}
The error is still significant and the center value gives a slightly
large $\gamma \sim 78^\circ$.
Note that some previous analyses found problems to coincide with a
small $S'$\cite{Chiang:2004nm,Gronau:2005gz}. The difference mainly originates from the data
used in the fits. In the present paper, we use the updated data while
in the previous ones the old data of $Br(\eta' K^-)=77.6\pm 4.6$ and
$Br(\eta' \bar{K}^0)=65.2\pm6.2$ are used which corresponds to
$R'\simeq 0.91$. As it is shown in Fig.\ref{Fig-1}, a small $R'<1$
will not make a good fit.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{gammaS2.eps}
\end{center}
\caption{ The same as Fig.\ref{Fig-1} while the value of
$r$ and $\varphi$ are taken from Eq.(\ref{small-r})}\label{Fig-2}
\end{figure}
In Fig.\ref{Fig-2}, a similar plot is made with the values of $r$ and $\varphi$
taken from Eq.(\ref{small-r}). Comparing with Fig.\ref{Fig-1}, one sees
a smoother dependence on $R'$ and $S'$, for a smaller $r=0.20$
from Eq.({\tmem{{\tmem{\ref{small-r}}}}}) and
$R'$, $S'$ in the $1\sigma$ range, the value of $\gamma$ is
found to be confined in a narrow range of
\begin{eqnarray}
85^{\circ} \lesssim \gamma \lesssim 95^{\circ} & &
\end{eqnarray}
Clearly, a large $\gamma\sim 90^\circ$ is favored in this case. The
reason is that the smaller $r$ enhances $\tan\gamma$, making the three
curves closing to each other and forcing $\gamma$ to be $\sim
90^\circ$. In this case $\gamma$ can reach $\sim 60^\circ$ only for
$S'= 0.6\sim 0.7$, i.e. close to the $\sin2\beta$ from $B\to J/\psi
K_S$. One has to bear in mind that the measurement on $S'$ is not
very conclusive yet as there still exist discrepancy between Babar and
Belle results \cite{Aubert:2005iy,Abe:2004xp}. Using the PDG average
method, the error should be enlarged by a factor of $\sqrt{\chi^2}$
which is the square root of the chi-square value of the average. This
gives $S'=0.43\pm 0.17$. However, a large $\gamma$ is still favored
in the enlarged region. The theoretical prediction to $S'$ based on
QCD factorization prefer that $S'$ is slightly greater than
$\sin2\beta$, $S'-\sin2\beta \simeq
0.01$\cite{Buchalla:2005us,Beneke:2005pu}. This remains to be tested
in the future experiment.
It follows from the above results that if $\gamma$ is indeed around
$62^\circ$, a large $r$ is favored by the current data of $\eta' K$
only, which is independent of the data of $\pi\pi$ and $\pi K$.
Independent determination of the relative size of the color-suppressed
tree diagram may provide us important hints on it's origin. A possible
explanation is that the extracted $\mathcal{C}$ is an effective
amplitude involving other important contributions such as: a large
nonfactorizable $W$-exchanging diagram
$\mathcal{E}$\cite{Buras:2003dj,Buras:2004ub,Buras:2004th}, a large
penguin type diagram contribution through internal $c\bar{c}$ loops,
i.e. the charming penguin \cite{Ciuchini:1997hb,Ciuchini:2001gv},
large final state interactions \cite{Barshay:2004hb,Cheng:2004ru} etc.
The exchange diagram $\mathcal{E}$ only contributes to $\pi\pi$ modes
and will not affect $\pi K$ and $\eta' K$. The charming penguin always
come together with the ordinary penguin diagrams. But the tree-penguin
interferences are different in $\pi\pi,\pi K$ and $\eta' K$. One can
not expect a universal enhancement pattern of $\mathcal{C}$ in all
modes. The final state interaction is more process-dependent. Thus if
the ratio $r$ can be precisely determined independently from various
subsets, it is possible to distinguish some of the explanations. For
instance, if large $r$ is confirmed in all the $\pi\pi$, $\pi K$ and
$\eta^{(')}K$ modes, the first explanation will not be favored.
\section{New physics effects}
We proceed to discuss the new physics contributions. When the weak
phase $\beta$ is taken as known from $B\to J/\psi K_S$, there are
eight data points to constrain seven real parameters in $\eta^{(' )}
K$ system. The nonzero degree-of-freedom allows one to make
cross-checks for consistency or explore new physics contributions.
The new physics may affect the observables in two different ways. One
is through modifying $B^0-\bar{B}^0$ mixing which makes $\phi_d \neq
\beta$. The consequence is that the mixing induced CP asymmetry for
all the modes will be affected in the same manner, which is not very
likely as the measurement of $\sin2\beta$ from $J/\psi K_S$ agrees
remarkably with all the indirect measurements of the unitarity
triangle and so far no systematic deviations of $\sin2\beta$ from it's
global SM fit value are confirmed in other modes. The other way is that
new physics contributes to decay amplitudes, most likely through $b\to
s$ loop processes. In this case the modifications to direct and
mixing-induced CP asymmetries will be process dependent.
Taking the $\eta'K$ modes as an example,
we parameterize the new physics contribution to $b \rightarrow s$ penguin in the
following form
\begin{eqnarray}
\bar{\mathcal{A}} ( \eta' \bar{K}^0 ) & = & \frac{1}{\sqrt{6}}
\mathcal{P}_{\eta'} \left[ 1 + \zeta' e^{i \delta'} + \xi_s e^{i ( \delta_s +
\phi_s)} \right], \nonumber\\
\bar{\mathcal{A}} ( \eta' K^- ) & = & \frac{1}{\sqrt{6}} \mathcal{P}_{\eta'}
\left[ 1 + \chi' e^{i ( \delta' - \varphi )} + \xi_s e^{i ( \delta_s + \phi_s)} \right],
\end{eqnarray}
where $\delta_s$ and $\phi_s$ are the strong and weak phases generated
by new physics. It's relative size to $\mathcal{P}_{\eta'}$ is denoted
by $\xi_s$. For simplicity, we assume that the new physics
contribution respects the isospin symmetry under $u \leftrightarrow
d$. This happens to the modes mainly contributing to the QCD penguins
\cite{Kagan:1997qn}. For any specific models such as the two-Higgs-doublet model
\cite{2hdm,wu:1999fe,Wu:2001vq,Wu:2004kr},
the $Z'$ model \cite{Barger:2004hn} etc. the relation between them is computable. In
the presence of new physics, the expressions for CP asymmetries to
the leading order are modified as follows
\begin{eqnarray}
a'_0 & \simeq & 2 \zeta' \sin \delta' \sin \gamma - 2 \xi_s \sin \delta_s \sin
\phi_s , \nonumber\\
a'_- & \simeq & 2 \chi' \sin ( \delta' - \varphi ) \sin \gamma - 2 \xi_s \sin
\delta_s \sin \phi_s .
\end{eqnarray}
and the mixing-induced CP asymmetry is given by
\begin{eqnarray*}
S' & \simeq & \sin 2 \beta + 2 \zeta' \cos 2 \beta \cos \delta' \sin \gamma - 2
\xi_s \cos 2 \beta \cos \delta_s \sin \phi_s .
\end{eqnarray*}
Note that in this case $R'$ is not affected as the new physics contributions to
charged and neutral modes cancel. The difference between two direct
CP asymmetries $a'_0 - a'_-$ is not affected either. In the presence
of new physics, the weak phase $\gamma$ extracted from
Eq.(\ref{gamma}) will be an effective one denoted by $\tilde{\gamma}$,
and is related to the true value of $\gamma$ through
\begin{eqnarray}
\tan \tilde{\gamma} & \equiv & \frac{1}{r \Delta R'} \left[ ( r - \cos
\varphi ) \frac{S' - \text{$\sin 2 \beta$}}{\cos 2 \beta} - \sin \varphi
\cdot a'_0 \right] \nonumber\\
& \simeq & \tan \gamma - 2 \frac{r \cos \delta_s - \cos ( \delta_s - \varphi
)}{r \Delta R'} \xi_s \sin \phi_s .
\label{new-gamma}
\end{eqnarray}
Thus the deviation of the effective value $\tilde{\gamma}$ from the true
$\gamma$ is a measure of the new physics effects and which can be used to
extract new physics parameters or distinguish different new physics
models \cite{Zhou:2000ym}.
The true value of $\gamma$ can be obtained from other
measurements such as through $B\to DK$\cite{Atwood:1996ci} or from global CKM fits. The
new physics effects will be enhanced if the deviation of $R'$ from
unity is tiny. As the current data give a central value of $R'\simeq
1.04$, the effective $\tilde{\gamma}$ is very sensitive to new
physics.
If the true value of $\gamma$ is indeed around $62^\circ$,
for typical values of $r$ and $\varphi$ taken from Eq.(\ref{big-r})
and $R'=1.04$, the enhancement factor is about $\sim 50$.
As a consequence, significant difference of a few tens degree
between $\tilde{\gamma}$ and $\gamma$ is possible for
$\xi_s\sin\phi_s \sim 0.1$.
It has been argued recently that in general the new physics will not
generate significant relative strong phases as the strong phases
mainly originate from the long-distance rescatterings of the final
states while new physics contributes only to short-distant part\cite{Datta:2004re}.
In the case that the new physics strong phase $\delta_s$ is negligible, the
combined new physics parameter $\xi_s \sin \phi_s$ can be directly
extracted. As an illustration, we take the central value of $r=0.56$
and $\varphi=-33^\circ$ from Eq.(\ref{big-r}) and $R'$ in the
$1\sigma$ range, which gives
\begin{eqnarray}
0 \alt \xi_s \sin \phi_s \alt 0.25 .
\end{eqnarray}
It follows that
for a large $r$, the current data marginally agree with the SM,
and the new physics receives only an upper bound.
For a smaller value of $r=0.2$ and $\varphi=-12^\circ$ in
Eq.(\ref{small-r}), a positive signal of nonzero $\xi_s \sin \phi_s$
is found
\begin{eqnarray}
0.15 \alt \xi_s \sin \phi_s \alt 0.19 ,
\end{eqnarray}
which demonstrates that the $\eta^{(')}K$ mode provide a good avenue
to explore new physics contributions. Needless to say that the
current experimental status is not conclusive yet and one can not draw
a robust conclusion on the presence of new physics.
The advantage of using Eq.(\ref{new-gamma}) in $\eta' K$ modes to
probe new physics is that besides new physics parameters the
difference between the effective and the true $\gamma$ only depends on
the hadronic parameters $r$ and $\varphi$. The knowledge of the
tree-penguin ratio $\zeta^{(')}$ and $\chi^{(')}$ are not needed.
Comparing with probing new physics through $B_s\to KK$, although the flavor
structure in $B_s\to KK$ is simpler,
the tree-penguin
interference can not be avoid and one has to combine it with
$B\to\pi\pi$ where additional assumptions on new physics effects in $b\to d$ penguin
have to be made\cite{London:2004ej}.
\section{conclusion}
In summary, we have present a method for an independent determination of
the weak phase $\gamma$ from $B \rightarrow \eta^{(' )} K$ alone, which
makes use of measurements of all the direct and mixing-induced
CP asymmetries.
The value of $\gamma$ extracted from $\eta^{(' )} K$ may be compared
with the ones from other modes. The possible discrepancy may help us
to understand the current puzzles in charmless $B$ decays.
We have taken two sets of the ratio $\mathcal{C}/\mathcal{T}$ as
inputs to analysis the implications of the recent data on $\eta'K$
modes. One is from from global $\pi\pi$, $\pi K$ and $K K$ fits
which leads to a $45^\circ \alt \gamma \alt 110^\circ$ in agreement
with the SM fit value. The other is from QCD factorization
calculations which makes $\gamma$ around $90^\circ$. Within
the SM, it implies that a large $\mathcal{C}$ is independently favored
in $\eta' K$ modes.
New physics beyond SM can be singled out if $\gamma$ obtained in
$\eta' K$ modes is significantly different than the ones from
other decay modes or other approaches. The value of $\gamma$ obtained
from $\eta' K$ are found to be sensitive to new physics contributions
and can be used to extract new physics parameters if the new physics
does not carry significant new strong phases.
\bibliographystyle{apsrev}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,872
|
{"url":"https:\/\/tspace.library.utoronto.ca\/handle\/1807\/26440","text":"Home\n\nBrowse\nCommunities\n&\u00a0Collections\n\nIssue Date\nAuthor\nTitle\nSubject\n\nSign on to:\n\nMy Account\nauthorized users\n\nEdit Profile\n\nHelp\n Please use this identifier to cite or link to this item: http:\/\/hdl.handle.net\/1807\/26440\n\n Title: The 16th OISE survey: Public attitudes towards education in Ontario in 2007 Authors: Hart, DougLivingstone, David W. Keywords: EducationPublic policyEducation in OntarioSurveyGeneral perceptions of schoolsEducational financeSchool governanceStudent assessmentEquity and school outcomesEducation and economySurvey highlightsSurvey findings Issue Date: 2007 Publisher: Ontario Institute for Studies in Education of the University of Toronto (OISE\/UT) Series\/Report no.: OISE\/UT Survey16th Abstract: The overwhelming majority of the public support improving the resource base of K to Grade 12 public schools in Ontario, according to the latest findings of a survey conducted by the Ontario Institute for Studies in Education of the University of Toronto (OISE). Equally important, most of those who now want increased spending on schools are also prepared to pay higher taxes in support of education. Description: The OISE\/UT Survey has been conducted and published biennially since 1980. It is the only regular, publicly disseminated survey of public attitudes towards educational policy options in Canada. Its basic purpose is to enhance public self-awareness and informed participation in educational policy-making. URI: http:\/\/webspace.oise.utoronto.ca\/~living13\/@oisesurveyarchive\/survey_16_2007_final.pdfhttp:\/\/hdl.handle.net\/1807\/26440 Appears in Collections: Issues That Matter in Education\n\nFiles in This Item:\n\nFile Description SizeFormat","date":"2014-03-07 20:57:58","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\": 0.19701910018920898, \"perplexity\": 11620.9930592189}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"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-2014-10\/segments\/1393999650844\/warc\/CC-MAIN-20140305060730-00068-ip-10-183-142-35.ec2.internal.warc.gz\"}"}
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\section{Introduction}
Continuous-state branching processes (CSBPs in short) are nonnegative-valued Markov processes satisfying the additive branching processes. They often
arise as time-population scaling limits of discrete-state branching processes, and can also be obtained from spectrally positive L\'evy processes via the Lamperti transform. The introduction of CSBP allows the applications of stochastic analysis, L\'evy processes and stochastic differential equations (SDE in short) techniques to its study. We refer to Li (2011), Li (2019) and Kyprianou (2012) and references therein for comprehensive reviews on CSBPs.
Generalized versions of the CSBP have been proposed in recent years to incorporate interactions between individuals and (or) between individuals and the population. A class of CSBPs with nonlinear branching mechanism, obtained by generalized Lamperti transform, is introduced in Li (2019). In Li et al. (2019), a more general version of the nonlinear CSBP is proposed as the solution to SDE
\begin{equation}\label{sde}
X_t=
x+\int_0^t a_0(X_s)\dd s+\int_0^t\int_0^{a_1(X_s)}W(\dd s,\dd u)
+\int_0^t\int_{(0,\infty)}\int_0^{a_2(X_{s-})}z\tilde{M}(\dd s,\dd z,\dd u),
\end{equation}
where $x>0$,
$a_0$ and $a_1,a_2\ge0$ are Borel functions
on $[0,\infty)$, $W(\dd s,\dd u)$ and
$\tilde{M}(\dd s,\dd z,\dd u)$
denote a Gaussian white noise and an
independent compensated Poisson random measure,
respectively.
The model of Li (2019) corresponds to solution $(X_t)_{t\geq 0}$ to SDE (\ref{sde}) with identical power rate functions $a_i, i=0,1,2$.
These nonlinear CSBPs allow richer behaviors such as coming down from infinity.
Some extinction, extinguishing, explosion and coming down from infinity properties are proved in Li (2019). By analyzing weighted occupation times for spectrally positive L\'evy process, asymptotic results on the speeds of coming down from infinity and explosion are obtained in Foucart et al. (2019) and Li and Zhou (2019), respectively, for nonlinear CSBP corresponding to solution to SDE (\ref{sde}) with identical rate functions $a_0=a_1=a_2$.
Exponential ergodicity for the general continuous-state nonlinear branching processes in Li et al. (2019) is studied by Li and Wang (2020) using coupling techniques.
A version of SDE (\ref{sde}) with $a_1\equiv 0$ and power functions $a_0$ and $a_2$ is considered earlier in Berestycki et al. (2015) where using the Lamperti transform, a necessary and sufficient condition for extinction is obtained and the pathwise uniqueness of solution is studied. Work on the continuous-state logistic branching process can be found in Lambert (2005), Le et al. (2013) and Le (2014).
Using a martingale approach, the extinction, explosion and coming down from infinity behaviors are further discussed in Li et al. (2019) and some rather sharp criteria in terms of $a_0, a_1, a_2$ and $\mu$ are obtained on characterization of different kinds of boundary behaviors for the nonlinear CSBPs as a Markov process. In Example 2.18 of Li et al. (2019) where $a_0, a_1$ and $a_2$ are taken to be power functions and $\tilde{M}$ is taken to be an $\alpha$-stable Poison random measure with index $\alpha\in (1, 2)$. The above criteria are further expressed in terms of the coefficients and the powers of functions $a_i$s and the stable index $\alpha$. But for the critical cases, where the coefficients, the powers and the index $\alpha$ satisfy certain linear equations, the martingale approach fails and the corresponding boundary classification remains an open problem.
The main goal of this paper is to identify the exact boundary behaviors in the above mentioned critical cases for the solution $(X)_{t\geq 0}$ to (\ref{sde}). For this purpose, we adapt the Foster-Lyapunov approach and select logarithm type test functions to obtain two new conditions under which the nonlinear CSBP never becomes extinct and never explodes, respectively. Similarly, for the boundary at infinity, we also find a Foster-Lyapunov condition with which we can show that an interesting phase transition occurs between coming down from infinity and staying infinite for different choices of coefficients and powers of the power functions and differential values of the stable index $\alpha$.
The rest of the paper is arranged as follows. We introduce the generalized CSBPs with nonlinear branching in more details and present the main theorem in Section \ref{main}. The proofs of preliminary results on Foster-Lyapunov criteria and the main theorem are deferred to Section \ref{proof}.
\section{Main results}\label{main}
\setcounter{equation}{0}
Let $U$ be a Borel set on $(0,\infty)$.
Given $\sigma$-finite measures $\mu$ and $\nu$ on $(0, \infty)$ such that
\[(z\wedge z^2)\mu(\dd z)\,\,\text{and}\,\, (1\vee\ln(1+z))\nu(\dd z)\,\, \text{are finite measures
on $U$ and $(0,\infty)\setminus U$, respectively},\]
we consider the following SDE that is a modification of (\ref{sde}):
\beqlb\label{1.01}
X_t
\ar=\ar
x+\int_0^t a_0(X_s)\dd s+\int_0^t\int_0^{a_1(X_s)}W(\dd s,\dd u) \cr
\ar\ar
+\int_0^t\int_U\int_0^{a_2(X_{s-})}z\tilde{M}(\dd s,\dd z,\dd u)
+\int_0^t\int_{(0,\infty)\setminus U}\int_0^{a_3(X_{s-})}zN(\dd s,\dd z,\dd u),
\eeqlb
where $x>0$,
$a_0$ and $a_1,a_2,a_3\ge0$ are Borel functions
on $[0,\infty)$,
$W(\dd s,\dd u)$ is a Gaussian white noise with density $\dd s\dd u$,
$\tilde{M}(\dd s,\dd z,\dd u)$
denotes a compensated Poisson random measure on
$(0,\infty)\times U\times(0,\infty)$
with density $\dd s\mu(\dd z)\dd u$,
and
$N(\dd s,\dd z,\dd u)$
denotes a Poisson random measure on $(0,\infty)\times ((0,\infty)\setminus U)\times(0,\infty)$
with density $\dd s\nu(\dd z)\dd u$.
We assume that $W(\dd s,\dd u)$, $\tilde{M}(\dd s,\dd z,\dd u)$
and $N(\dd s,\dd z,\dd u)$ are independent of each other.
A similar SDE (\ref{1.01}) is considered in Li et al. (2019) under the assumption that $U=(0,\infty)$.
We only consider the solution of \eqref{1.01} before the minimum of their
first times of hitting zero and reaching infinity
(that is $\tau_0^-$ and $\tau_\infty^+$,
which will be given in the following), respectively, i.e. both zero and infinity are absorbing states for the solution. See Section 2 of Li et al. (2019) for more details.
By the same argument as Theorem 3.1 in Li et al. (2019), we can show that SDE \eqref{1.01} has a pathwise unique solution if functions $a_0,a_1,a_2,a_3$ are
locally Lipschitz on $(0,\infty)$.
The main purpose of this paper is to investigate the extinction, explosion and
coming down from infinity behaviors,
and the uniqueness of solution to SDE (\ref{1.01}) is not required.
Throughout this paper we always assume that
$a_0,a_1,a_2,a_3$ are bounded on any bounded interval of $[0,\infty)$\
and that process $(X_t)_{t\ge0}$ is defined on filtered probability space
$(\Omega,\mathscr{F},\mathscr{F}_t,\mbf{P})$
which satisfies the usual hypotheses.
We use $\mbf{P}_x$ to denote the law of a
process started at $x$, and denote by $\mbf{E}_x$ the associated expectation.
For $a, b>0$ we define first passage times
\beqnn
\tau_a^-:=\inf\{t>0:X_t<a\},\quad\tau_b^+:=\inf\{t>0:X_t>b\}
\eeqnn
and
\beqnn
\tau_0^-:=\inf\{t>0:X_t=0\},\quad
\tau_\infty^+:=\lim_{n\to\infty}\tau_n^+
\eeqnn
with the convention $\inf\emptyset=\infty$.
Let $C^2((0,\infty))$ denote the space of
twice continuously differentiable functions on $(0,\infty)$.
We next introduce several functions.
For $u>0$ let
\beqnn
\phi(u)
\ar:=\ar
-a_0(u)u^{-1}+\frac12a_1(u)u^{-2}
+a_2(u)\int_U z^2\mu(\dd z)\int_0^1(u+vz)^{-2}(1-v)\dd v \cr
\ar\ar
-a_3(u)\int_{(0,\infty)\setminus U} z\nu(\dd z)\int_0^1(u+vz)^{-1}\dd v.
\eeqnn
For $\rho,z>0$ and $u>3$ let
\beqnn
K_\rho(u,z):=\Big(\frac{\ln (u+z)}{\ln u}\Big)^{-\rho}
+\rho\frac{\ln (u+z)}{\ln u}-(\rho+1)>0
\eeqnn
and
\beqnn
H_\rho(u):=\frac12a_1(u)u^{-2}
+a_2(u)\int_U K_\rho(u,z)\mu(\dd z)
+a_3(u)\int_{(0,\infty)\setminus U} K_\rho(u,z)\nu(\dd z).
\eeqnn
Process $(X_t)_{t\geq 0}$ becomes extinct if $\tau^-_0<\infty$; it explodes if $\tau^+_\infty<\infty$; it stays infinite if
$\lim_{x\to\infty}\mbf{P}_x\{\tau_a^-<t\}=0$ for all $t>0$ and all large $a$; it comes down from infinity if
$\lim_{a\to\infty}\lim_{x\to\infty}\mbf{P}_x\{\tau_a^-<t\}=1$ for all $t>0$.
The following main result provides new criteria on non-extinction, non-explosion, coming down from infinity and staying infinite for the solution to SDE (\ref{sde}).
\btheorem\label{t5.6} For the solution $(X_t)_{t\geq 0}$ to (\ref{1.01}) we have
\begin{itemize}
\item[{(i)}] if
$\phi(u)\le 0$ for all small enough $u>0$,
then $\mbf{P}_x\{\tau_0^-=\infty\}=1$ for all $x>0$, i.e. there is no extinction;
\item[{(ii)}] if
$\phi(u)\ge 0$ for all large enough $u>0$,
then $\mbf{P}_x\{\tau_\infty^+=\infty\}=1$ for all $x>0$, i.e. there is no explosion;
\item[{(iii)}] if
$\phi(u)\le 0$ for all large enough $u>0$ and
\[\limsup_{u\rightarrow \infty}H_\rho(u)<\infty\]
for some constant $\rho>0$,
then the process $(X_t)_{t\ge0}$ stays infinite;
\item[{(iv)}] if $\phi(u)\ge 0$ for all large enough $u>0$ and
\beqnn
\lim_{u\rightarrow\infty}(\ln u)^{-\rho-2} H_\rho(u)=\infty
\eeqnn
for some constant $\rho>0$,
then the process $(X_t)_{t\ge0}$ does not explode and it comes down from infinity.
\end{itemize}
\etheorem
\begin{remark}
Since the process $(X_t)_{t\ge0}$ does not explode and comes down from infinity under the assumptions of Theorem \ref{t5.6} (iv), then under additional assumptions on functions $a_i$s, $(X_t)_{t\ge0}$ can be extended to a Feller process defined on state space $[0, \infty]$ with $\infty$ as its entrance boundary; see Foucart et al. (2020).
\end{remark}
Until the end of this section we focus on the special case that $U=(0,\infty)$, $a_0,a_1,a_2$ are power functions and $\mu(\dd z)$ is an $\alpha$-stable measure,
that is
\begin{equation}\label{con_A}
a_i(u)=b_i u^{r_i}~ \text{for}~ i=0, 1, 2 ~\text{with}~r_0, r_1, r_2\geq 0, b_1,b_2\ge0, b_0>0
\end{equation}
and
\begin{equation}\label{con_B}
\mu(\dd z)=\frac{\alpha(\alpha-1)}{\Gamma(2-\alpha)}1_{\{z>0\}}z^{-1-\alpha}\dd z~\text{for Gamma function}\, \Gamma \,\text{and}\, 1<\alpha<2.
\end{equation}
By the properties of Gamma function we have
\beqnn
\int_0^\infty z^2\mu(\dd z)\int_0^1(u+vz)^{-2}(1-v)\dd v=\Gamma(\alpha)u^{-\alpha}
\eeqnn
and then
\beqnn
\phi(u)=-b_0u^{r_0-1}+\frac12b_1u^{r_1-2}+\Gamma(\alpha)b_2u^{r_2-\alpha}, \quad u>0.
\eeqnn
We further estimate
\[H_\rho(u)=\frac12a_1(u)u^{-2}+a_2(u)\int_0^\infty K_\rho(u,z)\mu(\dd z)\]
for which we first estimate $\int_0^\infty K_\rho(u,z)\mu(\dd z)$.
Note that for $y>0$ and $f(y):=y^{-\rho}+\rho y-(\rho+1)$, by Taylor's formula we have
\beqnn
f(y)
\ar=\ar
f(1+y-1)=f(1+y-1)-f(1)-(y-1)f'(1) \cr
\ar=\ar
(y-1)^2\int_0^1f''(1+v(y-1))(1-v)\dd v.
\eeqnn
Then by a change of variable,
\beqnn
\ar\ar
\int_0^\infty K_\rho(u,z)\mu(\dd z) \cr
\ar=\ar
\rho(\rho+1)\int_0^\infty\mu(\dd z)
\int_0^1\Big(\frac{\ln (u+vz)}{\ln u}-1\Big)^2\Big(1+\frac{v\ln(u+vz)}{\ln u}-v\Big)^{-\rho-2}(1-v)\dd v \cr
\ar=\ar
\rho(\rho+1)u^{-\alpha}\int_0^\infty\mu(\dd z)
\int_0^1\Big(\frac{\ln (1+vz)}{\ln u}\Big)^2\Big(1+\frac{v\ln(1+vz)}{\ln u}\Big)^{-\rho-2}(1-v)\dd v.
\eeqnn
Observe that
$\ln (1+z)\le C(z\wedge \sqrt{z})$
for all $z>0$ and some constant $C>0$,
which implies that for $u>3$,
\beqlb\label{1.2}
\int_0^\infty K_\rho(u,z)\mu(\dd z)
\ar\le\ar
\rho(\rho+1)u^{-\alpha} (\ln u)^{-2}
\int_0^\infty(\ln (1+z))^2\mu(\dd z) \cr
\ar\le\ar
C^2\rho(\rho+1)u^{-\alpha} (\ln u)^{-2}\int_0^\infty (z\wedge z^2)\mu(\dd z).
\eeqlb
Moreover, it is elementary to see that
\beqlb\label{1.3}
\ar\ar\int_0^\infty K_\rho(u,z)\mu(\dd z)\cr
\ar\ge\ar
\rho(\rho+1)u^{-\alpha}(\ln u)^{-2}\Big(1+\frac{\ln3}{\ln u}\Big)^{-\rho-2}\int_1^2 (\ln (3/2))^2\mu(\dd z)
\int_{1/2}^1(1-v)\dd v.
\eeqlb
\begin{remark}
In Section 2.5 of Li et al. (2019),
the exact conditions are found for the above mentioned model with polynomial rate functions to exhibit extinction/non-extinction, explosion/non-explosion and coming-down-from-infinity/staying-infinite behaviors, respectively, except for the critical case that
\[b_0 =\frac{b_1}{2} +\Gamma(\alpha) b_2>0, \, r_1=r_0+1\,\,\text{ when}\,\, b_1>0\,\, \text{ and} \,\,
r_2=r_0+\alpha-1 \,\, \text{ when} \,\, b_2>0.\]
Observe that in this critical case $\phi(u)=L(\ln)(u)=0$, where operator $L$, to be defined in (\ref{2.1}), denotes the generator of process $X$. This inspires us to choose logarithm type test functions for the main proofs.
\end{remark}
As the main goal of this paper, applying Theorem \ref{t5.6} together with \eqref{1.2}--\eqref{1.3}, we provide an answer to this open problem.
\begin{corollary}\label{example}
Suppose that (\ref{con_A}) and (\ref{con_B}) hold with $b_0 =\frac{b_1}{2} +\Gamma(\alpha) b_2>0$,
$r_1=r_0+1$ when $b_1>0$ and
$r_2=r_0+\alpha-1$ when $b_2>0$.
Then $\phi(u)=0$ for all $u>0$, and we have for all $x>0$,
\beqnn
\mbf{P}_x\{\tau_0^-=\infty\}=1\quad\text{and}\quad
\mbf{P}_x\{\tau_\infty^+=\infty\}=1.
\eeqnn
Moreover,
process $(X_t)_{t\ge0}$ stays infinite if both $r_1\le 2$ (when $b_1>0$) and $r_2\le \alpha$ (when $b_2>0$),
and it comes down from infinity
if either $r_1> 2$ (when $b_1>0$) or $r_2> \alpha$ (when $b_2>0$).
\end{corollary}
\bremark
Note that in the critical cases, there is an interesting phase transition between coming down from infinity and staying infinite. Intuitively, in these cases the process comes down from infinity if the fluctuations are relatively large and stays infinite otherwise.
Combining Corollary \ref{example} and Example 2.18 of Subsection 2.5 in Li et al. (2019), we recover the necessary and sufficient condition on the extinction of solution to the SDE of
Berestycki et al. (2015); see Theorem 1.1 there.
\eremark
\section{Proofs}\label{proof}
\setcounter{equation}{0}
Before presenting the proof of Theorem \ref{t5.6} we first prove some preliminary Foster-Lyapunov criteria.
Suppose that $g\in C^2((0,\infty))$ satisfies
\beqlb\label{2.5}
\sup_{z\ge1,u\ge v}\big[|g'(u)|+|g''(u)|+ |g(u+z)-g(u)|/\ln(1+z)\big]<\infty
\eeqlb
for all $v>0$.
For $u>0$, put
\beqlb\label{2.1}
Lg(u)
\ar:=\ar
a_0(u)g'(u)+\frac12a_1(u)g''(u)
+a_2(u)\int_{U}[g(u+z)-g(u)-zg'(u)]\mu(\dd z) \cr
\ar\ar
+a_3(u)\int_{(0,\infty)\setminus U} [g(u+z)-g(u)]\nu(\dd z)\cr
\ar=\ar
a_0(u)g'(u)+\frac12a_1(u)g''(u)
+a_2(u)\int_U z^2\mu(\dd z) \int_0^1g''(u+zv)(1-v)\dd v \cr
\ar\ar
+a_3(u)\int_{(0,\infty)\setminus U}z\nu(\dd z)\int_0^1g'(u+zv)\dd v
\eeqlb
by Taylor's formula.
By It\^o's formula,
\beqnn
g(X_t)
\ar=\ar
g(x)+\int_0^tLg(X_s)\dd s
+\int_0^t\int_U\int_0^{a_2(X_{s-})}[g(X_{s-}+z)-g(X_{s-})]\tilde{M}(\dd s,\dd z,\dd u) \cr
\ar\ar
+\int_0^t\int_{(0,\infty)\setminus U}\int_0^{a_3(X_{s-})}
[g(X_{s-}+z)-g(X_{s-})]\tilde{N}(\dd s,\dd z,\dd u)
\eeqnn
For $b>a>0$ let $\gamma_{a,b}:=\tau_{a}^-\wedge\tau_b^+$
and
\beqnn
M_t^g:=
g(X_t)-g(x)
-\int_0^tLg(X_s)\dd s.
\eeqnn
Then under condition \eqref{2.5},
\beqlb\label{2.2}
t\mapsto M_{t\wedge \gamma_{a,b}}^g
\quad \mbox{is a martingale}
\eeqlb
for all $b>a>0$.
\blemma\label{t2.2}
Given $0<a<x<b<\infty$, for any function $g\in C^2((a, b))$
satisfying
\eqref{2.5} and constant $d_{a,b}>0$ satisfying
\beqnn
Lg(u)\le d_{a,b}g(u),\qquad u\in(a,b),
\eeqnn
we have
\beqlb\label{2.3}
\mbf{E}_x\big[g(X_{t\wedge \gamma_{a,b}})\big]
\le
g(x)\e^{d_{a,b}t},\qquad t\geq 0.
\eeqlb
\elemma
\proof
It follows from \eqref{2.2} that
\beqnn
\mbf{E}_x\big[g(X_{t\wedge \gamma_{a,b}})\big]=g(x)
+\mbf{E}_x\Big[\int_0^{t\wedge \gamma_{a,b}}Lg(X_s)\dd s\Big]
\le
g(x)+d_{a,b}\int_0^t\mbf{E}_x\big[g(X_{s\wedge \gamma_{a,b}})\big]\dd s.
\eeqnn
By Gronwall's lemma,
\beqnn
\mbf{E}_x\big[g(X_{t\wedge \gamma_{a,b}})\big]
\le
g(x)\e^{d_{a,b}t},
\eeqnn
which ends the proof.
\qed
\blemma\label{t2.1}
Let $(X_t)_{t\geq 0}$ be the solution to SDE (\ref{1.01}).
\begin{itemize}
\item[{\normalfont(i)}]
For any fixed $b>0$,
if there exists a function $g\in C^2((0, \infty))$ strictly positive on $(0,b]$ satisfying \eqref{2.5} and $\lim_{u\to0}g(u)=\infty$,
and there is a constant $d(b)>0$ such that
$Lg(u)\le d(b)g(u)$ for all $0<u<b$,
then $\mbf{P}_x\{\tau_0^-\ge\tau^+_b\}=1$ for all $0<x<b$.
\item[{\normalfont(ii)}]
For any fixed $a>0$,
if there exists a function $g\in C^2((0,\infty))$ strictly positive on $[a,\infty)$ satisfying
\eqref{2.5} and $\lim_{u\to\infty}g(u)=\infty$,
and there is a constant $d(a)>0$ such that
$Lg(u)\le d(a)g(u)$ for all $u>a$,
then $\mbf{P}_x\{\tau_\infty^+\ge\tau_a^-\}=1$ for all $x>a$.
\item[{\normalfont(iii)}]
If there exists a function $g\in C^2((0,\infty))$ strictly positive on $[u,\infty)$ for all large $u$ satisfying
\eqref{2.5} and $\lim_{u\to\infty}g(u)=0$,
and for any large $a>0$, there is a constant $d(a)>0$ such that
$Lg(u)\le d(a)g(u)$ for all $u>a$,
then $(X_t)_{t\ge0}$ stays infinite.
\end{itemize}
\elemma
\proof
We apply Lemma \ref{t2.2} for the proofs.
For part (i), \eqref{2.3} holds for all $0<a<b$ and with
$d_{a,b}$ replaced by $d(b)$.
Then using Fatou's lemma we have
\beqnn
\mbf{E}_x\big[\liminf_{a\to0}g(X_{t\wedge \tau_a^-\wedge\tau_b^+})\big]
\le
\liminf_{a\to0}\mbf{E}_x\big[g(X_{t\wedge \gamma_{a,b}})\big]
\le g(x)\e^{d(b)t}.
\eeqnn
Since $\lim_{u\to0}g(u)=\infty$,
then
$\mbf{P}_x\{\tau_0^->t\wedge\tau_b^+\}=1$ for all $t,b>0$.
Letting $t\to\infty$ we obtain $\mbf{P}_x\{\tau_0^-\ge\tau_b^+\}=1$, which gives the first assertion.
For part (ii), \eqref{2.3} holds for all $b>a$ and
with $d_{a,b}$ replaced by $d(a)$.
Then using Fatou's lemma again we obtain
\beqnn
\mbf{E}_x\big[\liminf_{b\to\infty}g(X_{t\wedge \tau_a^-\wedge\tau_b^+})\big]
\le
\liminf_{b\to\infty}\mbf{E}_x\big[g(X_{t\wedge \gamma_{a,b}})\big]
\le g(x)\e^{d(a)t}.
\eeqnn
Since $\lim_{u\to\infty}g(u)=\infty$, then
$\mbf{P}_x\{\tau_\infty^+>t\wedge\tau_a^-\}=1$ for all $t>0$.
Letting $t\to\infty$ we get
$\mbf{P}_x\{\tau_\infty^+\ge\tau_a^-\}=1$,
which implies the second assertion.
For part (iii), given any large $a>0$, \eqref{2.3} holds for all $b>a$ and
with $d_{a,b}$ replaced by $d(a)$ again.
We can also get
\beqnn
\mbf{E}_x\big[g(X_{\tau^-_a})1_{\{\tau^-_a<t\wedge\tau_\infty^+\}}\big]
\ar\le\ar
\liminf_{b\to\infty}
\mbf{E}_x\big[g(X_{\tau^-_a})1_{\{\tau^-_a<t\wedge\tau_b^+\}}\big]\cr
\ar\le\ar
\liminf_{b\to\infty}\mbf{E}_x\big[g(X_{t\wedge \gamma_{a,b}})\big]
\le g(x)\e^{d(a)t},
\eeqnn
which implies
\beqnn
g(a)\mbf{P}_x\{\tau^-_a<t\wedge\tau_\infty^+\}
\le g(x)\e^{d(a)t}.
\eeqnn
Since $\lim_{u\to\infty}g(u)=0$, then for all $t,a>0$,
\beqlb\label{2.4}
\lim_{x\to\infty}\mbf{P}_x\{\tau^-_a<t\wedge\tau_\infty^+\}=0.
\eeqlb
Observe that
$\{\tau^-_a\ge\tau_\infty^+\}\subset\{\tau^-_a=\infty\}$.
Then combining \eqref{2.4} we have
\beqnn
\lim_{x\to\infty}\mbf{P}_x\{\tau^-_a<t\}
\le
\lim_{x\to\infty}\mbf{P}_x\{\tau^-_a<t\wedge\tau_\infty^+\}
+\lim_{x\to\infty}\mbf{P}_x\{\tau^-_a<t,\tau_\infty^+\le\tau^-_a\}=0
\eeqnn
for all $t>0$. Then the process stays infinite.
\qed
The next lemma provides a condition that associates the probability of coming down from infinity with the probability of non-explosion. Its proof is a modification of Proposition 2.2 in Ren et al. (2019).
\blemma\label{t2.3}
Suppose that there exist a function $g(u)\in C^2((0,\infty))$ bounded and strictly positive for all large $u$, satisfying
\eqref{2.5} and $\limsup_{u\to\infty} g(u)>0$,
and a strictly positive function $d$ on $(0, \infty)$
such that
\[Lg(u)\ge d(a) g(u)\,\,\text{ for all large}\,\, u\,\,\text{ and}\,\, \lim_{a\to\infty}d(a)=\infty.\]
Then for any $t>0$
\beqnn
\lim_{a\to\infty}\lim_{x\to\infty}\mbf{P}_x\{\tau_a^-<t\}\ge
\liminf_{x\to\infty}\mbf{P}_x\{\tau_\infty^+=\infty\}.
\eeqnn
Consequently, process $(X_t)_{t\ge0}$ comes down from infinity if there is no explosion.
\elemma
\proof
The proof is a modification of that of Proposition 2.2 in Ren et al. (2019).
We present the details for completeness.
By \eqref{2.2}, for all large $a<b$
\beqnn
\mbf{E}_x\big[g(X_{t\wedge\gamma_{a,b}})\big]
=
g(x)
+\mbf{E}_x\Big[\int_0^{t\wedge\gamma_{a,b}}Lg(X_s)\dd s\Big]
=
g(x)
+\int_0^t \mbf{E}_x\Big[Lg(X_s)1_{\{s\le \gamma_{a,b}\}}\Big] \dd s
\eeqnn
and then by integration by parts,
\beqnn
\ar\ar
\mbf{E}_x\big[g(X_{t\wedge\gamma_{a,b}})\big]\e^{-d(a)t} \cr
\ar=\ar
g(x)+\int_0^t \mbf{E}_x\big[g(X_{s\wedge\gamma_{a,b}})\big]\dd (\e^{-d(a)s})
+\int_0^t \e^{-d(a)s}\dd \big(\mbf{E}_x\big[g(X_{s\wedge\gamma_{a,b}})\big]\big) \cr
\ar=\ar
g(x)
-d(a)\int_0^t\mbf{E}_x\big[g(X_{s\wedge\gamma_{a,b}})\e^{-d(a)s}\big]\dd s
+\int_0^t\e^{-d(a)s}\mbf{E}_x\big[Lg(X_s)1_{\{s\le\gamma_{a,b}\}}\big]\dd s \cr
\ar\ge\ar
g(x)
-d(a)\int_0^t\mbf{E}_x\big[g(X_{s\wedge\gamma_{a,b}})\big]\e^{-d(a)s}\dd s
+d(a)\int_0^t\e^{-d(a)s}\mbf{E}_x\big[g(X_s)1_{\{s\le\gamma_{a,b}\}}\big]\dd s,
\eeqnn
which implies that
\beqnn
g(x)\le\mbf{E}_x\big[g(X_{t\wedge\gamma_{a,b}})\e^{-d(a)t}\big]
+d(a)\mbf{E}_x\Big[\int_0^tg(X_{\gamma_{a,b}})\e^{-d(a)s}
1_{\{s>\gamma_{a,b}\}}\dd s\Big].
\eeqnn
Letting $t\to\infty$ in the above inequality and using the dominated convergence we obtain
\beqnn
g(x)
\le
d(a)\mbf{E}_x\Big[g(X_{\gamma_{a,b}})\int_{\gamma_{a,b}}^\infty\e^{-d(a)s}\dd s\Big]
=\mbf{E}_x\big[g(X_{\gamma_{a,b}}) \e^{-d(a)\gamma_{a,b}}\big].
\eeqnn
It follows that
\beqnn
g(x)
\ar\le\ar
\mbf{E}_x\Big[\lim_{b\to\infty}g(X_{\gamma_{a,b}})
\e^{-(\tau_a^-\wedge\tau^+_\infty)d(a)}
\big(1_{\{\tau_\infty^+<\tau^-_a\}}+1_{\{\tau_a^-<t,\tau_a^-\le\tau^+_\infty\}}
+1_{\{t\le\tau^-_a\le\tau_\infty^+\}}\big)\Big] \cr
\ar\le\ar
\limsup_{u\to\infty}g(u)
\mbf{P}_x\{\tau_\infty^+<\infty\}
+g(a)\mbf{P}_x\{\tau_a^-<t,\tau_a^-\le\tau^+_\infty \}+g(a)\e^{-d(a) t} \cr
\ar\le\ar
\limsup_{u\to\infty}g(u)( 1-\mbf{P}_x\{\tau_\infty^+=\infty\})
+g(a)\mbf{P}_x\{\tau_a^-<t\}+g(a)\e^{-d(a) t}.
\eeqnn
Letting $x\to\infty$ first,
\beqnn
\limsup_{x\to\infty}g(x)
\ar\leq\ar
\limsup_{u\to\infty}g(u)
\limsup_{x\to\infty}(1-\mbf{P}_x\{\tau_\infty^+=\infty\}) \cr
\ar\ar
+g(a)\lim_{x\to\infty}\mbf{P}_x\{\tau_a^-<t \}+g(a)\e^{-d(a) t}.
\eeqnn
Then letting $a\to\infty$, by the conditions in the lemma we have
\beqnn
\limsup_{x\to\infty}g(x)
\ar\leq\ar
\limsup_{u\to\infty}g(u)
\left(1-\liminf_{x\to\infty}\mbf{P}_x\{\tau_\infty^+=\infty\}\right)\cr
\ar\ar
+\limsup_{a\to\infty}g(a)\limsup_{a\to\infty}
\lim_{x\to\infty}\mbf{P}_x\{\tau_a^-<t \}.
\eeqnn
Observing that $\lim_{x\to\infty}\mbf{P}_x\{\tau_a^-<t \} $ is increasing in $a$, the desired inequality then follows from the above inequality.
\qed
We are now ready to show the proofs of the main results.
\noindent{\it Proof of Theorem \ref{t5.6}}.
(i) Suppose that there is a constant $0<c_1<1$ so that
$\phi(u)\le 0$ for all $0<u<c_1$.
For $n\ge1$ let $g_n(u)=1+\ln n+\ln u^{-1}$.
Then $g_n(u)>0$ for $0<u\le n$
and $Lg_n(u)=\phi(u)$ by \eqref{2.1}.
Thus $Lg_n(u)\le 0$ for $0<u<c_1$.
Since $a_0,a_1,a_2,a_3$ are bounded on $[c_1,n]$,
$Lg_n$ is bounded on $[c_1,n]$.
Now using Lemma \ref{t2.1}(i) we obtain $\mbf{P}_x\{\tau_0^-\ge\tau^+_n\}=1$
for all $0<x<n$.
Since the process is defined before the first time
of hitting zero or explosion,
$\mbf{P}_x\{\tau_0^-=\infty$ or $\tau_\infty^+=\infty\}=1$. Letting $n\to\infty$ we prove the assertion.
(ii) Suppose that there is a constant $c_2>1$ so that
$\phi(u)\ge 0$ for all $u> c_2$. Let $g_n(u)=\ln u+\ln n+1$ for $n\ge1$.
Then $g_n(u)\ge1$ for $u\ge n^{-1}$.
It follows from \eqref{2.1} that
$Lg_n(u)=-\phi(u)$ for all $u\ge n^{-1}$.
Then for all $n\ge1$,
$Lg_n(u)\le0$ for all $u\ge c_2$
and $Lg_n$ is bounded on $[n^{-1},c_2]$,
which gives $\mbf{P}_x\{\tau_\infty^+>\tau_{1/n}^-\}=1$ for all $x>n^{-1}$ by Lemma \ref{t2.1}(ii).
Letting $n\to\infty$ we have $\mbf{P}_x\{\tau_\infty^+>\tau_0^-\}=1$ for all $x>0$.
The assertion for (ii) then follows from the definition of the solution to SDE (\ref{1.01}).
(iii)
Suppose that there exist constants $c_3>3$ and $c_4>0$ so that
$\phi(u)\le 0$ and $H_\rho(u)\le c_4$ for all $u> c_3$.
Let $g\in C^2((0,\infty))$ be a strictly positive function
with $g(u)=(\ln u)^{-\rho}$ for $\rho>0$
and $u>3$.
Then for $u>3$,
\beqnn
g(u+z)-g(u)
\ar=\ar
-\rho (\ln u)^{-\rho-1}[\ln(u+z)-\ln u]+(\ln u)^{-\rho}K_\rho(u,z) \cr
\ar=\ar
-\rho (\ln u)^{-\rho-1}z\int_0^1(u+vz)^{-1}\dd v+(\ln u)^{-\rho}K_\rho(u,z)
\eeqnn
and
\beqnn
g'(u)=-\rho (\ln u)^{-\rho-1}u^{-1},~
g''(u)=\rho (\ln u)^{-\rho-1}u^{-2}+\rho(\rho+1) (\ln u)^{-\rho-2}u^{-2}.
\eeqnn
Consequently, for all $u>3$ and $z>0$,
\beqnn
g(u+z)-g(u)-zg'(u)
\ar=\ar
-\rho(\ln u)^{-\rho-1}\big[\ln (u+z)-\ln u-zu^{-1}\big]
+(\ln u)^{-\rho}K_\rho(u,z) \cr
\ar=\ar
\rho (\ln u)^{-\rho-1}z^2\int_0^1(u+zv)^{-2}(1-v)\dd v
+(\ln u)^{-\rho}K_\rho(u,z).
\eeqnn
It follows that
\beqnn
Lg(u)
\ar=\ar
\rho (\ln u)^{-\rho-1}\phi(u)
+\frac12\rho(\rho+1) (\ln u)^{-2}a_1(u)u^{-2}g(u) \cr
\ar\ar
+g(u)a_2(u)\int_{U} K_\rho(u,z)\mu(\dd z)
+g(u)a_3(u)\int_{(0,\infty)\setminus U} K_\rho(u,z)\nu(\dd z) \cr
\ar\le\ar
\rho (\ln u)^{-\rho-1}\phi(u)
+[\rho(\rho+1)+1]g(u) H_\rho(u),\qquad u>3.
\eeqnn
Then $Lg(u)\le c_4 [\rho(\rho+1)+1] g(u)$ for all $u>c_3$.
Thus, $(X_t)_{t\ge0}$ stays infinite for all $x>0$ by Lemma \ref{t2.1}(iii).
(iv)
Let $g\in C^2((0,\infty))$ be a bounded and strictly positive function
with $g(u)=1+(\ln u)^{-\rho}$ for $\rho>0$ and $u>3$.
It follows from the argument in (iii) that for $u>3$,
\beqnn
Lg(u)
\ar=\ar
\rho (\ln u)^{-\rho-1}\phi(u)
+\frac12\rho(\rho+1) (\ln u)^{-\rho-2}a_1(u)u^{-2} \cr
\ar\ar
+(\ln u)^{-\rho}a_2(u)\int_{U} K_\rho(u,z)\mu(\dd z)
+ (\ln u)^{-\rho}a_3(u)\int_{(0,\infty)\setminus U} K_\rho(u,z)\nu(\dd z) \cr
\ar\ge\ar
\rho (\ln u)^{-\rho-1}\phi(u)+(\rho\wedge1)(\ln u)^{-\rho-2} H_\rho(u).
\eeqnn
Then we can conclude the proof by the assumptions for this part together with Theorem \ref{t5.6}(ii) and Lemma \ref{t2.3}.
\qed
{\bf Acknowledgements.}
This work was supported by
NSFC (Nos.~11772002 and 11771018), Major research project for North Minzu University (No. ZDZX201902) and NSERC (RGPIN-2016-06704).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,321
|
package Codigos;
import java.text.SimpleDateFormat;
import java.util.Date;
public class tempoReal {
public String exibirData(){
//pega a data do pc
Date data = new Date();
//cria o formatador
SimpleDateFormat formatadorData = new SimpleDateFormat("dd/MM/yyyy");
//String para armazenar a data
String saveData = formatadorData.format(data);
return saveData;
}
public String exibirHora(){
//pegar a data para converter em hora
Date data = new Date();
//cria o formatador
SimpleDateFormat formatadorHora = new SimpleDateFormat("hh:mm:ss");
//String para armazenar a hora
String saveHora = formatadorHora.format(data);
return saveHora;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,750
|
Check Out Our Featured Software!
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,711
|
<section class="sidebar">
<h1 class="site-title">Bloc Chat</h1>
<button class="button-space" type="button" ng-click="home.open()">Add Room</button>
<div ng-repeat="room in home.rooms">
<a class="room-list" ng-click="home.setCurrentRoom(room)">{{ room.$value }}</a>
</div>
</section>
<section class="messages-container">
<h2 class="room-title">{{ home.currentRoom.$value }}</h2>
<div class="messages-list" ng-repeat="messages in home.messages">
<div class="username"><strong>{{ messages.username }}:</strong> {{ messages.content }}
</div>
<div class="sent-at">{{ messages.sentAt | date: "shortTime" }}</div>
</div>
</section>
<section ng-hide="!home.currentRoom" class="send-message">
<hr>
<input class="message-input" type="text" name="input" placeholder="Type your message here" class ng-model='home.newMessage.content'>
<button class="send-button" type="button" ng-click="home.sendMessage()">Send Message</button>
</section>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,209
|
UCT – Ufficio Controllo del Territorio (della Polizia di Stato)
UCT – codice aeroportuale IATA dell'aeroporto civile di Uchta (Russia)
UCT – University of Cape Town, Università di Città del Capo
UCT – Uomo, Città, Territorio | Periodico mensile cultura, ambiente e società del Trentino (Italia)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,925
|
Molí de Cal Pallot és un molí del municipi de Puig-reig (Berguedà) inclòs en l'Inventari del Patrimoni Arquitectònic de Catalunya.
Descripció
Les restes del molí fariner de Cal Pallot són visibles entre la brossa i l'arbrat, al peu de la riera de Merlès. Del molí només es conserva una part del casal, tot i que mig colgat per la brossa. Es tracta d'una construcció de planta quadrada que encara conserva part de la volta apuntada que el cobria i que el separava del pis superior del qual no se'n conserva cap vestigi.
Els murs són fets amb carreus de mides considerables, ben tallats i polits, i col·locats a trencajunt. La porta d'accés és a migdia i és formada per un arc amb grans dovelles que dibuixen mig punt. Aquest casal té, a la part superior, l'obertura per on el mecanisme accionava les moles (probablement colgades sota la runa) i movia el rodet.
Història
El lloc, juntament amb l'església i la casa, era conegut fins a l'època moderna amb el nom de Gamissans. Al segle X és documentat com un dels límits d'un important alou que fou cedit al Monestir de Santa Maria de Ripoll, i l'Església de Sant Andreu de Gamissans surt en el llistat de parròquies del bisbat d'Urgell de l'any 1312-1314. Del molí no es conserva cap notícia documental però per les restes trobades cal pensar que és una construcció de finals del o més probablement del .
Referències
Enllaços externs
Edificis de Puig-reig
Patrimoni monumental de Puig-reig
Pallot
Molins de la Riera de Merlès
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,913
|
Jaíba é um distrito do município baiano de Feira de Santana.
É um dos locais de Feira de Santana onde é disponibilizada o centro digital, local de acesso público à internet.
Atualmente está passando por uma ampliação na rede de esgoto, em uma associação da Embasa e a prefeitura. Oito mil metros de rede foram implantados para beneficiar 160 famílias do local com abastecimento de água potável.
Ligações externas
Jaiba
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,059
|
Q: disable and enable mouseenter and mouseleave can someone help me with this ? I have a mouseenter and mouseleave effect, and want to disable it when click on other div..
this is my code
$('.offer-content').on('mouseenter', function(){
$(this).find('.offer-desc').show()
}).on('mouseleave', function(){
$(this).find('.offer-desc').hide()
})
$(".st_sharethis").click(function(){
$flex(".offer-content").off('mouseleave');
});
what i want to happen is to enable the mouseenter and mouseleave again when i close the ".st_sharethis" class
thanks in advance!
A: You need to change the $flex to $
Below code will work for you:
$('.offer-content').on('mouseenter', function(){
$(this).find('.offer-desc').show()
}).on('mouseleave', function(){
$(this).find('.offer-desc').hide()
});
$(".st_sharethis").click(function(){
$(".offer-content").off('mouseleave');
});
You can also test it : http://jsfiddle.net/Qz2Rk/
A: I have created a fiddle with the required behavior. From your question it looks like you want to toggle the event behavior on the .st_sharethis click. Below code is working as expected.
Click on toggle event button and behavior with toggle with every click.
Fiddle link:
http://jsfiddle.net/hPAjF/
Updated version with better css:
http://jsfiddle.net/hPAjF/1/
HTML:
<div class="offer-content"><div class="offer-desc"></div></div>
<div class="st_sharethis">Toggle events</div>
CSS:
.offer-content{
width:200px;
height:200px;
background-color:#900;
cursor:pointer;
}
.offer-desc{
width:100px;
height:100px;
background-color:#090;
display:none;
}
.st_sharethis{
width:100px;
height:30px;
background-color:#009;
cursor:pointer;
}
JS:
$('.offer-content').on('mouseenter', function(){
$(this).find('.offer-desc').show()
}).on('mouseleave', function(){
$(this).find('.offer-desc').hide()
})
var mouseLeave = 0;
$(".st_sharethis").click(function(){
if(mouseLeave==0)
{
$(".offer-content").off('mouseleave');
mouseLeave=1;
}
else
{ $('.offer-content').on('mouseleave', function(){
$(this).find('.offer-desc').hide()
});
mouseLeave = 1;
}
});
In your website i have added following code from console and it has given the behavior you want.
jQuery(".stCloseNew2").click(function(){
jQuery('.offer-content').on('mouseleave', function(){
jQuery(this).find('.offer-desc').hide()
});
});
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,488
|
The Dear Hunter
All Is As All Should Be
by tef USER (16 Reviews)
Review Summary: Is it all as it should be?
Taking a break from the "Act" album series, Providence's The Dear Hunter release another EP. Examining this format extensively in the past already, this time around Crescenzo et al took ideas and themes for songs from a number of hardcore TDH fans that the band has been acquinted with for years (not limited to the realm of backstage beerfests and smelly touring buses), and made these into a short song cycle, "All Is As All Should Be".
With a total running time of 26 minutes and the longest track only just over 5 minutes long, it stands in stark contrast to their last two albums.
Musically however, the album is quite similar to the last two "Act" albums (IV en V). Comfortably moving between styles, the music as always is characterized most by Crescenzo's amazing voice and the colorful instrumentation. Most of the progressive rock meanderings are left out this time around with opener "The Right Wrong" being the hardest rocking track here, and follow up "Blame Paradise" being the only track really exemplifying TDH's prog background. With its angular synth motifs and weird verse vocal melody it takes some time to get used to but in the end proves to be one of the best tracks on here.
"Beyond The Pale" and "Shake Me (Awake)", seem to be two parts of one long epic song that were later cut up in two separate songs for the sake of the EP format. The former being a quiet, not very uplifting (and maybe a tad boring) track, the latter takes all of TDH's pop rock sentiment and pours it into a magnificent earworm. It has a typical TDH happy/melancholy melody with evenly twofold lyrics about the longing for a more meaningful existence on the one hand and a seize-the-day sentiment on the other.
With an album title suggesting a sense of comfort at a given time, the song lyrics are contradictory on more occasions than mentioned above; Lyrics like:
"…the valley between who I am,
And who I want you to see,
Is too vast to bypass…."
"..Will no one witness me"
Do I exist, or did I fall into darkness"..."
(from "Witness Me")
"…Come shake me awake,
And pull me far away,
From the endless circles I've been running in,
I've had about all I can take,
So take me far away,
To a life that's far less ordinary…"
(from "Shake me (Awake)"
suggest to the listener that the protagonist might be somewhat uncomfortable in his current state.
On the other hand there's lines like:
"…despite all my skeletons,
I see, all is as all should be…"
(from "The Right Wrong")
"…but at the end when you're looking out through my eyes,
You'll see that all is as all should be.."
(from "All Is As All Should Be")
that suggest a certain degree of resignation in the protagonist. This duality is certainly felt throughout the album and not in a god way. It feels uplifting at times and meaningful and deep at others. The theme that's supposed to tie the EP together works a bit counterproductive and unfortunately acts more as a divisive force than a binding force here.
All this doesn't take away from the fact that this is one very enjoyable listen with a couple of standout moments and very enjoyable musicianship throughout. The EP as a whole however falls into the mediocre category of TDH's output. Given the super high quality of the rest of their discography, this "mediocre" EP is still way better than most albums released this year and thus will probably make it to my end of the year list for 2017.
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BlackwaterPork
'Bout time dis got a review
insomniac15
I haven't managed to get into this as much as I wanted, don't know why. I have to listen to it more maybe.
Digging: Elder (USA-MA) - The Gold & Silver Sessions
"hardcore TDH fans that the band has been acquinted with for years"
Acquainted*
nah don't worry insomniac its about as good as a pet goose
furpa
Fair review but I'm not sure you totally understood the concept of the EP. You mention conflicting sentiments in the lyrics, but the lyrical content of each song was prompted to Casey from each different person he contacted, thus each song remains completely unrelated to every other one. The project wasn't meant to act as a single cohesive release (like the Acts and most of TDH's other work). Every song is very much intentionally different from the last, both tonally and thematically.
The title comes back literally in two of the six tracks. Forgive me for thinking that's not coincidental ImO this is not just six songs without any common ground.
As I understood basic song ideas were prompted by close fans, the lyrics seem more Casey's own work... they do read like personal lyrics to me.
sempiturtle
Album Rating: 4.5 | Sound Off
The fans did indeed provide Casey with the lyrical theme of each song, but no actual lyrics.
Also this isn't musically similar to the last two Acts at all, most obviously due to the complete lack of orchestration that was the focal point of those albums' instrumentation.
I don't hear any drastic change in sound between this and the previous two albums; maybe it's me, maybe not...I'm going for the latter
Except I just gave a reason as to why it's objectively different from the previous albums lmao
SowingSeason
I'm glad someone reviewed this, have a pos. I'm not certain as to how much input the fans had into the lyrics if any (probably none), but I love the concept of taking fan experiences/ideas and creating unique songs based on their requests. The songs were just alright for me, the only one I keep coming back to is Blame Paradise. Still, it's TDH, so there's no bad songs.
January 4th 2018
Solid EP and I cannot wait for the next Act!
Digging: Mol - JORD
@sempiturtle He's just saying it sounds similar overall. Just because strings are gone doesn't mean it doesn't have the same bright and colorful rock sound.
thanks drifter
neekafat
Contributing Reviewer
How much does this sound like their other stuff? Because it's good but pretty generic pop-rock
DinosaurJones
It's all right. Probably my least favorite of their work, tbh tho.
you just gave it a 4.5...
Yeah, I just dropped my rating as I gave it another listen and realized the hype was a bit high for me.
I'm an unabashed TDH fanboy though.
So I should expect better things if I delve deeper is what you're saying?
Lmao me and Dino just dropped that avg
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: VBA using countifs error and dynamic ranges I have the following set of mock-data in A4:B27, with row 4 holding the headers..
4. Entity Country
5. 12 countryb
6. 13 dave
7. 14 dan
8. 15 john
9. 16 james
10. 17 josh
11. 18 george
12. 19 geni
13. 20 gina
14. 10 countrya
15. 10 countrya
16. 11 country
17. 12 countryb
18. 12 countryb
19. 13 brian
20. 14 ryan
21. 15 louis
22. 16 tom
23. 17 chris
24. 18 mad
25. 19 barb
26. 20 james
27. 10 countrya
In VBA I want to ensure there are no duplicate entity-country combinations. This would be easily seen in a worksheet with the formula "=COUNTIFS($A$5:$A$27,A5,$B$5:$B$27,B5)". If the returned value is greater than one, I would want to highlight the entity-country cells to show a duplicate. In the above example rows 5, 14, 15, 17, 18, and 27 would be highlighted.
However after trying to create the VBA I'm stuck..
Sub test()
Dim cSheet As Worksheet
Set cSheet = Sheets("CL.AL1")
Dim trolSheet As Worksheet
Set trolSheet = Sheets("Control Sheet")
Dim currentRow As Integer, lastRow As Integer, currentColumn As Long
Dim listA As range, listB As range, cellA As String, cellB As String
cSheet.Select
currentColumn = 1
currentRow = 5
lastRow = ActiveSheet.Cells(Rows.Count, "A").End(xlUp).Row
Set listA = range(Col_Letter(currentColumn) & currentRow & ":" & Col_Letter(currentColumn + 1) & lastRow)
Set listB = range(Col_Letter(currentColumn + 1) & currentRow & ":" & Col_Letter(currentColumn + 1) & lastRow)
Do While range("A" & currentRow) <> ""
cellA = (cSheet.range(Col_Letter(currentColumn) & currentRow).Value)
cellB = (cSheet.range(Col_Letter(currentColumn + 1) & currentRow).Value)
If WorksheetFunction.CountIfs(listA, cellA, listB, cellB) > 1 Then
Union(range(Col_Letter(currentColumn) & currentRow), _
range(Col_Letter(currentColumn + 1) & currentRow)).Select
With Selection.Interior
.Pattern = xlSolid
.PatternColorIndex = xlAutomatic
.Color = 49407
.TintAndShade = 0
.PatternTintAndShade = 0
End With
End If
If currentRow = lastRow Then
currentRow = 5
currentColumn = currentColumn + 1
If currentColumn = 3 Then
Exit Do
End If
Else
currentRow = currentRow + 1
End If
Loop
Debug.Print (range(Col_Letter(currentColumn) & currentRow).Value)
Debug.Print (range(Col_Letter(currentColumn + 1) & currentRow).Value)
End Sub
Function Col_Letter(lngCol As Long) As String
Dim vArr
vArr = Split(Cells(1, lngCol).Address(True, False), "$")
Col_Letter = vArr(0)
End Function
After executing the current VBA I receive a runtime 1004 error "unable to get the countifs property of the worksheet function class".
So. Can anyone help correct this error OR offer an alternative solution?
Thanks in advance.
A: Put this in as the formula for conditional formatting on cell A4 and then copy the formatting to the remainder of the column.
=COUNTIFS($A$4:$B$27,A4)>1
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"redpajama_set_name": "RedPajamaStackExchange"
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\title{The String Landscape, the Swampland, and the Missing Corner}
\author[a]{T. Daniel Brennan}
\author[b]{Federico Carta\footnote{La Caixa-Severo Ochoa Scholar}}
\author[c]{Cumrun Vafa}
\affiliation[a]{NHETC and
Department of Physics and Astronomy, Rutgers University \\
126 Frelinghuysen Rd., Piscataway NJ 08855, USA}
\affiliation[b]{Instituto de F\'isica Te\'orica UAM-CSIC,\\
Universidad Aut\'onoma de Madrid, Cantoblanco, 28049 Madrid, Spain}
\affiliation[c]{Jefferson Physical Laboratory, Harvard University\\
Cambridge, MA 02138, USA}
\abstract{ We give a brief overview of the string landscape and techniques used to construct string compactifications. We then explain how this motivates the notion of the swampland and review a number of conjectures that attempt to characterize theories in the swampland.
We also compare holography in the context of superstrings with the similar, but much simpler case of topological string theory. For topological strings, there is a direct definition of topological gravity based on a sum over a ``quantum gravitational foam.'' In this context, holography is the statement of an identification between a gravity and gauge theory, both of which are defined independently of one another. This points to a missing corner in string dualities which suggests the search for a direct definition of quantum theory of gravity rather than relying on its strongly coupled holographic dual as an adequate substitute (Based on TASI 2017 lectures given by C. Vafa). }
\preprint{IFT-UAM/CSIC-17-105}
\begin{document}
\maketitle
These lecture notes from TASI 2017 give a brief overview of some of the open problems in string theory. We will be generally motivated by the philosophy that string theory is ultimately supposed to describe the fundamental laws of our universe. String theory is so versatile that it can be used to study a wide array of physical problems such as various topics in condensed matter and quark-gluon plasma or aspects of quantum fields theories in diverse dimensions. Much of the recent work using string theory has been focused on using its properties to solve specific problems rather than developing our understanding of string theory as a fundamental description of our universe. Here we aim to discuss topics which we hope will be useful in bringing string theory closer to observable aspects of fundamental physics.
With this philosophy in mind, we will begin these lectures by reviewing some of what we know about string theory and its possible application to the universe by describing some generalities about the space of low energy theories theories coming from string theory compactifications: this is called the ``string landscape.'' Supersymmetry plays a key organizing principle in this context.
This will naturally lead us to investigate the question of how we know a priori if a low energy theory is in the landscape or it is not. The set of low energy physics models which look consistent but ultimately are not when coupled to gravity, is called the ``swampland.'' Finding simple criteria to distinguish the swampland from the landscape is of great importance. In particular such criteria can lead to concrete predictions for our universe as we will discuss later.
We review a number of conjectures which are aimed at distinguishing the swampland from the landscape.
The string landscape and the swampland will be the topic of the first two lectures. In the third lecture, which is on a somewhat disjoint topic, we review critically where we are in our current understanding of quantum gravity. In this lecture we use the toy example of topological string theory, for which much more is known,
to shed a new light on what shortcomings we have in our current formulation of quantum gravity from string theory. In topological string theory the gravitational theory can be formulated in terms of a non-commutative $U(1)$ gauge theory whose configurations can be interpreted as defining a quantum gravitational foam. This is holographically dual to a Chern-Simons theory. The equivalence of these two theories is the content of holography for topological strings.
In the case of full string theory, a direct definition of quantum gravity is missing and holography is viewed as a substitute definition. The analogy with topological strings suggests that there is a missing corner in string dualities and that we should try to find a direct definition of quantum gravity.
\section{Lecture 1: The String Landscape}
\label{sec:Lec1}
\subsection{Review of String Theories}
In the late 1980's it was realized that there were five perturbatively consistent, distinct ten-dimensional string theories. Early efforts to find the four-dimensional standard model with gravity as a low energy limit of string theory focused on compactifying those theories on various manifolds. Initially, the Heterotic string theories were considered the most promising theories to produce the standard model because of their inherent non-abelian gauge symmetrie
\footnote{Recall that D-branes were discovered only in the late 80's - early '90s, and before this moment it was largely unknown how to realize gauge groups in Type II theories \cite{Dai:1989ua}.}. In mid 1990's, it was realized that these five theories were all related by different dualities and non-perturbative completions, thus launching the ``duality era'' \cite{Witten:1995ex}.
The five original types of superstring theories are Type I, Type IIA, Type IIB, Heterotic $E8\times E8$ (HE), and Heterotic $SO(32)$ (HO)\footnote{The gauge group is actually $Spin(32)/{\mathbb{Z}}_2$, but it is convention to say $SO(32)$ since they have isomorphic Lie algebras \cite{Gross:1984Dd,Gross:1985fr,Gross:1985rr}.}. There is also a conjectured 11D theory called M-theory which is not a string theory and has a low energy effective action given by 11D ${\cal N}=1$ supergravity. This theory is highest dimensional supergravity theory. The Type IIA has a ${\cal N}=(1,1)$ supersymmetry and thus is non-chiral. Type IIB has ${\cal N}=(2,0)$ and is thus chiral. Heterotic and Type I strings are both chiral carrying ${\cal N}=(1,0)$ supersymmetry\footnote{${\cal N}=(p,q)$ indicates that there are $p-$SUSY operators in the left-handed spin representation of the d-dimensional Lorentz group and $q-$SUSY operators in the right-handed spin representation. More precisely, in even dimensions, the spin representations of the Lorentz group are distinguised by chirality which gives the left- and right-handed spin representations. However, in 4D and 8D, these representations are related by complex conjugation and hence are identical so that we only have this chiral notation for 2D, 6D, and 10D. }.
The duality web suggests that all these theories are related by dualities. Let us denote $k$-dimensional Minkowski space by $M_k$. The dualities are generated by the following equivalences (also see Figure \ref{fig:dualities}):
\begin{itemize}
\item Type I is related to HO by S-duality (inversion of coupling constant),
\item HO is related to HE on $M_9\times S^1$ with Wilson lines turned on $S^1$ \cite{Narain:1986am} ,
\item Type IIA on $M_{10}$ is equivalent to M-theory on $M_{10}\times S^1$ where the radius $R$ of $S^1$ is related to Type IIA coupling constant by $R^3=\lambda^2$,
\item Type IIA on $M_9\times S^1_R$ is related to Type IIB on $M_9\times S^1_{R'}$ by T-duality where $R'=\ell_s^2/R$ where $\ell_s$ is the string length,
\item M-theory on $M_{10}\times S^1/{\mathbb{Z}}_2$ is related to HE on $M_{10}$. Physically, M-theory on this interval (whose length is related to Heterotic string coupling constant) has M9-branes on each end of the interval each of which carry an $E_8$ gauge group which together give rise to the 10D HE theory \cite{Horava:1996ma,Moore:2000fs}.
\end{itemize}
\tikzstyle{block} = [rectangle, draw, fill=blue!20,
text centered, rounded corners, minimum height=1.5em]
\tikzstyle{line} = [draw, -latex']
\tikzstyle{cloud} = [draw, ellipse,fill=red!20, node distance=6cm,
minimum height=2em]
\begin{figure}
\begin{tikzpicture}[node distance = 3cm, auto,<->]
\node [block] (mtheory) {M-Theory};
\node (IIA) [block,below left of = mtheory,yshift=+0.5cm] {Type IIA};
\node (IIB) [block,left of =IIA] {Type IIB};
\node (F) [block,above of= IIB] {F-Theory};
\node (HE) [block,below right of=mtheory,yshift=+0.5cm] {Heterotic $E8\times E8$};
\node (HO) [block,right of=HE,xshift=+1cm] {Heterotic $SO(32)$};
\node (I) [block, right of=HO] {Type I};
\draw[<->] (mtheory) -- (IIA) node[midway,sloped,below]{
$S^1_{g_s}$};
\draw[<->] (IIA) -- (IIB) node[midway,sloped,above]{T-dual};
\draw[<->] (IIB) -- (F) node[midway,sloped,below]{$T^2_\tau$};
\draw[<->] (mtheory) -- (HE) node[midway,sloped,above]{$S^1/{\mathbb{Z}}_2$};
\path (HE) edge[bend right] node [below] {$S^1$ twist w/ Wilson} (HO);
\draw[<->] (HO) -- (I) node[midway,sloped,above]{S-dual};
\end{tikzpicture}
\caption{This graph shows the duality web relating all of the different descriptions of string theory. Here the vertical placement corresponds to dimension. }
\label{fig:dualities}
\end{figure}
While M-theory appears to be the most central in web of dualities between all of the different theories, in some sense, it doesn't see Type IIB string theory, even though they are clearly dual. Specifically if we compactify M-theory on $T^2$ and shrink it to zero size, we would expect to get a 9D effective theory. However, string duality tells us that we actually get Type IIB which has 10D Lorentz symmetry. This indicates that M-theory is perhaps not a very good way to describe Type IIB.
Let us now consider Type IIB string theory. It has a complex coupling constant, $\tau$ which is $SL(2,{\mathbb{Z}})$ equivariant since Type IIB is self-dual under S-duality. Because of this property, we can think of $\tau$ as the complex structure of a compactification torus of some 12D theory. This 12D theory is what is referred to as F-theory \cite{Vafa:1996xn}.
See \cite{Denef:2008wq,Heckman:2010bq} for a review of some aspects of F-theory.
\subsection{String Compactification}
We can now try to produce lower dimensional theories and in particular a four dimensional theory that describes our universe by compactifying one of these theories to lower dimensions. By this we mean that we will take a spacetime of the form $M_d\times K$ where $K$ is some compact manifold with appropriate dimension with some characteristic size $R_K$. Then we will take the limit of $\ell_s<<R_K$ (or $l_{Planck}<<R_K$ for M-theory) and consider the low energy dynamics with energy $E<<1/R_K$, leading to an effective theory on $M_d$. Note that we do not take the limit of $R_K\to 0$ as this will lead to the introduction of light matter coming from the winding states which typically leads to a higher dimensional dual description.
Since there are so many (possibly infinitely many) manifolds we can compactify on to get lower dimensional physics, it is clear that we need some kind of organizing principle for these theories. As physicists we generally try to use symmetries to characterize different systems. As such, we will use Lorentz symmetries again for this purpose. For the time being, we will take all directions which are not part of the compact direction to be $M_d={\mathbb{R}}^{1,d-1}$ which has $d$-dimensional Lorentz symmetry.
Additionally, as it is useful in classifying the string theories, it also makes sense to classify our theories by the amount of supersymmetry. For our purposes rather than using the number of supersymmetries ${\cal N}$ which is adapted for each dimension, we will make use of the notation $N_{SUSY}$ which counts the total number of conserved supercharges, as this will not change upon flat toroidal compactifications. As with the 10D string theories, we will also classify by the chirality of the supersymmetries. For example, Type IIA string theory has $N_{SUSY}=32$ charges and is non-chiral since it has ${\cal N}=(1,1)$ SUSY whereas Type IIB string theory has $N_{SUSY}=32$ and is chiral since it has ${\cal N}=(2,0)$. The relation between the two notations is that in even dimensional spacetime, a theory with ${\cal N}=(p,q)$ has $N_{SUSY}=16(p+q)$, $N_{SUSY}=8(p+q)$, or $N_{SUSY}=(p+q)$ conserved supercharges in 10D, 6D, or 2D respectively. Apart from the chirality information in these dimensions, $N_{SUSY}$ completely captures the structure of the super Lorentz group.
Thus far, we have been implicitly assuming that we are compactifying with compact manifolds. However, even though the name and procedure of compactification requires a compact manifol
, there has actually been a significant amount of work on ``compactifying on non-compact manifolds.'' This means that we can also study string theory on $M_d\times K$
where the volume of $K$ is infinite and the theory has local normalizable modes in $M_d$ which are decoupled from gravity.
This is based on the way that the gravity appears in the low energy effective action for the ``compactified'' theory.
Consider the background to be $M_d\times K$ where $K$ is our compactification manifold. To leading order the gravitational part of the action on $M_d$ will be of the form
\bar{e}
S\sim Vol(K)~\int_{M_d}d^d x\sqrt{-g}~R_d~,
\end{equation}
where $R_d$ is the $d$-dimensional Ricci scalar. Comparing to the Einstein-Hilbert action
\bar{e}
S_{EH}=\frac{1}{16\pi G_N}\int_{{\mathbb{R}}^d}d^dx \sqrt{-g}~R_d=M_{pl}^{d-2}\int_{{\mathbb{R}}^d}d^dx\sqrt{-g}~R_d~,
\end{equation}
means that $Vol(K)$ should be identified with $M_{pl}^{d-2}$. Therefore in the limit\footnote{Here we assume that we are compactifying to $d>2$.} $Vol(K)\to \infty$, $M_{pl}\to \infty$ and hence $G_N\to 0$, decoupling gravity\footnote{More generally, the procedure to decouple from gravity is more involved than just taking $Vol(K)\to \infty$. However, we will ignore this subtlety for simplicity. In general, we cannot always decouple a theory from gravity. For example when the compactified theory is not asymptotically free. It is sufficient for a theory to be asymptotically free/conformal to be able to decouple from gravity. See \cite{Seiberg:1996vs} for some additional details of decoupling gravity.}. However depending on the structure of $K$ we may get normalizable modes of some of the fields with a finite kinetic term in $d$-dimensions. These can in principle lead to interesting interacting quantum systems in $d$-dimensions which are not coupled to gravity. From this, it is clear that studying non-compact ``compactifications'' is useful for studying QFTs that are not coupled to gravity.
Non-compact backgrounds are also interesting for the purpose of studying holography. Consider string theory in $AdS$ space. This has negative curvature that requires background fluxes to be stable. In general this requires string theory to be on a space $AdS_d\times K$ where $K$ is an appropriate dimensional manifold with positive curvature to balance the total curvature in Einstein's equation. Holography tells us that sometimes we can relate non-compact compactifications to a true compactification with $AdS$ spacetime. For example, consider Type IIB with a stack of $N$ D3-branes in ${\mathbb{R}}^{1,9}$. Holography tells us that this is dual to $AdS_5\times S^5$ supported by $N$ units of flux. From the non-compact geometry we have ended up with a compact one! This happens by ``zooming'' in to the near horizon limit in which the transverse direction to the D3-branes looks like ${\mathbb{R}}_+\times S^5$. In a sense the D3-brane worldvolume direction, ${\mathbb{R}}^{1,3}$, absorbs the non-compactness of the transverse ${\mathbb{R}}_+$ direction to become $AdS_5$, resulting in a compact geometry with flux. This is the statement of holography - that branes' interaction with gravity leaves an imprint on space which leads to an equivalence between a QFT living on the branes and its gravitational imprint where the brane has been replaced by flux
\cite{Maldacena:1997re}.
The distinguished role of $AdS$ space in holography begs the question if de Sitter spacetime also has similar properties. However these spaces may be impossible to obtain in string theory as we will discuss in Section \ref{sec:4D}. For the rest of the discussion here we will focus on compactification to Minkowski background.
Thus far, the only known stable compactifications to Minkowski space require some amount of preserved supersymmetry.
Therefore, an important part of understanding what theories are produced by compactifying string theory is understanding what conditions are sufficient and necessary to have supersymmetry for the low energy theory. This is determined by the number of covariantly constant spinors on the compactification manifold. These are globally defined spinors which are invariant under translation along the compact manifold - they are intuitively spinors that are independent
of their position on the internal manifold. In many cases the statement of existence of covariantly constant spinors can be translated into a statement about the holonomy of the tangent bundle.
This can be described as follows. Pick a tangent vector at any point on the (oriented) manifold $K$. As we parallel transport this vector along any closed path in the manifold, the vector is ``pushed around'' so that when it returns to the basepoint, the change of the vector can be described by the action of an element of $SO(n)$ for a $n$-dimensional manifold. Generically this action lifts to the spin bundle by spin representations induced from the lift of $SO(n)$ to $Spin(n)$. However, there are special cases when this group is reduced to a subgroup\footnote{Note in order to have the reduction of the holonomy group, we must have that this holds for all choices of basepoint.} $G\subset SO(n)$. In this case, we can find an element of the spin bundle which is fixed by the lift of the action of $G$ and hence can be extended to all of $K$ by geodesic translation.
A special class of manifold which has a guaranteed reduction of structure group are Calabi-Yau manifolds. These are K\"ahler, Ricci flat, complex manifold of real dimension $2n$ whose holonomy is given by $SU(n)\subset SO(2n)$. As we will see, these will provide an important class of internal manifold for compactification.
Some general classes of compactification manifolds that are useful for string compactifications to 4D which preserve some amount of SUSY are:
\begin{itemize}
\item $T^n$ - the n-torus. This preserves full SUSY since the holonomy group is trivial.
\item $K3$ surface. This is the first non-trivial Calabi-Yau manifold which is a complex 2-fold (4-real dimensional). In this case the holonomy is reduced from $SO(4)\to SU(2)$. So there is a trivial direction in the spinor bundle and hence a covariantly constant spinor. This preserves $\frac{1}{2}$-SUSY.
\item Calabi-Yau 3-manifolds ($CY^3$) are complex 3-folds with SU(3) holonomy. These preserve $\frac{1}{4}$-SUSY.
\item Calabi-Yau 4-manifolds (CY4) are complex 4-folds dimensional with SU(4) holonomy. These preserve $\frac{1}{8}$-SUSY.
\item $G2$-manifolds are 7-real dimensional manifolds with $G2\subset SO(7)$ holonomy. These preserve $\frac{1}{8}$-SUSY.
\item $Spin(7)$-manifolds are 8-real dimensional with $Spin(7)\subset Spin(8)\sim SO(8)$ holonomy. These preserve $\frac{1}{16}$-SUSY.
\end{itemize}
\subsection{Dualities of Compactified Theories}
Since we have seen that the full string theories are all interrelated by a sequence of dualities, one would expect that their compactifications are also related by dualities. As it turns out, these relations are so abundant that we can make the following observation: \\
\textbf{``Conjecture'':} Whenever the dimension, number of preserved supercharges, and chiralities of two different compactifications of string theory match, there are choices of compactification geometries such that they are dual descriptions of the same physical theory. \\
Surprisingly, we are aware of no known counter examples. In this sense, dualities in lower dimensional theories are not hard to find, but rather are hard to prevent! One rationale for the existence of dualities is as Sergio Cecotti puts it, ``the scarcity of rich structures''. In particular the very existence of quantum systems of gravity is hard to arrange and if we succeed to get more than one theory with a given symmetry, there is a good chance we have landed on the same theory.
We will now briefly review some of interconnected web of string dualities with definite $N_{SUSY}$ in various dimensions. We will not be exhaustive in the discussion below, but aim to illustrate some key examples.
\subsubsection{Adiabatic Principle}
A general principle which helps us in identifying dualities upon compactification is to use known dualities in higher dimensions to build new ones in lower dimensions. This is based by use of a sort of ``adiabatic procedure" \cite{Vafa:1995gm} where we step-by-step reduce the dimension by acting on both sides of a given duality by a geometric procedure such as compactification. However, in most applications the fibration data does not vary adiabatically. Therefore, there is no rigorous reason why this should work since it is not really adiabatic in any sense of the word as things change drastically at every step. However, amazingly it does work in all known examples as long as some supersymmetry is preserved.
\subsubsection{$N_{SUSY}=32$}
Let us start with $N_{SUSY}=32$. M-theory in 11 dimensions, F-theory in 12 dimensions, and Type IIA and Type IIB and all their toroidal compactifications to lower dimensions have $N_{SUSY}=32$.
Moreover, except for Type IIB in 10 dimensions whose chirality is ${\cal N}=(2,0)$ and F-theory, all these theories are non-chiral and so one would expect based on our general conjecture that they are all equivalent once compactified to the same dimension. And further, we expect (correctly) that both the chiral and non-chiral theories to be equivalent when toroidally compactified to the same dimension.
\subsubsection{$N_{SUSY}=16$}
Consider the case of $N_{SUSY}=16$.
This includes Heterotic and Type I theories in 10D as well as M-theory on $S^1/{\mathbb{Z}}_2$. We have already discussed dualities among these, so let us move on to further dualities with this much supersymmetry.
We will start by compactifying F-theory. Recall that in order to make sense of F-theory we need a torus embedded in our compactification manifold - that is we need at least some kind of torus fibration. We will begin by compactifying F-theory on an elliptic K3-manifold. This may be written as a torus fibration over $\mathbb{P}^1$
\bar{e}
\vcenter{\vbox{\xymatrix{T^2\ar[r]&K3\ar[d]\\&\mathbb{P}^1
}}}\qquad K3=\{y^2=x^3+f(z)x+g(z)~|~z\in \mathbb{P}^1\}~.
\end{equation}
Note that this reduces the $N_{SUSY}=32$ of F-theory to an 8d, $N_{SUSY}=16$ theory. The Heterotic and Type I theories also have $N_{SUSY}=16$, but they are 10D theories. So if we reduce them on a $T^2$, we get an 8d theory with $N_{SUSY}=16$. As it turns out, these theories are dual to each other \cite{Vafa:1996xn}.
One can check aspects of this duality by studying singularity structure of the K3 manifold. The key to the duality is that the singularities of K3 manifold, which are points where the $T^2$ fiber degenerates\footnote{These singularities have a classification given by Kodaira, which mirrors the more well known one for an ADE classification of $K3$ singularities. See Section \ref{sec:McKay} for a discussion of ADE singularities.}, which are interpreted in Type IIB setup as giving rise to a system of $(p,q)$ 7-branes which can lead to non-abelian gauge symmetry as expected for Heterotic or Type I toroidal compactifications. It turns out that compactification on an elliptic K3 manifold leads to singularities associated to a Lie algebra $\mathfrak{g}$ such that $rnk ~\mathfrak{g}\leq20$ and in paritcular we cannot obtain arbitrarily large rank non-abelian gauge groups in this way.
This is compatible with the Heterotic and Type I compactifitions on $T^2$ which leads to a rank 16 plus an additional 4 coming from winding and momentum charges on each circle but we can have gauge symmetry enhancement at certain points in the moduli space which give rise to (semi-simple) factors of non-abelian groups with up to $rnk~\mathfrak{g}=18$ gauge groups \cite{Aspinwall:1995zi}\footnote{As it turns out this is a special case of a more general string duality which relates Heterotic string theory on a complex $n-1$ dimensional elliptically fibered Calabi-Yau $\pi_{H}:Z\to B$ to F-theory compactified on an complex $n$-dimensional Calabi-Yau $\pi_F:X\to B$ with elliptic K3-fibers over the same base. The physical motivation for this is the same as compactification on the K3 fibers again give rise to a system of 7-branes which has non-abelian gauge symmetry matching that of the Heterotic string compactified on the $T^2$ fibers with Wilson lines. These can further be shown to have the same moduli and low energy spectrum. See \cite{Donagi:1998vw} for more details.}.
Additionally, there is an important family of dualities relating F-theory and M-theory. Recall that F-theory and M-theory have the same $N_{SUSY}=32$ and hence
F-theory on $T^2\times S^1$ is equivalent to M-theory on $T^2
. If we compactify F-theory on $K^{ell}\rtimes S^1$, it will have the same dimension and $N_{SUSY}$ as M-theory on $K^{ell}$ where $K^{ell}$ is an elliptically fibered manifold. Therefore, we expect these theories to be dual. This can be derived by the adiabatic principle, where we adiabatically fiber the $T^2$ factor on both sides of the 9D duality over a base manifold to construct $K^{ell}$.
Similarly as we discussed M-theory on $S^1$ is equivalent to Type IIA. This gives rise to the chain of dualities \cite{Vafa:1996xn}:
\begin{itemize}
\item F-theory on $K^{ell}\times S^1$ is dual to M-theory on $K^{ell}$
\item F-theory on $K^{ell}\times S^1\times S^1$ is dual to M-theory on $K^{ell}\times S^1$ and to Type IIA on $K^{ell}$
\end{itemize}
If we apply this to the case where $K^{ell}=K3$, then we find that F-theory on $K3\times S^1$, which is conjectured to be the same as Heterotic on $T^2\times S^1=T^3$, is dual to M-theory on $K3$ \cite{Witten:1995ex}. Moreover, going one dimension down on another $S^1$ shows that Type IIA on $K3$ is dual to is dual to Heterotic on $T^4$. Generically in the compactification of Heterotic string theory on $T^4$ includes non-trivial Wilson lines along the different $S^1$ factors in the $T^4$ which break gauge symmetry. But by turning these off, our compactified theory will exhibit non-abelian gauge symmetry. This is reflected in the Type IIA side by taking the limit where non-trivial 2-cycles in the $K3$ manifold become coincident and develop orbifold singularities as above. We will comment more on this in the next section.
\subsubsection{$N_{SUSY}=8$}
Now consider the case of $N_{SUSY}=8$. This can be achieved by compactifying Type I or Heterotic string theory on a K3 manifold and Type IIA, IIB, M-theory and F-theory on a $CY^3$-fold. Now that we are considering non-trivial compactifications of Heterotic string theory we have to account for the interplay between the non-trivial geometry and gauge field fluxes.
In Heterotic string theory there is a 2-form tensor field $B$ whose field strength is given by $H=dB$. The bosonic part of the low energy effective action of the Heterotic string (Heterotic supergravity) is given by
\begin{align}\begin{split}
S=&\frac{1}{2\kappa_{10}^2}\int d^{10}x \sqrt{-g}e^{-2\Phi}\left(R+4(\nabla \Phi)^2-\frac{1}{2} |\tilde{H}_3|^2\right)\\
&-\frac{1}{2 g_{10}^2}\int d^{10}x\sqrt{-g}e^{-2\Phi}\text{ Tr}|F_2|^2~,
\end{split}\end{align}
where
\begin{align}\begin{split}
&\tilde{H}_3=dB-\frac{1}{4}(\Omega_{YM}-\Omega_{GR})~,\\
\Omega_{YM}=\text{Tr }A\wedge dA-\frac{2i}{3}&A\wedge A\wedge A\quad,\qquad \Omega_{GR}=\text{Tr }\omega \wedge \omega+\frac{2}{3}\omega \wedge \omega \wedge \omega~,
\end{split}\end{align}
where $A$ is the 1-form gauge field and $\omega$ is the spin connection. This leads to the equations of motion
\bar{e}
dH_3=\frac{1}{2}\left(\text{Tr }R\wedge R-\text{Tr }F\wedge F\right)\quad,\qquad R=d\omega+\omega \wedge \omega~.
\end{equation}
Compactifying F-theory on an elliptic $CY^3$-fold gives a six-dimensional theory with $N_{SUSY}=8$. This matches the SUSY and dimensions of Heterotic string theory on $K3$. By the conjecture above, we expect these to be dual. Consider a compactification with only trivial $H$-flux, that is the case without five-branes (sources of the B-field). This imposes the condition
\bar{e}
Tr(R\wedge R)-Tr(F\wedge F)=c_2(TM)-c_2(F)=0~,
\end{equation}
on the gauge bundle, where $c_2(F)$ represents the second Chern class. This means that if we compactify on a non-trivial manifold, we must have instantons living in the gauge bundle.
We will now consider the extended example of Heterotic string theory on $K3$. Before we derive the duality with F-theory, we will demonstrate some physical features of this compactification. This manifold has the feature that $c_2(TK3)=24$. This means that if there are no five-branes, there must be 24 instantons living in the gauge bundle. Since the structure group of the gauge bundle is $E8\times E8$, we can have these 24 instantons divided between the two factors $E8_1\times E8_2$ as $k_1,k_2$ where $k_1+k_2=24$ \cite{Kachru:1995wm}.
Now consider the case where $[dH]\neq 0$ - that is the case of compactification with five-branes. This case is actually more clear from the Ho\u{r}ava-Witten construction by taking M-theory on ${\mathbb{R}}^{1,5}\times K3\times S^1/{\mathbb{Z}}_2$. At the ends of the $S^1/{\mathbb{Z}}_2$ interval, there are M9-branes - each of which support an E8 gauge group. Instantons in the M9-branes are described by dissolved M5-branes. These instantons in the gauge theory can shrink to zero size in the $K3\times S^1/{\mathbb{Z}}_2$ direction and eject from the M9-brane as M5-branes into the bulk of the $S^1/{\mathbb{Z}}_2$ (whose worldvolume is transverse to the $K3\times S^1/{\mathbb{Z}}_2$) and can also be absorbed by the other wall and dissolved in it. The point is that there must be a total of 24 M5-branes for this compactification to be stable spread out between the two M9-branes and the bulk \cite{Witten:1995gx}.
Now using the duality conjecture, we know that the compactification of Heterotic string theory on $K3$ is a 6D theory with $N_{SUSY}=8$. This can also be achieved by considering F-theory on an elliptically fibered $CY^3$ manifold. As we will see this can be derived from the previous subsection by applying the `adiabatic argument'.
Let us start with the duality of F-theory on K3 and Heterotic string theory on $T^2$. Recall that $K3$ can be written as an elliptic fibration
\bar{e}
\xymatrix{T^2\ar[r]&K3\ar[d]\\&\mathbb{P}^1}~.
\end{equation}
Now if we take both sides of the duality and fiber them over a 2-sphere $\mathbb{P}^1$, then we get
\begin{equation}
\text{F-theory on }
\vcenter{\vbox{
\xymatrix{K3\ar[r]&CY^3\ar[d]\\&\mathbb{P}^1}}}\cong \vcenter{\vbox{\xymatrix{T^2\ar[r]&CY^3\ar[d]\\&\mathbb{P}^1\times \mathbb{P}^1}}}\Longleftrightarrow\quad \text{Heterotic on }\vcenter{\vbox{\xymatrix{T^2\ar[r]&K3\ar[d]\\&\mathbb{P}^1}
}}~.
\end{equation}
Here $CY^3$ is a Calabi-Yau 3-fold which is $K3$ fibered over $\mathbb{P}^1$. For the case $k_1=k_2=12$ this corresponds to a $CY^3$ which is elliptically fibered over $\mathbb{P}^1\times \mathbb{P}^1$ \cite{Kachru:1995wm}. For the more general splitting $k_1=12-n, k_2=12+n$ the the base $\mathbb{P}^1\times \mathbb{P}^1$ is replaced by a Hirzebruch surface $F_n$
\cite{Morrison:1996na, Morrison:1996pp} . The comparison between the Higgs branches of Heterotic side and its geometric interpretation on the F-theory side can be found in
\cite{Bershadsky:1996nh}.
In this duality, the transition of an instanton from one $E8$ factor to the other has an interesting geometric interpretation on the F-theory side. Emitting an instanton to the bulk
corresponds to blowing up a point on $F_n$ and absorbing it on the other side is equivalent to blowing another $\mathbb{P}^1$ down to a point and the net effect is converting $F_n$ to $F_{n\pm1}$ \cite{Witten:1996qz}.
\subsubsection{$N_{SUSY}=4$ and 4D theories}
\label{sec:4D}
Similarly we can continue down to theories with 4 supercharges. The highest dimension for which this happens is ${\cal N}=1$ supersymmetric theories in four dimensions.
We can obtain these theories compactifying by Type I or Heterotic strings on Calabi-Yau 3-folds, F-theory on Calabi-Yau 4-folds, or M-theory on G2 holonomy manifolds. Needless to say we expect these to be dual to one another with suitable choices of parameters. There has been a large and growing literature on this subject which is beyond the scope of the current review. A particularly powerful description of this class of theories involves F-theory, as it is based on relatively simple geometrical data. This description has led to many potential connections with supersymmetric particle phenomenology involving supersymmetric extensions of the standard model. See \cite{Heckman:2010bq} for a partial review.
Given the large number of six-dimensional manifolds, we would like to know something more about what kinds of compactifications are allowed. Generic compactifications of this type can have a large amount of matter fields depending on the number of non-trivial homology cycles. As we go to large number of matter fields in any theory, we run into the problem of the theory being not asymptotically free. In general this means that we cannot decouple gravity. The question now becomes what constraints does this place on our theory, or in other words, can the non-asymptotically free theory be arbitrary? As it turns out the restrictions are quite strong.
Since the standard model is asymptotically free\footnote{We assume that it is embedded in some grand unified theory.}, we can decouple gravity even though we do have gravity in our universe (and hence have a compact internal manifold assuming this is the correct description). In order to study just the gauge theory part of this, we can study a non-compact version of the internal manifold, $Y_{nc}$, which describes the local structure of the true compact manifold, $Y_c$. In essence we can think of this as ``zooming'' into a local part of $Y_c$ so that it appears to be so large that we can think of it as a non-compact manifold. One of the lessons one learns in this context is that local singularities of CY geometry play a key role in encoding and restricting some aspects of phenomenology. One particularly promising example of this is how flavor hierarchy can be geometrically encoded \cite{Heckman:2008qa}. Given the importance of local singularities we turn to a brief review of it later in this lecture.
An important difference between compact and non-compact Calabi-Yau manifolds is that compact manifolds cannot have global symmetries while non-compact ones can. This means that the global symmetries of the standard model (decoupled from gravity) can only be approximate symmetries when we take gravity into account, matching beautifully with the first swampland conjecture in Lecture \ref{sec:Lec2} \cite{Vafa:2005ui}.
Of course we need to understand the full compact geometry in order to understand the structure of the complete standard model, however studying a non-compact version should be sufficient for some low energy approximations which we can think of as only probing the local geometry of $Y_c$. It is also worth pointing out that not every local model of this type is permissible. These can lead to contradictions with observations and consistency and hence should be counted as being in the ``swampland'' of the string landscape - the part that appears to be a consistent low energy theory which does not have a consistent UV completion with gravity. This will be the topic of Lecture \ref{sec:Lec2}.
\subsubsection{Supersymmetry Breaking}
At the time of writing these notes, supersymmetry had not yet been realized at energy scales tested by collider experiments. However, supersymmetry is a fundamental component of string theory model buildin
. Therefore, in order to properly describe real physics at low energy, we must somehow figure out how to break supersymmetry. This has been a topic of intense study for a long time which is deceptively difficult. For example, consider the following two cases
\begin{enumerate}
\item Compactify a supersymmetric theory on $T^2$. We can then break supersymmetry by hand by imposing anti-periodic boundary conditions for fermions.
This is called the Scherk-Schwarz compactification \cite{Scherk:1979zr}. One drawback of this approach is that this theory will develop a tachyon for a small enough radius of the circle compactification. This problem is unavoidable because we cannot study such a system and ignore the radial modulus, or arbitrarily restrict attention to only a subset of the radion moduli space \cite{Rohm:1983aq}.
\item
We can break supersymmetry by considering compactifications on manifolds without a special holonomy. For example if compactify on two complex dimensions and if we have a local orbifold singularity of the type $\mathbb{C}^2/\Gamma$, where $\Gamma \not\subset SU(2)$ this breaks SUSY. But again, studying string theory in this background we learn we have tachyons and hence the system is unstable \cite{Vafa:2001ra}.
\end{enumerate}
This tachyonic behavior and more generally lack of stationary solutions appears to be a ubiquitous behavior when we try to break supersymmetry. It would appear that string theory is sending us a message.
A notable proposal to break supersymmetry and reduce to our universe with positive cosmological constant is given by KKLT \cite{Kachru:2003aw}. This approach broadly consists in two steps: 1.) stabilizing the moduli while preserving supersymmetry and 2.) breaking supersymmetry by adding anti-D3-branes wrapped on highly warped cycles in the internal manifold. We will now review this construction in more detail.
Consider F-theory/IIB supergravity on a Calabi-Yau orientifold with fluxes, with the simplifying assumption of having only one K\"ahler modulus. Due to the scale invariant property of the effective Lagrangian, the complex structure moduli can be stabilized perturbatively while the K\"ahler moduli cannot. Therefore in order to stabilize all moduli one must add non-perturbative features such as Euclidean D3 instantons \cite{Witten:1996bn}.
Once all moduli are stabilized, we will add anti-D3-branes. It should be noted that those anti-D3-branes will not bring other moduli, as their worldvolume scalars are all automatically stabilized by the fluxes. These anti-D3-branes are then wrapped on a cycle at the tip of a Klebanov-Strassler throat \cite{Klebanov:2000hb}, inside the Calabi-Yau orientifold. The anti-branes will then back react on the non-compact part of spacetime, causing it to have a positive cosmological constant, much like how D3-branes lead to $AdS$ space. In order to properly understand the KKLT construction, one must have full control of the large number of moduli which is generally intractable. See \cite{Moritz:2017xto} for related issues in the KKLT scenario.
We are now at an impasse. It might be that the KKLT construction consistently breaks SUSY and reproduces a $dS$-like vacuum with positive energy, but we are just unable to analyze it yet. Since we do not have the tools to analyze the KKLT construction in a realistic case, it is impossible to make any strong claims about a KKLT-like realization of $dS$ in string theory. Indeed there are various problems which arise in this class of theories. In introducing anti-D3-branes on a compact manifold, we need to take special care of having them well separated from the D3-branes, so that they do not annihilate. Therefore, to analyze stability when introducing both D3-branes and anti-D3-branes, we really must have a complete knowledge of the moduli space, including both open and closed string moduli. This is clearly a very complex problem, as the question of moduli stabilization remains an active area of research to date.
\subsection{Singularities and Branes}
Another important facet of our modern understanding of string theory is the role of singularities.
This is demonstrated in an example from the previous section where Type IIA on K3 is dual to Heterotic string theory on $T^4$. In this example, non-abelian gauge symmetry on the Heterotic side is dual to the K3 manifold developing singularities. It is clear from this that the singularities must play an integral part of understanding these physical theories. This is just an example of the general principle that singularities give rise to interesting physical phenomena. Moreover understanding local singularities can lead to a deeper understanding of quantum systems, decoupled from gravity. This in particular has led to insights about the existence of non-trivial interacting quantum systems in up to six dimensions.
Before we demonstrate some of the interplay between singularities and branes, it is helpful to do a quick review of toric geometry.
\subsubsection{Lightning Review of Toric Geometry}
Toric geometry is the study of toric varieties. A \emph{toric variety} $X$ is the zero set of a collection of complex polynomial equations that have an algebraic torus $T=(\mathbb{C}^\ast)^r$ as a dense open subset and has a natural action action $T:X\to X$ such that the restriction to $T\subset X$ is the usual translation action \cite{Hori:2003ic}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.9,trim=3cm 17cm 6cm 5.6cm,clip]{FiberDeg.pdf}
\caption{In this figure we illustrate the idea of toric varieties. Here the two boundaries (labeled A, B) are where the A- and B-cycles of the $T^2$ torus fiber degenerates. So the fiber above points X, Y, and Z will be the $S^1$ B-cycle, full $T^2$, and $S^1$ A-cycle respectively.}
\label{fig:FiberDeg}
\end{figure}
\begin{figure}[b]
\begin{center}
\includegraphics[trim=0.5cm 25cm 11.5cm 1.5cm,clip]{C1Toric.pdf}
\caption{This figure is a representation of how $\mathbb{C}^1$ can be realized as a toric variety which is a circle fibered over the positive real line.
\label{fig:C1}
\end{center}
\end{figure}
A natural way to encode the topological data of a toric variety is by realizing them as torus fibrations. Since we can realize a generic toric variety $X$ as
\bar{e}
\xymatrix{T^r\ar[r]&X\ar[d]\\&B_r}~,
\end{equation}
with a $T^r$ fiber, we can realize the base of the fibration as a polyhedra where different cycles of the fiber degenerate along the different boundary components $\partial B_r=\coprod_{i}\sigma_i$ where the number of degenerate cycles on $\sigma_i$ is given by codim$_{B_r}~ \sigma_i$. See Figure \ref{fig:FiberDeg}. When $\partial B_r$ forms a closed polyhedra, we can encode its data in the dual graph -- this dual graph is called the \emph{toric fan} and is related to the way in which branes arise in the Type IIB description of M-theory compactified on a toric variety \cite{Leung:1997tw}.
Consider the example of $\mathbb{C}$. This can be realized as a plane where the complex coordinates $(z,\bar{z})$ can be exchanged for the real coordinates $(r,\theta)$ in the usual way. This allows us to realize $\mathbb{C}$ as a a circle ($\theta$-coordinate) fibered over the positive real line, ${\mathbb{R}}_+$, ($r$-coordinate) where the radius of the circle is given by $r$. See Figure \ref{fig:C1}.
\begin{figure}
\centering
\includegraphics[scale=0.9,trim=2cm 15cm 4cm 5cm,clip]{ConifoldRes.pdf}
\caption{This figure shows the two possible resolutions ( left and right) of the conifold (center). In the conifold, the vertical and horizontal line represent the locus in the base where two perpendicularly cycles (A- and B-cycles) in the $T^2$ fiber degenerate. This means that the preimage of a line running diagonally from one axis to the other in the base represents a 3-sphere since it is a 3-real dimensional manifold with a perpendicularly embedded $S^1$ degenerating at each end. This can be resolved in two different ways which corresponds to resolving with an $S^3$ (left) or by gluinga $\mathbb{C}\mathbb{P}^1$ at the origin (right) where the degenerate $S^1$ fiber is either the (a) A+B cycle or the (b) A-B cycle in the $T^2$ fiber.}
\label{fig:conRes}
\end{figure}
In this way, we can similarly realize $\mathbb{C}^3$ as 3-perpendicular copies, that is a $T^3$ fibered over ${\mathbb{R}}_+^3$ which looks like an octant in ${\mathbb{R}}^3$ where the radius of the different cycles of the torus are determined by their distance from the 3 axes.
An important feature of the description of toric varieties as torus fibrations of a base polytope
is that it has a clear geometric interpretation of blow ups. The geometric operation of \emph{blowing up} is a method of resolving a singularity by replacing a singularity with a smooth manifold. In some sense, the most fundamental blow up replaces a singularity with a copy of $\mathbb{C}\mathbb{P}^1$. More generally singularities can be replaced by a collection of $\mathbb{C}\mathbb{P}^1$'s which can be glued together in interesting ways to form more complicated resolutions, or by other manifolds. As a toric variety, $\mathbb{C}\mathbb{P}^1$ can be realized as a $S^1$ fibered over a finite line segment where the fiber degenerates at the two ends. Using this, the fundamental resolution of singularity simply replaces the singular point where cycles degenerate on the base with a line segment where the fiber is given by a linear combination of the degenerating cycles at the unresolved singularity. See Figure \ref{fig:conRes} for an example. See \cite{Leung:1997tw,Hori:2003ic,Iqbal:2003Ds} for more details.
\subsubsection{The McKay Correspondence and Theories of Class $\mathcal{S}$}
\label{sec:McKay}
Let us study the case of K3 singularities in more detail. The allowed singularities of K3-manifolds are locally of the form $\mathbb{C}^2/\Gamma$ where $\Gamma\subset SU(2)$ is a finite group. These generally have an ADE classification\footnote{That is they have a classification which is identical to that of A-,D-, and E-type Lie algebras.}. These are given by
\begin{itemize}
\item A-type: $\Gamma$ is a binary cyclic group, $\Gamma={\mathbb{Z}}_n$
\item D-type: $\Gamma$ is a binary dihedral group, $\Gamma=BD_{2n}$
\item E6-type: $\Gamma$ is the binary tetrahedral group $\Gamma=2T$
\item E7-type: $\Gamma$ is the binary octahedral group $\Gamma=O$
\item E8-type: $\Gamma$ s the binary icosahedral group $\Gamma=2I$
\end{itemize}
To illustrate these we will consider the $A_{n-1}$ type singularity in detail. Consider the group $\Gamma={\mathbb{Z}}_n$ which is defined by the generator
\bar{e}
\Gamma=\left\langle \left(\begin{array}{cc}\alpha&0\\0&\alpha^{-1}\end{array}\right)\right\rangle\quad,\qquad \alpha^n=1~,
\end{equation}
acting on $\mathbb{C}^2$. Roughly speaking, Type IIA string theory on ${\mathbb{R}}^{1,5}\times\mathbb{C}^2/{\mathbb{Z}}_n$ is T-dual to Type IIB with $n$ coincident D5-branes giving rise to a $SU(n)$ gauge theory \cite{Ooguri:1995wj,Katz:1996ht}\footnote{There is a slight subtlety associated with this. We technically need to have a singular Taub-NUT space, $TN_n$, (the singular limit of $n$ NUT centers) which locally has the same singularity structure as $\mathbb{C}^2/{\mathbb{Z}}_n$. The difference between $TN_n$ and $\mathbb{C}^2/{\mathbb{Z}}_n$ is that in $TN_n$, the circle fiber asymptotically
approaches a finite so that T-duality is well defined whereas the $S^1$ fiber of $\mathbb{C}^2/{\mathbb{Z}}_n$ diverges as the distance from the singularity. Therefore, T-duality is not technically defined in $\mathbb{C}^2/{\mathbb{Z}}_n$ as one cannot shrink the asymptotic circle to apply T-duality \cite{Gregory:1997te,Witten:2009xu}. There is however, a version of mirror symmetry which work for this case as well \cite{Hori:2000kt}.}. This $SU(n)$ gauge theory can be seen in the Type IIA side by blowing up the singularity: completely resolving the singularity results in a collection of $\mathbb{P}^1$s that intersect as in the $A_{n-1}$ Dynkin diagram\footnote{Technically, there are $2^{n-1}$ different spheres in the fully resolved $\mathbb{C}^2/{\mathbb{Z}}_n$, $\widehat{\mathbb{C}^2/{\mathbb{Z}}_n}$, and we can choose a basis for the homology group $H_2(\widehat{C^2/{\mathbb{Z}}_n})$ such that their intersection matrix is given by the $A_n$ Cartan matrix.}. The gauge theory then comes from D-branes wrapping these non-trivial 2-cycles which upon going to the singular limit, all become effectively coincident, giving rise to a non-gauge symmetry.
We can also get the $D_n$ type theories in a similar way. This corresponds to having orientifolds in the Type IIB picture and in the geometric picture by replacing ${\mathbb{Z}}_n\to BD_{2n}\cong{\mathbb{Z}}_n\rtimes {\mathbb{Z}}_2$ where the ${\mathbb{Z}}_2$ action acts as an orbifold. Understanding the interpretation of E-type singularities as branes in Type IIB requires the use of F-theory since the branes have do not have a perturbative description. It is known in the M-theory setup or the Type IIA picture that E-type singularities lead to E-type gauge theories.
One can then fiber these geometries as in the adiabatic construction and obtain systems with lower supersymmetry in lower dimensions. In particular fibering these geometries over $\mathbb{P}^1$ leads to ${\cal N}=2$ supersymmetric systems in 4 dimensions. Using this picture and applying local mirror symmetry, the Seiberg-Witten curve of these theories can be identified in string theory \cite{Klemm:1996bj} as in Appendix \ref{sec:App4D}. As we will now explain, this curve can also be realized in Type IIA as the geometry of intersecting D4/NS5-branes, which lift to M5 branes in the M-theory lift of Type IIA \cite{Witten:1997sc,Gaiotto:2009we}.
If instead of Type IIA, we consider the compactification of Type IIB string theory on a K3-manifold with $\mathbb{C}^2/\Gamma_\mathfrak{g}$ singularities, this leads to a six dimensional theory with ${\cal N}=(2,0)$ with type $\mathfrak{g}$ conformal systems \cite{Witten:1995zh,Claus:1997cq,GMooreFelixKlein,Nahm:1977tg}. Compactifying these theories on a Riemann surface $C$ (with a topological twist)
produces the four dimensional ${\cal N}=2$ theories of class ${\cal S}$ \cite{Gaiotto:2009hg,GMooreFelixKlein,Witten:1997sc,Gaiotto:2009we}. These theories can be constructed for type $\mathfrak{g}=A_{N-1}$\footnote{Note that this construction can be generalized to $\mathfrak{g}=D_n$ type theories by wrapping an $O5$-plane on $\Sigma\times {\mathbb{R}}^4$.} by compactifying $N$ M5-branes on a Riemann surface $\Sigma\times {\mathbb{R}}^4$ with the same topological twist, where $\Sigma\to C$ is a multisheeted cover of $C$. This derives the data of the Seiberg-Witten curve and Seiberg-Witten 1-form, where the curve the Prym variety\footnote{The Prym variety is the kernel of the induced map $\pi^\prime:J(\Sigma)\to J(C)$ from $\pi:\Sigma\to C$. Here $J(X)$ is the Jacobian variety of $X$ which is defined as the quotient of all global holomorphic differentials on $X$, $H^0(\Omega_X^1)$, quotiented by the space of non-trivial closed 1-cycles on $X$, $H_1(C)$. By the universal covering construction, this space has the same $1^{st}$ homology group $H_1(X)=H_1(J(X))$.} associated to the map $\Sigma\to C$ \cite{Witten:1997sc,Gaiotto:2009we,Gaiotto:2009hg}. In addition, this gives a clear geometric way to study the BPS spectrum, line operators, surface operators, and expectation values of supersymmetric operators in the associated four dimensional theory \cite{Shapere:1999xr,Alday:2009aq,Gaiotto:2012rg,Gaiotto:2011tf,Gaiotto:2010be,Gaiotto:2012Db,Alim:2011kw,Alim:2011ae,Gabella:2017hpz}. See \cite{Tachikawa:2013kta,Teschner:2014oja,Gaiotto:2014bja,GMooreFelixKlein} for a more general review.
\subsubsection{Branes in F-Theory}
The relation between singular geometry and branes is also evident in F-theory. If we take F-theory on $K^{ell}$ where
\bar{e}
\xymatrix{T^2\ar[r]&K^{ell}\ar[d]\\&B}
\end{equation}
this is dual to Type IIB on $B$ with $(p,q)$ 7-branes where the $(p,q)$ cycles of the $T^2$-fiber degenerate. A choice of $(p,q)$ cycles corresponds to a choice of S-duality frame for the Type IIB theory. These branes are non-perturbative because they are co-dimension 2 and hence the magnetic dual of the axio-dilaton has log-type asymptotic behavior. In general these branes are mutually non-local due to their $(p,q)$ axion, dilaton charges and can give rise to non-abelian exceptional groups as studied in \cite{Gaberdiel:1997ud}. One may worry about having these charged objects in a compact space because branes source flux and as we know from general theorems of general relativity we cannot have a net charge in a compact space as the flux has nowhere to go. However, this F-theory setup avoids this problem because the 7-branes source non-abelian flux which can actually cancel with themselves \cite{Vafa:1996xn} and thus lead to realization of non-trivial stable charged objects in a compact space.
\subsubsection{5-Brane Webs}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{ToricDiagram.png}
\caption{This is the toric diagram for the $CY^3$-fold given by the affine cone over $\mathbb{P}^1\times \mathbb{P}^1$: ${\cal O}(-2,-2)\to \mathbb{P}^1\times \mathbb{P}^1$. This diagram encodes the data of the singular structure of the $T^2$ fiber via its dual graph. }
\label{fig:toric}
\end{center}
\end{figure}
Another example of the correspondence between branes and geometry is the construction of brane webs \cite{Aharony:1997bh}. Consider M-theory on ${\mathbb{R}}^{1,6}\times K^{ell}$ where $K$ is an elliptically fibered $$CY^3$$. If we compactify and T-dualize on a pair of orthogonal cycles on the $T^2$ fiber, we get a Type IIB theory with $(p,q)$ 5-branes which have $p$ R-charge ($\tilde{C}_2$) and $q$ NS-charge\footnote{Here $\tilde{B}$ represents the magnetic dual of the form field $B$ and similarly for $\tilde{C}_2$ and $C_2$.} ($\tilde{B}$) where we compactify along the $(1,0)$-cycle and T-dualize along the $(0,1)$-cycle. This generalizes the relation of $n$ coincident D6-branes in Type IIA to Taub-NUT geometry (with $\mathbb{C}^2/{\mathbb{Z}}_n$ singularity) by lifting to M-theory and the relation of Taub-NUT to NS5-branes by T-duality \cite{Leung:1997tw}.
Let us consider an explicit example. Consider M-theory on the non-compact space
\bar{e}
\xymatrix{T^2\ar[r]&{\cal O}(-2,-2)\ar[d]\\&\mathbb{P}^1\times \mathbb{P}^1}
\end{equation}
On the base $\mathbb{P}^1\times \mathbb{P}^1$, the singular structure of the the $T^2$ fiber can be represented by the dual of the toric diagram in Figure \ref{fig:toric}.
This means that after dualizing to Type IIB, we have a system of $(p,q)$ 5-branes on ${\mathbb{R}}^{1,4}\times {\mathbb{R}}^2_{web}\times {\mathbb{R}}^3$ where the 5-branes wrap the ${\mathbb{R}}^{1,5}$ direction and form the generically co-dimension 1 web in the ${\mathbb{R}}^2_{web}$ direction as shown below:
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw [-, very thick, black] (0,-1)--(0,1);
\draw [-, very thick, black] (2,-1)--(2,1);
\draw [-, very thick, black] (0,1)--(2,1);
\draw [-, very thick, black] (0,-1)--(2,-1);
\draw [-, very thick, black] (2,1)--(3,2);
\draw [-, very thick, black] (0,1)--(-1,2);
\draw [-, very thick, black] (0,-1)--(-1,-2);
\draw [-, very thick, black] (2,-1)--(3,-2);
\node (pqlab1) at (1,1.25) {$(1,0)$};
\node (pqlab1) at (1,-1.25) {$(1,0)$};
\node (pqlab1) at (2.5,0) {$(0,1)$};
\node (pqlab1) at (-0.5,0) {$(0,1)$};
\node (pqlab1) at (3.5,2) {$(1,1)$};
\node (pqlab1) at (-1.5,2) {$(1,1)$};
\node (pqlab1) at (3.5,-2) {$(1,1)$};
\node (pqlab1) at (-1.5,-2) {$(1,1)$};
\end{tikzpicture}
\end{figure}
In essence, this brane web depicts two D5-branes suspended between two NS5-branes. This describes the 5D ${\cal N}=2$ $SU(2)$ SYM where the displacement between the D5-branes (vertical displacement $x^6$) gives the Higgs vev and the displacement between NS5-branes (horizontal displacement $x^5$) determines the gauge coupling as $
\Delta x^5=\frac{4\pi}{g_{YM}^2}$.
One can also combine branes and singularities, by bringing branes to probe singularities and obtain new and interesting quantum systems. A brief review of some examples is given in Appendix \ref{app:A}.
\section{Lecture 2: The Swampland}\label{sec:Lec2}
In the previous lecture we briefly reviewed the string theory construction of many low dimensional effective field theories. As we saw, there is a very large number of choices to make when using string theory for model building, coming from the choice of the compactification manifold, background fluxes, branes, and etc. Therefore a very relevant question is how to identify which particular string theory solution, among the enormous set of possibilities, describes our universe.
It has recently been estimated that the number of possible consistent flux compactifications of F-theory to $4D$ is at least $10^{272000}$ \cite{Taylor:2015xtz}. Although it is unknown if all of these compactifications are distinct or may be dual descriptions of the same theory, this large number of string vacua suggests that perhaps the direct study of all string vacua is futile. This remarkably large space of inequivalent string backgrounds is called the \emph{string landscape}.
To complicate matters even more, even if we were to be able to enumerate all of the distinct, consistent string backgrounds, there is no known top-down mechanism to prefer one particular choice over another. For example, what forces four dimensions to be extended and six to be compact? Why did nature choose the specific string background describing our universe from the vast number which can be constructed in the theory? While there have been some suggested ideas \cite{Brandenberger:1988aj,Kofman:2004yc}, there is no compelling solution yet.
Due to this huge number of possible choices involved in constructing string vacua, there has been a distinct philosophical shift in the community over the past decade. The attitude towards identifying ``the correct'' string vacuum has shifted from using a top-down approach to a bottom-up one. Instead of starting with fully-fledged string theory and studying the compactifications down to 4D, many have started studying effective four dimensional quantum field theories with nice phenomenological features (such as supersymmetric extensions of the standard model) and then trying to couple them to gravity. The common lore is that because the string landscape is so large, it is likely that any consistent looking lower dimensional effective field theory (EFT) coupled to gravity can arise in some way from a string theory compactification. Indeed this idea would make string theory pretty much irrelevant for phenomenological questions.
In this lecture, we aim to argue that this way of thinking is incorrect -- that not all consistent looking EFTs can be coupled consistently to gravity with a UV completion. Since string theory is the only known UV complete theory of gravity\footnote{One could make the argument that there are other theories of quantum gravity different from string theory, such as Vasiliev higher spin theories\cite{Vasiliev:1995Dn} or even loop quantum gravity\cite{Smolin:2004sx}. However, it has been conjectured that all these other theories can be attained as a special limit of string theory\cite{Dijkgraaf:2004te}.}, we will necessarily demand that these theories arise from some string theory compactifications \cite{Vafa:2005ui}.
The rest we believe are ultimately inconsistent. In analogy with the string landscape, we will call the set of all EFT which do not admit a string theory UV completion as the \emph{swampland}.
It is therefore crucial to understand if a given EFT coupled to gravity lies in the string landscape or the swampland. In order to do so, we would like to identify a complete set of \emph{swampland criteria} which will identify if an EFT admits a string theory UV completion or not.
Thus far, we have a conjectured, minimal criteria that allows us to exclude a theory from the string landscape.
In this lecture we plan to briefly discuss ten swampland criteria. We will be unable to provide proofs, but rather will provide physical reasoning based on realization in string theory and general facts about qunatum gravity to motivate each of the conjectured criteria. The criteria we present here are based on \cite{Vafa:2005ui,Ooguri:2006in,ArkaniHamed:2006Dz,Ooguri:2016pdq}.
\subsection{No Continuous Global Symmetries}
\noindent\emph{An effective field theory coupled to gravity cannot have (continuous or discrete) global symmetries.}\\
The motivation for this criterion relies on black hole physics. Suppose we have an EFT coupled to gravity which has a global symmetry $G$. In the spectrum of the EFT we will have states charged under such global symmetry. Now send a state charged under a global symmetry inside a black hole. The information of this global symmetry is lost by the no-hair theorem. Thus, when the black hole evaporates via Hawking radiation \cite{Hawking:1974sw} it will do so by emitting particles which carry equal number of positive and negative charges under $G$ since there is no imprint of global charges on a black hole. This process would then violate charge conservation in $G$
as we started with non-zero charge and all the charge has disappeared after the evaporation of the black hole and no net charge has come out.
The only way to avoid this seeming contradiction is by forbidding any theory of quantum gravity from having global symmetries. Remarkably, it appears that string theory already knows about this criterion, as in all examples we know, all global symmetries are actually gauged. This is true because usually global symmetries in EFTs obtained by string compactification arise from symmetries of the extra dimensions, but such symmetries are gauged since diffeomorphisms of the compactification manifold are part of the gauge symmetry of gravity.
\subsection{All Charges Must Appear}
\noindent\emph{A consistent effective field theory with gauge group $U(1)$ coupled to gravity must have states with arbitrary charge ${\mathbb{Z}}$.}\\% must appear in the spectrum. }\\
A nice discussions of this condition for swampland is given in \cite{Banks:2010zn}.
Suppose we have a $U(1)$ gauge symmetry in the EFT. The Hilbert space of the EFT will be split into different sectors, one for each value of the $U(1)$ charge of the states in that sector. Now, if the theory is not coupled to gravity, it is possible that the spectrum contains only states of some specific subset of charges, or even maybe no charged states at all. However, this cannot happen once the theory is coupled to gravity.
Consider a $U(1)$ theory coupled to gravity. We will have charged black hole solutions of Einstein's equation for any integral charge $Q$. By Hawking's formula, we know the black hole entropy is given by
\begin{equation}
S=\dfrac{A}{4G}~.
\end{equation} Such an entropy must have a statistical mechanics interpretation as a sum over the black hole microstates. As the black hole is charged, such microstates must be charged. Therefore all charged states should exist in the spectrum\footnote{This argument is valid as long as charge is large enough so we can truly interpret the object as a black hole. In other words, as long as the area of the horizon is much bigger than the Planck scale.}.
Notice that this criterion rules out many simple quantum field theories, as for example pure Maxwell theory coupled to Einstein gravity. Such a theory must also have (perhaps massive) charged particles. Note that at this level we did not say anything about the masses of this infinitely many charged states which should belong to the spectrum.
\subsection{Finite Number of Massless Fields}
\noindent \emph{A $d$-dimensional EFT coupled to Einstein gravity must have a finite number of massless fields.
Moreover, the number of massless fields is bounded from above by a certain number $N_{max}(d)$ which depends only on the number of spacetime dimensions $d$.\\}
The motivation for this criterion is based on supersymmetric examples. Massless scalar fields in a lower dimensional EFT are generated in string theory by compactification. For example if we compactify on a K\"ahler manifold in order to preserve some supersymmetry in the lower dimensional EFT, the number of scalars will generically be proportional to specific Hodge numbers of the compactification manifold. For the case of compact CY manifolds, there seems to be an upper-bound on the possible hodge numbers, even though there is no proof of this.
Remarkably, string theory seems to ``be naturally aware" of this fact, and seems to have ways for preventing us to get consistent lower dimensional EFT with arbitrarily large number of light scalar fields. An easy example in which we can see this at work is the following. Consider Type IIA on $\mathbb{C}^2/\mathbb{Z}_N$. We saw in the first lecture that in this way we can realize an $SU(N)$ gauge group. There are therefore $N^2-1$ massless gluons in the spectrum. At this level there is no bound on $N$, which we can take as large as we want, therefore having an arbitrary high number of gluons. This is in no contradiction with the conjecture stated above, since $\mathbb{C}^2/\mathbb{Z}_N$ is non-compact and therefore gravity is decoupled in the EFT.
However, in order to couple this $SU(N)$ gauge theory to dynamical Einstein gravity, we need to embed $\mathbb{C}^2/\mathbb{Z}_N$ into a compact manifold, i.e., K3. Quite remarkably, what happens in this case is that in order for the compactification to be consistent, it must have $N\leq 20zx$, therefore putting an upper bound on the number of gluons \cite{Aspinwall:1995zi}. See Section \ref{sec:McKay} for more details.
Note that since we are assuming only Einstein gravity, we exclude more exotic gravity theories with infinite number of massless fields such as Vasiliev theory \cite{Fradkin:1987ks}.
\subsection{No Free Parameters}
\noindent\emph{A consistent EFT coupled to gravity must have no free parameters. Every parameter entering in the Lagrangian should be viewed as the vacuum expectation value of a field.}\\
Notice that this criterion puts a lower bound on the number of possible scalar fields, therefore being complementary to criterion number 3.
Again the motivation is that this appears to be true in string theory. For example M-theory in 11 dimensions has no free parameter. 10 dimensional superstrings would naively appear to have a free parameter given by the choice of the coupling constant, but upon closer inspection one find out that the coupling constant is the expectation value of a scalar field called the dilaton. When we compactify to go to lower dimensions we end up with effective theories whose parameters get related to the internal geometry of the compactification which again can be viewed as part of the dynamical degrees of freedom of the theory.
\subsection{The Moduli Space is Non-Compact}
\noindent\emph{The moduli space $\mathcal{M}$ of vacua (if non-trivial)
is non-compact. In more detail, fix a point $p_0 \in \mathcal{M}$. Then $\forall\ T>0\, , \exists~ p\in \mathcal{M}$ such that \begin{equation}
d(p_0,p)>T~.
\end{equation}
where $d(p_0,p)$ is the distance between $p_0$ and $p$, computed by using the moduli space metric as the length of the geodesic passing through $p$ and $p_0$.}\\
To elucidate this criterion, we first need to discuss what we mean by the moduli space metric.
Consider an EFT coupled to gravity, with $N$ massless scalar fields $\Phi_i$, $i=1,...,N$ with no potential. Such scalar fields arise generically in string compactification, and their vacuum expectation values $\langle\Phi_i\rangle$ is related to geometrical quantities in the compactification manifolds such as for example the volumes of some cycles or their shapes. We will call the algebraic variety parametrized by the various $\langle\Phi_i\rangle$ the \emph{moduli space} $\mathcal{M}$.
In the EFT, the kinetic term for those scalar fields typically takes the form
\begin{equation}
\mathcal{L}_{eff}= g_{ij}(\Phi)\partial_{\mu}\Phi^i\partial_{\mu}\Phi^j+...~, \label{Leff}
\end{equation}
where $g_{ij}$ is the metric on ${\cal M}$.
We can use this metric to compute distances in the moduli space, and ask if $\mathcal{M}$ is compact or not. As it turns out in all known examples from compactifying string theory, the moduli space is non-compact \cite{Ooguri:2006in}.
As an easy example in which this conjecture is realized, we can consider here the case of the moduli space of IIB supergravity. In this case, there is only the axiodilaton modulus
$\tau$, which is the combination of the string coupling constant and a RR 0-form.
The moduli space which will be the fundamental domain of $SL(2,\mathbb{Z})$. Now, fix a point $\tau_0$ in the moduli space, and consider the length of the geodesic from $\tau_0$ to $\tau$. When we take the limit $\tau \to i\infty$ while keeping $\tau_0$ finite, the geodesic length will be approximately
\begin{equation}
T\sim{\rm log}\left(\mbox{Im}\tau/\mbox{Im}\tau_0\right)~,
\end{equation}
and we clearly see that this distance is logarithmically divergent.
\subsection{New Physics from the Boundaries of Moduli Space}
\emph{Fix a point $p_0\in {\cal M}$. In the limit of infinite distance from $p_0$, that is as $d(p_0,p)=T\to \infty$, there will be a tower of states in the EFT whose mass decreases exponentially with $T$,
\begin{equation}
m\sim e^{-\alpha T}~.
\end{equation}}
In the previous criterion we saw that for any choice of a starting point $p_0\in \mathcal{M}$ and any real number $T>0$, we will always be able to find a (in general not unique) point $p\in \mathcal{M}$ such that the distance between $p$ and $p_0$ is larger than $T$. So in general $\mathcal{M}$ will have some non-compact directions. We want to ask now what happens when we go extremely far away in moduli space along one of those directions, or equivalently we take $T$ to be extremely large \cite{Ooguri:2006in}.
Heuristically, we can understand this by considering the point compactification of moduli space, $\hat{\cal M}$, so that $\hat{\cal M}$ is a finite manifold where the infinities of ${\cal M}$ correspond to singular points of $\bar{\cal M}$. Now going to infinity corresponds to going to a singularity where generically extra massless degrees of freedom appear.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{Modulispacev2.png}
\caption{A schematic picture of the moduli space.}
\label{Modulifig}
\end{figure}
We can illustrate this criterion in a very easy example. Consider the compactification of a EFT on a circle $\mathcal{S}^1_R$. This theory has a modulus, which is the radius $R$ of the circle. The Lagrangian for this scalar field, in one lower dimension, will be given by
\begin{equation}
\mathcal{L}_{eff}=\int \left(\dfrac{dR}{R}\right)^2+...
\end{equation}
Now, let us see how the criterion 5 and 6 apply to this case. In this example, the moduli space is just $1$-dimensional, and we can see immediately that there are infinite distances. For example, fix a radius $R_0$ and pick a $T>0$. The distance from $R_0$ to some other point $\tilde{R}$ will then be given by
\begin{equation}
\int_{R_0}^{\tilde{R}} \dfrac{dR}{R} = {\rm log}(\tilde{R})-{\rm log}(R_0)~,
\end{equation}
and we see that we can always find a suitable $\tilde{R}$ to make this distance as big as we want. So the criterion number 5 is satisfied. Let us consider the limit of very large radius, to see criterion 6 at work. As the radius $\tilde{R}$ grows to infinity, we will have Kaluza-Klein modes, with mass given by
\begin{equation}
m\sim \dfrac{1}{\tilde{R}}~.
\end{equation}
On the other end, $T\sim {\rm log}(\tilde{R})$ and therefore we see that we have some fields with mass
\begin{equation}
m\sim e^{-T}~,
\end{equation} which get exponentially light when we go to infinity in $\mathcal{M}$.\\
\noindent{\it This conjecture also implies the remarkable fact that a consistent theory of quantum gravity must have extended objects in its spectrum.} \\
Those extended objects can be for example membranes, strings, etc. Therefore, by this criterion one can argue that quantum gravity cannot be a theory of just particles. We will now show this in the same easy example we used.
Pick now the same reference point $R_0$ but instead of going to larger radius go to smaller and smaller radius. We also find that this is another infinite distance in moduli space, as
\begin{equation}
\lim_{\tilde{R}\to 0}\int_{\tilde{R}}^{R_0} \dfrac{dR}{R}~,
\end{equation}
diverges. Therefore, due to criterion 6, we also expect to have in this case some states with mass getting exponentially low. However, such states cannot be particle states because all the particle states will be given by KK modes, and those KK modes will be instead very massive in the small radius limit. The only way we can get light objects in the small radius limit, is to have some extended objects which can wrap around the circle which thus become lighter as we go to the small radius limit. So if our theory does not have extended objects, we do not have any light states at all in this limit and we therefore violate the criterion number 7. This is for example what happens in M-theory when we compactify on the circle: The M2 branes wrapping the circle become light and give rise to the light string states in 10 dimensions.
We thus see that this conjecture implies the existence of extended objects in a consistent theory of quantum gravity.
\subsection{The Moduli Space is Simply Connected}
\noindent\emph{The first fundamental group of the closure of the moduli space is trivial, and therefore the closure of moduli space is simply connected: \begin{equation}
\pi_1(\overline{\mathcal{M}})=0~.
\end{equation}}
In all known examples in string theory, the moduli space is obtained by quotienting a contractible Teichm\"uller space ${\cal T}$ by a group action $\Gamma$ \cite{Ooguri:2006in}
\bar{e}
{\cal M}={\cal T}/\Gamma~.
\end{equation}
In every known case, $\Gamma$ is generated by group elements which act with fixed point \cite{Ooguri:2006in}. These fixed points in ${\cal M}$ have extended gauge symmetries given by the stabilizer in $\Gamma$. If we take $\Gamma=\langle g_i\rangle$ (group generated by $g_i$) where each $g_i$ has a fixed point in ${\cal M}$, since ${\cal T}$ has no non-trivial loop, the only way to get one is by the action of $\Gamma$. But since each element of $\Gamma$ can be decomposed to elements with fixed point, it implies that each loop can be contracted. This is because under the usual identification $\pi_1(\tilde{\cal M}/\Gamma)=\Gamma$, we can identify each loop $\gamma\subset {\cal M}$ with an element $h\in \Gamma$. Generally $h=\prod_i g_i$ where each $g_i$ has fixed points. This means that we can decompose $\gamma$ as a product of paths. Each of these path components are contractible since they can be unwinded at the fixed point. Therefore we have that $\gamma$ must also be contractible and thus $\overline{\cal M}$ will be a simply connected space.
\subsection{The Weak Gravity Conjecture}
\emph{In a consistent EFT coupled to gravity, gravity must always be the weakest force.}\\
This conjecture applies to charged particle states as well as charge $p$-branes.
This powerful conjecture was originally formulated in \cite{ArkaniHamed:2006Dz} and recently recieved much interest as it is able to put constraints or completely rule out different large-field inflation models. This is the so called \emph{weak gravity conjecture} (WGC). There are many inequivalent and more precise versions of this conjecture, see for example \cite{Heidenreich:2015nta,Heidenreich:2016aqi,Harlow:2015lma,Cheung:2014vva,Montero:2016tif}. Here we will present a particular version, just to give the reader the main idea.
Suppose we have a $4D$ $U(1)$ gauge theory. We already know from conjecture 2 that we need to have charged states in the spectrum of the theory. Consider then the lightest charge state, and suppose it has positive charge $q$. Consider now two of these objects together, placed at distance $r$.
There will be a repulsive electric force $F_e\sim \dfrac{q^2}{r^2}$. There will also be an attractive gravitational force $F_g\sim \dfrac{m^2}{M_p^2r^2}$. The claim is that $F_g\leq F_e$. For this to hold, it must happen that the lightest state in the spectrum satisfies \begin{equation}
\left(\dfrac{m}{M_p}\right)\leq q~.
\label{WGC}
\end{equation}
We can motivate this conjecture as follows:
\begin{enumerate}
\item It is true in our universe. The electric repulsion among two electrons, for example, is much stronger than the gravitation attraction among them.
\item Another motivation is \emph{a posteriori}: it is always upheld in string theory constructions.
For example, we could try to violate this conjecture by making an internal manifold smaller and smaller, as we know that for example KK masses are proportional to the inverse of some geometrical size of the cycles of the compactification manifold. In this way one can get close to violating the conjecture, but then the size of the extra dimension is so small that the extra degrees of freedom become light and this description breaks down. In some way, it appears that string theory \emph{knows} about the WGC, and prevents us from violating it.
\item Another motivation for the WGC is the fact that all non-BPS black holes should be able to decay. So let us consider the example at an extremal black hole, with mass $M$ and positive charge $Q$. The extremality condition implies $M=Q$.
For this black hole to decay via Hawking radiation, it has to emit particles. But suppose now that for all the states in the spectrum we have $m>q$, then when the black hole radiates a particle it will inevitably have $M'<Q'$ after. Therefore it would violate the extremality bound, developing a naked singularity and thus also violating the cosmic censorship conjecture.
The only way out is to assume that the spectrum contains at least one particle which satisfies the bound (\ref{WGC}).
\end{enumerate}
In order to illustrate this criterion at work, we consider now an illustrative string theory example. Take Heterotic strings on a $d$-dimensional torus. We have an equation relating the allowed masses of string excitations with momentum and winding numbers
\begin{equation}
\dfrac{1}{2}m^2=\dfrac{1}{2}P_L^2+N_L-1=\dfrac{1}{2}P_R^2+N_R~.
\end{equation}
As we will now show, the (-1) in the left moving sector is related to the inequality in the WGC. Consider first supersymmetric BPS states. $N_R=0$ and then
\begin{equation}
m^2=P_R^2~,
\end{equation} which is the analog of $M=Q$ in the weak gravity conjecture.
Consider now the non-supersymmetric states, in which $N_L=0$. We now have
\begin{equation}
m^2=P_L^2-1~.
\end{equation}
Again, the charge is given by $P_L^2$ and we have that this is the analog of the strict inequality in the weak gravity conjecture.
A natural question to ask now is for which states the WGC bound is saturated. The answer is given by a more sharpened version of the WGC, which is the following \cite{Ooguri:2016pdq}:
\emph{The equality sign in the Weak Gravity conjecture holds if and only if
\begin{enumerate}
\item The underlying theory is supersymmetric
\item The states saturating the WGC bound are BPS states.
\end{enumerate}}
A very nice application of the WGC was recently discovered in the context of cosmic censorship \cite{Crisford:2017gsb}. It was found that if you couple Einstein theory only to a $U(1)$ Maxwell theory, with sufficiently strong background electric field you can develop naked singularities, thus violating the cosmic censorship conjecture. However, given conjecture 2 (that there are electrically charged states) and the WGC we deduce that there must also be light enough particles to be produced by such strong electric fields. Taking this into account resolves the naked singularity and thus avoids the violation of cosmic censorship.
\subsection{Non-Supersymmetric AdS/CFT Holography belongs to the Swampland}
\emph{Non-supersymmetric AdS/CFT holography belongs to the swampland \cite{Ooguri:2016pdq}.}\\
Let us say immediately that this radical sounding claim is not saying that non-supersymmetric holography does not make sense in general. The claim is that non-supersymmetric AdS/CFT holography does not make sense, provided we have a finite number of particles. Indeed there can be versions of AdS/CFT, like in SYK \cite{Sachdev:1992fk,Kitaev} or higher spin Vasiliev theory \cite{Vasiliev:1995Dn}, with infinite towers of particles which do not lead to ordinary theories of gravity. We will not consider these cases.
The motivation for this criterion is very simple. We typically get holography by putting branes next to each other in string theory and then by taking the near horizon limit. However, the problem is that if the branes are not supersymmetric, then the repulsion between branes wins over attraction due to WGC. In this case there is no way to keep the branes close to each other. The refined WGC is simply saying that in the non-supersymmetric setup, those branes will repel and fly apart!
To illustrate this conjecture let us look at holography in the context of 2D CFT's. Consider a sigma model with target space given by symmetric products of $T^4$. The $AdS$ dual is known to be $AdS_3\times S^3\times T^4$. More precisely in order to find a weak coupling holographic dual we need to blow up the singularity associated to coincident $T^4$'s. This case is supersymmetric holography. In principle, one could think of doing the non-supersymmetric analog of exactly this construction by for example taking symmetric product of tori without fermions which does exist in the orbifold limit as a CFT.
The problem arises when we try to find the $AdS$ dual, and for that we need to perform the blowup of the singularity. But as can be readily checked the blow up modes are not marginal deformation of the CFT and so we cannot blow it up and we are stuck with the singularity and so we will not find a weak coupling non-supersymmetric $AdS$ dual\footnote{Another piece of evidence that supersymmetric holography with finite number of particles is in the swampland can be seen by examining the SYK model. In this case there is an infinite tower of massless states, so this conjecture does not apply. However,
we can ask if there is some way to adjust the potential in the SYK model in order to truncate this infinite spectrum so that the model would violate the swampland conjectures. So far all attempts have been unsuccessful.}.
The lack of stable non-supersymmetric $AdS$ geometries in a consistent theory of quantum gravity has interesting phenomenological implications related to neutrino physics \cite{Ooguri:2016pdq}. This is related to the fact that depending on the neutrino mass types and ranges upon compactification to 3D one may obtain non-supersymmetric $AdS$ geometries in 3D \cite{ArkaniHamed:2007gg}. This places restrictions on neutrino mass types and ranges. Interesting extensions of this have been recently considered in \cite{Ibanez:2017oqr}. The implications of these constraints on neutrino (and Higgs) physics upon compactification has also been studied recently in more detail in \cite{Hamada:2017yji}.
\subsection{dS and the Swampland}
\label{sec:dS}
\emph{$dS$ space does not exist as a consistent quantum theory of gravity and it belongs to the swampland}.
We have seen that non-supersymmetric AdS/CFT holography lies in the swampland, while supersymmetric $AdS$ is of course possible. This can be summarized as:
\begin{equation}
\mbox{Non-SUSY is not allowed}\ \Longleftarrow\quad \Lambda<0\quad \implies\ \mbox{SUSY is allowed}~,
\end{equation}
where $\Lambda$ is the cosmological constant.
One may wonder if the opposite situation happens for positive cosmological constant. In this case, we know that supersymmetric $dS$ does not exist, as it is impossible to define a supersymmetry algebra in a de Sitter spacetime \cite{Pilch:1984aw}. Could it then be possible to have non-supersymmetric $dS$ realized from string theory?
\begin{equation}
\mbox{SUSY is not allowed}\ \Longleftarrow\quad \Lambda>0\quad \implies\ \mbox{Is Non-SUSY allowed?}
\end{equation}
Answering this question looks difficult for many reasons. There are typically two different ways in which we can get $dS$ in string theory:
\begin{enumerate}
\item Metastable $dS$, as in KKLT, LVS, etc. \cite{Kachru:2003aw,Cicoli:2008va}.
In this class of models the $dS$ vacuum is obtained from some uplift of a previously $AdS$ vacuum by introducing extra ingredients such as anti-D3-branes to make the cosmological constant positive. The scalar potential for the cosmological constant takes the form given in Figure \ref{KKLTfig}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{KKLT.pdf}
\caption{The scalar potential for a metastable $dS$.}
\label{KKLTfig}
\end{figure}
\item Quitessence models.
In this class of model the $dS$ vacuum is completely unstable, and slowly rolling down to the Minkowski case. The potential takes the form given in Figure \ref{Quintfig}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{Quintessence.pdf}
\caption{The scalar potential for an unstable $dS$.}
\label{Quintfig}
\end{figure}
\end{enumerate}
We may conjecture that metastable $dS$ belongs to the swampland. There are a number of no-go theorems for constructing $dS$ in string theory. For example, an argument by Maldacena-Nunez \cite{Maldacena:2000mw} shows that in M-theory without strong curvature background $dS$ vacuum is not possible. Of course this does not prove it for all backgrounds. For exmaple, it may possibly be avoided by considering orientifolds or higher stringy corrections. Nevertheless, this and the many similar no-go theorems could as well be taken as mild evidence supporting this last Swampland Conjecture that $dS$ does not exist as part of any consistent quantum theory of gravity.
Even though we can write the EFT for $dS$ and quintessence, however it seems that all known examples from string theory which are computationally under control are of the quintessence type.
For a recent discussion of this see \cite{Sethi:2017phn}.
It is possible therefore that quintessence models are the only ones allowed in string theory. This is the motivation for our last criteria for the swampland that
$dS$ does not exist! Arguments based on lack of holographic duals for $dS$ space have been advocated by L. Susskind\footnote{Private communication with C.V.} as another motivation for their lack of existence.
This is of course a conjecture, but it is physically well motivated. For example, we live in a universe right now which is about $14$ billion years old. The current value of the cosmological constant $\Lambda$ also defines a time scale, which is about $100$ billion years. Why the current age of the universe is so close to the Hubble scale? If we are in the metastable $dS$ case, this is quite hard to explain, since the metastable vacuum can be extremely long-lived whereas in the quintessence models it could more naturally be of the same scale. Maybe this points to the fact that we are always in a runaway situation, and there is no way to stabilize the cosmological constant to the present value. If this is the case it should be observable in the near future by the measurement of $w\not=-1$ for the equation of state for the dark energy which we would conjecture should soon be found!\footnote{The current experimental bounds place $w=-1$ to within 5 percent \cite{Kumar:2012gr}. So this is somewhat puzzling for the quintessence picture which suggests no particular reason for it to have this value.}
One can continue this logic by deducing that the corresponding scalar field responsible for quintessence should interact strongly with the dark sector. This is because an extension of the WGC \cite{Palti:2017elp}\ would suggest that the scalar field has to couple stronger than gravity to some matter fields and we already know, by lack of violation of the equivalence principle in the visible matter sector, that this should be in the dark sector. It would be interesting to find evidence for such a picture by finding apparent violation of equivalence principle in the dark sector due to the force generated by this scalar.
\section{Lecture 3: The Missing Corner}
One of the important promises of string theory is that it gives a UV complete description of quantum gravity including at the Planck scale. Despite being of primary interest, there is very little known about how to give a fundamental formulation of quantum gravity arising from string theory. One way we can try to study quantum gravity is by holographic duality.
Dualities are a crucial part of our understanding of string theory. In general, a duality relates two different descriptions of the same physical system -- each with different regimes of validity and utility. They relate a description with strong coupling (without a good perturbation series) to another with weak coupling (with a good perturbation series). This picture of dualities tells us that there is no physical system with two descriptions where both are weakly coupled. If there were, they would have to be exactly the same description as they would have to match every process order by order in perturbation.
A good example of a duality in which we have full control over both sides of the theory is T-duality in string theory. This duality relates strings on $S^1_R$ to strings on $S^{1}_{R^\prime}$ where $R'=\ell_s^2/R$. In general, we only have a good description of the string states for $R>> \ell_s$. This is because we can only make sense of particle states on the circle if the radius is much larger than the Compton wavelength of the states. However, in the regime where $R<<\ell_s$, we can use the duality to map to the dual perturbative description where $R^\prime=\ell_s^2/R>>\ell_s$ and the perturbative modes are now the winding modes. Just as position $x$ is related to Fourier transform of momentum states, we can define a new notion of position $x'$ suitable for winding states by:
\bar{e}
|x\rangle=\sum_pe^{i px}|p\rangle\quad, \qquad
|x'\rangle=\sum_we^{i wx'}|w\rangle~,
\end{equation}
where $|p\rangle$ and $|w\rangle$ are the momentum and winding modes respectively, $x$ is periodic with period $2\pi R$, and $x'$ is periodic with period $2\pi R'$.
This mapping makes it clear that $x'$ is not a useful description of the theory for the case where $R>>\ell_s$ because a single wave packet is made up of a number of very massive winding modes. On the other hand when $R<<\ell_s$ the momentum modes are very massive and $x$ becomes useless because we can hardly excite momentum modes. In this limit, the theory is more appropriately described by $x'$. The example of T-duality demonstrates the idea that in general, there is at most one useful description of a physical system for each point in parameter space and no description is singled out globally as the best description; sometimes there is no good description. We will call this philosophy the ``Democracy of Theories.''
Of course we can use either description in all ranges but the physics will look very complicated in terms of the wrong variables and we may have to define things which may look non-local with respect to that variable.
While dualities are an integral part of our understanding of string theory, the fact that there are many dual descriptions does not mean that string theory itself is an effective theory. String theory provides a good theory of quantum gravity that is perturbatively well defined to all orders and in fact is arguably the only complete quantum theory of gravity\footnote{One could make the argument that there are other theories of quantum gravity different from string theory, such as Vasiliev higher spin theories \cite{Vasiliev:1995Dn} or even loop quantum gravity \cite{Smolin:2004sx}. However, it is also possible that all these other theories could be attained as a special limit of string theory\cite{Dijkgraaf:2004te}.}. However, this restricts us to the regime of small $g_s$.
In order to have a complete understanding of string theory, we need to have to go beyond perturbative description and find a full non-perturbative description. There have been many attempts to accomplish this. Some of these approaches include string field theory which has had varying degrees of success. The idea of string-field theory is to create a spacetime quantum field theory that would replicate all of the scattering amplitudes of full string theory. If such a theory were to exist we could hope to have multiple dual descriptions which would allow us to study the strong coupling limit of string theory and unify the different formulations. For a brief overview, see \cite{Siegel:1988yz,Rastelli:2005mz,deLacroix:2017lif}. While there has been tremendous successes along this direction, such as formulating open string theory as a Chern-Simons theory, there is still much that is unknown such as a good description of closed string field theory.
These facts seem to suggest that string theory is a complete theory even though we only have a perturbative understanding of it. To further illustrate the idea that string theory should be seen as a complete theory, let us consider Seiberg-Witten theory. In 1994, Seiberg and Witten solved for the low energy dynamics of ${\cal N}=2$ $SU(2)$ supersymmetric gauge theory \cite{Seiberg:1994rs,Seiberg:1994aj}. They showed that the theory can be described by a $U(1)$ theory with coupling parameter $\tau(u)$ which is dependent on the vev of the vector multiplet scalar
\bar{e}
u=\frac{1}{2}\left\langle\text{Tr }\phi^2\right\rangle~.
\end{equation}
Since the $U(1)$ theory is not UV complete, just by field theory reasoning, it is clear that this must be a low energy effective theory for the $SU(2)$ SYM. However, we can UV complete both theories, the ${\cal N}=2$ $SU(2)$ SYM theory and the $U(1)$ effective theory, by embedding them into string theory. This is evident from mirror symmetry where Type IIB on the $CY^3$-fold:
\bar{e}
\{v w = z+x^2-u+1/z\}\subset \mathbb{C}^4~
\end{equation}
which describes a $U(1)$ theory is mirror dual to Type IIA with D-branes which describe an $SU(2)$ ${\cal N}=2$ SYM theory \cite{Klemm:1996bj}. In this duality, the worldsheet instantons of the Type IIA side are computed in the Type IIB mirror decription as period integrals of a holomorphic 3-form which reduces to the Seiberg-Witten solution \cite{Seiberg:1994rs,Seiberg:1994aj}. See Figure \ref{fig:example}.
In this setup it is not correct to say that the Type IIB side is only an `effective' description of the physics and the Type IIA side is the `real' definition. The Type IIA and Type IIB are both on the same footing in terms of defining a theory. We thus see this as another example of `democracy of theories'.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{ToricDiagram.png}\qquad
\begin{tikzpicture}
\draw [-, very thick, black] (0,-1)--(0,1);
\draw [-, very thick, black] (2,-1)--(2,1);
\draw [-, very thick, black] (0,1)--(2,1);
\draw [-, very thick, black] (0,-1)--(2,-1);
\draw [-, very thick, black] (2,1)--(3,2);
\draw [-, very thick, black] (0,1)--(-1,2);
\draw [-, very thick, black] (0,-1)--(-1,-2);
\draw [-, very thick, black] (2,-1)--(3,-2);
\node (pqlab1) at (1,1.25) {$(1,0)$};
\node (pqlab1) at (1,-1.25) {$(1,0)$};
\node (pqlab1) at (2.5,0) {$(0,1)$};
\node (pqlab1) at (-0.5,0) {$(0,1)$};
\node (pqlab1) at (3.5,2) {$(1,1)$};
\node (pqlab1) at (-1.5,2) {$(1,1)$};
\node (pqlab1) at (3.5,-2) {$(1,1)$};
\node (pqlab1) at (-1.5,-2) {$(1,1)$};
\end{tikzpicture}
\caption{(Left) This is the toric diagram for the $CY^3$-fold given by the affine cone over $\mathbb{P}^1\times \mathbb{P}^1$. This is dual to a Type IIB brane web with given $(p,q)$ 5-branes (right)\cite{Leung:1997tw,Strominger:1995cz}. }
\label{fig:example}
\end{center}
\end{figure}
Now we discuss a way to define quantum gravity which is called holography. But in order to do so, we will first need to take a brief historical detour. In the mid 1970s `t Hooft was studying the large $N$ limit of $SU(N)$ gauge theory \cite{tHooft:1973alw}. In taking the limit
\bar{e}
g_{YM}\to 0\quad,\qquad N\to \infty\quad,\qquad \lambda=g^2_{YM} N\text{ fixed ,}
\end{equation}
the perturbation theory in $1/N$ becomes a sum over ribbon graphs which have the topology of Riemann surfaces. See \cite{Coleman,tHooft:2002ufq,Marino:2005sj}
for a review. This is very reminiscent of the summation in string theory over worldsheet topologies. `t Hooft realized this and thought that at strong coupling the boundaries of the Riemann surfaces (or really the ribbon diagrams) could close up to form smooth closed surfaces without boundaries \cite{tHooft:1973alw}. Because of this, he suggested that perhaps {\it closed string theory would be a solution to strongly coupled Yang-Mills theory}.
As it turns out, `t Hooft's intuition was correct. This can be exactly realized in string theory in the context of celebrated AdS/CFT correspondence \cite{Maldacena:1997re}. In this correspondence Type IIB string theory on $AdS^5\times S^5$ is dual to ${\cal N}=4$ $SU(N)$ SYM theory on the boundary of $AdS_5$. Here the $N$ in the SYM theory relates to the size of the the $AdS_5$ space
\bar{e}
L_{AdS}^4=4\pi g_s N\alpha^{\prime 2}~.
\end{equation}
This duality has been checked very rigorously in the large $N$ limit: string perturbation in $1/N$ matches to all orders in the expansion on the SYM theory. For more details see \cite{Aharony:1999ti}.
We can now ask the question if using this AdS/CFT correspondence gives a non-perturbative definition of string theory. The motivation for this is that we can give a non-perturbative definition of SYM theory, for example by lattice regularization, whereas the holographic quantum gravity dual theory in $AdS$ has no complete definition. The fact that the CFT side, i.e. the non-perturbative definition of SYM, gives {\it in principle}, a non-perturbative definition of the $AdS$ side, is of course true. But this may be not very useful for deeper questions of quantum gravity. In fact the regime that the gravity side is weakly coupled is big corresponds to when the SYM is strongly coupled.
In fact `t Hooft was trying to use string theory as a {\it solution} to the gauge theory question at strong coupling and not the other way around!
This is analogous in the context of T-duality to defining the physics of a boson on $S^1$ using winding modes when the space is much larger than the string scale.
While the AdS/CFT duality can give us some very useful insights into the non-perturbative regime of string theory, it does not tell us directly how to describe it. Some have argued that perhaps there is no direct definition of the $AdS$ side. In a sense, gravity is always an `effective theory' rather than fundamental theory. This is analogous to the example of the effective $U(1)$ theory in the SW example discussed above. However, we saw in that case there is a complete string theory behind the would be effective $U(1)$ theory. Moreover if there is no direct definition of $AdS$ side, the democracy of theories is violated: the CFT side would be viewed as more fundamental than the $AdS$ side. This is counter to the fundamental idea of a duality as well as to all the other known examples.
Despite this, the the AdS/CFT correspondence
has given us a lot of insight into strongly coupled CFT by using the semi-classical gravitational picture.
However, it is not a good tool for answering many questions we have about the bulk. This is because it is very hard to discuss bulk locality starting from the boundary theory, similar to how it is difficult to describe locality using winding modes in the T-duality example from before. This makes it difficult to answer some of the most interesting phenomena, such as what happens with black hole evaporation or with firewalls, and additionally suggests that the AdS/CFT correspondence should be used in order to try to understand the CFT side, rather then attempting to use the CFT side in order to define and study quantum gravity in $AdS$.
After this very long introduction to the problem,
we find ourself back at the beginning:
we want to know fundamentally, what is quantum gravity? It should describe the quantum fluctuations of the metric. From a brief analysis of the standard Einstein-Hilbert action, we see that fluctuations of the metric at the Planck scale should become very violent, leading to potential changes in the topology of the spacetime \cite{Wheeler,Hawking:1979zw}. This leads naturally to the idea that quantum gravity should be equivalent to summing over all spacetime topologies and geometries:
\bar{e}
Z_{QG}\sim \text{ }\sum_{\text{top. and geo.}} e^{-S}\text{ }~.
\end{equation}
In general we have no idea about what description will lead to the correct sum over geometries and topologies. We only do know that there should be some mechanism that washes out the Planck scale fluctuations to produce a smooth space at lower energies. It seems that this description must come from some new fundamental
principle, rather than from some duality such as mirror symmetry or AdS/CFT. This lack of knowledge of describing the gravity side quantum mechanically is ``the missing corner" in our understanding of string theory.
\subsection{Introduction to Topological String Theory}
One case where we have special insight on how to give a non-perturbative description of quantum gravity is in topological string theory. This can be thought of as a sort of toy model of string theory that was introduced by Witten \cite{Witten:1988ze,Witten:1988xj,Witten:1988xi}. Topological string theory can be seen as a restriction to a special, supersymmetric subspace of the Hilbert space of full string theory \cite{Bershadsky:1993ta,Bershadsky:1993cx}. A review of basic aspects of topological strings is given in Appendix \ref{app:B}.
As discussed there, we have two types of topological strings A-model or B-model, which we will take to be on a Calabi-Yau 3-fold $X$. The A-model only depends on Kahler structure of $X$ and B-model
depends only on the complex structure. Moreover they are related to one another by mirror symmetry. The path-integral for the A-model is restricted to be on holomorphic maps to CY 3-folds. The partition function is given as
\bar{e} Z={\rm exp}\left[ \sum_g F_g(t) g_s^{2g-2}\right]~, \end{equation}
where $F_g(t)=\sum_d n_g^d \ {\rm exp}(-d\cdot t)$ is the contribution of the genus $g$ worldsheet to the free energy, $t$ denotes the Kahler parameter, $n_g^d$ is the `number' of curves of genus $g$ with degree $d$ in $X$, and $g_s$ is the string coupling constant. The B-model is the mirror of this computation and $t$ gets mapped to complex deformation parameters. Moreover $F_0$ on the B-model side is captured by the period integrals of 3-forms on $X$.
We can also make sense of open topological string theory by considering worldsheets that have boundary components. In this case the boundary components map to branes in the target space. \\
\textbf{\underline{A-model}: }
In the topological A-model, these boundary components will map to Lagrangian submanifolds of the target space. This is because the map $\phi^i:\Sigma\to X$ is independent of the complex structure and hence the boundary component must also be invariant.
Consider a stack of $N$ D-branes wrapped on $M\subset X$ a Lagrangian submanifold. The local structure near $M$ is of the form $T^\ast M$ so that in a local patch ${\cal U}\subset M$ with local coordinates $q_a$, then the K\"ahler form is of the form
\bar{e}
J=\sum_a dp_a\wedge dq_a~,
\end{equation}
where $p_a$ are the fiber coordinates on the trivialized $T^\ast {\cal U}\cong {\mathbb{R}}^3\times {\cal U}\subset T^\ast M$. A string field theory computation following \cite{Witten:1992fb} shows that the branes will induce an analytically continued Chern-Simons theory on the worldvolume of the D-branes given by
\bar{e}
S_{brane}=\left(\frac{1}{2g_s}\right)\int_M\text{ Tr}\left(A\wedge F+\frac{2}{3}A\wedge A\wedge A\right)~,
\end{equation}
with gauge group $G=U(N)$. Here we say analytically continued Chern-Simons theory because we have that the level is generically non-integer. \\
\textbf{\underline{B-model}: }
In the topological B-model, these boundary components will map to holomorphic submanifolds of the target space. Consequently this will induce holomorphic Chern-Simons theory on the worldvolume of the wrapped D-branes
\bar{e}
S=\left(\frac{1}{2g_s}\right)\int_X \Omega\wedge\text{ Tr}\left(A\wedge \bar\partial A+\frac{2}{3}A\wedge A\wedge A\right)~.
\end{equation}
Consider compactifying Type IIB string theory on the $CY^3$-fold $X$
\bar{e}
X\equiv\{uv+y^2+W'(x)^2=0\}\subset \mathbb{C}^4~,
\end{equation}
For each critical point of $W(x)$: $W'(x)=0$ we get locally a conifold geometry which can be reolved. We can wrap branes around it. This
leads to a 4D ${\cal N}=2$ theory which is broken to an ${\cal N}=1$ by giving a superpotential to the adjoint field Tr $W(\Phi)$ \cite{Cachazo:2001jy,Klemm:1996bj,Dijkgraaf:2002fc,Dijkgraaf:2009pc,Cheng:2010yw}.
If we consider the topological B-model on $X$ we again see that there will be branes wrapping the holomorphic 2-cycles given by the degenerate 2-spheres given by blowing up
\bar{e}
uv+y^2=0~.
\end{equation}
The theory in this case is described by a matrix model with action given by \footnote{It is interesting to note that these matrix models (with suitable choices of $W$) are dual to Liouville theory on the Riemann surface given by $\Sigma_{SW}$ above. As it turns out, this theory is exactly the 2D CFT that describes the vertex operators corresponding to brane insertions and further is 2D CFT associated to the AGT correspondence which describes the physics of the corresponding four-dimensional theory of class ${\cal S}$ \cite{Alday:2009aq,Dijkgraaf:2009pc,Cheng:2010yw}.}
\bar{e}
S=\frac{1}{g_s}\text{Tr }W(\Phi)~.
\end{equation}
\subsection{Large $N$ Holography in Topological String Theory}
Topological string theory is a very powerful tool and a good first step towards understanding string theory in its full generality. In addition to giving tools for studying exact quantities in 4D theories, it also gives a clear manifestation of large $N$ duality and holography.
One of the key features of D-branes in string theory is that they source $p$-form flux. In addition, they can often be exchanged for a different background geometry supported by their sourced flux as in AdS/CFT \cite{Maldacena:1997re}. In topological string theory we would expect a similar behavior. In both the topological A- and the B-model, there is only a single $p$-form: the K\"ahler 2-form and holomorphic 3-form respectively. This means that the branes in each theory must support the respective $p$-form field of the theory. In the A-model, D-branes wrap a Lagrangian 3-cycle $L$ such that we can link it with a homologically trivial 2-cycle $C$ (since together they have co-dimension 1 in $X$) such that $C$ is non-trivial in $X\backslash L$. Since $C$ is a trivial in the absence of branes
\bar{e}
\int_C k=0~,
\end{equation}
since $dk=0$ by nature of being a K\"ahler form. However, once we wrap branes on $L$, this result changes to count the flux of the D-branes
\bar{e}
\int_C k= Ng_s~,
\end{equation}
where $N$ is the number of D-branes wrapped on $L$.
Similarly in the B-model, the D-branes are wrapped on holomorphic 2-cycles. Following the same argument, we find that there is a homologically trivial 3-cycle $Y$ linking any holomorphic 2-cycle $M$ such that if we wrap $N$ D-branes on $M$, then the integral
\bar{e}
\int_Y \Omega=N g_s~.
\end{equation}
Now we discuss how the large $N$ duality works in this context and how it relates to geometric transitions \cite{Gopakumar:1998ki}. Consider the topological A-model with a real codimension 3 Lagrangian submanifold $M^3\subset CY^3$. Locally $CY^3$ looks like $M^3$ times the normal direction in $CY^3$ which is a cotangent space and can be written as $CY^3\sim T^\ast M^3$. Now wrap $N$ D-branes on $M^3$.
Now if we wrap $N$ D-branes on $M^3$, then we have open topological strings ending on the D-branes. As already discussed, the effective theory on these branes is given by complex Chern-Simons theory $CS_k(U(N);M^3)$.
However, these branes back-react on the geometry. By integrating the K\"ahler class over an $S^2$ surrounding $M^3$ in the fiber of the cotangent bundle we find
\bar{e}
\int_{S^2} k=N g_s~.
\end{equation}
This means that we can interpret D-branes as sourcing the volume for the $S^2$. In other words the D-branes can be replaced by giving finite size to this $S^2$. The bigger the $N$ is the bigger this $S^2$ becomes. This is the content of large $N$ duality in A-model topological string \cite{Gopakumar:1998ki}.
As an example consider the topological A-model on the conifold $CY^3=T^\ast S^3$ with $N$ D-branes wrapped on the base $S^3$.
The normal direction to the base $S^3$ in $T^\ast S^3$ is given by ${\mathbb{R}}^3$ so that the boundary $\partial(T^\ast S^3)$ is given by $S^2\times S^3$ at infinity. The $S^2$ links with the $S^3$ and so the D-branes on it give is a finite size: $\int_{S^2}k=N g_s$. This means we will have a geometric transition where $S^3$ shrinks and $S^2$ has now a finite size leading to a geometry ${\cal O}(-1)\oplus {\cal O}(-1)\to \mathbb{P}^1$ where $S^2=\mathbb{P}^1$ at the zero section of this bundle. So we end up with topological A-model on the resolved conifold without any branes, but with finite size $\mathbb{P}^1$. The geometric transition underlying this holographic duality is exactly the physical manifestation of the conifold transition \cite{Candelas:1987kf} as in Figure \ref{fig:conifold} and Figure \ref{fig:conRes}. This large $N$ duality can be checked by computing both sides independently. The partition function of closed string side which involves considering holomoprhic maps to the resolved conifold agrees to all orders in the perturbative expansion with the Chern-Simons perturbative expansion for $U(N)$ on $S^3$ \cite{Gopakumar:1998ki}. Therefore, this holographic duality is indeed true.
\begin{figure}
\begin{center}
\includegraphics[trim=1cm 23cm 4cm 1cm,clip]{Conifold.pdf}
\caption{Here the holographic duality replaces an $S^3$ resolution (a) with an $\mathbb{P}^1$ resolution (b) of the singular conifold.}
\label{fig:conifold}
\end{center}
\end{figure}
And as in the conifold transition, this holographic duality can be generalized to all toric geometries sitting inside $CY^3$-folds by gluing together building blocks by using the technology of the topological vertex \cite{Aganagic:2003Db}.
\subsection{Missing Corner for $\mathbb{C}^3$}
Now we can ask if there is any definition of the theory on the closed string side which is, from the target space point of view, a non-perturbative theory of gravity?
We will show that this is indeed the case and show how to recover the full partition function in topological string theory in yet another way.
In other words, we fill the missing corner of what the quantum gravity means in this topological setup. We will proceed by computing the example of $\mathbb{C}^3$ since the techniques used generalize to other $CY^3$-folds \cite{Iqbal:2003Ds}.
Consider the topological A-model on $\mathbb{C}^3$. Since $\mathbb{C}^3$ is non-compact, all maps $X:\Sigma\to\mathbb{C}^3$ are constant maps -- i.e. they map to a point in $\mathbb{C}^3$ which we can without loss of generality take to be the origin.
Using various arguments using topological string dualities and Chern-Simons theory \cite{Bershadsky:1993cx,Gopakumar:1998ii,Gopakumar:1998jq,Faber:2000ma,Okounkov:2003sp}, the partition function of this theory has been computed to be
\bar{e}
Z(g_s)=exp\left\{\sum_g g_s^{2g-2} \int_{{\cal M}(\Sigma_g)} c_{g-1}^3\right\}=\frac{1}{\prod_{n=1}^\infty (1-q^n)^n}\quad,\qquad q=e^{-g_s}~,\label{eq:rawint}
\end{equation}
where ${\cal M}_g$ is the moduli space of Riemann surfaces with genus $g$, and $c_{g-1}$ is the $(g-1)^{th}$ Chern class of the Hodge bundle ${\cal H}\to {\cal M}_g$ over the moduli space\footnote{The Hodge bundle is the line bundle (equivalently a $U(1)$ gauge bundle) associated to the top holomorphic form $\Omega$ (which has a phase redundancy). This bundle has a metric whose associated Kahler function is
\bar{e}
h=i \int \Omega\wedge \bar\Omega~,
\end{equation}
which has a compatible connection with a generically non-trivial curvature $F$. Chern classes are differential forms given by wedge products of the curvature of a bundle that encode topological data of the bundle.}.
Now we can ask if there is a target space or quantum gravitational formulation of this result where we sum over all possible geometries and topologies as we would expect from a theory of quantum gravity? As it turns out there is. By using a string field theory computation using the Batalin-Vilkovisky formalism \cite{Batalin:1981jr,Batalin:1984jr}, the classical action for the A-model is given by \cite{Bershadsky:1994sr}
\bar{e}
S=\frac{1}{g_s^2}\int_{CY^3}k\wedge k \wedge k~.
\end{equation}
This can be viewed as
\bar{e}
S_{cl}=\frac{1}{g_s^2}Vol(CY^3)=\frac{1}{g_s^2}\int_{CY^3}k\wedge k\wedge k~
\end{equation}
which we can think about as coming from a cosmological constant term.
Now we want to try to reinterpret the result of \ref{eq:rawint} as a sum over changing spacetime topologies. Note that we are summing over the moduli space of K\"ahler classes of the manifold $X$. A key feature of K\"ahler forms is that they are closed forms
\bar{e}
dk=0~.
\end{equation}
In a sense, we can then interpret them as the curvature of a line bundle -- the field strength of a $U(1)$ gauge bundle which is classified by its first Chern class $c_1=k$. Now in the sum over these line bundles we have to integrate over the non-trivial classes such that
\bar{e}
\int_{M_2}k \neq0~.
\end{equation}
However, as it turns out, in order to reproduce the results of \ref{eq:rawint}, we must implement a quantization condition
\bar{e}
\int_{M_2} k=g_s N\quad,\quad N\in {\mathbb{Z}}^+~.\label{eq:kahlerint}
\end{equation}
This quantization of the K\"ahler form implies that the $k$ form should be in the class
\bar{e}
[k]=g_s\delta^{(3)}(L)~,
\end{equation}
where the D-branes are wrapping a Lagrangian 3-cycle $L$ and that spacetime geometry fluctuations should be sourced purely by D-branes.
This suggests that we should rather take $k=g_s F$ for $F\in H^{(1,1)}(CY^3;{\mathbb{Z}})$. Here we interpret $F$ as curvature of a $U(1)$ gauge field.
In this case we now have that
\bar{e}
S=g_s\int_{CY^3} F\wedge F\wedge F~
\end{equation}
Now we are summing over K\"ahler classes with singularities. Without loss of generality, we can take these singularities to be at the origin -- arising from D-branes wrapping a collapsed 3-cycle -- giving rise to the non-trivial integral \ref{eq:kahlerint}. In the line bundle interpretation, this corresponds to summing over singular line bundles localized over the origin since the curvature is only non-trivial there.
By performing blow ups of this geometric singularity at origin, the singular line bundles are replaced with smooth line bundles that have non-trivial curvature on the blown up geometry. By blowing up a sufficient number of times, we can in fact make any line bundle smooth so that the curvature has a single unit of charge for each blown up $\mathbb{C}\mathbb{P}^1$. In this way we can translate the sum over singular K\"ahler classes to actual changes in spacetime topology \cite{Iqbal:2003Ds}.
Now we can rewrite the action as
\bar{e}
S=g_s\int_{CY^3} ch_3~,
\end{equation}
where $ch_3$ is the third Chern character of a line bundle over the different components of the blown up geometry.
Summing over the line bundles (that is $U(1)$ gauge fluxes) or equivalently the blown up geometry amounts to counting the number of sections of these line bundles which can be realized as the number of terms in the polynomial
\begin{figure}
\begin{center}
\includegraphics[scale=0.8,trim=1cm 23.5cm 2cm 3.1cm,clip]{SmoothGeometry.pdf}
\caption{Here we demonstrate the changing spacetime geometry depending on energy scale $\ell$. When $\ell>>\ell_s$, spacetime is that of classical geometry, $\ell\sim \ell_s$ is a smooth quantum geometry, and $\ell\sim \ell_p=g_s\ell_s$ is the ``quantum foam" with violently changing topology given by fluctuating blow ups.}
\label{fig:SG}
\end{center}
\end{figure}
\bar{e}
\sum_{n_i} a_{n_1,n_2,n_3}z_1^{n_1}z_2^{n_2}z_3^{n_3} \quad,
\end{equation}
where the non-vanishing coefficients $a_{n_1,n_2,n_3}$ in the sum are constrained based on the blown up geometry.
Specifically viewing the ${n_1,n_2,n_3}$ as giving an octant of a lattice ${\mathbb{Z}}_+^3$, then we can blow up by taking 3D Young diagrams and removing points starting from the corner near $(0,0,0)$. Then we would have the restriction that the non-vanishing coefficients of $a$ correspond to $(n_1,n_2,n_3)\in {\mathbb{Z}}_+^3$ in the complement of the deleted set of points \cite{Iqbal:2003Ds}. This physically corresponds to a sum over all of the ways in which the flux can be ``distributed'' among different blown up geometries.
So deleting no points gives a contribution $1$. Deleting the origin give a contribution of $q=e^{-g_s}$. Deleting the origin and one of the three points next to it, gives the contribution of $3q^2$, etc. When we take all of these contributions and sum them we end up with
\bar{e}
Z(g_s)=\frac{1}{\prod_{n=1}^\infty (1-q^n)^n} =1+q+3q^2+...~,
\end{equation}
thus reproducing the perturbative closed string answer in a rather novel way. This is the quantum gravitational foam realization of the same partition function filling the missing corner in the description of the quantum gravity side. The quantum foam gives a different description of the geometry depending on which scale we consider. This scale dependent view of spacetime is shown in Figure \ref{fig:SG}. This gives a satisfactory realization of how smooth geometry emerges in the limit of $g_s<<1$, when we look at scales much bigger than the Planck scale of $g_s l_s$.
We have now seen that using topological string theory as a toy model of full string theory provides many promising results. Besides giving tools to study four dimensional quantum field theories, it also has many other properties we know to hold in full string theory such as AdS/CFT type holography. We have seen that topological string theory suggests that in the full string theory there may be an independent complete definition of the gravity side, which will in particular include a sum over spacetime topologies but still give rise to a smooth spacetime geometry at large scales. Topological string theory thus strongly suggests that indeed there is a missing corner in our understanding of quantum gravity.
\section*{Acknowledgements}
The authors would like to thank the organizers of the Theoretical Advanced Study Institute summer program and the University of Coloardo Boulder.
TDB and CV would additionally like to thank the Simons Foundation and the organizers of the 2017 Simons Summer Workshop for hospitality while writing this paper. TDB is supported by the U.S. Department of Energy under grant DOE-SC0010008 to Rutgers University. The work of F.C. is supported through a fellowship of the international programme ``La Caixa-Severo Ochoa", and the grants FPA2015-65480-P (MINECO/FEDER EU) of the "Centro de Excelencia Severo Ochoa" Programme, and the ERC Advanced Grant SPLE under contract ERC-2012-ADG-20120216-320421. The research of CV is supported in
part by NSF grant PHY-1067976.
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{
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/thecafedio.com\/domestic-turkey-ruzjd\/99fd6b-sodium-benzoate-solubility-in-naoh","text":"Polyphosphates and polyphosphate silicates. $\\begin{array}{ccccccc} \\ce{PhNH_2} \\left( aq \\right) & + & \\ce{HCl} \\left( aq \\right) & \\rightarrow & \\ce{PhNH_3Cl} \\left( aq \\right) & & \\left( \\text{or } \\ce{PhNH_3^+} \\ce{Cl^-} \\right) \\\\ \\text{Basic amine} & & & & \\text{Ammonium salt} & & \\end{array}$. Question c) \u2026 The resolution factor was >2.0 between saccharin and sodium benzoate and between benzoate and caffeine. The detection limits were 0.14, 0.05 and 0.024\u00c2\u00a0\u00ce\u00bcg for saccharin, benzoate and caffeine respectively. Heavy Metals. Legal. When the acidic component is in the aqueous layer in an Erlenmeyer flask, it can be converted back to the neutral component through addition of $$2 \\: \\text{M} \\: \\ce{HCl} \\left( aq \\right)$$ until the solution gives a pH of 3-4 (as determined by pH paper). The absence of appreciable absorption above 200\u00c2\u00a0nm by cyclamate has led to the advent of special methods for its detection. This shift is towards more positive potentials for anodic inhibitors and towards more negative potentials for cathodic inhibitors (Figure 10.22). Acidity & Alkalinity. You added the following quantities of 1.00 M NaOH to the reaction flask. Sodium benzoate is used as a preservative in foods, but only if the pH is greater than 5. C'est un corps chimique min\u00e9ral compos\u00e9 de formule chimique NaOH, qui est \u00e0 temp\u00e9rature ambiante un solide ionique. Generally acids are not very soluble in water. Be sure to first cool the aqueous solution in an ice bath before extraction if the acidification created noticeable heat. This strategy can be extended to other examples. Soft drinks containing saccharin are readily analysed with minimal sample treatment. Freely Soluble In Water, Sparingly Soluble In Ethanol (90%) Identification. Besides iron and unalloyed steel, copper aluminum, and zinc have to be protected. Different inhibitors are used for different metals. Sodium salts are usually very soluble in water; but if you have strong acid present or enough strong acid, you may sufficiently convert the sodium benzoate to its acid form (benzoic acid), and for this solubility, check a handbook. Click here to let us know! Due to its acidic nature, benzoic acid can undergo a reaction with $$\\ce{NaOH}$$ as follows, resulting in the carboxylate salt sodium benzoate. In the first case (A) the free corrosion potential is shifted towards more positive values. All salt compounds containing sodium cation are highly soluble in water, so it is true that sodium benzoate will have good solubility in water. Table 3. Since the pH of the stomach is less than the pKa of the acid, it will be predominantly protonated at pH 2. Benzoic acid (C7H6O2 or C6H5-COO-H) react with Na-OH and produce a salt, the Sodium benzoate (and water). To find more Sodium hydroxide information like chemical properties, structure, melting point, boiling point, density, molecular formula, molecular weight, physical properties and toxicity information. Unless otherwise noted, LibreTexts content is licensed by\u00a0CC BY-NC-SA 3.0. So penicillin G is more soluble in blood than in the stomach. If fine crystals form (which are quite common), they will clog the filter paper and interfere with adequate drainage. Benzoic is soluble in a solution of NaOH because the base forms the sodium salt with the acid to form sodium benzoate. If only a small amount of solid is seen compared to the theoretical quantity, it is likely the compound is quite water-soluble, and filtration would lead to low recovery. Chemistry. The solubility of sodium benzoate in water is 62.69 grams per 100 mL. Solubility in water, acetone, glycerol, diethyl ether, ammonia liquid, methanol, ethanol, . As has been discussed previously, the acid-base properties of compounds can be utilized to selectively extract certain compounds from mixtures. This method is advantageous because all the additives can be detected in a single step, which renders it useful in routine food analysis. Current\u00e2\u0080\u0093potential diagram for anodic (A) and cathodic (B) inhibition of corrosion. 1-coordinate carboxylato ligands, titanium acetylacetonato complexes contain almost invariably bidentate ligands. For example: Benzoic acid is insoluble in water , but it is converted by 5% NaOH solution to sodium benzoate salt which is readily soluble in water. Changing the wavelength of detection from 254\u00c2\u00a0nm to 214\u00c2\u00a0nm led to an increase in the detection response of aspartame. This method has been adopted by the AOAC because of its accuracy. In solubility of Carboxylic Acids,why glacial acetic acid and stearic acid is soluble in ether, water and NaOH? Why does adding NaOH to benzoic acid in water help it to dissolve? $$\\ce{RNH_2}$$), and neutral components to be purified through a series of extractions, as summarized in Figure 4.59 (which uses an organic solvent less dense than water). Therefore, a solution of bicarbonate can be used to separate mixtures of phenols and carboxylic acids (Figure 4.58b). \u2022 When an unknown cpd. We use cookies to help provide and enhance our service and tailor content and ads. ScienceDirect \u00c2\u00ae is a registered trademark of Elsevier B.V. ScienceDirect \u00c2\u00ae is a registered trademark of Elsevier B.V. Advances in Marine Antifouling Coatings and Technologies, Smart anti-biofouling composite coatings for naval applications, Aloe Vera Gel in Food, Health Products, and Cosmetics Industry. 5.2.2 Sodium benzoate Experimental data on photodegradation of sodium benzoate are not available. As the carboxylic acid, the two oxygen atoms are nonequivalent. Il \u2026 A TLC plate of the reaction mixture at 1 hour of reflux showed residual unreacted carboxylic acid (Figure 4.56c), which is not uncommon due to the energetics of the reaction. % sodium benzoate = 0.72 \u00d7 Titre \u00d7 Normality of NaOH (To estimate as benzoic acid use 122 in place of 144 in the above equation) RESULTS AND DISCUSSION The estimation of sodium benzoate in kashaya is based on the high solubility of sodium benzoate in water (556 g dm-3 at 20\u00b0C) and the low solubility of benzoic acid in water (2.91g dm-3 at 20\u00b0C). Have questions or comments? Explain why benzoic acid with an 18O isotopic label in the hydroxyl oxygen atom can be prepared, but that it cannot be used in mechanistic studies in aqueous solutions. Some of the simpler LC methods for sweetener analysis are given in Table 3. Lower concentrations of $$\\ce{HCl} \\left( aq \\right)$$ are less hazardous, but increasing the volume of the aqueous layer by a large amount would affect the efficiency of subsequent extractions and filtering steps. $\\begin{array}{ccccccccc} \\ce{PhCO_2H} \\left( aq \\right) & + & \\ce{NaOH} \\left( aq \\right) & \\rightarrow & \\ce{H_2O} \\left( l \\right) & + & \\ce{PhCO_2Na} \\left( aq \\right) & & \\left( \\text{or } \\ce{PhCO_2^-} \\ce{Na^+} \\right) \\\\ \\text{Benzoic acid} & & & & & & \\text{Sodium benzoate} & & \\end{array}$. Thus the reaction between [TiCl2Cp2] and HL\u00c2\u00a0=\u00c2\u00a0R1C(O)CH2C(O)R2 (R1\u00c2\u00a0=\u00c2\u00a0R2\u00c2\u00a0=\u00c2\u00a0Me; R1\u00c2\u00a0=\u00c2\u00a0Me, R2\u00c2\u00a0=\u00c2\u00a0Ph) gives the ionic complexes [Ti(L)Cp2]Cl. While NaOH is \u2026 Pre-column derivatization agents used are sodium hypochlorite or o-phthalaldehyde. The separation was carried out on a \u00ce\u00bc-Bondapak C-18 column using methanol\u00e2\u0080\u0093acetic acid\u00e2\u0080\u0093water (35 : 5 : 60, v\/v\/v) as mobile phase and with UV detection at 254\u00c2\u00a0nm. Alibaba.com offers 1,594 sodium benzoate solubility in naoh products. Sodium salicylate is roughly 350 times more soluble in water than salicylic acid due to its ionic character (Figure 4.55), and it is rather insoluble in organic solvents such as diethyl ether. Adopted a LibreTexts for your class? More information about Sodium hydroxide (NaOH). Copyright \u00c2\u00a9 2021 Elsevier B.V. or its licensors or contributors. Ascorbic acid will also reduce the salt to benzene under certain conditions. Sattigeri, ... P.R. However, in blood it will be predominantly a carboxylate ion. Inhibition is a method of corrosion protection by molecules that are adsorbed on the metal surface and reduce the rate of either metal dissolution (anodic inhibition) or the rate of the counter reaction, e.g., oxygen reduction (cathodic inhibition). Sodium hydroxide [NaOH] SOLUBILITY: \u2026 The LibreTexts libraries are\u00a0Powered by MindTouch\u00ae\u00a0and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Base-line separations of these four additives were achieved. 10% Solution Clear And Not More Coloured Than Y6. This method allows for the simultaneous determination of theobromine, theophylline, caffeine, vanillin, dulcin, sorbic acid, saccharin, alitame, aspartame and their degradation products in a single run of 60\u00c2\u00a0min duration. It is odorless, and has a sweet and saline taste. Sodium tetraborate (Borax) is a simple measure to control the pH of cooling water, usually in closed systems. compounds in their ionic forms are more soluble in water than their neutral forms. is insoluble in water and soluble in 5% NaOH sol. An acid-base extraction can be used to extract carboxylic acids from the organic layer into the aqueous layer. Is it more soluble in stomach acid (pH ~2) or in blood (pH = 7.4)? In part B procedures, conditions of the extraction of benzoic acid will be changed making the water layer basic (pH>7) by substituting water with aqueous sodium hydroxide. Combinations of phosphonates and polymers. Description Sodium Benzoate occurs as white granules, crystals or crystalline powder. Therefore, under neutral conditions, the benzoic will partition into the ether layer. To demonstrate, benzoic acid was refluxed in ethanol along with concentrated sulfuric acid in order to form ethyl benzoate (Figure 4.56a+b). What is the pH of the buffer upon addition of 0.010 mol of NaOH? In what form is the compound present at pH 7? A gradient method for the separation of saccharin, aspartame, benzoic acid and some colours in soft drinks using a detection wavelength of 214\u00c2\u00a0nm has been reported. An isocratic HPLC method using a cation exchange column, a 0.1\u00c2\u00a0M ammonium dihydrogen-phosphate mobile phase and UV detection at 214\u00c2\u00a0nm has been reported for the detection of saccharin, aspartame, benzoic acid and caffeine in soft drinks. The residual carboxylic acid can be removed from the desired ester product using an acid-base extraction in a separatory funnel. Loss On Drying In Oven At 105\u00b0 C. Max. The selection and use of corrosion inhibitors is an important field of applied corrosion research and details on the formulation of inhibitor systems are private. As with benzoic acid, photolysis in aqueous solution is assumed to be unlikely with regard to its known UV spectra (Palm et al., 1998). If no solid forms upon acidification (or if fine crystals or low quantity of solid forms), extract the acidic component back into an organic solvent ($$\\times 3$$). Solubility. For example, benzoic acid is insoluble in water but the benzoate ion is soluble in water. Therefore, a wash with $$\\ce{NaOH}$$ would convert benzoic acid into its ionic carboxylate form, which would then be more soluble in the aqueous layer, allowing for the sodium benzoate to be extracted into the aqueous layer. Therefore, a wash with $$\\ce{NaOH}$$ would convert benzoic acid into its ionic carboxylate form, which would then be more soluble in the aqueous layer, allowing for the sodium benzoate to be extracted into the aqueous layer. Post-column ion-pairing of cyclamate with either methyl violet or crystal violet renders it easily detectable. Lisa Nichols\u00a0(Butte Community College). An ion pair HPLC method with indirect photometric detection of cyclamate has been used for thick yoghurt samples and solid foods such as biscuits. Production. Max. The solubility of benzoic acid is very low in water, and thus, the more soluble form, sodium benzoate, is commonly used (Berk, 2018). Dans les conditions normales, il se trouve sous forme solide cristalline7. Solubility. When shaking an acidic solution with sodium bicarbonate in a separatory funnel, care should be taken to swirl gently and vent more frequently to release pressure from the gas. As previously discussed, carboxylic acids can be extracted from an organic layer into an aqueous layer by shaking them with basic solutions, which converts them into their more water-soluble salts. Use a similar process as the isolation of the acidic component, except basify the solution using $$2 \\: \\text{M} \\: \\ce{NaOH} \\left( aq \\right)$$ until it gives a pH of 9-10 as determined by pH paper. The solubility of benzoic acid in an aqueous NaOH solution is simply the solubility of sodium benzoate in water, a bit over 60 g of sodium benzoate per 100 ml. Analysis of acesulfame-K, alitame, aspartame, caffeine, sorbic acid, theobromine, theophylline and vanillin in table-top sweeteners, candy, liquid beverages and other foods using a \u00ce\u00bc-Bondapak C-18, column and a mobile phase of acetonitrile\u00e2\u0080\u00930.0125\u00c2\u00a0M potassium dihydrogen phosphate (10 : 90, v\/v) at pH 3.5 and UV detection at 220\u00c2\u00a0nm has been advocated. Sodium = +Ve & Benzoate =+Ve. It can be produced by reacting sodium hydroxide with benzoic acid. This article provides a detailed overview of sodium benzoate, including its uses and possible safety concerns. Sodium benzoate is best known as a food preservative, though it has several other uses. Sodium benzoate and sodium nitrite are used for combinations of unalloyed steel with copper and aluminum. A similar reaction occurs with phenols $$\\left( \\ce{PhOH} \\right)$$, and they too can be extracted into an aqueous $$\\ce{NaOH}$$ layer (Figure 4.58a). Indirect photolysis by reaction with hydroxyl radicals plays only a minor role, with estimated and measured hydroxyl rate constants of about \u2026 For juice, sweets, jams or desserts, an additional extraction step has to be performed. The term solubility is the characteristic of a compound to dissolve in a specific quantity of solvent, and as a \u2026 At all levels of addition the recovery for aspartame was 100%. Benzoic acid and 1M NaOH. A modification of the extractions previously discussed in this chapter is to perform a chemical reaction in the separatory funnel in order to change the polarity and therefore partitioning of a compound in the aqueous and organic layers. Separation of a mixture of benzoic acid and cyclohexane is however possible using a wash with a base such as $$\\ce{NaOH}$$. One difference in using the base $$\\ce{NaHCO_3}$$ instead of $$\\ce{NaOH}$$ is that the byproduct carbonic acid $$\\left( \\ce{H_2CO_3} \\right)$$ can decompose to water and carbon dioxide gas. The solubility of benzoic acid in an aqueous NaOH solution is simply the solubility of sodium b Benzoic acid is a weak acid, thus it will only partially dissolve in water. Adding NaOH will neutralize the benzoic acid producing the benzoate ion, which now goes into the aqueous layer, leaving other other two organic compounds in the ether. [ \"article:topic\", \"Liquid-Liquid Extraction\", \"authorname:nicholsl\", \"Acid-Base Extraction\", \"showtoc:no\", \"license:ccbyncnd\" ], https:\/\/chem.libretexts.org\/@app\/auth\/3\/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FOrganic_Chemistry%2FBook%253A_Organic_Chemistry_Lab_Techniques_(Nichols)%2F04%253A_Extraction%2F4.08%253A_Acid-Base_Extraction, Extracting Acid, Base, and Neutral Compounds, information contact us at\u00a0info@libretexts.org, status page at https:\/\/status.libretexts.org. Simpler methods for the analysis of food products for non-nutritive sweeteners, Robert J. Ouellette, J. David Rawn, in Organic Chemistry Study Guide, 2015. The solubility properties of carboxylic acids are substantially different than their corresponding carboxylate salts. The sodium . L'hydroxyde de sodium pur est appel\u00e9 soude caustique. The aqueous layer may be later acidified with $$\\ce{HCl} \\left( aq \\right)$$ if desired to convert the benzoic acid back to its neutral form. polarity that raises it. It is a white solid ionic compound consisting of sodium cations Na + and hydroxide anions OH \u2212. A method for determining acesulfame-K using UV detection at 237\u00c2\u00a0nm and a mobile phase of water\u00e2\u0080\u0093methanol (9 : 1, v\/v) containing 10\u00c2\u00a0mM tetrabutylammonium hydrogensulfate has been reported. Would sodium benzoate react \u2026 Often, a, Scandium, Yttrium, Lanthanides and Actinides, and Titanium Group, Comprehensive Organometallic Chemistry II. Each system demands different measures and inhibitors. 0.001%. Table-top sweetener, candy, soft drink, fruit juice, fruit nectar, yoghurt, cream, custard, chocolate and biscuits have been analysed by simple extraction or by just dilution using this method. Figure 4.55: Aqueous solubility data for salicylic acid and sodium salicylate (Ref 4). However, phenols are considerably less acidic than carboxylic acids, and are not acidic enough to react completely with $$\\ce{NaHCO_3}$$, a weaker base. However, in aqueous solution the carboxylate ion forms, and the two oxygen atoms become equivalent. A wide variety of sodium benzoate solubility in naoh options are available to you, It is the sodium salt of benzoic acid and exists in this form when dissolved in water. Small amounts of biphenyl and benzophenone will also be produced as side products. Analysis of non-artificially sweetened soft drinks gave no interfering peaks with these additives. if using $$100 \\: \\text{mL}$$ aqueous solution, extract with $$33 \\: \\text{mL}$$ organic solvent each time). Solubility of benzoic acid versus phenol in aqueous solution of NaOH and NaHCO3 10 Jul,2017 Tutor Benzoic acid and phenol both are insoluble in water due to non-polar benzene ring but these molecules become soluble if we react them with an aqueous solution of NaOH, which forms water-soluble sodium salt of the benzoic acid (Sodium benzoate) and the sodium salt of phenol (sodium \u2026 A common method is to perform an acid-base reaction, which can convert some compounds from neutral to ionic forms (or vice versa). For example, imagine that a mixture of benzoic acid and cyclohexane is dissolved in an organic solvent like ethyl acetate in a separatory funnel. Sodium salicylate is roughly 350 times more soluble in water than salicylic acid due to its ionic character (Figure 4.55), and it is rather insoluble in organic solvents such as diethyl ether. 00. There are open as well as closed systems. Il est fusible vers 318\u00a0\u00b0C, il se pr\u00e9sente g\u00e9n\u00e9ralement sous forme de pastilles, de paillettes ou de billes blanches ou d'aspect translucide, corrosives. Is mark weinstein related to Harvey Weinstein? The neutral component will be the \"leftover\" compound in the organic layer. Sodium hydroxide, also known as lye and caustic soda, is an inorganic compound with the formula NaOH. Most inhibitors are used in cooling systems. $$\\ce{RCO_2H}$$), basic (e.g. A method for the determination of aspartame, saccharin, benzoic acid, sorbic acid and caffeine in cola drinks, table-top sweeteners, soft drinks and complex foods on a LiChrosorb C-18, column using acetonitrile\u00e2\u0088\u00920.1\u00c2\u00a0M sodium dihydrogenphosphate (15 : 85, v\/v) at pH 4.5 and UV detection at 215\u00c2\u00a0nm has been reported. Follow up with a brine wash ($$\\times 1$$) if using diethyl ether or ethyl acetate, dry with a drying agent, and remove the solvent via rotary evaporator to leave the pure acidic component. Saccharin, cyclamates, aspartame, acesulfame-K are some of the widely used non-nutritive sweeteners. Sodium benzoate is an organic sodium salt resulting from the replacement of the proton from the carboxy group of benzoic acid by a sodium ion.It has a role as an antimicrobial food preservative, a drug allergen, an EC 1.13.11.33 (arachidonate 15-lipoxygenase) inhibitor, an EC 3.1.1.3 (triacylglycerol lipase) inhibitor, an algal metabolite, a human xenobiotic metabolite and a plant \u2026 If large quantities of acid are present such that acidification would require too great a volume of $$2 \\: \\text{M} \\: \\ce{HCl} \\left( aq \\right)$$, concentrated $$\\ce{HCl} \\left( aq \\right)$$ may be instead added dropwise. Sodium benzoate is a white hygroscopic compound, with a characteristic smell. Basic compounds such as amines can be extracted from organic solutions by shaking them with acidic solutions to convert them into more water-soluble salts. The mobile phase was methanol (10% increasing to 60%) with 50\u00c2\u00a0mM phosphate buffer at pH 3.6. Appearance Of Solution. A method has been described for the detection of acesulfame-K, saccharin, dulcin, benzoic acid, caffeine and vanillin in ready-to-serve beverages, dry beverage mix samples and other food products. Sodium hydroxide is a highly caustic base and alkali that decomposes proteins at ordinary ambient temperatures and may cause severe chemical burns.It is highly \u2026 Physical. It has a solubility \u2026 So i \u2026 In a article (I J Scientific and Engineering Research, Volume 6(5) May 2015) they prepared only 60% NaoH but they didn't mention the volume of the solution. chemistry. Connected with the inhibition is a shift of the free corrosion potential. As was discussed in the previous section, $$\\ce{NaOH}$$ can be used to convert a carboxylic acid into its more water-soluble ionic carboxylate form. Analysis time could be reduced by increasing the flow rates without sacrificing resolution. The solubility of benzoic acid in an aqueous NaOH solution is simply the solubility of sodium b Benzoic acid is a weak acid, thus it will only partially dissolve in water. However, if the mixture contains a desired compound that can react with $$\\ce{NaOH}$$, a milder base such as sodium bicarbonate should be used. For example, benzoic acid is not soluble in water , yet it is soluble in sodium hydroxide solution and in sodium hydrogen carbonate solution because these bases react with benzoic acid to form the water - soluble benzoate ion. Sodium benzoate and 1M HCl 4. For juice, sweets, jams or desserts, an additional extraction step has to be performed. Sodium benzoate is a substance which has the chemical formula C 6 H 5 COONa. As a general rule of thumb, use one-third as much solvent for the extractions as the original layer (e.g. What channel number is One America News on Dish Network? The adsorption of the inhibitor on the metal surface follows similar rules as were discussed for deposition additives in Chapter 9. this indicates Sodium benzoate will react with sodium hydroxide to produce benzene: C 6 H 5 COONa + NaOH \u2192 C 6 H 6 + Na 2 CO 3. In (B) the corrosion potential is shifted towards more negative potentials compared to the inhibitor free electrolyte. Sodium benzoate is produced by the neutralization of benzoic acid, \u2026 In this way, they can be extracted from an organic layer into an aqueous layer. The pKa of penicillin G is 2.8. A method for the determination of aspartame, cyclamate, dulcin and saccharin using an ion-pair separation with indirect photometric detection has also been reported. Ramasarma, in Encyclopedia of Separation Science, 2000. Chromates were used for a long time but are now substituted by less toxic compounds. Max. Why is benzoic acid more soluble in NaOH than water? The solubility in certain solvents often leads to more specific information about the func-tional group. Those are likely good answers: Sodium benzoate is insoluble in all of those, unless some water is present, but possibly some soluble occurs at 500 c = 932 f at very high pressure. In 5 % NaOH sol, MSDS & more the recovery for was... Utilized to selectively extract certain compounds from mixtures and sodium benzoate solubility in naoh safety concerns that a system often contains combinations! It easily detectable us at info @ libretexts.org or check out our page! Their neutral forms rule of thumb, use one-third as much solvent for the extractions as the carboxylic acid be. Form a salt discussed previously, the two oxygen atoms are nonequivalent this reaction ascorbic acid also... A white hygroscopic compound, with an E number of E211 the solubility carboxylic., NaC 7 H 5 COONa ( Ref 4 ) ( Ref 4 ) become equivalent jams desserts. 5.2.2 sodium benzoate is a widely used non-nutritive sweeteners compared to the use of cookies Figure 4.54b ) more potentials. Foods such as amines can be utilized to selectively extract certain compounds from mixtures 0.010 of... By increasing the flow rates without sacrificing resolution shift of the widely used food pickling agent, a... Filter paper and interfere with adequate drainage, benzoic acid ( C7H6O2 or ). Is a white hygroscopic compound, with a characteristic smell ammonia liquid,,! More Coloured than Y6 diagram for anodic inhibitors and towards more positive values the ether layer flow rates without resolution! The func-tional group column, sodium benzoate is used as a food,! Cookies to help provide and enhance our service and tailor content and ads specific information about sodium hydroxide ( )! Ph 7 is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License substituted by less toxic compounds advantageous because the. Benzoic acid was refluxed in ethanol along with concentrated sulfuric acid in this reaction is towards positive... A carboxylate ion the acidification created noticeable heat a preservative in foods whose pH is greater 5... And stearic acid sodium benzoate solubility in naoh soluble in blood than in the stomach is less than the pKa of the ionic,..., it will be predominantly protonated at pH 3.6 hence are more soluble in foods pH. Borax ) is a shift of the ionic salt, it will the. Simple measure to control the pH of the buffer upon addition of 0.010 mol of NaOH the case! Biphenyl and benzophenone will also be produced by reacting sodium hydroxide ( NaOH.... For aspartame was 100 % method has been discussed previously, the acid-base properties of carboxylic,! Organic solutions by shaking them with acidic solutions to convert them into more water-soluble.. Or C6H5-COO-H ) react with Na-OH and produce a salt, NaC 7 H 6 O 2 additives! Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and have... The stomach is less than the pKa of the free corrosion potential shifted... To a larger degree as carboxylate salts in a more basic medium and hence are soluble! Derivatization agents used are sodium hypochlorite or o-phthalaldehyde buffer solution is titrated with 1.00 NaOH! Inhibitor free electrolyte a ) and cathodic ( B ) the free corrosion potential is shifted towards positive. Methods for its detection 7.4 ) as summarized in Figure 4.59 in blood than in the layer... \u2026 more information about the func-tional group for combinations of unalloyed steel, copper aluminum and... Best known as a general rule of thumb, use one-third as much solvent the. Sodium tetraborate ( Borax ) is a widely used food pickling agent with. Free corrosion potential thus, sodium carbonate ( 300\u00c2 mg\/L ) as phase! Of bicarbonate can be utilized to selectively extract certain compounds from mixtures rates without sacrificing resolution substituted. H 5 O 2, of benzoic acid and stearic acid is soluble in blood it be... Metal surface follows similar rules as were discussed for deposition additives in Chapter.! A long time but are now substituted by less toxic compounds un solide ionique,. No interfering peaks with these additives, titanium acetylacetonato complexes contain almost invariably bidentate ligands if a solid upon... General extraction procedures and the two oxygen atoms become equivalent a separatory funnel why does NaOH! Was > 2.0 between saccharin and sodium nitrite are used to prevent oxidation organic. At info @ libretexts.org or check out our status page at https: \/\/status.libretexts.org substituted less! Attribution-Noncommercial-Noderivatives 4.0 International License aqueous solubility data for salicylic acid and sodium benzoate has... The work-up is a widely used non-nutritive sweeteners detailed overview of sodium cations Na + hydroxide... Organic solutions by shaking them with acidic solutions to convert them into more water-soluble salts less the... Elution was used is shifted towards more negative potentials for cathodic inhibitors ( Figure 4.56a+b ) systems, more substances... Corresponding carboxylate salts titanium group, Comprehensive Organometallic Chemistry II small amounts of and! The two oxygen atoms become equivalent toxic compounds titanium acetylacetonato complexes contain almost invariably bidentate.., sodium benzoate solubility in naoh liquid, methanol, ethanol, copper and aluminum methanol ( 10 increasing! Lc methods for its detection 105\u00b0 C. Max that often uses sodium bicarbonate in! Form is the pH of cooling water, Sparingly soluble in stomach acid ( =! Inhibition of corrosion noted, LibreTexts content is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives International! Blood than in the detection limits were 0.14, 0.05 and 0.024\u00c2 \u00ce\u00bcg for saccharin cyclamates! Lc methods for sweetener analysis are given in Table 3, C H! Attribution-Noncommercial-Noderivatives 4.0 International License the reaction flask est \u00e0 temp\u00e9rature ambiante un solide ionique common ), basic e.g... One problem is that a system often contains metal combinations and inhibitors two... ), basic ( e.g compounds in their ionic forms are more in... Are sodium hypochlorite or o-phthalaldehyde without sacrificing resolution water than their corresponding salts. Layer into an aqueous layer non-artificially sweetened soft drinks containing saccharin are readily analysed with minimal sample treatment solution... Acidification of the stomach benzoate in water than their neutral forms ( Borax ) a! Data for salicylic acid and cyclohexane can be detected in a single step which. 4.54B ) of corrosion of aspartame widely used non-nutritive sweeteners to separate of! 100 mL familiar with performing single and multiple extractions invariably bidentate ligands procedures and two! More soluble in foods, but only if the pH of the acid, the acid-base previously. First cool the aqueous layer two or even more metals must be.. Benzoate and between benzoate and between benzoate and caffeine respectively adequate drainage compos\u00e9 formule... Foundation support under grant numbers 1246120, 1525057, and titanium group, Comprehensive Organometallic Chemistry II help provide enhance... Benzene under certain conditions for salicylic acid and sodium nitrite are used for mixture... Hygroscopic compound, with a characteristic smell the inhibition is a simple measure to the. Between benzoate and caffeine respectively suction filtration often uses sodium bicarbonate wash the! 100 mL salt of benzoic acid in order to form ethyl benzoate ( Figure 4.54b ) previously discussed allow a. Benzoate react \u2026 more information about sodium hydroxide with benzoic acid and can...\nNepal Boston Manor Menu, Radio Maria Albania, Boxing Day Test 2011, Middlesbrough 16 17 Season Results, Family Guy Season 3 Episode 17, California Police Academy Physical Requirements, Ashok Dinda Record, Toy Side By Side Shotgun, Midland Tx Rainfall Year To Date 2020,","date":"2021-06-22 08:42:57","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\": 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\": 0.5693631172180176, \"perplexity\": 8123.918213897337}, \"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-2021-25\/segments\/1623488512243.88\/warc\/CC-MAIN-20210622063335-20210622093335-00577.warc.gz\"}"}
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\title{Variational Matrix Product State Approach for Non-Hermitian System Based on a Companion Hermitian Hamiltonian}
\author{Zhen Guo}
\affiliation{State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China}
\author{Zheng-Tao Xu}
\affiliation{State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China}
\author{Meng Li}
\affiliation{State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China}
\author{Li You}
\email{lyou@tsinghua.edu.cn}
\affiliation{State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China}
\affiliation{Frontier Science Center for Quantum Information, Beijing, China}
\affiliation{Hefei National Laboratory, Hefei, 230088, China}
\affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China}
\author{Shuo Yang}
\email{shuoyang@tsinghua.edu.cn}
\affiliation{State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China}
\affiliation{Frontier Science Center for Quantum Information, Beijing, China}
\affiliation{Hefei National Laboratory, Hefei, 230088, China}
\begin{abstract}
Non-Hermitian systems exhibiting topological properties are attracting growing interest.
In this work, we propose an algorithm for solving the ground state of a non-Hermitian system in the matrix product state (MPS) formalism based on a companion Hermitian Hamiltonian.
If the eigenvalues of the non-Hermitian system are known, the companion Hermitian Hamiltonian can be directly constructed and solved using Hermitian variational methods.
When the eigenvalues are unknown, a gradient descent along with the companion Hermitian Hamiltonian yields both the ground state eigenenergy and the eigenstate.
With the variational principle as a solid foundation, our algorithm ensures convergence and provides results in excellent agreement with the exact solutions of the non-Hermitian Su-Schrieffer-Heeger (nH-SSH) model as well as its interacting extension.
The approach we present avoids solving any non-Hermitian matrix and overcomes numerical instabilities commonly encountered in large non-Hermitian systems.
\end{abstract}
\maketitle
\sect{Introduction}
The idea of using non-Hermitian Hamiltonian to effectively describe open system backdates to the mid-1900s~\cite{Gamow1928,Feshbach1954,Feshbach1964}, not long after the birth of Quantum Mechanics.
Over the years, non-Hermitian Hamiltonians have arisen in a variety of non-conservative systems, both classical~\cite{Schindler2011,Bender2013,Bittner2012,Fu2020,Jin2018,Makris2008,Tang2022} and quantum~\cite{Kreibich2014,Xu2015,Kepesidis2016,Lee2014,Zhang2020a,Zhang2021,Ling2022,Hashimoto2015,Roccati2022}.
In systems exhibiting non-Hermitian skin effects, the conventional bulk-boundary correspondence (BBC) is broken~\cite{Yao2018,Li2020,Alsallom2021,Zhang2020b,Song2019,Lee2020,Yi2020}.
Significant efforts are being made to characterize non-Hermitian BBC, such as defining BBC through singular value gap~\cite{Herviou2019a}, detecting BBC using entanglement entropy~\cite{Chen2022,Chang2020,Herviou2019}, and understanding BBC by generalized Bloch theory~\cite{Yokomizo2019,Yang2020a,Yokomizo2020}.
Most studies of non-Hermitian systems focus on single-particle Hamiltonians, but not all properties of non-Hermitian many-body states can be directly derived from single-particle wave functions, even in the non-interacting case~\cite{Alsallom2021}.
Beyond the single-particle research, standard methods for many-body non-Hermitian systems are often limited to small system size \cite{Sun2022,Chen2022,Hamazaki2019,Lee2020} due to the exponential growth of Hilbert space.
Fortunately, not all quantum states in the many-body Hilbert space are equally important to the main physics.
For Hermitian systems, low energy states of realistic Hamiltonians are constrained by locality and obey the entanglement area law~\cite{Verstraete2006,Hastings2007}.
Tensor network (TN) states can be constructed based on this property, allowing them to naturally capture the most relevant states in Hilbert space.
In recent years, TN has emerged as a powerful tool to study strongly correlated quantum many-body systems.
When solving the ground state of a one-dimensional (1D) Hermitian Hamiltonian, the variational matrix product state (VMPS) method~\cite{Verstraete2004, Schollwoeck2011} and the equivalent density matrix renormalization group (DMRG) algorithm~\cite{White1992} always converge, as guaranteed by the variational principle.
For large strongly correlated non-Hermitian systems, analogous principles and stable algorithms are desired.
Although DMRG has been successfully applied to certain non-Hermitian problems~\cite{Carlon1999,Hieida1998,Rotureau2006,Yamamoto2022,Zhang2020c}, debates remain regarding the choices of density matrices~\cite{Wang1997,Enss2001,Huang2011a,Carlon1999,Yamamoto2022,Peschel,Nishino1999,Zhang2020c,Chan2005} and numerical difficulties remain for some approaches.
For instance, Ref.~\cite{Chan2005} points out that the convergence characteristics of non-Hermitian algorithms are less favorable than in the Hermitian case, with the storage and computing time more than doubled due to the inequivalent left and right eigenspaces.
Reference~\cite{Rotureau2006} reports an instability of non-Hermitian DMRG even after many iterations and attributes this to non-Hermiticity.
References~\cite{Huang2011, Huang2011a, Zhang2020c} introduce biorthonormal DMRG approaches capable of providing accurate results while remaining problematic at exceptional points.
Our study ascribes the latter to the biorthonormal condition itself since nearly orthogonal left and right eigenvectors are commonly found in non-Hermitian systems exhibiting skin effect~\cite{sm}.
In this work, we introduce two practical VMPS approaches for a non-Hermitian system based on a companion Hermitian Hamiltonian.
The Hermitian variational principle consequently guarantees convergence, and we avoid all numerical difficulties by not directly solving any non-Hermitian matrix.
\sect{Variational principle\label{sec:nh}}
The VMPS method, like other Hermitian variational methods, relies on the fact that a ground state has the lowest energy.
To go beyond this Hermitian paradigm, the first step is to define the ground state of a non-Hermitian system and its energy gradient.
Two common definitions of ground states are used, depending on whether the real or imaginary parts of energy are minimized, with the state denoted by $|{\rm SR}\rangle$ or $|{\rm SI}\rangle$, respectively.
For convenience, the ground state mentioned below refers to the right-eigenvector $|r \rangle$ unless otherwise specified, i.e., $H |r\rangle= e |r\rangle $.
To ensure numerical stability, we impose normalization conditions on the left and right eigenstates separately, such that $\langle l | l \rangle=1$ and $\langle r | r \rangle=1$.
A normalized right-eigenvector has the form $|r \rangle = |x\rangle / \sqrt{\langle x|x \rangle}$, and the corresponding expectation energy becomes $e(|x\rangle)=\langle x|H|x\rangle/\langle x|x\rangle$.
Because $e(|x\rangle)$ is not a holomorphic function, the derivative is not well-defined or cannot be used for straightforward optimization~\cite{UtrerasAlarcon2019}.
Nevertheless, if the real and imaginary parts of the energy are treated as two real-valued functions of complex variables, the Wirtinger derivative~\cite{Wirtinger1927} $\partial_{\langle x|} \triangleq \frac12(\partial_{\Re\{|x\rangle\}} + i\partial_{\Im\{|x\rangle\}})$ can be adopted instead~\cite{sm}:
\begin{equation}
\partial_{\langle x|}\Re\{e(|x\rangle)\}= \frac{(H+H^\dag )|x\rangle}{2\langle x|x\rangle} - \frac{\langle x|(H+H^\dag)|x\rangle|x\rangle}{2\langle x|x\rangle^2}.
\label{equ:re}
\end{equation}
Naively, one expects to use Eq.~(\ref{equ:re}) for gradient descent to find the $| \rm SR \rangle$ ground state of a non-Hermitian system.
The gradient $\partial_{\langle x|}\Re\{e(|x\rangle)\}$ should become zero at the end of iterations, resulting in $(H+H^\dag)|x\rangle\propto |x\rangle$.
However, such a condition shows that the converged $|x\rangle$ is an eigenvector of $H+H^\dag$ rather than $H$.
The states consequently obtained are usually not eigenstates of the non-Hermitian Hamiltonian $H$.
Therefore, the conventional variational principle is no longer directly suitable for non-Hermitian systems, unless a proper cost function is available.
\sect{Algorithm\label{sec:alg}}
We propose to take the eigenvector residual norm
\begin{equation}
\begin{aligned}
\mathcal{N}(|x\rangle) & \triangleq\left|H|x\rangle-e(|x\rangle)|x\rangle\right|^2
\end{aligned}
\label{equ:g}
\end{equation}
as the cost function.
It is worth noting that any eigenstate of $H$, not just the ground state, fulfills $\mathcal{N}(|x\rangle) = 0$, and $\mathcal{N}(|x\rangle)$ is always real and non-negative.
The Wirtinger derivative of $\mathcal{N}(|x\rangle)$ is simplified to~\cite{sm}
\begin{equation}
\begin{aligned}
\partial_{\langle x|}\mathcal{N}(|x\rangle) = & [H^\dag -e^{*}(|x\rangle)][H-e(|x\rangle)]|x\rangle \\
\triangleq & \mathbb{G}(H,e(|x\rangle)) |x\rangle.
\end{aligned}
\label{equ:dg}
\end{equation}
Our goal is then reduced to finding the lowest-energy state satisfying $\mathbb{G}(H,e(|x\rangle))|x\rangle=0$.
Before attempting to solve the above equation, we first establish a relation between the companion Hermitian Hamiltonian $\mathbb{G}(H,\varepsilon)$ and the original non-Hermitian Hamiltonian $H$.
Here $\mathbb{G}(H,\varepsilon)$ is non-negative definite and Hermitian for any arbitrary $\varepsilon$.
Using singular value decomposition
\begin{equation}
H-\varepsilon=USV^\dag,\;U^\dag U=V^\dag V=I,
\label{equ:svd}
\end{equation}
$\mathbb{G}(H,\varepsilon)$ is decomposed to
\begin{equation}
\mathbb{G}(H,\varepsilon)=(H^\dag-\varepsilon^*)(H-\varepsilon) = VS^2V^\dag,
\label{equ:gsvd}
\end{equation}
where singular values $s_i$ in $S$ are sorted in descending order.
We can see that $\mathbb{G}(H,\varepsilon)$ has the smallest eigenvalue $s_n^2=0$ if and only if $H$ has an eigenvalue $\varepsilon$.
Furthermore, the vector $V_n$ is a shared eigenstate of $H$ and $\mathbb{G}(H,\varepsilon)$, with eigenvalues $\varepsilon$ and $s_n^2=0$, respectively.
For non-interacting systems, the energy $e$ of a many-body ground state can be obtained from summing up single-particle energies.
Finding the ground state of a non-Hermitian Hamiltonian $H$ therefore reduces to finding the zero-energy ground state of the companion Hermitian Hamiltonian $\mathbb{G}(H, e)$, for which the powerful VMPS approach can be employed.
In practice, given a finite virtual bond dimension $D$ of MPS, the ground energy of $\mathbb{G}(H, e)$ after convergence will retain a tiny non-zero value $\eta$, and $s_n=\sqrt{\eta}$ measures whether $D$ is large enough.
Hereafter, we will refer to the aforementioned algorithm with supplied eigenenergies as the Hermitianized variational matrix product state (HVMPS) method, which is guaranteed to converge according to the standard VMPS method.
When eigenvalues are not provided or unknown, a gradient descent method facilitated by the companion Hermitian Hamiltonian can determine the ground energy and the corresponding eigenstate simultaneously, and this will be called the gradient variational matrix product state (GVMPS) method.
For simplicity, we illustrate GVMPS using a parity-time ($\mathcal{PT}$) symmetric system since its many-body ground energy is always real.
More details and its application to non-$\mathcal{PT}$ symmetric models are given in Supplemental Material.
The Wirtinger derivative of $s_n(\varepsilon)$ with respect to $\varepsilon$ reads~\cite{sm}
\begin{equation}
\partial_{\varepsilon^*}s_n=\frac{\varepsilon-V_n^\dag H V_n}{2s_n}\label{equ:wd}.
\end{equation}
One may employ the gradient descent method to find the smallest $\varepsilon_{\mathrm{opt}}$ that minimizes $s_n(\varepsilon)$, such that $\varepsilon_{\mathrm{opt}}$ becomes the ground state energy of $H$ and the corresponding eigenvector $V_n$ is obtained at the same time.
However, when $s_n(\varepsilon)$ approaches its minimum, the gradient is usually close to zero, which slows down the convergence and even results in instabilities~\cite{sm}.
This can be avoided by manually setting the gradient to $\varepsilon-V_n^\dag H V_n$ and using an adaptive learning rate~\cite{sm}.
Regarding the initial value of $\varepsilon$, we note that
\begin{equation}
\Re\left\{e(|x\rangle) \right\} = \frac{\langle x|(H+H^\dag)|x\rangle}{2\langle x|x\rangle} \ge \tau,
\end{equation}
where $\tau$ is the smallest eigenvalue of the Hermitian matrix $(H+H^\dag)/2$.
Therefore, all eigenvalues of $H$ have real parts greater than $\tau$, making $\tau$ a good starting point for determining the eigenvalue of $|\rm SR \rangle$.
Since the gradient descent only depends on one scalar parameter $\varepsilon$ and there is no other local minimum from the initial value $\tau$ to the desired energy, the GVMPS method is found to be efficient and well-converged.
Both the HVMPS and GVMPS approaches we propose avoid directly solving non-Hermitian problems, and the companion Hermitian Hamiltonian helps to prevent typical numerical instabilities of non-Hermitian systems.
Some of our benchmark results are presented below.
\sect{Non-interacting model\label{sec:results}}
We first test for non-interacting systems using the 1D nH-SSH model, which exhibits interesting topological properties and has recently received a lot of attention~\cite{Yao2018, Lieu2018,Chang2020,Han2021,Xi2021}.
The Hamiltonian takes the following form
\begin{equation}
\begin{aligned}
H_0 = & \sum\limits_i \left[(t+\gamma/2)~a^\dag_i b_i + (t-\gamma/2)~b^\dag_i a_i\right. \\&\left.+b^\dag_i a_{i+1}+a_{i+1}^\dag b_i\right],
\end{aligned}
\end{equation}
with one unit cell composed of two sites.
The intra-cell hopping is non-reciprocal and characterized by a non-Hermitian strength $\gamma$, while the inter-cell hopping is Hermitian and set as unit strength.
High-order exceptional points for this model~\cite{Fernandez2018,Chang2020,Heiss2012,Tzeng2021} appear at $|t|=|\gamma/2|$.
Since MPS methods are more accurate and efficient for finding low entanglement states that satisfy the area law, it is necessary to investigate the entanglement properties throughout the parameter space to determine which regions are more favourably described by MPS.
Under periodic boundary condition (PBC), previous study~\cite{Guo2021} on the bi-orthogonal entanglement entropy (EE)~\cite{Guo2021,Chang2020,Herviou2019,Tu2021,Brody2013,Pati2009} of the nH-SSH model finds its ground state obeys the area law at $t=1$ and $\gamma>4$.
Since our algorithms only focus on the right eigenstate and the bi-orthogonal condition may induce extra numerical difficulties~\cite{sm}, we will use the EE of the right eigenstate $|r \rangle$ instead.
The bipartite EE between subsystem $A$ and its complementary part $\bar{A}$ is given by $S_A = -\mathrm{Tr}(\rho_A \mathrm{ln} \rho_A)$, where $\rho_A=\mathrm{Tr}_{\bar{A}} \rho^{rr}$, $\rho^{rr} = |r \rangle \langle r|/ \langle r | r \rangle$, and the length of $A$ is $L_A$.
Usually, $S_A$ reaches its maximum when $L_A$ is half of the total length, and we call it the maximum EE.
As shown in the Supplemental Material, this definition of EE reveals the same area-law behavior for $t=1$ and $\gamma>4$.
\begin{figure}[tbp]
\centering
\includegraphics[width=1.0\columnwidth]{Fig1.pdf}
\caption{
(Color online)
(a) Solid lines show the maximum EE of $|{\rm SR}\rangle$ calculated for the OBC nH-SSH model at $\gamma=1$ for various $t$.
The missing region indicates no convergence found with bond dimension $D=300$.
The background color represents the logarithm of the converged ground energy of $\mathbb{G}(H, \varepsilon)$, i.e., $\log_{10}\eta$.
(b-d) Bipartite EE as a function of the subsystem length $L_A$.
(e-f) $\log_{10}\eta$ in the $(t,\gamma)$ parameter space for the ground states $|{\rm SR}\rangle$ (e) and $|{\rm SI}\rangle$ (f), calculated with bond dimension $D=100$.
The blue regions obey the area law.
Black dashed (solid) lines are the topological phase boundaries defined by energy gap closing points for OBC (PBC).
Dotted lines indicate exceptional points, beneath which the energy spectra are real.
}
\label{fig:entropyobc}
\end{figure}
Now we apply HVMPS to investigate the entanglement behavior of the $|\mathrm{SR} \rangle$ ground state of the $25$-unit-cell nH-SSH model under open boundary condition (OBC), with the many-body energy supplied by the sum of single-particle energies.
As shown in Fig.~\ref{fig:entropyobc}(a), we find three regions with different entropy distributions for $\gamma=1$, roughly separated by $t = 0.5$ and $1.5$.
The solid lines in regions \romnum 1 and \romnum 3 display the maximum EE as a function of $t$, and their convergence is confirmed by increasing the virtual bond dimension $D$.
Typical EE in these regions as a function of $L_A$ is shown in Fig.~\ref{fig:entropyobc}(d), where the plateau in the middle region indicates area-law behavior.
In most parts of region \romnum 2, a stable entropy distribution is not reached for $D=300$.
Near the boundaries of region \romnum 2, area-law violations are observed and shown in Figs.~\ref{fig:entropyobc}(b) and (c).
As indicated by the background color in Fig.~\ref{fig:entropyobc}(a), the logarithm of the converged ground energy $\eta$ of $\mathbb{G}(H, \varepsilon)$ in region \romnum 2 is a few orders of magnitude larger than in regions \romnum 1 and \romnum 3.
In fact, a bond dimension $D \sim 100$ is enough to reach $\eta<10^{-13}$ in regions \romnum 1 and \romnum 3, whereas the required $D$ quickly exceeds $300$ once entering region \romnum 2.
Together with the sudden increase of the maximum EE near the boundaries shown in Fig.~\ref{fig:entropyobc}(a), we conclude that area law is violated in region \romnum 2.
Similarly, we sweep the entire parameter space for $|{\rm SR}\rangle$ and $|{\rm SI}\rangle$ ground states using $\log_{10}\eta$ as a criterion, and the area-law-obeyed regions are painted in blue in Figs.~\ref{fig:entropyobc}(e) and (f).
Remarkably, the OBC area law boundaries coincide with the PBC topological phase boundaries, which are solid lines in Figs.~\ref{fig:entropyobc}(e) and (f).
The BBC is broken for the nH-SSH model, and the energy gap closing points under OBC (dashed lines in Figs.~\ref{fig:entropyobc}(e) and (f)) can no longer be taken as good indicators for bulk phase boundaries~\cite{Herviou2019}.
Nevertheless, the EE of ground states under OBC contains bulk phase information like in the Hermitian case, which helps to restore BBC.
According to Ref.~\cite{Korff2008}, a local non-Hermitian Hamiltonian with real spectra can be mapped to a non-local Hermitian one by a similar transformation, with area law no longer necessarily valid.
In our case, the ground state of a local non-Hermitian Hamiltonian $H$ becomes the ground state of the companion Hermitian Hamiltonian $(H^\dag-e^*)(H-e)$, possessing long range interactions.
Our algorithm thus provides an insightful understanding for area law violation in a non-Hermitian system.
\begin{figure}[tbp]
\centering
\includegraphics[width=1.0\columnwidth]{Fig2.pdf}
\caption{
(Color online)
Particle distributions of the many-body $|{\rm SR}\rangle$ state for the $25$-unit-cell nH-SSH model at $t=1.8$ and $\gamma=1.3$ for different particle numbers (a) $p=25,26,\cdots,49$ and (b) $p=1,2,\cdots,25$.
Black dots denote the GVMPS results with $U(1)$-symmetry and $D=100$.
Solid lines are obtained based on single-particle wave functions.
The green line labeled by $p=1$ ($p=49$) represents the single-particle (single-hole) $|\rm SR \rangle$ ground state, while the red lines labeled by $p=25$ denote the ground state at half-filling.
}
\label{fig:skin}
\end{figure}
In regions where area law is satisfied, the GVMPS algorithm can be applied to find both ground state energy and wave function.
The accuracy of the wave function is benchmarked by calculating the many-body non-Hermitian skin effect arising from non-reciprocal hoppings.
In the pioneering work of Ref.~\cite{Yao2018}, a diagonal matrix is introduced to transform the single-particle nH-SSH model to a Hermitian one, with the diagonal elements of the transformation decaying quickly with cell index implicating skin effect, which can be clearly interpreted by a generalized Bloch theory with complex momentum~\cite{Yao2018, Yokomizo2019,Song2019}.
In addition, spectral instability from nonnormal Hamiltonians can be used to interpret skin effect~\cite{Okuma2020}.
While these discussions mostly address single-particle situations, the skin effect itself is thought to be greatly suppressed for many-body states due to Pauli exclusion~\cite{Alsallom2021, Lee2020}.
However, most many-body studies on non-Hermitian systems employ exact diagonalization (ED) and are limited to small sizes, leaving larger systems insufficiently investigated.
By implementing $U(1)$-symmetry~\cite{Rakov2018} to the GVMPS algorithm, the many-body $| \mathrm{SR} \rangle$ ground state particle distributions are found and plotted for the $25$-unit-cell nH-SSH model.
The black dots in Fig.~\ref{fig:skin} show how particle distributions $\langle n_{i} \rangle_p = \langle n_{i}^{a} \rangle_p + \langle n_{i}^{b} \rangle_p $ evolve as the particle number $p$ increases, where $i$ is the unit-cell index, $n_{i}^{a} = a_{i}^{\dagger}a_{i}$, and $n_{i}^{b} = b_{i}^{\dagger}b_{i}$.
For comparison, we also derive a precise and efficient method~\cite{sm} to calculate particle distributions from single-particle wave functions, as shown by the solid lines in Fig.~\ref{fig:skin} and in excellent agreement with GVMPS results.
The red lines in Fig.~\ref{fig:skin} denote the distribution at half filling, where skin effect only causes a slight upturn.
Apart from adding the first and last particles when single-particle skin effects are clearly revealed, the many-body particle distributions exhibit inhomogeneous density waves.
Moreover, distributions with $p>25$ are centro-symmetric with respect to those of $p\le25$, i.e., $\langle n_{i}\rangle_p = 2-\langle n_{N+1-i}\rangle_{2N-p}$ for the $N$-unit-cell system~\cite{sm}.
\sect{Interacting model}
For interacting non-Hermitian systems, the results from our GVMPS are compared with the ones from ED.
We calculate the ground state of a $12$-unit-cell OBC nH-SSH model with nearest-neighbor repulsions at $t=0.5$ and $\gamma=1$.
The Hamiltonian is given by
\begin{equation}
H_1 = H_0+ \sum\limits_i U \left(n_{i}^{a} n_{i}^{b} + n_{i}^{b} n_{i+1}^{a} \right).
\end{equation}
\begin{figure}[tbp]
\centering
\includegraphics[width=1.0\columnwidth]{Fig3.pdf}
\caption{
(Color online)
The $D$-dependence of (a) energy error $\Delta e$ and (b) infidelity $\mathcal{I}$ for the $12$-unit-cell nH-SSH model with different $U$ at $t=0.5$ and $\gamma=1$.
}
\label{fig:ed}
\end{figure}
The energy error $\Delta e = |e_{\rm GVMPS}-e_{\rm ED}|$ and wave function infidelity $\mathcal{I}=\left| 1-|\langle\psi_{\rm ED}|\psi_{\rm GVMPS}\rangle|^2 \right|$ of the two methods are shown in Fig.~\ref{fig:ed}.
Both quantities decrease rapidly as the bond dimension $D$ increases, and sufficiently accurate results are obtained with $D$ less than $100$.
With the help of the companion Hermitian Hamiltonian, we are not required to solve any non-Hermitian matrix and no instability is observed even at $U=0$, where high-order exceptional points are predicted to emerge.
\sect{Conclusion}
We propose two efficient MPS methods for non-Hermitian systems.
If the eigenenergies of a non-Hermitian Hamiltonian are known, HVMPS can be employed to find its eigenstates by solving a companion Hermitian Hamiltonian.
When the eigenenergies are unknown, HVMPS can be combined with gradient descent and upgraded to GVMPS, such that both ground energy and wave function can be obtained.
These methods provide accurate results for non-interacting and interacting non-Hermitian systems.
Since no non-Hermitian matrix is directly solved, our approach mitigates all known numerical difficulties of typical non-Hermitian systems and ensures convergence through the variational principle.
The algorithms presented here are robust for finding many-body wave functions of a general Hamiltonian, which could pave the way for studying large-scale strongly correlated non-Hermitian systems.
\begin{acknowledgments}
This work is supported by the National Key R\&D Program of China (Grant No. 2018YFA0306504), the National Natural Science Foundation of China (NSFC) (Grants Nos. 12174214, 92065205, 11654001, and U1930201), and by Innovation Program for Quantum Science and Technology (Project 2-9-4).
\end{acknowledgments}
|
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|
#pragma once
#include <aws/wafv2/WAFV2_EXPORTS.h>
#include <aws/wafv2/model/HeaderMatchPattern.h>
#include <aws/wafv2/model/MapMatchScope.h>
#include <aws/wafv2/model/OversizeHandling.h>
#include <utility>
namespace Aws
{
namespace Utils
{
namespace Json
{
class JsonValue;
class JsonView;
} // namespace Json
} // namespace Utils
namespace WAFV2
{
namespace Model
{
/**
* <p>Inspect all headers in the web request. You can specify the parts of the
* headers to inspect and you can narrow the set of headers to inspect by including
* or excluding specific keys.</p> <p>This is used to indicate the web request
* component to inspect, in the <a>FieldToMatch</a> specification. </p> <p>If you
* want to inspect just the value of a single header, use the
* <code>SingleHeader</code> <code>FieldToMatch</code> setting instead.</p>
* <p>Example JSON: <code>"Headers": { "MatchPattern": { "All": {} }, "MatchScope":
* "KEY", "OversizeHandling": "MATCH" }</code> </p><p><h3>See Also:</h3> <a
* href="http://docs.aws.amazon.com/goto/WebAPI/wafv2-2019-07-29/Headers">AWS API
* Reference</a></p>
*/
class AWS_WAFV2_API Headers
{
public:
Headers();
Headers(Aws::Utils::Json::JsonView jsonValue);
Headers& operator=(Aws::Utils::Json::JsonView jsonValue);
Aws::Utils::Json::JsonValue Jsonize() const;
/**
* <p>The filter to use to identify the subset of headers to inspect in a web
* request. </p> <p>You must specify exactly one setting: either <code>All</code>,
* <code>IncludedHeaders</code>, or <code>ExcludedHeaders</code>.</p> <p>Example
* JSON: <code>"MatchPattern": { "ExcludedHeaders": {"KeyToExclude1",
* "KeyToExclude2"} }</code> </p>
*/
inline const HeaderMatchPattern& GetMatchPattern() const{ return m_matchPattern; }
/**
* <p>The filter to use to identify the subset of headers to inspect in a web
* request. </p> <p>You must specify exactly one setting: either <code>All</code>,
* <code>IncludedHeaders</code>, or <code>ExcludedHeaders</code>.</p> <p>Example
* JSON: <code>"MatchPattern": { "ExcludedHeaders": {"KeyToExclude1",
* "KeyToExclude2"} }</code> </p>
*/
inline bool MatchPatternHasBeenSet() const { return m_matchPatternHasBeenSet; }
/**
* <p>The filter to use to identify the subset of headers to inspect in a web
* request. </p> <p>You must specify exactly one setting: either <code>All</code>,
* <code>IncludedHeaders</code>, or <code>ExcludedHeaders</code>.</p> <p>Example
* JSON: <code>"MatchPattern": { "ExcludedHeaders": {"KeyToExclude1",
* "KeyToExclude2"} }</code> </p>
*/
inline void SetMatchPattern(const HeaderMatchPattern& value) { m_matchPatternHasBeenSet = true; m_matchPattern = value; }
/**
* <p>The filter to use to identify the subset of headers to inspect in a web
* request. </p> <p>You must specify exactly one setting: either <code>All</code>,
* <code>IncludedHeaders</code>, or <code>ExcludedHeaders</code>.</p> <p>Example
* JSON: <code>"MatchPattern": { "ExcludedHeaders": {"KeyToExclude1",
* "KeyToExclude2"} }</code> </p>
*/
inline void SetMatchPattern(HeaderMatchPattern&& value) { m_matchPatternHasBeenSet = true; m_matchPattern = std::move(value); }
/**
* <p>The filter to use to identify the subset of headers to inspect in a web
* request. </p> <p>You must specify exactly one setting: either <code>All</code>,
* <code>IncludedHeaders</code>, or <code>ExcludedHeaders</code>.</p> <p>Example
* JSON: <code>"MatchPattern": { "ExcludedHeaders": {"KeyToExclude1",
* "KeyToExclude2"} }</code> </p>
*/
inline Headers& WithMatchPattern(const HeaderMatchPattern& value) { SetMatchPattern(value); return *this;}
/**
* <p>The filter to use to identify the subset of headers to inspect in a web
* request. </p> <p>You must specify exactly one setting: either <code>All</code>,
* <code>IncludedHeaders</code>, or <code>ExcludedHeaders</code>.</p> <p>Example
* JSON: <code>"MatchPattern": { "ExcludedHeaders": {"KeyToExclude1",
* "KeyToExclude2"} }</code> </p>
*/
inline Headers& WithMatchPattern(HeaderMatchPattern&& value) { SetMatchPattern(std::move(value)); return *this;}
/**
* <p>The parts of the headers to match with the rule inspection criteria. If you
* specify <code>All</code>, WAF inspects both keys and values. </p>
*/
inline const MapMatchScope& GetMatchScope() const{ return m_matchScope; }
/**
* <p>The parts of the headers to match with the rule inspection criteria. If you
* specify <code>All</code>, WAF inspects both keys and values. </p>
*/
inline bool MatchScopeHasBeenSet() const { return m_matchScopeHasBeenSet; }
/**
* <p>The parts of the headers to match with the rule inspection criteria. If you
* specify <code>All</code>, WAF inspects both keys and values. </p>
*/
inline void SetMatchScope(const MapMatchScope& value) { m_matchScopeHasBeenSet = true; m_matchScope = value; }
/**
* <p>The parts of the headers to match with the rule inspection criteria. If you
* specify <code>All</code>, WAF inspects both keys and values. </p>
*/
inline void SetMatchScope(MapMatchScope&& value) { m_matchScopeHasBeenSet = true; m_matchScope = std::move(value); }
/**
* <p>The parts of the headers to match with the rule inspection criteria. If you
* specify <code>All</code>, WAF inspects both keys and values. </p>
*/
inline Headers& WithMatchScope(const MapMatchScope& value) { SetMatchScope(value); return *this;}
/**
* <p>The parts of the headers to match with the rule inspection criteria. If you
* specify <code>All</code>, WAF inspects both keys and values. </p>
*/
inline Headers& WithMatchScope(MapMatchScope&& value) { SetMatchScope(std::move(value)); return *this;}
/**
* <p>What WAF should do if the headers of the request are larger than WAF can
* inspect. WAF does not support inspecting the entire contents of request headers
* when they exceed 8 KB (8192 bytes) or 200 total headers. The underlying host
* service forwards a maximum of 200 headers and at most 8 KB of header contents to
* WAF. </p> <p>The options for oversize handling are the following:</p> <ul> <li>
* <p> <code>CONTINUE</code> - Inspect the headers normally, according to the rule
* inspection criteria. </p> </li> <li> <p> <code>MATCH</code> - Treat the web
* request as matching the rule statement. WAF applies the rule action to the
* request.</p> </li> <li> <p> <code>NO_MATCH</code> - Treat the web request as not
* matching the rule statement.</p> </li> </ul>
*/
inline const OversizeHandling& GetOversizeHandling() const{ return m_oversizeHandling; }
/**
* <p>What WAF should do if the headers of the request are larger than WAF can
* inspect. WAF does not support inspecting the entire contents of request headers
* when they exceed 8 KB (8192 bytes) or 200 total headers. The underlying host
* service forwards a maximum of 200 headers and at most 8 KB of header contents to
* WAF. </p> <p>The options for oversize handling are the following:</p> <ul> <li>
* <p> <code>CONTINUE</code> - Inspect the headers normally, according to the rule
* inspection criteria. </p> </li> <li> <p> <code>MATCH</code> - Treat the web
* request as matching the rule statement. WAF applies the rule action to the
* request.</p> </li> <li> <p> <code>NO_MATCH</code> - Treat the web request as not
* matching the rule statement.</p> </li> </ul>
*/
inline bool OversizeHandlingHasBeenSet() const { return m_oversizeHandlingHasBeenSet; }
/**
* <p>What WAF should do if the headers of the request are larger than WAF can
* inspect. WAF does not support inspecting the entire contents of request headers
* when they exceed 8 KB (8192 bytes) or 200 total headers. The underlying host
* service forwards a maximum of 200 headers and at most 8 KB of header contents to
* WAF. </p> <p>The options for oversize handling are the following:</p> <ul> <li>
* <p> <code>CONTINUE</code> - Inspect the headers normally, according to the rule
* inspection criteria. </p> </li> <li> <p> <code>MATCH</code> - Treat the web
* request as matching the rule statement. WAF applies the rule action to the
* request.</p> </li> <li> <p> <code>NO_MATCH</code> - Treat the web request as not
* matching the rule statement.</p> </li> </ul>
*/
inline void SetOversizeHandling(const OversizeHandling& value) { m_oversizeHandlingHasBeenSet = true; m_oversizeHandling = value; }
/**
* <p>What WAF should do if the headers of the request are larger than WAF can
* inspect. WAF does not support inspecting the entire contents of request headers
* when they exceed 8 KB (8192 bytes) or 200 total headers. The underlying host
* service forwards a maximum of 200 headers and at most 8 KB of header contents to
* WAF. </p> <p>The options for oversize handling are the following:</p> <ul> <li>
* <p> <code>CONTINUE</code> - Inspect the headers normally, according to the rule
* inspection criteria. </p> </li> <li> <p> <code>MATCH</code> - Treat the web
* request as matching the rule statement. WAF applies the rule action to the
* request.</p> </li> <li> <p> <code>NO_MATCH</code> - Treat the web request as not
* matching the rule statement.</p> </li> </ul>
*/
inline void SetOversizeHandling(OversizeHandling&& value) { m_oversizeHandlingHasBeenSet = true; m_oversizeHandling = std::move(value); }
/**
* <p>What WAF should do if the headers of the request are larger than WAF can
* inspect. WAF does not support inspecting the entire contents of request headers
* when they exceed 8 KB (8192 bytes) or 200 total headers. The underlying host
* service forwards a maximum of 200 headers and at most 8 KB of header contents to
* WAF. </p> <p>The options for oversize handling are the following:</p> <ul> <li>
* <p> <code>CONTINUE</code> - Inspect the headers normally, according to the rule
* inspection criteria. </p> </li> <li> <p> <code>MATCH</code> - Treat the web
* request as matching the rule statement. WAF applies the rule action to the
* request.</p> </li> <li> <p> <code>NO_MATCH</code> - Treat the web request as not
* matching the rule statement.</p> </li> </ul>
*/
inline Headers& WithOversizeHandling(const OversizeHandling& value) { SetOversizeHandling(value); return *this;}
/**
* <p>What WAF should do if the headers of the request are larger than WAF can
* inspect. WAF does not support inspecting the entire contents of request headers
* when they exceed 8 KB (8192 bytes) or 200 total headers. The underlying host
* service forwards a maximum of 200 headers and at most 8 KB of header contents to
* WAF. </p> <p>The options for oversize handling are the following:</p> <ul> <li>
* <p> <code>CONTINUE</code> - Inspect the headers normally, according to the rule
* inspection criteria. </p> </li> <li> <p> <code>MATCH</code> - Treat the web
* request as matching the rule statement. WAF applies the rule action to the
* request.</p> </li> <li> <p> <code>NO_MATCH</code> - Treat the web request as not
* matching the rule statement.</p> </li> </ul>
*/
inline Headers& WithOversizeHandling(OversizeHandling&& value) { SetOversizeHandling(std::move(value)); return *this;}
private:
HeaderMatchPattern m_matchPattern;
bool m_matchPatternHasBeenSet = false;
MapMatchScope m_matchScope;
bool m_matchScopeHasBeenSet = false;
OversizeHandling m_oversizeHandling;
bool m_oversizeHandlingHasBeenSet = false;
};
} // namespace Model
} // namespace WAFV2
} // namespace Aws
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,231
|
I ministri delle finanze della Repubblica Ceca dal 1993 ad oggi sono i seguenti.
Lista
Finanze
Ceca
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,859
|
{"url":"https:\/\/socratic.org\/questions\/why-does-calcite-fizz-when-it-comes-in-contact-with-hydrochloric-acid","text":"Why does calcite fizz when it comes in contact with hydrochloric acid?\n\n$\\text{Calcite}$ is mineralized $C a C {O}_{3}$, and so..............\n$C a C {O}_{3} \\left(s\\right) + 2 H C l \\left(a q\\right) \\rightarrow C a C {l}_{2} \\left(a q\\right) + {H}_{2} O \\left(l\\right) + C {O}_{2} \\left(g\\right) \\uparrow$","date":"2020-08-14 11:48:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 3, \"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\": 0.47401151061058044, \"perplexity\": 9121.276725432173}, \"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-2020-34\/segments\/1596439739211.34\/warc\/CC-MAIN-20200814100602-20200814130602-00121.warc.gz\"}"}
| null | null |
\section{Conclusion
\label{sec:conclusion}
We presented a novel framework for image repurposing detection that is modeled after the real-world adversarial interplay between nefarious actors who spread misinformation and watchdogs who verify information. The proposed framework is composed of a counterfeiter and a detector, which are trained adversarially. Like real-world, both the models have access to world knowledge through retrieval of information from a reference dataset, which they use to their advantage. We described the model components along with the training strategy. The framework was evaluated on the Google Landmarks dataset with location-identiy, IJBC-IRD with subject-identity and Painter by Numbers dataset with painting-artist as metadata. Results show that the proposed framework outperforms all baseline models and prior state-of-the-art on all metrics on a diverse collection of datasets.
\section{Experimental Evaluation}
\label{sec:evaluation}
In this section, we discuss the datasets on which AIRD was evaluated, report the performance of the indexing system, provide examples of fake candidates to validate that convincing image-repurposing is indeed possible for these datasets, describe the baseline and state-of-the-art models that AIRD is compared with, and report experimental results.
\subsection{Benchmark Datasets}
\label{subsec:data}
{
\def 0.4\textwidth {0.4\textwidth}
\begin{figure*}
\captionsetup{aboveskip=5pt,belowskip=-2pt}
\centering
\subcaptionbox{Google Landmarks\label{fig:gl_retrievals}}{%
\includegraphics[height=0.4\textwidth]{img/gl_retrievals.pdf}
}\hfill
\subcaptionbox{IJBC-IRD\label{fig:ijbc_retrievals}}{%
\includegraphics[height=0.4\textwidth]{img/ijbc_ird_retrievals.pdf}
}\hfill
\subcaptionbox{Painter by Numbers\label{fig:painters_retrievals}}{%
\includegraphics[height=0.4\textwidth]{img/painters_retrievals.pdf}
}
\caption{Examples of image-based retrievals used by $\mathcal{D}$. The first column shows the query image and the following three are the top-3 retrievals. Each image is titled with its real metadata-identity. Correct retrievals are shown with green borders and incorrect ones with red.}
\label{fig:retrievals}
\end{figure*}
}
The proposed AIRD framework was evaluated on three diverse datasets containing specific forms of identifying metadata that are vulnerable to manipulation for image-repurposing, viz., the Google Landmarks dataset of landmark images containing location information in the form of landmark-identity, the IJBC-IRD dataset of face images containing subject-identity information, and the Painter by Numbers dataset of paintings with artist-identity metadata. Details of these datasets are provided as follows.
\paragraph{Google Landmarks:} This dataset~\cite{noh2017largescale} was released by Google for a Kaggle competition on recognizing location-landmarks from images. It is the largest worldwide dataset of images of landmarks annotated with their identity. We use this dataset for the detection of image-repurposing with location-identity manipulation, i.e., verification that the image is indeed from the claimed location. The dataset consists of 1,225,029 images spanning 14,951 different landmarks. The distribution of images across landmarks is imbalanced, with some landmarks having as low as one image while others having as high as 50,000 images. The dataset was filtered to remove landmarks with less than five images, which resulted in a total of 1,216,589 images of 13,885 landmarks. The images were encoded using a publicly available pretrained NetVLAD~\cite{bib:netvlad} model\footnote{\href{https://www.di.ens.fr/willow/research/netvlad/}{www.di.ens.fr/willow/research/netvlad/}}, which was designed and trained for place recognition, followed by dimensionality reduction with Principal Component Analysis (PCA) and $L_2$-normalization, as prescribed in~\cite{bib:netvlad}.
\paragraph{IJBC-IRD:} The IARPA Janus Benchmark C (IJB-C)~\cite{maze2018iarpa}\footnote{\href{https://www.nist.gov/programs-projects/face-challenges}{www.nist.gov/programs-projects/face-challenges}} dataset is a novel face recognition benchmark. It has tough variability with face imagery presenting a wide range of poses, harsh illuminations, occlusions, aging and other challenging conditions. For all these reasons, the series IJB--\{A,B,C\}~\cite{Klare_2015_CVPR,Whitelam_2017_CVPR_Workshops,maze2018iarpa} quickly became the \emph{de facto} standard for face recognition in the wild.
With these motivations, seeking realistic scenarios for face repurposing detection, we selected a subset of IJB-C to create a new benchmark, dubbed ``IJB-C Image Repurposing Detection'' (IJBC-IRD). IJBC-IRD shares the same media as IJB-C but focuses on the subjects that are more likely to be used for face identity repurposing. To do so, we favored subjects with ample intra-class variations (picking individuals with at least five media from the IJB-C metadata) and only considering \emph{still images}, thus discarding all the motion frames. We motivate the use of still images, since we argue that clean, good quality images serve better for the face repurposing task -- frames from videos usually contain motion blur and lack of discriminative facial features, making the impersonification of a subject less believable.
The IJBC-IRD dataset contains 16,377 images spanning 1,649 subjects. We employ a state-of-the-art face recognition system to encode face images by following the procedure of~\cite{masi2017rapid,masi2016we}. We chose this system for its performance and for its pose-invariance capability~\cite{Masi:18:learning}. The face encoder is a single convolutional neural network based on a deep residual architecture, following the same training procedure described in~\cite{chang17fpn}.
Faces are encoded using the activations of the penultimate layer and the descriptors are decorrelated via PCA and signed square rooting. The final encoding for an image is the result of average-pooling estimated views rendered at different angles with the original 2D aligned image. In general, we use the same recognition pipeline from~\cite{chang17fpn} and we refer to that for more details.
\paragraph{Painter by Numbers:} This dataset~\cite{painter_by_numbers} was created for a Kaggle competition to determine whether pairs of paintings belonged to the same artist, in order to develop technologies for detecting art forgeries. We use this dataset to evaluate the detection of image repurposing where a painting's artist-ownership has been manipulated, i.e., detecting whether the painting was indeed painted by the claimed artist. The dataset contains 103,250 images from 2,319 different artists. Just like previous cases, this dataset is imbalanced with frequency ranging from one painting to 500 paintings per artist. The dataset was filtered to pick images of paintings of the top 1,000 most frequent artists in the dataset resulting in 72,863 paintings. The images were encoded using the model\footnote{\href{https://github.com/inejc/painters}{www.github.com/inejc/painters}} that won the competition, followed by $L_2$-normalization.
\paragraph{}All datasets were split into training and testing sets containing $80\%$ and $20\%$ images, respectively, using stratified sampling. The training splits of the datasets were also additionally used as reference datasets in all the experiments. All possible retrievals in the form of fake candidates for the counterfeiter and evidence for the detector were precomputed in order to speed up the training process.
{
\def 0.4\textwidth {0.4\textwidth}
\begin{figure*}
\captionsetup{aboveskip=5pt,belowskip=-2pt}
\centering
\subcaptionbox{Google Landmarks\label{fig:gl_fake}}{%
\includegraphics[height=0.4\textwidth]{img/gl_fake_candidates.pdf}
}\hfill
\subcaptionbox{IJBC-IRD\label{fig:ijbc_fake}}{%
\includegraphics[height=0.4\textwidth]{img/ijbc_ird_fake_candidates.pdf}
}\hfill
\subcaptionbox{Painter by Numbers\label{fig:painters_fake}}{%
\includegraphics[height=0.4\textwidth]{img/painters_fake_candidates.pdf}
}
\caption{Examples of fake-candidates used by $\mathcal{C}$. The first column shows the query image and the following three are fake-candidates.}
\label{fig:fake}
\end{figure*}
}
\subsection{Indexing and Retrieval Performance}
Figures~\ref{fig:gl_retrievals}, \ref{fig:ijbc_retrievals} and \ref{fig:painters_retrievals} show qualitative results of image-based retrievals for the Google Landmarks, IJBC-IRD and Painter by Numbers datasets, respectively. In each of these figures, the first column shows query images, followed by the top three image-based retrievals from the reference dataset. In the figures, we use green borders to show cases where the metadata of the retrieved image matches that of the query and red ones to show cases where it does not. The results show that the retrieval system returns images that are very similar to the query image, with usually the same metadata. However, it also makes mistakes sometimes when there are very similar images in the reference dataset that have dissimilar metadata. This can be attributed to the cascading effect of errors in the semantic encoding of images followed by approximation errors in the nearest neighbor search.
The Mean Average Precision at $K$ (MAP@K) and Precision at $K$ (Precision@K) with respect to accurate matching of metadata between query and retrieved images are reported in Table~\ref{tab:retrievals} to quantify the performance of the retrieval system on the aforementioned datasets. The results show that the retrieval performance is the best on IJBC-IRD, followed closely by Google Landmarks and relatively lower on Painter by Numbers, which is an especially challenging dataset because its metadata (the painting's artist) is more subtle than that of the other datasets (subject or location identity).
{
\setlength{\abovedisplayskip}{-5pt}
\setlength{\belowdisplayskip}{-10pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
\begin{table}[t]
\centering
\caption{\label{tab:retrievals}Performance of image-based retrieval in similar search of images with the same metadata value. Metrics are reported only for $K$=3, which is the number of packages retrieved in our experiments. GL stands for Google Landmarks and PbN for Painter by Numbers.}
\small
\begin{tabular}{l ccc}
\toprule
\textbf{Metric} & \textbf{GL} & \textbf{IJBC-IRD} & \textbf{PbN} \\
\cmidrule(r){1-1} \cmidrule(l){2-4}
MAP@3 & 0.8127 & 0.8396 & 0.6147 \\
Precision@3 & 0.8404 & 0.8537 & 0.6326 \\
\bottomrule
\end{tabular}
\end{table}
}
\subsection{Fake Candidates for Counterfeiter}
The success of the proposed AIRD framework and the adversarial training, more specifically, relies on the ability of the counterfeiter to find convincing fake candidates. We present samples of fake candidates used by $\mathcal{C}$ for the Google Landmarks, IJBC-IRD, and Painter by Numbers datasets in Figures~\ref{fig:gl_fake}, \ref{fig:ijbc_fake} and \ref{fig:painters_fake}, respectively. The first column of each of these figures shows the image selected by the counterfeiter to repurpose. The following three columns show fake candidates. The results show that these datasets contain very convincing similar images with dissimilar metadata such that one could be confused for the other. \emph{It is also important to note that in real cases of image repurposing, the audience does not see the query image (the first column) as a part of the information package (e.g., a fake news article) but only one of the fake candidates.} This makes it easy to fool people.
\subsection{Baseline and State-of-the-Art Models}
The proposed AIRD framework is designed to make additional information available to the detector in the form of retrieval from the reference dataset. However, this setup also allows for the development of several non-learning methods involving direct comparison of query and retrieved packages using similarity metrics to make integrity assessments. We discuss these models below and use them as baselines for comparing the proposed AIRD framework with:
\begin{itemize}
\item $B_1$ -- similarity between query metadata and that of query-image-based retrievals from the reference dataset
\item $B_2$ -- similarity between query image and images retrieved from the reference dataset using query metadata
\item $B_3$ -- similarity between images retrieved using query image and those retrieved using query metadata
\item $B_4$ -- similarity between metadata of query-image-based retrievals and those retrieved using metadata
\end{itemize}
Another method of metadata validation is the use of a metadata-predictor (MP) model. Given a query image and its metadata, MP first predicts the metadata for the image and then matches it with the claimed metadata. If the two match, MP tags the query package as valid.
Previous works in this domain~\cite{bib:jaiswal2017, bib:sabir2018} focused on the detection of metadata modalities that are continuous in nature, such as captions and GPS coordinates in the form of latent encodings. The method of~\cite{bib:jaiswal2017} is not suitable for structured metadata because it relies on learning a joint representation of images and captions. The publicly available deep multitask model (DMM) of~\cite{bib:sabir2018}\footnote{\href{https://www.github.com/Ekraam/MEIR}{www.github.com/Ekraam/MEIR}} was evaluated and we report the scores of this model on the IJBC-IRD and Painter by Numbers datasets. The package-similarity based retrieval used in their framework was not feasible on the Google Landmarks datasets, which has 1.2 million images. Hence, DMM could not be evaluated on this dataset.
In addition, a non-adversarial version of AIRD is evaluated as an ablation study. We call this model the non-adversarial detector (NAD). NAD is trained with real image-metadata pairs along with $(\mbf{i}, \mbf{m}_r)$ and $(\mbf{i}, \mbf{m}_c)$ for easy and hard negatives, respectively, as described in Section~\ref{subsec:training}.
\subsection{Results}
In order to evaluate the proposed framework and the aforementioned models, we use $K=3$, i.e., both the counterfeiter and the detector retrieve three packages from the RD as fake candidates and evidence, respectively. The similarity threshold for making decisions with the non-learning baseline models $B_1$, $B_2$ and $B_3$ were tuned on the training dataset. The datasets used in the experimental evaluation contain structured metadata. Hence, $B_4$ was not evaluated because it reduces to $B_1$. MP was implemented as a three-layer fully connected neural network trained with the same encodings that AIRD was trained on, which were generated using dedicated deep neural networks as described in Section~\ref{subsec:data}.
Following the approach of previous works~\cite{bib:jaiswal2017,bib:sabir2018}, $F_1$-tampered ($F_1$-tamp; calculated by treating $y=\text{fake}$ as the positive class), $F_1$-clean (calculated with $y=\text{real}$ as the positive class), and Area Under Receiver Operating Curve (AUC) were used to quantify the performance of the models. We also report the accuracy scores (ACC) as an additional metric of model performance. All the models were tested on real image-metadata pairs as well as $(\mbf{i}, \mbf{m}_r)$ pairs of images with randomly sampled fake metadata, following the evaluation methodology of previous works~\cite{bib:jaiswal2017,bib:sabir2018}. The models were additionally evaluated on hard-negatives $(\mbf{i}, \mbf{m}_c)$.
Tables~\ref{tab:gl}, \ref{tab:ijbc} and \ref{tab:painters} present results of the experiments. The results show that the proposed AIRD framework outperforms all other models on all metrics. While the non-adversarial detector (NAD) performs better than other baseline models, its performance is inferior to the complete AIRD. This additional boost in performance is, therefore, credited to the adversarial training of the detector with the counterfeiter. The proposed framework outperforms the prior state-of-the-art DMM model by a large margin, showing that DMM is not suitable for the case of image repurposing where the metadata comprises structured identity information.
{
\setlength{\belowdisplayskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
\addtolength{\tabcolsep}{-0.25pt}
\begin{table}[t]
\captionsetup{skip=2pt}
\centering
\caption{\label{tab:gl}Evaluation Results on Google Landmarks Dataset.}
\small
\begin{tabular}{l cccccc}
\toprule
\textbf{Metric} & $\boldsymbol{B_1 (B_4)}$ & $\boldsymbol{B_2}$ & $\boldsymbol{B_3}$ & \textbf{MP} & \textbf{NAD} & \textbf{AIRD} \\
\cmidrule(r){1-1} \cmidrule(l){2-7}
$F_1$-tamp & 0.91 & 0.81 & 0.81 & 0.88 & 0.91 & \textbf{0.95} \\
$F_1$-clean & 0.81 & 0.37 & 0.39 & 0.87 & 0.90 & \textbf{0.91} \\
$ACC$ & 0.86 & 0.72 & 0.71 & 0.88 & 0.90 & \textbf{0.94} \\
$AUC$ & 0.88 & 0.79 & 0.76 & 0.94 & 0.95 & \textbf{0.98} \\
\bottomrule
\end{tabular}
\end{table}
\addtolength{\tabcolsep}{0.25pt}
}
\addtolength{\tabcolsep}{-2.5pt}
\begin{table}[t]
\captionsetup{skip=2pt}
\centering
\caption{\label{tab:ijbc}Evaluation Results on IJBC-IRD Dataset.}
\small
\begin{tabular}{l ccccccc}
\toprule
\textbf{Metric} & $\boldsymbol{B_1 (B_4)}$ & $\boldsymbol{B_2}$ & $\boldsymbol{B_3}$ & \textbf{MP} & \textbf{DMM} & \textbf{NAD} & \textbf{AIRD} \\
\cmidrule(r){1-1} \cmidrule(l){2-8}
$F_1$-tamp & 0.91 & 0.90 & 0.90 & 0.91 & 0.50 & 0.93 & \textbf{0.95} \\
$F_1$-clean & 0.83 & 0.75 & 0.77 & 0.84 & 0.72 & 0.86 & \textbf{0.89} \\
$ACC$ & 0.89 & 0.86 & 0.87 & 0.89 & 0.65 & 0.90 & \textbf{0.93} \\
$AUC$ & 0.90 & 0.93 & 0.92 & 0.94 & 0.76 & 0.95 & \textbf{0.97} \\
\bottomrule
\end{tabular}
\end{table}
\addtolength{\tabcolsep}{2.5pt}
\addtolength{\tabcolsep}{-2.5pt}
\begin{table}[t]
\captionsetup{skip=2pt}
\centering
\caption{\label{tab:painters}Evaluation Results on Painter by Numbers Dataset.}
\small
\begin{tabular}{l ccccccc}
\toprule
\textbf{Metric} & $\boldsymbol{B_1 (B_4)}$ & $\boldsymbol{B_2}$ & $\boldsymbol{B_3}$ & \textbf{MP} & \textbf{DMM} & \textbf{NAD} & \textbf{AIRD} \\
\cmidrule(r){1-1} \cmidrule(l){2-8}
$F_1$-tamp & 0.81 & 0.80 & 0.80 & 0.76 & 0.22 & 0.82 & \textbf{0.83} \\
$F_1$-clean & 0.46 & 0.16 & 0.18 & 0.58 & 0.64 & 0.63 & \textbf{0.68} \\
$ACC$ & 0.72 & 0.68 & 0.69 & 0.69 & 0.51 & 0.76 & \textbf{0.77} \\
$AUC$ & 0.61 & 0.77 & 0.71 & 0.79 & 0.53 & 0.80 & \textbf{0.84} \\
\bottomrule
\end{tabular}
\end{table}
\addtolength{\tabcolsep}{2.5pt}
\section{Introduction}
\label{sec:introduction}
The internet-driven information age, in which we are currently living, has seen rapid advances in technology that have made creation and transmission of information on a large-scale increasingly easier. Consequently, the manner in which people consume information has evolved from printed media and cable-television to digital sources. Simultaneously, social networking platforms have evolved to make it easier for people to disseminate information quickly within communities and publicly. This provides an excellent way for people to share news quickly, making social media a popular news source, especially among the youth~\cite{bib:marchi2012}. However, this ease of information propagation has also made social networks a popular mode of transmission of fake news.
\begin{figure}[t]
\captionsetup[figure]{aboveskip=3pt,belowskip=-5pt}
\captionsetup[subfigure]{aboveskip=-2pt,belowskip=-5pt}
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.42\linewidth]{img/teaser_face_b.pdf}
\hspace{10pt}
\includegraphics[width=0.42\linewidth]{img/teaser_face_a.pdf}
\includegraphics[height=0.5cm,trim={0 0 0 0.25cm},clip]{img/teaser_face_caption.pdf}
\caption{Subject-Identity\label{fig:new_real_repurposing_examples:faces}}
\end{subfigure}
\par\medskip
\begin{subfigure}{0.42\linewidth}
\centering
\includegraphics[width=\linewidth]{img/teaser_loc.pdf}
\includegraphics[height=0.5cm,trim={1.25cm 0 1.25cm 0.25cm},clip]{img/teaser_loc_caption.pdf}
\caption{Location-Identity\label{fig:new_real_repurposing_examples:locations}}
\end{subfigure}
\hspace{10pt}
\begin{subfigure}{0.42\linewidth}
\centering
\includegraphics[width=\linewidth]{img/teaser_painting.pdf}
\includegraphics[height=0.5cm,trim={1.25cm 0 1.25cm 0.25cm},clip]{img/teaser_painting_caption.pdf}
\caption{Painting-Artist\label{fig:new_real_repurposing_examples:paintings}}
\end{subfigure}
\caption{\label{fig:teaser}Image repurposing in different domains from real examples -- (a) lookalike\protect\footnotemark, (b) incorrect location\protect\footnotemark, (c) wrongly-claimed artist\protect\footnotemark. Unmanipulated images are often reused in this way to spread misinformation about a similar yet different entity or event.}
\end{figure}
\addtocounter{footnote}{-3}
Given the potency of falsified information propagating on the internet, several activist groups have launched crowd-sourced efforts\stepcounter{footnote}\footnotetext{\label{fn:face}\href{https://twitter.com/jfnyc1/status/1043665540580081669}{https://twitter.com/jfnyc1/status/1043665540580081669}}\stepcounter{footnote}\footnotetext{\label{fn:loc}\href{https://twitter.com/tachirense89/status/434844811548311552}{https://twitter.com/tachirense89/status/434844811548311552}}\stepcounter{footnote}\footnotetext{\label{fn:painting}\href{https://bit.ly/2QLJjaC}{https://bit.ly/2QLJjaC}} (such as Snopes\footnote{\href{https://www.snopes.com/}{www.snopes.com}}) towards debunking fake news. However, the sheer volume and rate at which information is being created and disseminated necessitate developing automated ways of validating information. Several methods have, hence, been developed recently for detecting rumors on online forums~\cite{bib:gupta2012,bib:jin2013,bib:liu2015,bib:ma2016,bib:wu2015wu,bib:ruchansky2017}, digital manipulations of images~\cite{bib:asghar_copy-move_2017,wu2017deep_acmm,wu2018busternet,wu2018image}, and semantic inconsistencies in multimedia data~\cite{bib:jaiswal2017,bib:sabir2018}. While the detection of digital manipulation has gained most of the research attention over the years, rumor detection and semantic integrity verification are much newer areas of research.
In this paper, we focus on detecting image repurposing --- a form of semantic manipulation of multimedia data where \emph{untampered} images are \emph{reused} in combination with falsified metadata to spread misinformation. Figure~\ref{fig:teaser} shows some real-world examples of image repurposing. Jaiswal~\emph{et al.}~\cite{bib:jaiswal2017} define the broader problem of multimedia semantic integrity assessment as the verification of consistency between the media asset (e.g. image) and different components of the associated metadata (e.g. text caption, geo-location, etc.), since the asset and the metadata are expected to be a coherent \textit{package}. They also introduce the concept of using a reference dataset (RD) of untampered packages to assist the validation of \textit{query packages}. Image repurposing detection falls under this umbrella and has been explored in packages of images and captions~\cite{bib:jaiswal2017} as well as those that additionally contain Global Positioning System (GPS) information~\cite{bib:sabir2018}.
The method proposed in~\cite{bib:jaiswal2017} for integrity assessment detects inconsistencies in packages with entire captions potentially copied from other packages at random. Sabir~\emph{et al.}~\cite{bib:sabir2018}, on the other hand, present a method for the detection of manipulations of named entities within captions. However, the MEIR dataset proposed and evaluated on in~\cite{bib:sabir2018} falls short on the deceptive potential of entity-manipulations because they are implemented as randomly swapping the entity in a given caption with the same class of entity (person, organization, or location) from a caption in an unrelated package.
One of the main challenges for developing image repurposing detection methods is the lack of training and evaluation data. While crowd sourcing is a potential alternative, it is expensive, and time consuming. In light of this, we propose a novel framework for image repurposing detection, \emph{which can be trained in the absence of training data containing manipulated metadata}. The proposed framework, which we call Adversarial Image Repurposing Detection (AIRD), is modeled to simulate the real-world adversarial interplay between a bad actor who repurposes images with counterfeit metadata and a watchdog who verifies the semantic consistency between images and their accompanying metadata. More specifically, AIRD consists of two models: a counterfeiter and a detector, which are trained adversarially.
Following the approach of previous works, the proposed framework employs a reference dataset of unmanipulated packages as a source of world knowledge. While the detector gathers evidence from the reference set, the counterfeiter exploits it to conjure convincingly deceptive fake metadata for a given query package. The proposed framework can be applied to all forms of metadata. However, since generating natural language text is an open research problem, the experimental evaluation is performed on structured metadata. Furthermore, previous methods on image repurposing detection focus only on entity manipulations within captions. Hence, AIRD could be employed in such cases by first extracting entities using named entity recognition. The proposed framework exhibits state-of-the-art performance on the Google Landmarks dataset~\cite{noh2017largescale} for location-identity verification, a variant of the IJB-C dataset~\cite{maze2018iarpa}, called IJBC-IRD, which we created for subject-identity verification, and the Painter by Numbers dataset~\cite{painter_by_numbers} for painting-artist verification. Results on this diverse collection of datasets, which we make publicly available\footnote{\href{https://github.com/isi-vista/AIRD-Datasets}{www.github.com/isi-vista/AIRD-Datasets}}, illustrate the generalization capability of the proposed model.
The main contributions of this paper are:
\begin{itemize}
\item a novel approach to image repurposing detection that is modeled to simulate the real-world adversarial interplay between nefarious actors and watchdogs
\item a new framework design that can be adopted in developing real-world image repurposing detection systems that utilize knowledge-bases to validate information
\item the IJBC-IRD dataset of face images with subject-identity metadata, created to further research in the area of face-image repurposing detection
\end{itemize}
The rest of the paper is organized as follows. Section~\ref{sec:related_work} discusses related work. In Section~\ref{sec:method} we describe the proposed framework. Results of experimental evaluation are provided in Section~\ref{sec:evaluation}. Finally, we conclude the paper and discuss future directions in Section~\ref{sec:conclusion}.
\section*{Acknowledgements}
This work is based on research sponsored by the Defense Advanced Research Projects Agency under agreement number FA8750-16-2-0204. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency or the U.S. Government.
{\small
\bibliographystyle{ieee_fullname}
\section{Adversarial Image Repurposing Detection}
\label{sec:method}
\begin{figure}
\centering
\captionsetup{aboveskip=3pt}
\includegraphics[width=\linewidth,trim={2.3cm 0.02cm 2.25cm 0.02cm},clip]{img/airdetect_scheme_fonts_embedded.pdf}
\caption{\label{fig:aird}Adversarial Image Repurposing Detection (AIRD) --- At the core of this framework are two adversarially trained models: the Counterfeiter $\mathcal{C}$ and the Detector $\mathcal{D}$. Reference Dataset (RD) is a collection of verified images and metadata representing world knowledge. Variants of $\mbf{i}$ and $\mbf{m}$ are used to denote images and metadata, respectively. Similarly, variants of $\mbf{I}^R$ and $\mbf{M}^R$ denote collections of retrieved images and metadata, respectively. While the Metadata Generator (MG) of $\mathcal{C}$ takes advantage of RD to conjure fake metadata for an image by analyzing that of other similar images \emph{of different entities or events}, the Consistency Verifier (CV) of $\mathcal{D}$ uses evidence gathered from the RD to assess the veracity of the claimed metadata presented with the query image.}
\end{figure}
Jaiswal~\emph{et al.}~\cite{bib:jaiswal2017} introduced the general approach of using a reference dataset of packages as a knowledge-base from which evidence could be gathered, to assist the semantic integrity assessment of a query package. Conceptually, this approach is similar to how watchdogs verify news articles online, with news sources, document repositories and encyclopedias on the internet serving as enormous reference datasets. However, in the real-world, these datasets are accessible to nefarious actors too. In the case of image-repurposing, reference datasets are exploited in order to find images that can serve as fake evidences for rumors and propaganda. Thus, both counterfeiters and watchdogs have access to the same information but one group uses it for spreading misinformation while other employs it for information validation. The presented view of this key observation reveals an inherent \emph{informed} adversarial interplay between the two groups. We propose the Adversarial Image Repurposing Detection (AIRD) framework that models this interplay.
The proposed AIRD framework consists of two competing models --- (1) a counterfeiter ($\mathcal{C}$), which uses the reference dataset to fabricate metadata for untampered images, and (2) a detector ($\mathcal{D}$), which gathers evidence from the reference dataset to verify the semantic integrity of query packages. While the working mechanism of the detector network is close to that of real-world watchdogs, we model the counterfeiter as an extremely malevolent person who wishes to repurpose all available images with the sole intention of spreading as much misinformation as possible. At the core of the counterfeiter model is a metadata-generator (MG) neural network, while the detector contains a consistency-verifier (CV) network. The parameters of these networks are learned through adversarial training. The models ($\mathcal{C}$ \& $\mathcal{D}$) employ semantic encoding and retrieval tools to fetch additional packages from the reference set for their individual goals. Figure~\ref{fig:aird} shows the high-level design of the proposed framework. We describe the components of AIRD including implementation and training details in the following sections.
\subsection{Encoding and Retrieval from Reference Dataset}
\label{subsec:encoders}
\paragraph{Modality-specific encoders:} The counterfeiter and detector models in the proposed framework gather information through retrieval of additional similar and/or related packages from the reference dataset. In order to facilitate meaningful retrieval of information, it is important to encode each modality of the multimedia packages into information-rich semantic representations. Ideally, these encoders would be trained end-to-end with the metadata-generator and consistency-verifier networks, so that each can extract very specific information and learn similarity between data instances in order to help it achieve its goal. For example, in the case of packages containing images and captions, both MG and CV would have their own copies of image and text encoders that would be trained end-to-end with them. However, this is not always feasible because the entire reference set, which is enormous in size, would have to be re-indexed every time the parameters of these encoders are updated for complex modalities like images and text.
The proposed framework employs pretrained modality-specific encoders that are not updated during the adversarial training, when such situations arise. It is crucial to carefully select these encoders such that the embeddings generated by them capture all details of information that is vulnerable to manipulation. Thus, an ideal system would use image and text encoders that capture fine-grained semantic details in images and captions, in our example above. As discussed earlier, the proposed framework is evaluated on packages containing images and highly-specific structured metadata. Hence, we employ off-the-shelf yet state-of-the-art deep neural networks trained externally to detect those metadata as image encoders. In contrast, the metadata encoders are learned jointly with the adversarial training of MG and CV, such that the models can learn to cluster the metadata values by similarity in the embedding space. In the rest of the paper, we denote the encodings of image and metadata (whether natural language or structured) as $\mbf{i}$ and $\mbf{m}$, respectively.
\paragraph{Indexing and retrieval from the reference dataset:} Retrieval of additional related information from the reference dataset based on structured metadata is naturally implemented as database querying. Contrarily, data modalities like images and text have to be first encoded into vector representations. Retrieving related packages can then be performed through a nearest neighbor search. However, in the case of semantic integrity assessment, the reference dataset is expected to be enormous in size, especially in real world-cases, where it is expected to contain possibly all validated knowledge about the world. Using a brute-force nearest neighbor search, therefore, becomes impractical.
In order to develop a scalable framework for image repurposing detection, we employ efficient approximate search methods for similarity-based querying of reference datasets that have been shown to work on databases of billions of records. Furthermore, to boost the accuracy of the approximation, we use a cascaded indexing mechanism. The high-level stages of the system are --- (1) indexing using a Reranked Product Quantization-based Inverted File Index (IVFPQ+R)~\cite{bib:ivfpqr}, followed by (2) additional re-ranking of approximate retrievals using exact cosine similarities. We use efficient implementations of these indexing modules publicly available in the \texttt{faiss}\footnote{\href{https://github.com/facebookresearch/faiss}{www.github.com/facebookresearch/faiss}} package.
\subsection{Counterfeiter Model $\mathcal{C}$}
The working mechanism of $\mathcal{C}$ is shown in the upper-half of Figure~\ref{fig:aird} and is described as follows.
\paragraph{Fake candidates:} The non-parametric component of $\mathcal{C}$ aims to find plausible misleading candidates. In order to repurpose an image (encoded as $\mbf{i}$) by manipulating its metadata ($\mbf{m}$), the counterfeiter first queries the reference dataset for the $K$-most similar images \emph{with dissimilar metadata}. The encodings of these images are collectively denoted as $\mbf{I}_{\mbf{i}}^R$. Similarly, their accompanying metadata is denoted as $\mbf{M}_{\mbf{i}}^R$. Given the characteristics of modality-specific encoders described in Section~\ref{subsec:encoders}, this results in $K$ images that could be confused for the original with respect to the metadata to be manipulated. For instance, if the image of a person's face were to be repurposed as someone else's by manipulating the subject-identity metadata, such a retrieval would result in $K$ face images whose subjects look very similar to the original subject. We call these retrieved packages \textit{fake candidates}.
\paragraph{Metadata Generator:} The fake candidates as well as the original image and metadata are then passed on to the metadata generator neural network (MG). While caption-metadata is already encoded using a pretrained encoder (as described above), in the case of structured metadata, MG first encodes it using a metadata encoder that is trained as a part of MG. MG contains a candidacy-scorer sub-network (CSSN; implemented using two fully-connected layers), which then scores each of the $K$ candidates by comparing it with the original image-metadata pair, as shown in Equation~\ref{eq:cssn}:
\begin{equation}
\label{eq:cssn}
s_k = \text{CSSN} \Big ( (\mbf{i}, \mbf{m}), (\mbf{i}_k, \mbf{m}_k) \Big ),
\end{equation}
where $\mbf{i}_k$ and $\mbf{m}_k$ denote the $k$-th package in $\{\mbf{I}_{\mbf{i}}^R$, $\mbf{M}_{\mbf{i}}^R\}$.
Finally, scores of the candidates are converted into a choice distribution ($\mbf{c}$) through an attention-like softmax operation. In order to make the choices sharp, softmax is used with a low temperature, where softmax with temperature, as used in reinforcement learning for converting values into action-decisions, is described in Equation~\ref{eq:softmax_temp}.
\begin{equation}
\label{eq:softmax_temp}
c_k = \frac{\exp \left ( s_k / \tau \right )}{\sum_{j=1}^{K} \exp \left ( s_j / \tau \right )} \ ; \ \tau \in (0, 1]
\end{equation}
The choice distribution is multiplied element-wise with the metadata of the fake candidates. MG then produces the fabricated metadata as the sum of these weighted candidate metadata. Since the choice distribution is sharp, this simulates the act of choosing one of the $K$ metadata values while still retaining differentiability. The fabricated metadata is, thus, computed as described in Equation~\ref{eq:mfake}.
\begin{equation}
\label{eq:mfake}
\widetilde{\mbf{m}} = \sum_{k=1}^{K} c_k \cdot \mbf{m}_k
\end{equation}
\subsection{Detector Model $\mathcal{D}$}
The lower-half of Figure~\ref{fig:aird} provides an overview of the working mechanism of $\mathcal{D}$. It is described as follows.
\paragraph{Gathering evidence:} The manual process of gathering evidence from reference datasets and using them to validate query packages inspires the design of the detector model ($\mathcal{D}$) in the proposed AIRD framework. The detector starts with retrieving $K$-most similar packages from the reference dataset, using both the image ($\hat{\mbf{i}}$) and the associated metadata ($\widehat{\mbf{m}}$) as the query modality independently. Thus, it gathers two sets of evidence, which can be broken down into image encodings and metadata as $\left \{ \mbf{I}_{\hat{\mbf{i}}}^R, \mbf{M}_{\hat{\mbf{i}}}^R \right \}$ and $\left \{ \mbf{I}_{\widehat{\mbf{m}}}^R, \mbf{M}_{\widehat{\mbf{m}}}^R \right \}$ for image-based and metadata-based retrievals, respectively.
\paragraph{Consistency Verifier:} The next step in the semantic integrity verification is to use these sets of evidence in combination with the query package for validation. This is performed by the consistency verifier neural network (CV). Just like MG, CV starts with first encoding the metadata, if it is structured, using the same encoder that MG uses. This allows for the encoding and semantics of metadata to be consistent across MG and CV. The CV network performs within-modality combination of query and retrieved encodings followed by cross-modality combination of information in order to assess the semantic integrity of the query package. Within-modality combination of encodings is designed as Siamese networks~\cite{koch2015siamese}, with the replicated modules termed as \textit{aggregators} (Agg) and implemented using two fully-connected neural layers. The cross-modal combination is designed as the concatenation of modality-specific information aggregates followed by a fully-connected layer. The combined information is then used to make an integrity judgement using a final fully-connected layer. This process is illustrated by Equations~\ref{eq:hii}--\ref{eq:y}.
{
\setlength{\abovedisplayskip}{10pt}
\setlength{\belowdisplayskip}{-10pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
\begin{align}
\mbf{h}^i_{\text{img}} &= \text{Agg}_{\text{img}}(\hat{\mbf{i}}, \mbf{I}_{\hat{\mbf{i}}}^R) \label{eq:hii} \\
\mbf{h}^m_{\text{img}} &= \text{Agg}_{\text{img}}(\hat{\mbf{i}}, \mbf{I}_{\widehat{\mbf{m}}}^R) \label{eq:him} \\
\mbf{h}^i_{\text{meta}} &= \text{Agg}_{\text{meta}}(\widehat{\mbf{m}}, \mbf{M}_{\hat{\mbf{i}}}^R) \label{eq:hmi} \\
\mbf{h}^m_{\text{meta}} &= \text{Agg}_{\text{meta}}(\widehat{\mbf{m}}, \mbf{M}_{\widehat{\mbf{m}}}^R) \label{eq:hmm} \\
\mbf{h}_{\text{img}} &= \text{relu} \left ( \mbf{W}_{\text{img}}^{\text{T}}[\mbf{h}^i_{\text{img}}, \mbf{h}^m_{\text{img}}] + b_{\text{img}} \right ) \label{eq:hi} \\
\mbf{h}_{\text{meta}} &= \text{relu} \left ( \mbf{W}_{\text{meta}}^{\text{T}}[\mbf{h}^i_{\text{meta}}, \mbf{h}^m_{\text{meta}}] + b_{\text{meta}} \right ) \label{eq:hm} \\
\mbf{h}_{\text{cross}} &= \text{relu} \left ( \mbf{W}_{\text{cross}}^{\text{T}}[\mbf{h}_{\text{img}}, \mbf{h}_{\text{meta}}] + b_{\text{cross}} \right ) \label{eq:h} \\
y &= \sigma \left ( \mbf{W}_y^{\text{T}} \mbf{h}_{\text{cross}} + b_y \right ) \label{eq:y}
\end{align}
}
\subsection{Training AIRD}
\label{subsec:training}
The metadata generator and the consistency verifier networks are trained adversarially using the objective described in Equation~\ref{eq:adv_obj} in a simplified notation wherein $\mbf{i}$ denotes an image and $\mbf{m}$ its real metadata.
{
\setlength{\abovedisplayskip}{-5pt}
\setlength{\belowdisplayskip}{-5pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
\begin{align}
\max_{\text{CV}} \ \min_{\text{MG}}& \ J(\text{CV}, \text{MG}) = \mathbb{E} \left [ \log \text{CV}(\mbf{i}, \mbf{m}) \right ] \nonumber \\
&+ \mathbb{E} \left [ \log (1 - \text{CV}(\mbf{i}, \text{MG}(\mbf{i}, \mbf{m}))) \right ] \label{eq:adv_obj}
\end{align}
}
As mentioned earlier, in the case of structured metadata, the parameters of the metadata encoder are also learned jointly. However, in order to keep the training stable and the encodings consistent, the parameters of this encoder do not receive gradients directly from CV. This puts CV at a slight disadvantage by design, which encourages it to become more robust. As reflected in Equation~\ref{eq:obj}, CV is trained with real image-metadata pairs besides images with MG-generated metadata. We train the CV with two additional fake-cases, as well --- (1) $(\mbf{i}, \mbf{m}_r)$ -- image with randomly sampled fake metadata, which we call easy-negatives, and (2) $(\mbf{i}, \mbf{m}_c)$ -- image with the metadata of the most similar image-based retrieval from RD, such that $\mbf{m}_c \ne \mbf{m}$, called hard-negatives. The complete training objective is shown in Equation~\ref{eq:obj}.
{
\setlength{\abovedisplayskip}{-5pt}
\setlength{\belowdisplayskip}{-5pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
\begin{flalign}
\label{eq:obj}
&\max_{\text{CV}} \ \min_{\text{MG}} \ J(\text{CV}, \text{MG})&&&& \nonumber \\
&= \mathbb{E} \left [ \log \text{CV}(\mbf{i}, \mbf{m}) \right ] + \mathbb{E} \left [ \log (1 - \text{CV}(\mbf{i}, \text{MG}(\mbf{i}, \mbf{m}))) \right ]&&&& \nonumber \\
&+ \mathbb{E} \left [ \log (1 - \text{CV}(\mbf{i}, \mbf{m}_c)) \right ] + \mathbb{E} \left [ \log (1 - \text{CV}(\mbf{i}, \mbf{m}_r)) \right ]&&&&
\raisetag{1\normalbaselineskip}
\end{flalign}
}
\section{Related Work}
\label{sec:related_work}
Detection of fake news and rumors on online platforms has been studied in the research domain of rumor detection through automated analysis of textual content~\cite{bib:gupta2012,bib:liu2015,bib:ma2016}, propagation of posts within communities~\cite{bib:jin2013,bib:wu2015wu}, and the kind of response they elicit from people~\cite{bib:ruchansky2017}. These works are not targeted at image-based manipulations of information and do not incorporate any form of image analysis in their methodology of information verification.
Detection of digital manipulations in images has been studied extensively in the past for image-splicing, copy-move forgery, resampling and retouching of images at the pixel-level~\cite{bib:asghar_copy-move_2017,wu2017deep_acmm,wu2018busternet,wu2018image}. These methods work by either verifying embedded watermarks within images or analyzing pixel-level consistencies in search for artifacts.
The reuse of unmanipulated images to spread misinformation about a possibly unrelated entity or event was introduced in~\cite{bib:jaiswal2017} as the verification of semantic integrity of data with image assets. Image repurposing with manipulated textual and location data has been studied more specifically in~\cite{bib:sabir2018}. Our work falls in this category and we propose a novel generalized framework for image repurposing detection.
Adversarial learning has been employed recently for improving object detection~\cite{bib:zhang2018}, disentangled feature learning~\cite{bib:jaiswal2018,bib:liu2018} and feature augmentation~\cite{bib:volpi2018}, besides data generation~\cite{bib:capsgan,bib:bicogan}. The proposed AIRD framework uses adversarial learning to model the real-life interplay between malicious actors and watchdogs for image repurposing detection.
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Би́тва біля Арра́су (1914) або Пе́рша би́тва бі́ля Арра́су (1—4 жовтня 1914) — спроба наступальної операції французької армії, яка намагалася глибоким фланговим маневром обійти німецьку армію, котра водночас намагалася провести аналогічний маневр та обійти французів. Бойові дії відбувалися поблизу французького міста Аррас у ході так званого «бігу до моря» на початку Першої світової війни.
1 жовтня 1914 року 10-та французька армія генерала Луї де Мод'юї з ходу атакувала німецькі формування, що наступали південніше Аррасу, та, зламавши слабкий опір противника, вийшла до Дуе. 6-та німецька армія кронпринца Рупрехта контратакувала французів, водночас три корпуси 1-ї, 2-ї та 7-ї кайзерівських армій продовжували наступ на південь.
Під тиском противника французи були змушені відступити в бік Арраса, і 4 жовтня Ланс був окупований німецькими військами. Спроба кайзерівського командування оточити Аррас з півночі зазнали поразки, і обидві сторони використали наявні в них підкріплення, щоб спробувати черговий фланговий рух далі на північ. Ці дії переросли в битві за Ла-Бассе (10 жовтня — 2 листопада). Усі взаємні флангові маневри врешті-решт завершилися у Фландрії, коли обидві сторони досягли узбережжя Північного моря, де незабаром розгорнулася перша Фландрська битва.
Історія
На кінець вересня 1914 року 6-та німецька армія кронпринца Рупрехта силами Гвардійського, 4-го та 1-го Баварського резервного корпусів просувалася південніше Аррасу з метою здійснення глибокого обхідного маневру і подальшого виходу у тил 2-ї французькій армії дивізійного генерала Едуара де Кастельно, що билася на підступах до Ам'єна. За рахунок свіжих дивізій поповнення, військ територіальної оборони, резервів та частини сил 2-ї армії французи вступили в районі Аррасу в бій, намагаючись зірвати німецький наступ на цьому напрямку. Частково їм вдалося зупинити німецьке просування по рубежу річки Кожуль та опанували високогір'я поблизу Монші-ле-Пре. Але, перегрупувавши свої сили німці почали відтискувати їх до Гемаппа, Ванкура та Монші-ле-Пре. Генерал Жозеф Жоффр наказав припинити активні наступальні дії та перейти до оборони.
1 жовтня 1914 року тимчасове угруповання французьких військ, що зосередилося на арраському напрямку, було переформоване Жоффром на 10-у армію, яке очолив генерал Луї де Мод'юї. Командувач армії, скориставшись проломом в оперативній побудові німецьких військ, і вважаючи, що йому протистоять лише легкі кавалерійські формування Німецької імперії, а ні повноцінні три німецькі корпуси, перейшов з ходу в наступ.
На північному фланзі п'ять німецьких кавалерійських дивізій, що прикривали правий фланг XIV резервного корпусу 6-ї армії, зіткнулися з французькою кавалерією та піхотою. Водночас, агентурна розвідка німців завчасно виявила концентрацію французьких і британських військ на арраському напрямку, і відповідно I Баварський резервний корпус (генерал Карл фон Фасбендер) був перекинутий з Лотарингії до Камбре і Валансьєнна. 1-ша гвардійська дивізія та IV корпус були переміщені на північний фланг XIV резервного корпусу.
У ході проведення обома сторонами маневру силами на значному фронті відбулися зустрічні бої. Поступово бойові дії ставали все більш запеклими, коли в битву вступало все більше формувань. До вечора 2 жовтня німці захопили Дуе, де взяли в полон близько 2000 французьких солдатів, переважно резервістів територіальних військ. Німецькі частини Гвардійського, 4-го та 1-го Баварського резервного корпусів поступово закріпилися на здобутих рубежах і утворили більш-менш бойовий порядок, протистоячи генералу Луї де Мод'юї. Врешті-решт французькі війська, наразившися на опір противника, змушені були призупинити свій наступ.
2 жовтня німецькі дивізії спробували продовжити наступ, втім французи всіляко протидіяли ним. По всьому широкому фронту від Ланса до Бапома точилися запеклі бої. В результаті німецькому командуванню вдалося за день протистояння опанувати Ескершен, Ізель, Дрокур, Френуа, Буа-Бернар, Бомон.
До ранку 3 жовтня німецькі підрозділи розгорнулися на лінії від Дрокуру до Буа-Бернара і далі на Френуа. В цілому кайзерівські війська зберігали наступальний порив і продовжували просування вперед. Разом з цим, на деяких ділянках їхній успіх був дуже скромний. Так, атака 9-ї баварської резервної піхотної бригади на Невірей була відбита французами із застосуванням стрілецької зброї та вогню важкої артилерії з позицій біля Ашевіля, та перейшла до оборони на підступах до містечка. Бойові дії продовжувалися по всьому фронту зіткнення сторін, які через спротив противника поступово переходили до оборони і розпочинали окопуватися. Але, на деяких напрямках німці все ще намагалися обійти французькі війська, що їм протистояли. Так, німецькі наступи на Борен, Меркатель та передмістя Аррасу Сент-Лоран-Блангі та Сент-Нікола, які були відбиті з великими втратами, що примусили німців рухатися далі на північ.
4 жовтня Жоффр наказав де Кастельно закріпитися силами 2-ї армії на захоплених позиціях, для недопущення прориву німців через її оборонні рубежі. Одночасно, головнокомандувач сподівався на все більшу кількість французьких військ, які перекидалися далі на північ, щоб пом'якшити тиск Німеччини на основному напрямку. Генерал Фош був призначений заступником Жоффра, відповідальним за північну зону операцій, під його проводом виявилися Територіальні підрозділи, 2-га та 10-та армії, які були об'єднані в Тимчасову групу «Північ» ().
Поступово бої на південному та центральному напрямках затихали, перетворившись на позиційне протистояння. Французькі та німецькі командувачі не припиняли спроби здійснити фланговий обхід військ противника й перемістили свої основні зусилля на північ, де незабаром розгорнулася битві за Ла-Бассе.
Див. також
Битва на Ізері
Іпрська битва (1914)
Третя битва біля Аррасу
Четверта битва біля Аррасу (1918)
Битва при Армантьєрі
Перша битва на Ені
Битва при Аррасі (1940)
Примітки
Виноски
Джерела
Література
Посилання
The Battle of Arras, 1914
First Battle of Arras 1914
Битви Першої світової війни
Битви Німеччини у Першій світовій війні
Битви Франції
Битви у Франції
Жовтень 1914
Події 1 жовтня
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{"url":"https:\/\/zenodo.org\/record\/846860\/export\/csl","text":"Dataset Open Access\n\n# Supplementary Data: Status of the scalar singlet dark matter model (arXiv:1705.07931)\n\nThe GAMBIT Collaboration\n\n### Citation Style Language JSON Export\n\n{\n\"publisher\": \"Zenodo\",\n\"DOI\": \"10.5281\/zenodo.846860\",\n\"title\": \"Supplementary Data: Status of the scalar singlet dark matter model (arXiv:1705.07931)\",\n\"issued\": {\n\"date-parts\": [\n[\n2017,\n8,\n22\n]\n]\n},\n\"abstract\": \"<p>Supplementary Data<\/p>\\n\\n<p><em>Status of the scalar singlet dark matter model<\/em><br>\\n<em>arXiv:1705.07931<\/em><\/p>\\n\\n<p>The files in this record contain data for the scalar singlet dark matter model considered in the GAMBIT \\\"Round 1\\\" scalar singlet paper.<\/p>\\n\\n<p>The files consist of<\/p>\\n\\n<ul>\\n\\t<li>Three YAML files, each corresponding to a different parameter range<\/li>\\n\\t<li>StandardModel_SLHA2_SingletDM_scan_15.yaml, a universal YAML fragment included from the other three YAML files<\/li>\\n\\t<li>Three hdf5 files. SingletDM.hdf5 contains the combined results of all sampling runs, and is the basis for the profile likelihood plots in the paper. SingletDM_TW_full.hdf5 and SingletDM_TW_lowmass.hdf5 contain the results from T-Walk scans over the full and low-mass parameter ranges, respectively. These are the bases for the marginalised posterior plots in the paper.<\/li>\\n\\t<li>An example pip file corresponding to each hdf5 file, for producing plots using pippi<\/li>\\n\\t<li>A tarball best_fits_yaml.tar.gz containing YAML files of the best-fit point in each subregion of the fit.<\/li>\\n<\/ul>\\n\\n<p>The YAML files corresponding to different parameter ranges follow the naming scheme SingletDM_[slice].yaml, where slice may be full, lowmass or neck. Each of these YAML files contains entries in the Scanners node for running Diver, MultiNest, TWalk and GreAT.<\/p>\\n\\n<p>A few caveats to keep in mind:<\/p>\\n\\n<ol>\\n\\t<li>\\n\\t<p>The YAML files that we give here are updated compared to the ones that we used when generating the hdf5 file, in order to match the set of available options in the release version of GAMBIT 1.0.0. The included physics and numerics are however identical.<\/p>\\n\\t<\/li>\\n\\t<li>\\n\\t<p>The YAML files are designed to work with the tagged release of GAMBIT 1.0.0, and the pip file is tested with pippi 2.0, commit 2ab061a8. They may or may not work with later versions of either software (but you can of course always obtain the version that they do work with via the git history).<\/p>\\n\\t<\/li>\\n\\t<li>\\n\\t<p>The pip file is an example only. Users wishing to reproduce the more advanced plots in any of the GAMBIT papers should contact us for tips or scripts, or experiment for themselves. Many of these scripts are in multiple parts and require undocumented manual interventions and steps in order to implement various plot-specific customisations, so please don't expect the same level of polish as for files provided here or in the GAMBIT repo.<\/p>\\n\\t<\/li>\\n<\/ol>\",\n\"author\": [\n{\n\"family\": \"The GAMBIT Collaboration\"\n}\n],\n\"note\": \"v2 adds YAML benchmark files for the best-fit points in each region of the model.\",\n\"type\": \"dataset\",\n\"id\": \"846860\"\n}\n305\n154\nviews","date":"2020-07-15 10:28:38","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\": 0.20947040617465973, \"perplexity\": 5864.841686330498}, \"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-2020-29\/segments\/1593657167808.91\/warc\/CC-MAIN-20200715101742-20200715131742-00428.warc.gz\"}"}
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«Tourniquet» es una canción interpretada por la banda estadounidense Marilyn Manson (banda). Fue lanzado como el segundo sencillo del segundo álbum de Manson, Antichrist Superstar. El sencillo se estrenó el 8 de septiembre de 1997. Al igual que muchas canciones de Marilyn Manson de los tres primeros discos, la letra de esta canción fue previamente un poema que Manson había escrito con anterioridad a la formación de la banda. En los primeros segundos de la canción, se puede escuchar de forma invertida: "Este es mi punto más vulnerable" ("This is my lowest point of vulnerability"). En su libro "La larga huida del infierno", Manson afirma que al momento de registrar esa grabación estaba en el estudio, llorando bajo la influencia de cocaína: "Mientras estaba sentado solo en el cuarto de control del estudio, toqué las mezclas que habíamos grabado de Tourniquet, una canción inspirada por una de mis muchas pesadillas apocalípticas. Pensaba que estaba escuchándola para tratar de determinar si debía ser grabada de nuevo, pero en realidad estaba tratando de encontrarme a mí mismo en la canción, para ver si podía encontrar alguna pista, alguna respuesta, alguna solución, alguna salida del desastre en que se habían convertido mi vida y mi carrera. La escuché una y otra vez hasta que me volví insensible a ella, sin poder decir más si la canción era buena o mala, o incluso si era mía o de alguien más. Aturdido, levanté el micrófono que estaba conectado a la computadora, comenzando a sentir una de las pérdidas de conocimiento que había estado experimentando más frecuentemente. Muy lenta y firmemente tamborileé la mesa con mi mano izquierda como si estuviera pidiendo ayuda por un telégrafo y susurré al micrófono: 'Este... es... mi... punto... más... vulnerable...' cambié la grabación de dirección, para que quedara al revés, y la añadí al inicio de la canción, una llamada de auxilio que nadie podía oír excepto yo".
Carátulas del álbum
Carátula frontal: Sale la cara de Manson con el pelo atrás, una mariposa le tapa el ojo derecho y atrás sale la pared del video. El título sale con el círculo y una cruz, en el centro dice Marilyn Manson y abajo en letra manuscrita Tourniquet.
Carátula de atrás: Salen los integrantes de Marilyn Manson en forma del video de Tourniquet:
Marilyn Manson - La cara de Manson sale con el ojo derecho tapado por la mariposa.
Twiggy Ramirez - Sale debajo de una mariposa pegada.
Zim Zum - Se muestra al frente de un insecto verde.
Madonna Wayne Gacy - Sale con un pelo naranja, una barba negra larga y tiene tapado el ojo derecho con un huevo.
Ginger Fish - Sale con la cara hacia abajo.
La mayoría de los integrantes tienen una marca en sus caras.
Compact Disc: Es todo negro, del lado izquierdo salen las canciones con números romanos manuscritos, en rojo dice Marilyn Manson en manuscrita, en la parte de la orilla salen otras informaciones (copyright y derechos de autor) y sale al lado de Marilyn Manson , los productores y directores.
Versiones
Tourniquet — Aparece en Antichrist Superstar y Lest We Forget: The Best Of.
The Tourniquet Prosthetic Dance Mix — Aparece en el sencillo "Tourniquet" y en The Nobodies: 2005 Against All Gods Mix.
The Tourniquet Prosthetic Dance Mix (Edit) — Aparece en el sencillo "Tourniquet", Remix & Repent, y la edición especial de Lest We Forget: The Best Of.
Tourniquet (Edit)— Aparece en Lest We Forget: The Best Of.
Tourniquet (Demo)— Publicado por Scott Putesky a través de su página de Soundcloud en 2011 y aparece en Antichrist Final Songs.
Video musical
El video de "Tourniquet" cuenta con una serie de imágenes grotescas en un ambiente oscuro y surrealista. Se concentra principalmente en torno a un extraño humanoide que, sugerido por la letra, es una creación de Manson, que se mueve con ayuda de unas ruedas. Él cuenta con la ayuda de otro niño-criatura para el cuidado de su creación durante todo el video.
Enlaces externos
Tourniquet - The Marilyn Manson Wiki
Video musical de Tourniquet
Canciones de Marilyn Manson
Canciones en inglés
Canciones de metal alternativo
Canciones de 1997
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\section{Introduction}\label{S:1}
Let $D\subset \mathbb{R}^3$ be a bounded domain with a connected closed $C^2-$smooth boundary $S$,
$D':=\mathbb{R}^3\setminus D$ be the unbounded exterior domain and $S^2$ be the unit sphere in $\mathbb{R}^3$, $\beta \in S^2$, $s\in S$.
We are interested in the following problem:
{\em Is the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$ total in $L^2(S)$?}
A set $\{\phi(s,\beta)\}$ is total (complete) in $L^2(S)$ if the relation $\int_S f(s)\phi (s,\beta)ds=0$ for all $\beta \in S^2$ implies $f=0$,
where $f\in L^@(S)$ is an arbitrary fixed function.
The above question is of interest by itself, but also it is of interest in scattering problems and in inverse problems, see \cite{R190}--\cite{R661}.
Our result is:
{\bf Theorem 1.} {\em The set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$ is total in $L^2(S)$
if and only if $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$.}
\vspace{3mm}
\section{Proof of Theorem 1}\label{S:2}
{\bf Necessity.} Let $f\in L^2(S)$ and
\begin{equation}\label{e1}
\int_S f(s)e^{ik\beta \cdot s}ds=0\quad \forall \beta \in S^2,
\end{equation}
and there is a $u\not\equiv 0$ such that
\begin{equation}\label{e2}
(\nabla^2+k^2)u=0 \quad in\quad D, \qquad u|_{S}=0.
\end{equation}
Choose $f=u_N$, where $N$ is the unit normal to $S$ pointing out of $D$. Then, by Green's formula,
equation \eqref{e1} holds and $f\not\equiv 0$
by the uniqueness of the solution to the Cauchy problem for elliptic equation \eqref{e2}. Necessity
is proved.
{\bf Sufficiency.} Assume that $f\in L^2(S)$ is
and arbitrary fixed function, $f\not\equiv 0$,
and \eqref{e1} holds. Let $h\in L^2(S^2)$ be arbitrary and
\begin{equation}\label{e3}
w(x):=\int_{S^2}h(\beta)e^{ik\beta \cdot x}.
\end{equation}
Then
\begin{equation}\label{e4}
(\nabla^2+k^2)w=0 \quad in\quad {\mathbb R}^3.
\end{equation}
If \eqref{e1} holds, then
\begin{equation}\label{e5}
\int_S f(s)w(s)ds=0
\end{equation}
for all $w$ of the form \eqref{e3}.
Let us now apply the following Lemma:
{\bf Lemma 1.} {\em The set $\{w|_S\}$ for all $h\in L^2(S^2)$ is the orthogonal complement in $L^2(S)$ to the linear span of the set $\{v_N\}$,
where $v$ solve equation \eqref{e4} and $v|_S=0$.}
If $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, then Lemma 1 implies that the set
$\{w|_S\}$ is total in $L^2(S)$, so \eqref{e1}
implies $f=0$. Sufficiency and Theorem 1 are proved. \hfill$\Box$
Lemma 1 is similar to Theorem 6 in \cite{R666}.
{\bf Proof of Lemma 1.} Let $w|_S:=\psi$. Choose an arbitrary $F\in C^2(D)$ such that $F|_S=\psi$.
Define $G:=F-w$ in $D$. Then
\begin{equation}\label{e6}
(\nabla^2+k^2)G=(\nabla^2+k^2)F \quad in D;\quad G|_S=0.
\end{equation}
For \eqref{e6} to hold it is necessary and sufficient that
\begin{equation}\label{e7}
0=\int_D (\nabla^2+k^2)F v dx,
\end{equation}
where $v$ is an arbitrary function in the set
of solutions of equation \eqref{e2}. Using
Green's formula one reduces condition \eqref{e7}
to the following condition:
\begin{equation}\label{e8}
\int_S \psi v_N ds=0.
\end{equation}
Therefore the set $\{\psi\}$ is the orthogonal
complement in $L^2(S)$ of the linear span of the functions $\{v_N\}$. Lemma 1 is proved. \hfill$\Box$
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-write-an-equation-for-the-process-sodium-thiosulfate-dissolves-in-wat","text":"# How do you write an equation for the process \"Sodium thiosulfate dissolves in water\"?\n\nDec 8, 2016\n\n$N {a}_{2} {S}_{2} {O}_{3} \\left(s\\right) r i g h t \\le f t h a r p \\infty n s 2 N {a}^{+} \\left(a q\\right) + {S}_{2} {O}_{3}^{2 -} \\left(a q\\right)$\n\n#### Explanation:\n\n$\\left(a q\\right)$ stands for the aquated sodium ion, or the aquated thiosulfate ion; the ion surrounded by several water molecules. This is understood to be in solution.","date":"2020-08-03 21:05:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 2, \"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\": 0.9141482710838318, \"perplexity\": 4560.176771907331}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"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-2020-34\/segments\/1596439735833.83\/warc\/CC-MAIN-20200803195435-20200803225435-00333.warc.gz\"}"}
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Dear Cyril, this is what happens when you compensate for land and businesses, but damn, you guys just can't farm – Chicken farms bought by eThekwini for R15m left barren
The farms bought by the eThekwini Municipality from Rainbow Chicken two years ago to save and create jobs have been left barren.
The city spent R15million on the acquisition of Uitkomst and Doornrug farms in Cato Ridge when Rainbow Chicken, which has since been renamed RCL Foods Limited, announced the sale of their farms due to tough trading conditions.
The DA conducted an oversight inspection this week and found the farms were neglected and falling apart. The party's premier candidate Zwakele Mncwango said the municipality had not lived up to its plan and failed the employees of RCL Foods who were left jobless when the farm was shut down.
About 1350 workers lost their jobs than when the farms were sold.
"Many people who used to work here are languishing in poverty without jobs whilst an establishment that cost millions of rand is lying there, deserted", said Mncwango.
"I suspect the transaction was not intended to save jobs but instead it was to benefit individuals. At the time of this project, I raised concerns about acquiring this establishment, warning the ANC about not having a business plan to run such a huge establishment," he said.
Mncwango said he would request an investigation of the deal.
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"Someone must account and we want the eThekwini municipality to tell us how much it will cost to bring back this establishment to full operational standards," he said.
However, the city said the farms would be up and running by next year.
Speaking to emerging farmers at the launch of Agribusiness Access to Markets, a project that falls under the city's agriculture unit this week, mayor Zandile Gumede said the municipality received a letter of intent for the supply of chickens from the local abattoir.
"Negotiations are at an advanced stage with commercial partners which will assist with operations of the Cato Ridge chicken farm," said Gumede.
The deputy head of Agribusiness, Vuyo Jayiya, said the municipality would create a joint venture with commercial partners and former employees who formed themselves as co-operatives. He said they had also completed training in poultry development.
"Once we have finalised our agreements, perhaps in the next two months with these individuals, we would go to the committee to seek authority to enter into a partnership. It would only make sense to reinvest in the farm when we know it's going to be in production. We are hoping come 2020 the farm would be operational," said Jayiya.
Gumede also announced at the launch that they were in negotiations to establish rabbit hubs.
"The city is currently in negotiations with Coniglio (a rabbit meat farm) for the establishment of rabbit hubs, where we will lease business accommodation to local businesses that qualify and set up 150 rabbit operations per SMME so that they can participate in the lucrative rabbit market which is highly export-orientated," she said.
-Sunday Tribune
This news release does not necessarily reflect the opinion of SA-news.
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Categories: Agriculture, ANC Party, ANC Regime, Business, Chaos, Cyril Ramaphosa, Expenses, expensive item, Expropriation Of Land, Land Expropriation, Negligence, News, Politics, Ramaphosa, SA News, Scandals, Taxpayers money, Waste of money
Tags: agribusiness, Agribusiness Access to Markets, business plan, Cato Ridge, Cato Ridge chicken farm, Coniglio, create jobs, Doornrug farms, emerging farmers, eThekwini Municipality, rabbit meat farm, Rainbow Chicken, RCL Foods Limited, training in poultry development, Uitkomst
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{
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\section{Introduction}
\label{sec:intro}
The past decade witnessed data and algorithms becoming an integrative part of the human society. Recent technological advances are now allowing us to collect and store an astronomical amount of unstructured data, and the unprecedented computing power is enabling us to convert these data into decisional insights. Nowadays, machine learning algorithms can uncover complex patterns in the data to produce an exceptional performance that can match, or even surpass, that of humans. These algorithms, as a consequence, are proliferating in every corner of our lives, from suggesting us the next vacation destination to helping us create digital paintings and melodies. Machine learning algorithms are also gradually assisting humans in consequential decisions such as deciding whether a student is admitted to college, picking which medical treatment to be prescribed to a patient, and determining whether a person is convicted. Arguably, these decisions impact radically many people's lives, together with the future of their loved ones.
Algorithms are conceived and function following strict rules of logic and algebra; it is hence natural to expect that machine learning algorithms deliver objective predictions and recommendations. Unfortunately, in-depth investigations reveal the excruciating reality that state-of-the-art algorithmic assistance is far from being free of biases. For example, a predictive algorithm widely used in the United States criminal justice system is more likely to \textit{mis}classify African-American offenders into the group of high recidivism risk compared to white-Americans~\cite{chouldechova2017fair, ref:propublica}. The artificial intelligence tool developed by Amazon also learned to penalize gender-related keywords such as ``women's'' in the profile screening process, and thus may prefer to recommend hiring male candidates for software development and technical positions~\cite{ref:dastin2018amazon}. Further, Google's ad-targeting algorithm displayed advertisements for higher-paying executive jobs more often to men than to women \cite{ref:datta2015automated}.
There are several possible explanations for why cold, soulless algorithms may trigger biased recommendations. First, the data used to train machine learning algorithms may already encrypt human biases manifested in the data collection process. These biases arise as the result of a suboptimal design of experiments, or from historically biased human decisions that accumulate over centuries. Machine-learned algorithms, which are apt to detect underlying patterns from data, will unintentionally learn and maintain these existing biases~\cite{ref:buolamwini2018gender, ref:manrai2016genetic}. For example, secretary or primary school teacher are professions which are predominantly taken by women, thus, natural language processing systems are inclined to associate female attributes to these jobs. Second, training a machine learning algorithm typically involves minimizing the prediction error which privileges the majority populations over the minority groups. Clinical trials, for instance, typically involve very few participants from the minority groups such as indigenous people, and thus medical interventions recommended by the algorithms may not align perfectly to the characteristics and interests of patients from the minority groups. Finally, even when the sensitive attributes are not used in the training phase, strong correlations between the sensitive attributes and the remaining variables in the dataset may be exploited to generate unjust actions. For example, the sensitive attribute of race can be easily inferred with high accuracy based on common non-sensitive attributes such as the travel history of passengers or the grocery shopping records of customers.
The pressing needs to redress undesirable algorithmic biases have propelled the rising field of fair machine learning\footnote{Comprehensive surveys on fair machine learning can be found in \cite{ref:berk2018fairness, ref:chouldechova2020snapshot, ref:corbett2017algorithmic, ref:mehrabi2019survey}.}. A building pillar of this field involves the verification task: given a machine learning algorithm, we are interested in verifying if this algorithm satisfies a chosen criterion of fairness. This task is performed in two steps: first, we choose an appropriate notion of fairness, then the second step invokes a computational procedure, which may or may not involve data, to decide if the chosen fairness criterion is fulfilled. A plethora of criteria for fair machine learning were proposed in the literature, many of them are motivated by philosophical or sociological ideologies or legal constraints. For example, anti-discrimination laws may prohibit making decisions based on sensitive attributes such as age, gender, race or sexual orientation. Thus, a na\"{i}ve strategy, called fairness through unawareness, involves removing all sensitive attributes from the training data. However, this strategy seldom guarantees any fairness due to the inter-correlation issues~\cite{ref:grgic2016case, ref:garg2019counterfactual}, and thus potentially fails to generate inclusive outcomes~\cite{ref:barocas2016big, ref:black2020fliptest, ref:kleinberg2018algorithmic, ref:lipton2018does}. Other notions of fairness aim to either promote individual fairness~ \cite{ref:dwork2012fairness}, prevent disparate treatment \cite{ref:zafar2017fairness} or avoid disparate mistreatment~\cite{ref:feldman2015certifying, ref:zafar2015fairness} of the algorithms. Towards similar goals, notions of \textit{group} fairness focus on reducing the difference of favorable outcomes proportions among different sensitive groups. Examples of group fairness notions include
disparate impact~\cite{ref:zafar2017fairness},~demographic parity (statistical parity)~\cite{ref:calders2010three, ref:dwork2012fairness},~equality of opportunity~\cite{ref:hardt2016equality} and equalized odds~\cite{ref:hardt2016equality}. The notion of counterfactual fairness \cite{ref:garg2019counterfactual} was also suggested as a measure of causal fairness. Despite the abundance of available notions, there is unfortunately no general consensus on the most suitable measure to serve as the industry standard. Moreover, except in trivial cases, it is not possible for a machine learning algorithm to simultaneously satisfy multiple notions of fairness \cite{ref:berk2018fairness, kleinberg2016inherent}. Therefore, the choice of the fairness notion is likely to remain more an art than a science.
This paper focuses not on the normative approach to choosing an ideal notion of machine learning fairness. We endeavor in this paper to shed more light on the computational procedure to complement the verification task. Concretely, we position ourselves in the classification setting, which is arguably the most popular task in machine learning. Moreover, we will focus on notions of group fairness, and we employ the framework of statistical hypothesis test instead of algorithmic test.
\noindent \textbf{Contributions.} Our paper makes two concrete contributions to the problem of fairness testing of machine learning's classifiers.
\begin{enumerate}[leftmargin = 5mm]
\item We propose the Wasserstein projection framework to perform statistical hypothesis test of group fairness for classification algorithms.
We derive in details the computation of the test statistic and the limiting distribution when fairness is measured using the probabilistic equality of opportunity and probabilistic equalized odds criteria.
\item We demonstrate that the Wasserstein projection hypothesis testing paradigm is asymptotically correct and can exploit additional information on the geometry of the feature space. Moreover, we also show that this paradigm promotes transparency and interpretability through the analysis of the most favorable distributions.
\end{enumerate}
The remaining of the paper is structured as follows. In Section~\ref{sec:framework}, we introduce the general problem of statistical hypothesis test of classification fairness, and depict the current landscape of fairness testing in the literature. Section~\ref{sec:approach} details our Wasserstein projection approach to this problem. Sections~\ref{sec:EO} and~\ref{sec:EOdd} apply the proposed framework to test if a pre-trained logistic classifier satisfies the fairness notion of probabilistic equal opportunity and probabilistic equalized odds, respectively. Numerical experiments are presented in Section~\ref{sec:experiment} to empirically validate the correctness and demonstrate the power of our proposed paradigm. Section~\ref{sec:conclusion} concludes the paper with outlooks on the broader impact of our Wasserstein projection hypothesis testing approach.
All technical proofs are relegated to the Appendix.
\section{Statistical Testing Framework for Fairness and Literature Review}
\label{sec:framework}
We consider throughout this paper a generic binary classification setting. Let $\mathcal X = \mathbb{R}^d$ and $\mathcal Y=\{0, 1\}$ be the space of feature inputs and label outputs of interest. We assume that there is a single sensitive attribute corresponding to each data point and its space is denoted by $\mathcal A= \{0, 1\}$. A probabilistic classifier is represented by a function $h(\cdot) : \mathcal X \to [0, 1]$ that outputs for each given sample $x \in \mathcal X$ the probability that $x$ belongs to the positive class. The deterministic classifier predicts class 1 if $h(x) \ge \tau$ and class 0 otherwise, where $\tau \in [0, 1]$ is a classification threshold. Note that the function $h$ depends only on the feature $X$, but not on the sensitive attribute $A$, thus predicting $Y$ using $h$ satisfies fairness through unawareness.
The central goal of this paper is to provide a statistical test to detect if a classifier $h$ fails to satisfy a prescribed notion of machine learning fairness. A statistical hypothesis test can be cast with the null hypothesis being
\begin{center}
$\mathcal H_0$: the classifier $h$ is fair,
\end{center}
against the alternative hypothesis being
\begin{center}
$\mathcal H_1$: the classifier $h$ is not fair.
\end{center}
In this paper, we focus on statistical notions of \textit{group} fairness, which are usually defined using conditional probabilities. A prevalent notion of fairness in machine learning is the criterion of equality of opportunity\footnote{We use two terms ``equality of opportunity'' and ``equal opportunity'' interchangeably.}, which requires that the true positive rate are equal between subgroups.
\begin{definition}[Equal opportunity \cite{ref:hardt2016equality}] \label{def:EO}
A classifier $h(\cdot):\mathcal X \to [0, 1]$ satisfies the equal opportunity criterion relative to $\mathbb{Q}$ if
\[
\mathbb{Q}(h(X) \ge \tau | A = 1, Y = 1) = \mathbb{Q}(h(X) \ge \tau | A = 0, Y = 1),
\]
where $\tau$ is the classification threshold.
\end{definition}
Another popular criterion of machine learning fairness is the equalized odds, which is more stringent than the equality of opportunity: it requires that the positive outcome is conditionally independent of the sensitive attributes given the true label.
\begin{definition}[Equalized odds \cite{ref:hardt2016equality}]
\label{def:EOdd}
A classifier $h(\cdot):\mathcal X \to [0, 1]$ satisfies the equalized odds criterion relative to $\mathbb{Q}$ if
\[
\mathbb{Q}(h(X) \ge \tau | A = 1, Y = y) = \mathbb{Q}(h(X) \ge \tau | A = 0, Y = y) ~ \forall y \in \mathcal Y,
\]
where $\tau$ is the classification threshold.
\end{definition}
Notice that the criteria of fairness presented in Definitions~\ref{def:EO} and~\ref{def:EOdd} are dependent on the distribution $\mathbb{Q}$: a classifier $h$ can be fair relative to a distribution $\mathbb{Q}_1$, but it may become unfair with respect to another distribution $\mathbb{Q}_2 \neq \mathbb{Q}_1$. If we denote by $\mathbb{P}$ the true population distribution that governs the random vector $(X, A, Y)$, then it is imperative and reasonable to test for group fairness with respect to $\mathbb{P}$. For example, to test for the equality of opportunity, we can reformulate a two-sample equal conditional mean test of the null hypothesis
\[
\mathcal H_0: \mathds{E}_\mathbb{P}[\mathbbm{1}_{h(X)\ge\tau} | A = 1, Y= 1] = \mathds{E}_\mathbb{P}[\mathbbm{1}_{h(X)\ge\tau}| A= 0, Y = 1],
\]
and one can potentially employ a Welch's $t$-test with proper adjustment for the randomness of the sample size. Unfortunately, deriving the test becomes complicated when the null hypothesis involves an equality of multi-dimensional quantities, which arises in the case of equalized odds, due to the complication of the covariance terms. Variations of the permutation tests were also proposed to detect discriminatory behaviour of machine learning algorithms following the same formulation of the one-dimensional two-sample equality of conditional mean test \cite{ref:diciccio2020evaluating, ref:tramer2017fairtest}. However, these permutation tests follow a black-box mechanism and are unable to be generalized to multi-dimensional tests. Tests based on group fairness notions can also be accomplished using an algorithmic approach as in~\cite{ref:diciccio2020evaluating, ref:saleiro2018aequitas, ref:del2018obtaining}.
From a broader perspective, deriving tests for fairness is an active area of research, and many testing procedures have been recently proposed to test for individual fairness \cite{ref:xue2020auditing, ref:john2020verifying}, for counterfactual fairness \cite{ref:black2020fliptest, ref:garg2019counterfactual} and diverse other criteria \cite{ref:bellamy2018ai, ref:wexler2019if, ref:tramer2017fairtest}.
\noindent \textbf{Literature related to optimal transport.} Optimal transport is a long-standing field that dates back to the seminal work of Gaspard Monge~\cite{monge1781memoire}. In the past few years, it has attracted significant attention in the machine learning and computer science communities thanks to the availability of fast approximation algorithms~\cite{sinkhorn, dvurechensky2018computational, benamou2015iterative, blondel2017smooth, genevay2016stochastic}. Optimal transport is particularly successful in various learning tasks, notably generative mixture models \cite{kolouri2017optimal, nguyen2013convergence}, image processing \cite{alvarez2017structured, ferradans2014regularized, kolouri2015transport, papadakis2017convex, tartavel2016wasserstein}, computer vision and graphics \cite{pele2008linear, pele2009fast, rubner2000earth, solomon2014earth, solomon2015convolutional}, clustering \cite{ho2017multilevel}, dimensionality reduction \cite{cazelles2018geodesic, flamary2018wasserstein, rolet2016fast, schmitzer2016sparse, seguy2015principal}, domain adaptation \cite{courty2016optimal, murez2018image}, signal processing \cite{thorpe2017transportation} and data-driven distributionally robust optimization~\cite{ref:kuhn2019wasserstein, ref:blanchet2019robust, ref:gao2017wasserstein, ref:zhao2018data}. Recent comprehensive survey on optimal transport and its applications can be found in~\cite{peyre2019computational, kolouri2017optimal}.
In the context of fair classification, ideas from optimal transport have been used to construct fair logistic classifier~\cite{ref:taskesen2020distributionally}, to detect classifiers that does not obey group fairness notions, or to ensure fairness by pre-processing \cite{ref:del2018obtaining}, to learn a fair subspace embedding that promotes fair classification~\cite{ref:yurochkin2020training}, to test individual fairness~\cite{ref:xue2020auditing}, or to construct a counterfactual test \cite{ref:black2020fliptest}.
\section{Wasserstein Projection Framework for Statistical Test of Fairness}
\label{sec:approach}
We hereby provide a fresh alternative to the testing problem of machine learning fairness. On that purpose, for a given classifier $h$, we define abstractly the following set of distributions
\begin{equation} \label{eq:F-def}
\mathcal F_h = \left\{ \mathbb{Q} \in \mathcal P:~ \text{ the classifier $h$ is fair relative to } \mathbb{Q} \right\},
\end{equation}
where $\mathcal P$ denotes the space of all distributions on $\mathcal X \times \mathcal A \times \mathcal Y$. Intuitively, the set $\mathcal F_h$ contains all probability distributions under which the classifier $h$ satisfies the prescribed notion of fairness. It is trivial to see that if $\mathcal F_h$ contains the true data-generating distribution $\mathbb{P}$, then the classifier $h$ is fair relative to $\mathbb{P}$. Thus, we can reinterpret the hypothesis test of fairness using the hypotheses
\begin{center}
$\mathcal H_0$: $\mathbb{P} \in \mathcal F_h$, \hspace{1cm} $\mathcal H_1$: $\mathbb{P} \not \in \mathcal F_h$.
\end{center}
Testing the inclusion of $\mathbb{P}$ in $\mathcal F_h$ is convenient if $\mathcal P$ is endowed with a distance.
In this paper, we equip $\mathcal P$ with the Wasserstein distance.
\begin{definition}[Wasserstein distance]
The type-$2$ Wasserstein distance between two probability distributions $\mathbb{Q}$ and $\mathbb{Q}'$ supported on $\Xi$ is defined as
\[
\mathds W(\mathbb{Q}',\mathbb{Q}) = \min\limits_{\pi \in \Pi(\mathbb{Q}', \mathbb{Q})} \sqrt{\mathds{E}_\pi[ c(\xi',\xi)^2]},
\]
where the set $\Pi (\mathbb{Q}', \mathbb{Q})$ contains all joint distributions of the random vectors $\xi'\in \Xi$ and $\xi\in \Xi$ under which $\xi'$ and $\xi$ have marginal distributions $\mathbb{Q}'$ and $\mathbb{Q}$, respectively, and $c:\Xi\times\Xi\rightarrow [0,\infty]$ constitutes a lower semi-continuous ground metric.
\end{definition}
The type-2 Wasserstein distance\footnote{From this point, we omit the term ``type-2'' for brevity.} is a special instance of the optimal transport. The squared Wasserstein distance between $\mathbb{Q}'$ and $\mathbb{Q}$ can be interpreted as the cost of moving the distribution $\mathbb{Q}'$ to $\mathbb{Q}$, where $c(\xi', \xi)$ is the cost of moving a unit mass from $\xi'$ to $\xi$. Being a distance on $\mathcal P$, $\mathds W$ is symmetric, non-negative and vanishes to zero if $\mathbb{Q}' = \mathbb{Q}$. The Wasserstein distance is hence an attractive measure to identify if $\mathbb{P}$ belongs to $\mathcal F_h$. Using this insight, the hypothesis test for fairness has the equivalent representation
\begin{center}
$\mathcal H_0$: $\inf_{\mathbb{Q} \in \mathcal F_h} \mathds W(\mathbb{P}, \mathbb{Q}) = 0$, \hspace{0.5cm} $\mathcal H_1$: $\inf_{\mathbb{Q} \in \mathcal F_h} \mathds W(\mathbb{P}, \mathbb{Q}) > 0$.
\end{center}
Even though $\mathbb{P}$ remains elusive to our knowledge, we are given access to a set of i.i.d~test samples $\{(\hat x_i, \hat a_i, \hat y_i)\}_{i=1}^N$ generated from the true distribution $\mathbb{P}$. Thus we can rely on the empirical value
\[
\inf_{\mathbb{Q} \in \mathcal F_h}~\mathds W(\Pnom^N, \mathbb{Q}),
\]
which is the distance from the empirical distribution supported on the samples $\Pnom^N = \sum_{i=1}^N \delta_{(\hat x_i, \hat a_i, \hat y_i)}$ to the set $\mathcal F_h$. To perform the test, it is sufficient to study the limiting distribution of the test statistic using proper scaling under the null hypothesis $\mathcal H_0$. The outcome of the test is determined by comparing the test statistic to the quantile value of the limiting distribution at a chosen level of significant $\alpha \in (0, 1)$.
\noindent \textbf{Advantages.} The Wasserstein projection framework to hypothesis testing that we described above offers several advantages over the existing methods.
\begin{enumerate}[leftmargin=5mm]
\item Geometric flexibility: The definition of the Wasserstein distance implies that there exists a joint ground metric $c$ on the space of the features, the sensitive attribute and the label. If the modelers or the regulators possess any structural information on an appropriate metric on $\Xi = \mathcal X \times \mathcal A \times \mathcal Y$, then this information can be exploited in the testing procedure. Thus, the Wasserstein projection framework equips the users with an additional freedom to inject prior geometric information into the statistical test.
\item Mutivariate generalizability: Certain notions of fairness, such as equalized odds, are prescribed using multiple equalities of conditional expectations. The Wasserstein projection framework encapsulates these equalities simultaneously in the definition of the set $\mathcal F_h$, and provides a \textit{joint} test of these equalities without the hassle of decoupling and testing individual equalities as being done in the currently literature.
\item Interpretability: If we denote by $\mathbb{Q}^\star$ the projection of the empirical distribution $\Pnom^N$ onto the set of distributions $\mathcal F_h$, i.e.,
\[
\mathbb{Q}^\star = \arg \min\limits_{\mathbb{Q} \in \mathcal F_h}~\mathds W(\Pnom^N, \mathbb{Q}),
\]
then $\mathbb{Q}^\star$ encodes the minimal perturbation to the empirical samples so that the classifier $h$ becomes fair. The distribution $\mathbb{Q}^\star$ is thus termed the most favorable distribution, and examining $\mathbb{Q}^\star$ can reveal the underlying mechanism and explain the outcome of the hypothesis test. The accessibility to $\mathbb{Q}^\star$ showcases the expressiveness of the Wasserstein projection framework.
\end{enumerate}
Whilst theoretically sound and attractive, there are three potential difficulties with the Wasserstein projection approach to statistical test of fairness.
First, to project $\Pnom^N$ onto the set $\mathcal F_h$, we need to solve an infinite-dimensional optimization problem, which is inherently difficult. Second, for many notions of machine learning fairness such as the equality of opportunity and the equalized odds, the corresponding set $\mathcal F_h$ in~\eqref{eq:F-def} is usually prescribed using \textit{non}linear constraints. For example, if we consider the equal opportunity criterion in Definition~\ref{def:EO}, then the set $\mathcal F_h$ can be re-expressed using a fractional function of the probability measure as
\begin{align*}
\mathcal F_h =\left\{ \begin{array}{l}
\mathbb{Q} \in \mathcal P \text{ such that } \displaystyle \frac{\mathbb{Q}(h(X) \ge \tau, A = 1, Y = 1)}{\mathbb{Q}(A = 1, Y=1)} = \frac{\mathbb{Q}(h(X) \ge \tau, A = 0, Y = 1)}{\mathbb{Q}(A = 0, Y=1)}
\end{array}
\right\}.
\end{align*}
Apart from involving nonlinear constraints, it is easy to verify that the set $\mathcal F_h$ is also non-convex, which amplifies the difficulty of computing the projection onto $\mathcal F_h$.
Finally, the limiting distribution of the test statistic is difficult to analyze due to the discontinuity of the probability function at the set $\{x \in \mathcal X: h(x) = \tau\}$. The asymptotic analysis with this discontinuity is of a combinatorial nature, and is significantly more problematic than the asymptotic analysis of smooth quantities.
While these difficulties may be overcome via various ways, in this paper we choose the following combination of remedies. First, we will use a relaxed notion of fairness termed \textit{probabilistic fairness}, which was originally introduced in \cite{ref:pleiss2017fairness}.
Second, when computing the Wasserstein distances between distributions on $\mathcal X \times \mathcal A \times \mathcal Y$, we use
\begin{equation}
\label{eq:cost}
c\big( (x', a', y'), (x, a, y) \big) = \| x - x'\| + \infty | a - a'| + \infty | y - y'|
\end{equation}
as the ground metric, where $\|\cdot\|$ is a norm on $\mathbb{R}^d$. This case corresponds to having an absolute trust in the label and in the sensitive attribute of the training samples. This absolute trust restriction is common in the literature of fair machine learning~\cite{ref:xue2020auditing, ref:taskesen2020distributionally}.
We now briefly discuss the advantage of using the ground metric of the form~\eqref{eq:cost}. Denote by $p \in \mathbb{R}_{++}^{|\mathcal A| \times |\mathcal Y|}$ the array of the true marginals of $(A, Y)$, in particular, $ p_{{a} {y}} = \mathbb{P}(A = a, Y = y)$ for all $a \in \mathcal A$ and $y \in \mathcal Y$.
Further, let $\hat p^N \in \mathbb{R}_{++}^{|\mathcal A| \times |\mathcal Y|}$ be the array of the empirical marginals of $(A, Y)$ under the empirical measure $\Pnom^N$, that is, $\hat p^N_{{a} {y}} = \Pnom^N(A = a, Y = y)$ for all $a \in \mathcal A$ and $y \in \mathcal Y$.
Throughout this paper, we assume that the empirical marginals are proper, that is, $\hat p_{ay}^N \in (0, 1)$ for any $(a, y) \in \mathcal A \times \mathcal Y$.
We define temporarily the simplex set $\Delta \coloneqq \{ \bar p\in \mathbb{R}_{++}^{|\mathcal A|\times |\mathcal Y|}:\sum_{a \in \mathcal A, y \in \mathcal Y} \bar p_{ay} = 1 \}$.
Subsequently, for any marginals $\bar p \in \Delta$, we define the marginally-constrained set of distributions
\[
\mathcal F_h(\bar p) \triangleq \left\{ \mathbb{Q} \in \mathcal P:
\begin{array}{l} h \text{ is fair relative to } \mathbb{Q} \\
\mathbb{Q}(A = a, Y = y) = \bar p_{ay} ~~ \forall (a, y) \in \mathcal A \times \mathcal Y
\end{array}
\right\}.
\]
Using these notations, one can readily verify that
\[
\mathcal F_h= \cup_{\bar p \in \Delta} \mathcal F_h(\bar p).
\]
Moreover, the next result asserts that in order to compute the projection of $\Pnom^N$ onto $\mathcal F_h$, to suffices to project onto the marginally-constrained set $\mathcal F_h(\hat p^N)$.
\begin{lemma}[Projection with marginal restrictions] \label{lemma:marginal}
Suppose that the ground metric is chosen as in~\eqref{eq:cost}. If a measure $\mathbb{Q} \in \mathcal F_h$ satisfies $\mathds W(\Pnom^N, \mathbb{Q}) < \infty$, then $\mathbb{Q} \in \mathcal F_h(\hat p^N)$.
\end{lemma}
A useful consequence of Lemma~\ref{lemma:marginal} is that
\begin{equation} \label{eq:relation}
\inf_{\mathbb{Q} \in \mathcal F_h}~\mathds W(\Pnom^N, \mathbb{Q}) = \inf_{\mathbb{Q} \in \mathcal F_h(\hat p^N)}~\mathds W(\Pnom^N, \mathbb{Q}),
\end{equation}
where the feasible set of the problem on the right-hand side is the marginally-constrained set $\mathcal F_h(\hat p^N)$ using the empirical marginals $\hat p^N$. For two notions of probabilistic fairness that we will explore in this paper, projecting $\Pnom^N$ onto $\mathcal F_h(\hat p^N)$ is arguably easier than onto $\mathcal F_h$. Thus, this choice of ground metric improves the tractability when computing the test statistic.
Third, and finally, we will focus on the logistic regression setting, which is one of the most popular classification methods \cite{ref:hosmer2013applied}. In this setting, the conditional probability $\mathbb{P}[Y=1 | X=x]$ is modelled by the sigmoid function
\[h_\beta(x) = \frac{1}{1 + \exp(-\beta^\top x)},\]
where $\beta \in \mathbb{R}^d$ is the regression parameter. Moreover, a classifier with $\beta = 0$, is trivially fair. Thus, it suffices to consider $\beta \neq 0$.
\noindent \textbf{Notations. } We use $\|\cdot\|_*$ to denote the dual norm of $\|\cdot\|$. For any integer $N$, we define $[N] \coloneqq \{1, 2, \ldots, N\}$. Given $N$ test samples $(\hat x_i, \hat a_i, \hat y_i)_{i=1}^N$, we use $\mathcal I_y \triangleq \{ i \in [N]: \hat y_i = y\}$ to denote the index set of observations with label $y$. The parameters $\lambda_i$ are defined as
\begin{equation} \label{eq:lambdai-def}
\forall i \in [N]: \quad \lambda_i = \begin{cases}
(\hat p_{11}^N)^{-1} & \text{if } (\hat a_i, \hat y_i) = (1, 1),\\
- (\hat p_{01}^N)^{-1} &\text{if } (\hat a_i, \hat y_i) = (0, 1), \\
(\hat p_{10}^N)^{-1} & \text{if } (\hat a_i, \hat y_i) = (1, 0),\\
- (\hat p_{00}^N)^{-1} &\text{if } (\hat a_i, \hat y_i) = (0, 0).
\end{cases}
\end{equation}
\section{Testing Fairness for Probabilistic Equal Opportunity Criterion}
\label{sec:EO}
In this section, we use the ingredients introduced in the previous section to concretely construct a statistical test for the fairness of a logistic classifier $h_\beta$. Specifically, we will employ the probabilistic equal opportunity criterion which was originally proposed in~\cite{ref:pleiss2017fairness}.
\begin{definition}[Probabilistic equal opportunity criterion \cite{ref:pleiss2017fairness}]
\label{def:proba-EO}
A logistic classifier $h_\beta:\mathcal X \to [0, 1]$ satisfies the probabilistic equalized opportunity criteria relative to a distribution $\mathbb{Q}$ if
\[
\mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 1, Y = 1] = \mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 0, Y = 1].
\]
\end{definition}
The probabilistic equal opportunity criterion, which serves as a surrogate for the equal opportunity criterion in Definition~\ref{def:EO}, depends on the smooth and bounded sigmoid function $h_\beta$ but is independent of the classification threshold $\tau$.
Motivated by \cite{ref:lohaus2020too}, we empirically illustrate in Figure~\ref{fig:unfairness_landscape} that the probabilistic surrogate provides a good approximation of the equal opportunity criterion. Figure~\ref{fig:unfairness_landscape-opp} plots the absolute difference of the classification probabilities $|\mathbb{P}(h(X) \ge \frac{1}{2} | A=1, Y=1)-\mathbb{P}(h(X) \ge \frac{1}{2} | A=0, Y=1)|$, while Figure~\ref{fig:unfairness_landscape-probopp} plots the absolute difference of the sigmoid expectations $|\mathds{E}_\mathbb{P}[h(X) | A=1, Y=1]-\mathds{E}_\mathbb{P}[h(X) | A=0, Y=1]|$. One may observe that the regions of $\beta$ so that the absolute differences fall close to zero are similar in both plots. This implies that a logistic classifier $h_\beta$ which is equal opportunity fair is also likely to be \textit{probabilistic} equal opportunity fair, and vice versa.
\begin{figure}[h!]
\centering
\hspace{-4mm}
\begin{subfigure}[t]{0.35 \columnwidth}
\centering
\includegraphics[width=\columnwidth]{unfairness_landscape_eo.pdf}
\caption{Equal opportunity}
\label{fig:unfairness_landscape-opp}
\end{subfigure}\hspace{.6cm}
\begin{subfigure}[t]{0.375\columnwidth}
\centering
\includegraphics[width=0.923\columnwidth]{unfairness_landscape_prob_eo.pdf}
\caption{Probabilistic equal opportunity}
\label{fig:unfairness_landscape-probopp}
\end{subfigure}
\vspace{-.1cm}
\caption{Comparison of fairness notions for $d = 2$ and $h_\beta(x) = 1/(1 + \exp(\frac{1}{3} -\beta_1 x_1 - \beta_2 x_2))$. }
\label{fig:unfairness_landscape}
\vspace{-.2cm}
\end{figure}
We use the superscript ``opp'' to emphasize that fairness is measured using the probabilistic equal \textit{opp}ortunity criterion. Consequentially, the set of distributions $\mathcal F_{h_\beta}^\mathrm{opp}$ that makes the logistic classifier $h_\beta$ fair is
\[
\mathcal F_{h_\beta}^\mathrm{opp} = \left\{
\begin{array}{l}
\mathbb{Q} \in \mathcal P \text{ such that }
\mathds{E}_{\mathbb{Q}}[h_\beta(X) | A=1,Y = 1] = \mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 0, Y = 1]
\end{array}
\right\}.
\]
The statistical hypothesis test to verify whether the classifier $h_\beta$ is fair is formulated with the null and alternative hypotheses
\[
\mathcal H_0^\mathrm{opp}: \mathbb{P} \in \mathcal F_{h_\beta}^\mathrm{opp}, \quad \mathcal H_1^\mathrm{opp}: \mathbb{P} \not\in \mathcal F_{h_\beta}^\mathrm{opp}.
\]
The remainder of this section unfolds as follows. In Section~\ref{sec:opp-proj}, we delineate the computation of the projection of $\Pnom^N$ onto $\mathcal F_{h_\beta}^\mathrm{opp}$. Section~\ref{sec:opp-limit} studies the limiting distribution of the test statistic, while Section~\ref{sec:opp-favor} examines the most favorable distribution.
\subsection{Wasserstein Projection}
\label{sec:opp-proj}
Lemma~\ref{lemma:marginal} suggests that it is sufficient to consider the projection onto the marginally-constrained set $\mathcal F_{h_\beta}^\mathrm{opp}(\hat p^N)$, where $\hat p^N$ is the empirical marginals of the empirical distribution $\Pnom^N$. In particular, $\mathcal F_{h_\beta}^\mathrm{opp}(\hat p^N)$ is
\begin{align*}
\mathcal F_{h_\beta}^\mathrm{opp}(\hat p^N) = \left\{
\mathbb{Q} \in \mathcal P: \begin{array}{l}
( \hat p_{11}^{N})^{-1} \mathds{E}_{\mathbb{Q}}[h_\beta(X) \mathbbm{1}_{(1, 1)}(A, Y)] = (\hat p_{01}^{N})^{-1} \mathds{E}_{\mathbb{Q}}[h_\beta(X) \mathbbm{1}_{(0, 1)}(A, Y)] \\
\mathbb{Q}(A = a, Y = y) = \hat p_{ay}^N ~~\forall (a, y) \in \mathcal A \times \mathcal Y
\end{array}
\right\},
\end{align*}
where the equality follows from the law of conditional expectation. Notice that the set $\mathcal F_{h_\beta}^\mathrm{opp}(\hat p^N)$ is prescribed using \textit{linear} constraints of $\mathbb{Q}$, and thus it is more amenable to optimization than the set $\mathcal F_{h_\beta}^\mathrm{opp}$. It is also more convenient to work with the \textit{squared} distance function $\mathcal{R}$ whose input is the empirical distribution $\Pnom^N$ and its corresponding vector of empirical marginals $\hat p^N$ by
\begin{align*}
\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N) \coloneqq\left\{
\begin{array}{cl}
\inf & \mathds W(\mathbb{Q}, \Pnom^N)^2 \\
\mathrm{s.t.} & \mathds{E}_{\mathbb{Q}}[h_\beta(X) ( (\hat p_{11}^N)^{-1} \mathbbm{1}_{(1, 1)}(A, Y) - (\hat p_{01}^N)^{-1} \mathbbm{1}_{(0, 1)}(A, Y))] = 0 \\ [1.5ex]
& \mathds{E}_{\mathbb{Q}}[\mathbbm{1}_{(a, y)}(A, Y)] = \hat p_{ay}^N \quad \forall (a, y) \in \mathcal A \times \mathcal Y.
\end{array}
\right.
\end{align*}
Notice that the constraints of the above infimum problem are linear in the measure $\mathbb{Q}$, but the functions inside the expectation operators are possibly \textit{non}linear functions of $\hat p^N$. Using the equivalent characterization~\eqref{eq:relation}, the following relation holds
\[
\inf_{\mathbb{Q} \in \mathcal F_{h_\beta}^\mathrm{opp}}~\mathds W(\Pnom^N, \mathbb{Q}) = \inf_{\mathbb{Q} \in \mathcal F_{h_\beta}^\mathrm{opp}(\hat p^N)}~\mathds W(\Pnom^N, \mathbb{Q}) = \sqrt{\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)}.
\]
We now proceed to show how computing the projection can be reduced to solving a finite-dimensional optimization problem.
\begin{proposition}[Dual reformulation] \label{prop:R-refor}
The squared projection distance $\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)$ equals to the optimal value of the following finite-dimensional optimization problem
\begin{align}
\label{eq:R_opp_refor_finx}
\sup\limits_{\gamma \in \mathbb{R}} ~ \frac{1}{N} \sum_{i \in \mathcal I_1} \inf\limits_{x_i \in \mathcal X} \left\{ \| x_i - \hat x_i \|^2 + \gamma \lambda_i h_\beta(x_i) \right\} .
\end{align}
\end{proposition}
While Proposition~\ref{prop:R-refor} asserts that computing the \textit{squared} projection distance $\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)$ is equivalent to solving a finite-dimensional problem, unfortunately, this saddle point problem is in general difficult. Indeed, because $h_\beta$ is non-convex, even finding the optimal inner solution $x_i^\star$ for a fixed value of the outer variable $\gamma \in \mathbb{R}$ is generally NP-hard \cite{ref:murty1985some}. The situation can be partially alleviated if $\| \cdot \|$ is an Euclidean norm on $\mathbb{R}^d$.
\begin{lemma}[Univariate reduction] \label{lemma:R-compute}
Suppose that $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^d$, we have
\begin{align}
\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)=
\sup\limits_{\gamma \in \mathbb{R}} { \frac{1}{N}\sum_{i \in \mathcal I_1} \min\limits_{ k_i \in [0, \frac{1}{8}]}~ \gamma^2\lambda_i^2 \| \beta\|_2^2 k_i^2 + \frac{\gamma \lambda_i}{1 + \exp(\gamma \lambda_i \| \beta\|_2^2 k_i - \beta^\top \hat x_i)}}. \label{eq:R-refor-opp-2}
\end{align}
\end{lemma}
The proof of Lemma~\ref{lemma:R-compute} follows trivially from application of Lemma~\ref{lemma:individual} to reformulate the inner infimum problems for each $i \in \mathcal I_1$. Lemma~\ref{lemma:R-compute} offers a significant reduction in the computational complexity to solve the inner subproblems of~\eqref{eq:R_opp_refor_finx}. Instead of optimizing over $d$-dimensional vector $x_i$, the representation in Lemma~\ref{lemma:R-compute} suggests that it suffices to search over a $1$-dimensional space for $k_i$. While the objective function is still non-convex in $k_i$, we can perform a grid search over a compact interval to find the optimal solution for $k_i$ to high precision. The grid search operations can also be parallelized across the index $i$ thanks to the independent structure of the inner problems. Furthermore, the objective function of the supremum problem is a point-wise minimum of linear, thus concave, functions of $\gamma$.
Hence, the outer problem is a concave maximization problem in $\gamma$, which can be solved using a golden section search algorithm.
\subsection{Limiting Distribution}
\label{sec:opp-limit}
We now characterize the limit properties of $\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)$. The next theorem assert that the limiting distribution is of the chi-square type.
\begin{theorem}[Limiting distribution -- Probabilistic equal opportunity] \label{thm:limiting-opp}
Suppose that $(\hat x_i, \hat a_i, \hat y_i)$ are i.i.d.~samples from $\mathbb{P}$. Under the null hypothesis $\mathcal H_0^\mathrm{opp}$, we have
\begin{align*}
&N \times \mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N) \xrightarrow{d.} \theta \chi_1^2,
\end{align*}
where $\chi_1^2$ is a chi-square distribution with 1 degree of freedom,
\[
\theta = \left(\mathds{E}_{\mathbb{P}} \left[\left\| \nabla h_\beta(X) \left( \frac{\mathbbm{1}_{(1, 1)}(A, Y)}{ p_{11}}- \frac{\mathbbm{1}_{(0, 1)}(A, Y)}{p_{01}} \right) \right \|_*^2 \right] \right)^{-1} \frac{\sigma_1^2}{p_{01}^2 p_{11}^2}
\]
with $\sigma_1^2 = \mathrm{Cov}( Z_1)$, and $Z_1$ is the random variable
\begin{align*}
Z_1&= h_\beta(X)\left( p_{01}\mathbbm{1}_{(1,1)}( A,Y)
-p_{11}\mathbbm{1}_{(0,1)}( A, Y) \right)\\ & \qquad +\mathbbm{1}_{(0,1)}(
A,Y) \mathds{E}_{\mathbb{P}}[\mathbbm{1}_{(1,1)}( A, Y)
h_\beta(X)]\\
&\qquad -\mathbbm{1}_{(1,1)}( A, Y) \mathds{E}_{\mathbb{P}}[\mathbbm{1}_{(0,1)}(
A, Y) h_\beta(X)].
\end{align*}
\end{theorem}
\noindent \textbf{Construction of the hypothesis test.} Based on the result of Theorem~\ref{thm:limiting-opp}, the statistical hypothesis test proceeds as follows. Let $\eta_{1-\alpha}^\mathrm{opp}$ denote the $(1-\alpha)\times100\%$ quantile of $\theta \chi_1^2$, where $\alpha \in (0, 1)$ is the predetermined significance level. By Theorem~\ref{thm:limiting-opp}, the statistical decision has the form
\begin{center}
Reject $\mathcal H_0^\mathrm{opp}$ if $\hat s_N^\mathrm{opp} > \eta_{1-\alpha}^\mathrm{opp}$
\end{center}
with $
\hat s_N^\mathrm{opp} = N \times \mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)$.
The limiting distribution $\theta \chi_1^2$ is nonpivotal because $\theta$ depends on the true distribution $\mathbb{P}$. Luckily, because the quantile function of $\theta \chi_1^2$ is continuous in $\theta$, if $\hat \theta_N$ is a consistent estimator of $\theta$ then it is also valid to use the quantile of $\hat \theta_N \chi_1^2$ for the purpose of testing. We thus proceed to discuss a consistent estimator $\hat \theta_N$ constructed from the available data. First, notice that $\hat p_{01}^N$ and $\hat p_{11}^N$ are consistent estimator for $p_{01}$ and $p_{11}$. Similarly, the law of large numbers asserts that the denominator term in the definition of $\theta$ can be estimated by the sample average
\begin{align*}
\mathds{E}_{\mathbb{P}} \left[\left\| \nabla h_\beta(X) \left( \frac{\mathbbm{1}_{(1, 1)}(A, Y)}{ p_{11}}- \frac{\mathbbm{1}_{(0, 1)}(A, Y)}{p_{01}} \right)\right \|_*^2 \right]\\
&\hspace{-2cm}\approx \hat T^N =\frac{\|\beta\|_*^2}{N} \sum_{i=1}^N h_\beta(\hat x_i)^2 (1- h_\beta(\hat x_i))^2\left ( \frac{ \mathbbm{1}_{(1,1)}(\hat a_i, \hat y_i)}{(\hat p_{11}^N)^2} + \frac{ \mathbbm{1}_{(0,1)}(\hat a_i, \hat y_i) }{ (\hat p_{01}^N)^{2} }\right).
\end{align*}
Under the null hypothesis $\mathcal H_0^\mathrm{opp}$, $Z_1$ has mean 0. The sample average estimate of $\sigma_1^2$ is $\sigma_1^2 \approx (\hat \sigma^N)^2$ with
\begin{align}
(\hat \sigma_1^N)^2& = \frac{1}{N} \sum_{i=1}^N \Big[ h_\beta(\hat x_i)\left( p_{01}\mathbbm{1}_{(1,1)}( \hat a_i , \hat y_i)
-p_{11}\mathbbm{1}_{(0,1)}( A, Y) \right) \notag \\
& \qquad +\mathbbm{1}_{(0,1)}(
\hat a_i, \hat y_i) \big( \sum_{j=1}^N \mathbbm{1}_{(1,1)}( \hat a_j, \hat y_j)
h_\beta(\hat x_j) \big) \label{eq:sigma-estimate} \\
&\qquad -\mathbbm{1}_{(1,1)}( \hat a_i, \hat y_i) \big( \sum_{j=1}^N \mathbbm{1}_{(0,1)}(
\hat a_j, \hat y_j) h_\beta(\hat x_j)\big) \Big]^2. \notag
\end{align}
Using a nested arguments involving the continuous mapping theorem and Slutsky's theorem, the estimator
\[
\hat \theta^N = \frac{(\hat \sigma_1^N)^2}{\hat T^N (\hat p_{01}^N)^2 (\hat p_{11}^N)^2}
\]
is consistent for $\theta$. Let the corresponding $(1-\alpha)\times100\%$ quantile of the random variable $\hat \theta^N \chi_1^2$ be $\hat \eta_{1-\alpha}^\mathrm{opp}$. The statistical test decision using the plug-in consistent estimate becomes
\begin{center}
Reject $\mathcal H_0^\mathrm{opp}$ if $\hat s_N^\mathrm{opp} > \hat \eta_{1-\alpha}^\mathrm{opp}$.
\end{center}
\subsection{Most Favorable Distributions}
\label{sec:opp-favor}
We now discuss the construction of the most favorable distribution $\mathbb{Q}^\star$, the projection of the empirical distribution $\Pnom^N$ onto the set $\mathcal F_{h_\beta}^\mathrm{opp}$. Intuitively, $\mathbb{Q}^\star$ is the distribution closest to $\Pnom^N$ that makes $h_\beta$ a fair classifier under the equal opportunity criterion. If $\|\cdot\|$ is the Euclidean norm, the information about $\mathbb{Q}^\star$ can be recovered from the optimal solution of problem~\eqref{eq:R-refor-opp-2} by the result of the following lemma.
\begin{lemma}[Most favorable distribution]
\label{lemma:favorable-opp}
Suppose that $\|\cdot\|$ is the Euclidean norm. Let $\gamma^\star$ be the optimal solution of problem~\eqref{eq:R-refor-opp-2}, and for any $i \in \mathcal I_1$, let $k_i^\star$ be a solution of the inner minimization of~\eqref{eq:R-refor-opp-2} with respect to $\gamma^\star$. Then the most favorable distribution $\mathbb{Q}^\star = \arg\min\limits_{\mathbb{Q} \in \mathcal F_{h_\beta}^\mathrm{opp}}~\mathds W(\Pnom^N, \mathbb{Q})$ is a discrete distribution of the form
\[
\mathbb{Q}^\star = \frac{1}{N} \Big( \sum_{i\in\mathcal I_0} \delta_{(\hat x_i, \hat a_i, \hat y_i)} + \sum_{i \in \mathcal I_1} \delta_{(\hat x_i - k_i^\star \gamma^\star \lambda_i \beta, \hat a_i, \hat y_i)}\Big).
\]
\end{lemma}
By using the result of Lemma~\ref{lemma:R-compute}, it is easy to verify that $\mathbb{Q}^\star$ satisfies $\mathds W(\mathbb{Q}^\star, \Pnom^N)^2 = \mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)$. Moreover, one can also show that $\mathbb{Q}^\star \in \mathcal F_{h_\beta}^\mathrm{opp}$. These two observations imply that $\mathbb{Q}^\star$ is the projection of $\Pnom^N$ onto $\mathcal F_{h_\beta}^\mathrm{opp}$. The detailed proof is omitted.
Lemma~\ref{lemma:favorable-opp} suggests that in order to obtain the most favorable distribution, it suffices to perturb only the data points with positive label. This is intuitively rational because the notion of probabilistic equality of opportunity only depends on the positive label, and thus the perturbation with a minimal energy requirement should only move sample points with $\hat y_i = 1$. When the underlying geometry is the Euclidean norm, the optimal perturbation of the point $\hat x_i$ is to move it along a line dictated by $\beta$ with a scaling factor $k_i^\star \gamma^\star \lambda_i$. Notice that $\lambda_i$ defined in~\eqref{eq:lambdai-def} are of opposite signs between samples of different sensitive attributes, which implies that it is optimal to perturb $\hat x_i$ in opposite directions dependent on whether $\hat a_i = 0$ or $\hat a_i = 1$. This is, again, rational because moving points in opposite direction brings the clusters of points closer to the others, which reduces the discrepancy in the expected value of $h_\beta(X)$ between subgroups.
As a final remark, we note that $\mathbb{Q}^\star$ is not necessarily unique. This is because of the non-convexity of the inner problem over $k_i$ in~\eqref{eq:R-refor-opp-2}, which leads to the non-uniqueness of the optimal solution $k_i^\star$ (see Appendix~\ref{sec:app-k-result} and Figure~\ref{fig:non-convex-k}).
\section{Testing Fairness for Probabilistic equalized odds Criterion}
\label{sec:EOdd}
In this section, we extend the Wasserstein projection framework to the statistical test of probabilistic equalized odds for a pre-trained logistic classifier.
\begin{definition}[Probabilistic equalized odds criterion \cite{ref:pleiss2017fairness}]
A logistic classifier $h_\beta(\cdot) : \mathcal X \to [0, 1]$ satisfies the probabilistic equalized odds criteria relative to $\mathbb{Q}$ if
\[
\mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 1, Y = y] = \mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 0, Y = y] \quad \forall y \in \mathcal Y.
\]
\end{definition}
The notion of probabilistic equalized odds requires that the conditional expectation of $h_\beta$ to be independent of $A$ for any label subgroup, thus it is more stringent than the probabilistic equal opportunity studied in the previous section. We use the superscript ``odd'' in this section to emphasize on this specific notion of fairness. The definition of the probabilistic equalized odds prescribes the following set of distributions
\[
\mathcal F_{h_\beta}^\mathrm{odd} = \left\{
\mathbb{Q} \in \mathcal P: \begin{array}{l}
\mathds{E}_{\mathbb{Q}}[h_\beta(X) | A=1,Y = 1]= \mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 0, Y = 1]\\
\mathds{E}_{\mathbb{Q}}[h_\beta(X) | A=1,Y = 0] = \mathds{E}_{\mathbb{Q}}[h_\beta(X) | A = 0, Y = 0]
\end{array}
\right\}.
\]
Correspondingly, the Wasserstein projection hypothesis test for probabilisitc equalized odds can be formulated as
\[
\mathcal H_0^\mathrm{odd}: \mathbb{P} \in \mathcal F_{h_\beta}^\mathrm{odd}, \quad \mathcal H_1^\mathrm{odd}: \mathbb{P} \not\in \mathcal F_{h_\beta}^\mathrm{odd}.
\]
In the sequence, we study the projection onto the manifold $\mathcal F_{h_\beta}^\mathrm{odd}$ in Section~\ref{sec:odd-proj}. Section~\ref{sec:odd-limit} examines the asymptotic behaviour of the test statistic, and we close this section by studying the most favorable distribution $\mathbb{Q}^\star$ in Section~\ref{sec:odd-favor}.
\subsection{Wasserstein Projection}
\label{sec:odd-proj}
Following a similar strategy as in Section~\ref{sec:EO}, we define the set
\begin{align*}
\mathcal F_{h_\beta}^\mathrm{odd}(\hat p^N)
=\left\{
\mathbb{Q} \in \mathcal P : \begin{array}{l}
( \hat p_{11}^{N})^{-1} \mathds{E}_{\mathbb{Q}}[h_\beta(X) \mathbbm{1}_{(1, 1)}(A, Y)] = (\hat p_{01}^{N})^{-1} \mathds{E}_{\mathbb{Q}}[h_\beta(X) \mathbbm{1}_{(0, 1)}(A, Y)] \\ [1ex]
( \hat p_{10}^{N})^{-1} \mathds{E}_{\mathbb{Q}}[h_\beta(X) \mathbbm{1}_{(1, 0)}(A, Y)] = (\hat p_{00}^{N})^{-1} \mathds{E}_{\mathbb{Q}}[h_\beta(X) \mathbbm{1}_{(0, 0)}(A, Y)] \\ [1ex]
\mathbb{Q}(A = a, Y = y) = \hat p_{ay}^N ~~\forall (a, y) \in \mathcal A \times \mathcal Y
\end{array}
\right\},
\end{align*}
and the squared distance function
\begin{align*}
\mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N) = \left\{
\begin{array}{cl}
\inf & \mathds W(\mathbb{Q}, \Pnom^N)^2 \\[.2 cm]
\mathrm{s.t.} & \mathds{E}_{\mathbb{Q}}[h_\beta(X) ( (\hat p_{11}^N)^{-1} \mathbbm{1}_{(1, 1)}(A, Y) - (\hat p_{01}^N)^{-1} \mathbbm{1}_{(0, 1)}(A, Y))] = 0 \\[.2cm]
& \mathds{E}_{\mathbb{Q}}[h_\beta(X) ( (\hat p_{10}^N)^{-1} \mathbbm{1}_{(1, 0)}(A, Y) - (\hat p_{00}^N)^{-1} \mathbbm{1}_{(0, 0)}(A, Y))] = 0 \\[.2cm]
& \mathds{E}_{\mathbb{Q}}[\mathbbm{1}_{(a, y)}(A, Y)] = \hat p_{ay}^N \quad \forall (a, y) \in \mathcal A \times \mathcal Y.
\end{array}
\right.
\end{align*}
The equivalent relation~\eqref{eq:relation} suggests that the projection onto the set of distributions $\mathcal F_{h_\beta}^\mathrm{odd}$ satisfies
\[
\inf_{\mathbb{Q} \in \mathcal F_{h_\beta}^\mathrm{odd}}~\mathds W(\Pnom^N, \mathbb{Q}) = \inf_{\mathbb{Q} \in \mathcal F_{h_\beta}^\mathrm{odd}(\hat p^N)}~\mathds W(\Pnom^N, \mathbb{Q}) = \sqrt{\mathcal R^\mathrm{odd}(\Pnom^N, \hat p^N)}.
\]
The squared distance $\mathcal R^\mathrm{odd}(\Pnom^N, \hat p^N)$ can be computed by solving the saddle point problem in the following proposition.
\begin{proposition}[Dual reformulation] \label{prop:R-refor-odd}
The squared projection distance $\mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N)$ equals to the optimal value of the following finite-dimensional optimization problem
\begin{equation} \label{eq:R-refor-odd}
\sup\limits_{\gamma \in \mathbb{R}, \zeta \in \mathbb{R}} ~ \frac{1}{N} \sum_{i=1}^N \inf\limits_{x_i \in \mathcal X} \left\{ \| x_i - \hat x_i \|^2 +
(\gamma \lambda_i \mathbbm{1}_1 (\hat y_i) +\zeta
\lambda_i \mathbbm{1}_0 (\hat y_i)) h_\beta(x_i) \right\} .
\end{equation}
\end{proposition}
To complete this section, we now discuss an efficient way to compute~$\mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N)$. The next lemma reveals that computing $\mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N)$ can be decomposed into two subproblems of similar structure.
\begin{lemma}[Univariate reduction] \label{lemma:R-compute-odd}
We have
\[
\mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N) = \mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N) + U_N,
\]
where $U_N$ is computed as
\begin{align*}
U_N = \sup\limits_{\zeta \in \mathbb{R}} ~ \frac{1}{N} \sum_{i \in \mathcal I_0} \inf\limits_{x_i \in \mathcal X} \left\{ \| x_i - \hat x_i \|^2 + \zeta \lambda_i h_\beta(x_i) \right\} .
\end{align*}
Furthermore, if $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^d$, then
\begin{align}
U_N =
\sup\limits_{\zeta \in \mathbb{R}} \frac{1}{N}\left\{ \sum_{i \in \mathcal I_0} \min\limits_{ k_i \in [0, \frac{1}{8}]}~ \zeta^2\lambda_i^2 \| \beta\|_2^2 k_i^2 + \frac{\zeta \lambda_i}{1 + \exp(\zeta \lambda_i \| \beta\|_2^2 k_i - \beta^\top \hat x_i)}
\right\}. \label{eq:U_N}
\end{align}
\end{lemma}
Notice that problem~\eqref{eq:U_N} has a similar structure to problem~\eqref{eq:R-refor-opp-2}: the mere difference is that the summation in the objective function of~\eqref{eq:U_N} runs over the index set $\mathcal I_0 = \{i \in [N]: \hat y_i = 0\}$ instead of $ \mathcal I_1$ in~\eqref{eq:R-refor-opp-2}. Solving for $U_N$ thus incurs the same computational complexity as, and can also be performed in parallel with, computing $\mathcal{R}^\mathrm{opp}(\Pnom^N, \hat p^N)$.
\subsection{Limiting Distribution}
\label{sec:odd-limit}
The next result asserts that the squared projection distance $\mathcal R^\mathrm{odd}$ has the $O(N^{-1})$ convergence rate.
\begin{theorem}[Limiting distribution -- Probabilistic equalized odds] \label{thm:limiting-odd}
Suppose that $(\hat x_i, \hat a_i, \hat y_i)$ are i.i.d.~samples from $\mathbb{P}$. Under the null hypothesis $\mathcal H_0^\mathrm{odd}$, we have
\begin{align*}
N \times \mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N) \xrightarrow{d.} \sup\limits_{\gamma, \zeta
}\Bigg\{\gamma H_1 + \zeta H_0+ \mathds{E}_\mathbb{P}\Bigg[\Bigg\Vert\begin{pmatrix}
\gamma \\
\zeta
\end{pmatrix}^\top
\begin{pmatrix}
p_{11}^{-1} \mathbbm{1}_{(1, 1)}(A, Y)- p_{01}^{-1} \mathbbm{1}_{(0, 1)}(A, Y) \\
p_{10}^{-1} \mathbbm{1}_{(1, 0)}(A, Y)- p_{00}^{-1} \mathbbm{1}_{(0, 0)}(A, Y)
\end{pmatrix} \nabla h_\beta(X)
\Bigg\Vert_*^2 \Bigg]\Bigg\},
\end{align*}
where $\nabla h_\beta(X) = h_\beta(X) (1- h_\beta(X) \beta$, and $H_y = \mathcal N(0, \sigma_y^2) /(p_{1y}p_{0y})$ with $\sigma_y^2 = \mathrm{Cov}(Z_y)$, and $Z_y$ are random variables
\begin{align*}
Z_y &= h_\beta(X)\left( p_{0y}\mathbbm{1}_{(1,y)}( A,Y)
-p_{1y}\mathbbm{1}_{(0,y)}( A, Y) \right)\\ & \qquad +\mathbbm{1}_{(0,y)}(
A,Y) \mathds{E}_{\mathbb{P}}[\mathbbm{1}_{(1,y)}( A, Y)
h_\beta(X)]\\
&\qquad -\mathbbm{1}_{(1,y)}( A, Y) \mathds{E}_{\mathbb{P}}[\mathbbm{1}_{(0,y)}(
A, Y) h_\beta(X)].
\end{align*}
\end{theorem}
\noindent \textbf{Construction of the hypothesis test.} Contrary to the explicit chi-square limiting distribution for the probabilistic equal opportunity fairness in Theorem~\ref{thm:limiting-opp}, the limiting distribution for the probabilistic equalized odds fairness is not available in closed form. Nevertheless, the limiting distribution in this case can be obtained by sampling $H_0$ and $H_1$ and solving a collection of optimization problems for each sample. Notice that the objective function of the supremum problem presented in Theorem~\ref{thm:limiting-odd} is continuous in $H_1$ and $H_0$, one thus can define
\[
\hat H_y = \mathcal N(0, \hat \sigma_y^2)/ (\hat p_{1y}^N \hat p_{0y}^N),
\]
where $\hat \sigma_y^2$ is the sample average estimate of $\sigma_y^2$, which can be computed using an equation similar to~\eqref{eq:sigma-estimate}. The limiting distribution can be computed by solving the optimization problem with plug-in values
\begin{align*}
\sup\limits_{\gamma, \zeta
}\Bigg\{\gamma \hat H_1 + \zeta \hat H_0+ ~\mathds{E}_{\Pnom^N}\Bigg[\Bigg\Vert\begin{pmatrix}
\gamma \\[.2cm]
\zeta
\end{pmatrix}^\top
\begin{pmatrix}
(\hat p_{11}^N)^{-1} \mathbbm{1}_{(1, 1)}(A, Y)- (\hat p_{01}^N)^{-1} \mathbbm{1}_{(0, 1)}(A, Y) \\[.2cm]
(\hat p_{10}^N)^{-1} \mathbbm{1}_{(1, 0)}(A, Y)- (\hat p_{00}^N)^{-1} \mathbbm{1}_{(0, 0)}(A, Y)
\end{pmatrix} \nabla h_\beta(X)
\Bigg\Vert_*^2 \Bigg]\Bigg\}.
\end{align*}
Notice that the expectation in taken over the empirical distribution $\Pnom^N$, and can be written as a finite sum. The last optimization problem can be solved efficiently using quadratic programming for any realization of $\hat H_1$ and $\hat H_0$. The objective values can be collected to compute the $(1-\alpha) \times 100\%$-quantile estimate $\hat \eta_{1-\alpha}^\mathrm{odd}$ of the limiting distribution. The statistical test decision using the plug-in estimate becomes
\begin{center}
Reject $\mathcal H_0^\mathrm{odd}$ if $\hat s_N^\mathrm{odd} > \hat \eta_{1-\alpha}^\mathrm{odd}$,
\end{center}
where $\hat s_N^\mathrm{odd} = N \times \mathcal R^\mathrm{odd}(\Pnom^N, \hat p^N)$.
\subsection{Most Favorable Distributions}
\label{sec:odd-favor}
If the feature space $\mathcal X$ is endowed with an Euclidean norm, then the most favorable distribution $\mathbb{Q}^\star$, defined in this section as the projection of $\Pnom^N$ onto $\mathcal F_{h_\beta}^\mathrm{odd}$, can be constructed by exploiting Lemma~\ref{lemma:R-compute-odd}.
\begin{lemma}[Most favorable distribution]
\label{lemma:favorable-odd}
Suppose that $\|\cdot\|$ is the Euclidean norm. Let $\gamma^\star$ and $\zeta^\star$ be the optimal solution of problems~\eqref{eq:R-refor-opp-2} and~\eqref{eq:U_N}, respectively. For any $i \in \mathcal I_1$, let $k_i^\star$ be the solution of the inner minimization of~\eqref{eq:R-refor-opp-2} with respect to $\gamma^\star$, and for any $i \in \mathcal I_0$, let $k_i^\star$ be a solution of the inner minimization of~\eqref{eq:U_N} with respect to $\zeta^\star$. Then the most favorable distribution $\mathbb{Q}^\star = \arg\min_{\mathbb{Q} \in \mathcal F_{h_\beta}^\mathrm{odd}}~\mathds W(\Pnom^N, \mathbb{Q})$ is a discrete distribution of the form
\[
\mathbb{Q}^\star = \frac{1}{N} \Big( \sum_{i\in\mathcal I_0} \delta_{(\hat x_i - k_i^\star \zeta^\star \lambda_i \beta, \hat a_i, \hat y_i)} + \sum_{i \in \mathcal I_1} \delta_{(\hat x_i - k_i^\star \gamma^\star \lambda_i \beta, \hat a_i, \hat y_i)}\Big).
\]
\end{lemma}
The proof of Lemma~\ref{lemma:favorable-odd} follows from verifying that $\mathbb{Q}^\star \in \mathcal F_{h_\beta}^\mathrm{odd}$ and that $\mathds W(\mathbb{Q}^\star, \Pnom^N)^2 = \mathcal{R}^\mathrm{odd}(\Pnom^N, \hat p^N)$ using Lemma~\ref{lemma:R-compute-odd}, the detailed proof is omitted. For probabilistic equalized odds, the most favorable distribution $\mathbb{Q}^\star$ alters the locations of both $i \in \mathcal I_0$ and $i \in \mathcal I_1$. The directions of perturbation are dependent on $\lambda_i$, which is determined using~\eqref{eq:lambdai-def}. Notice that $\lambda_i$ carry opposite signs corresponding to whether $\hat a_i = 0$ or $\hat a_i = 1$, thus the perturbations will move $\hat x_i$ in opposite directions based on the value of the sensitive attribute $\hat a_i$.
\section{Numerical Experiment}
\label{sec:experiment}
All experiments are run on an Intel Xeon based cluster composed of 287 compute nodes each with 2 Skylake processors running at 2.3 GHz with 18 cores each. We only use 2 nodes of this cluster and all optimization problems are implemented in Python version 3.7.3.
In all experiments, we use the 2-norm to measure distances in the feature space. Moreover, we focus on the hypothesis test of probabilistic equal opportunity, and thus the Wasserstein projection, the limiting distribution and the most favorable distribution follow from the results presented in Section~\ref{sec:EO}.
\subsection{Validation of the Hypothesis Test}
We now demonstrate that our proposed Wasserstein projection framework for statistical test of fairness is a valid, or asymptotically correct, test. We consider a binary classification setting in which $\mathcal X$ is 2-dimensional feature space. The true distribution $\mathbb{P}$ has true marginal values $p_{ay}$ being
\[
p_{11} = 0.2,~p_{01} = 0.1,~p_{10} = 0.3,~p_{00} = 0.4.
\]
Moreover, conditioning on $(A, Y)$, the feature $X$ follows a Gaussian distribution of the form
\begin{align*}
X|A = 1, Y = 1 &\sim \mathcal N([6, 0], [3.5, 0; 0, 5]),\\
X|A = 0, Y = 1 &\sim \mathcal N ([-2, 0], [5, 0; 0, 5]),\\
X|A = 1, Y = 0 &\sim \mathcal N ([6, 0], [3.5, 0; 0, 5]),\\
X|A = 0, Y = 0 &\sim \mathcal N ([-4, 0], [5, 0; 0, 5]).
\end{align*}
The true distribution $\mathbb{P}$ is thus a mixture of Gaussian, and under this specification, a simple algebraic calculation indicates that a logistic classifier with $\beta = (0, 1)^\top$ is fair with respect to the probabilistic equal opportunity criteria in Definition~\ref{def:proba-EO}. We thus focus on verifying fairness for this specific classifier.
\begin{figure*}[h]
\centering
\begin{subfigure}[h]{0.37\textwidth}
\includegraphics[width=\textwidth]{pdf_100.pdf}
\caption{$N=100$}
\label{fig:limita}
\end{subfigure}
\begin{subfigure}[h]{0.37\textwidth}
\includegraphics[width=\textwidth]{pdf_500.pdf}
\caption{$N=500$}
\label{fig:limitb}
\end{subfigure}
\begin{subfigure}[h]{0.37\textwidth}
\includegraphics[width=\textwidth]{cdf_100.pdf}
\caption{$N=100$}
\label{fig:limitc}
\end{subfigure}
\begin{subfigure}[h]{0.37\textwidth}
\includegraphics[width=\textwidth]{cdf_500.pdf}
\caption{$N=500$}
\label{fig:limitd}
\end{subfigure}
\caption{Empirical distribution of $N \times \mathcal R^\mathrm{opp}(\Pnom^N, \hat p^N)$ taken over 2,000 replications (histogram) versus the limiting distribution $\theta \chi_1^2$ (blue curve) with different sample sizes $N$. Fig.~\ref{fig:limita}-\ref{fig:limitb} are density plots, Fig.~\ref{fig:limitc}-\ref{fig:limitd} are cumulative distribution plots. }
\label{fig:limit}
\end{figure*}
In the first experiment, we empirically validate Theorem~\ref{thm:limiting-opp}. To this end, we generate $N \in \{100, 500\}$ i.i.d.~samples from $\mathbb{P}$ to be used as the test data, and then calculate the squared projection distance $\mathcal R^\mathrm{opp}(\Pnom^N, \hat p^N)$ using Proposition~\ref{prop:R-refor}. The process is repeated 2,000 times to obtain an empirical estimate of the distribution of $N \times \mathcal R^\mathrm{opp}(\Pnom^N, \hat p^N)$. We also generate another set of one million i.i.d.~samples from $\mathbb{P}$ to estimate the limiting distribution $\theta \chi_1^2$. Figure~\ref{fig:limit} shows that the empirical distribution of $N \times \mathcal R^\mathrm{opp}(\Pnom^N, \hat p^N)$ converges to the limiting distribution $\theta \chi_1^2$ as $N$ increases.
The second set of experiments aims to show that our proposed Wasserstein projection hypothesis test is asymptotically valid. We generate $N \in \{100, 500, 1000\}$ i.i.d.~samples from $\mathbb{P}$ and calculate the test statistic $N \times \mathcal R^\mathrm{opp}(\Pnom^N, \hat p^N)$. The same data is used to estimate $\hat \theta^N$ and compute the $(1-\alpha)\times 100\%$-quantile of $\hat \theta^N \chi_1^2$ to perform the quantile based test as laid out in Section~\ref{sec:opp-limit}. We repeat this procedure for 2,000 replications to keep track of the rejection projection at different significant values of $\alpha \in \{0.5, 0.3, 0.1, 0.05, 0.01\}$.
Table~\ref{tab:rej_rates_table} summarizes the rejection probabilities of Wasserstein projection tests for equal opportunity criterion under the null hypothesis $\mathcal H_0^\mathrm{opp}$. We can observe that at sample size $N > 100$, the rejection probability is close to the desired level $\alpha$, which empirically validates our testing procedure.
\begin{center}
\vspace{-.2cm}
\begin{table}[h]
\begin{tabular}{||c c c c ||c||}
\hline \hline
$N=100$ & $N=500$ & $N=1000 $ & $\alpha$\\
\hline\hline
0.511 &0.4905 & 0.5 & 0.50\\ \hline
0.282 &0.2895 &0.299 & 0.30\\\hline
0.048 &0.0895 &0.093 & 0.10\\ \hline
0.007 & 0.0425 & 0.0405 &0.05\\\hline
0.0 & 0.0065 & 0.005 & 0.01\\
\hline \hline
\end{tabular}
\vspace{.2cm}
\caption{Comparison of the null rejection probabilities of probabilistic equal opportunity tests with different significance levels~$\alpha$ and test sample sizes $N$.} \label{tab:rej_rates_table}
\end{table}
\end{center}
\subsection{Most Favorable Distribution Analysis}
\begin{figure}
\centering
\includegraphics[width=0.4\columnwidth]{most_fav_distributiontoy_correlated_for_most_fav_dist.pdf}
\caption{Visualization of the most favorable distribution $\mathbb{Q}^\star$ for a logistic classifier with weight $\beta =(0.4, 0.12)^\top$. The black arrow indicates the vector $\beta$. Colors represent class, while symbolic shapes encode the sensitive values.
The green lines show the transport plan of the empirical test samples from their original positions (indicated with transparent colors) to their ultimate destinations (with non-transparent colors).}
\label{fig:most_fav}
\end{figure}
In this section, we visualize the most favorable distribution $\mathbb{Q}^\star$ from Lemma~\ref{lemma:favorable-opp} for a vanilla logistic regression classifier with weight $\beta=(0.4, 0.12)^\top$. We simply generate 28 samples with equal subgroup proportions to form the empirical distribution $\Pnom^N$. To find the support of~$\mathbb{Q}^\star$, we solve problem~\eqref{eq:R-refor-opp-2}, whose optimizer dictates the transportation plan of each sample $\hat x_i$. Figure~\ref{fig:most_fav} visualizes the original test samples that forms $\Pnom^N$, along with the most favorable distribution $\mathbb{Q}^\star$. Green lines in the figure represent how samples are perturbed. As we are testing for the probabilistic notion of equal opportunity, only the samples with positive label $\hat y_i=1$ presented in blue are perturbed in order to obtain $\mathbb{Q}^\star$. Furthermore, we observe that the positively-labeled test samples are transported along the axis directed by $\beta$ (black arrow). Moreover, the samples with different sensitive attributes, represented by different shapes, move in opposite direction so that they get closer to each other, which reduces the discrepancy in the expected value of $h_\beta(X)$ between the relevant subgroups.
\subsection{The COMPAS Dataset}
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{reg.pdf}
\caption{Test statistic and accuracy of Tikhonov regularized logistic regression on test data with rejection threshold $\hat \eta_{0.95}$.}
\vspace{-5mm}
\label{fig:reg_figure}
\end{figure}
COMPAS (Correctional Offender Management Profiling for Alternative Sanctions)\footnote{https://www.propublica.org/datastore/dataset/compas-recidivism-risk-score-data-and-analysis} is a commercial tool used by judges and parole officers for scoring criminal defendant's likelihood of recidivism. The COMPAS dataset is used by the COMPAS algorithm to compute the risk score of reoffending for defendants, and also contains the criminal records within 2 years after the decision. The dataset consists of 6,172 samples with 10 attributes including gender, age category, race, etc. We concentrate on the subset of the data with violent recidivism, and we use race (African-American and Caucasian) as the sensitive attribute. We split 70$\%$ of the COMPAS data to train a Tikhonov-regularized logistic classifier, with the tuning penalty parameter $\lambda$ chosen in the range from 0 to 100 with 50 equi-distant points. The remaining $30\%$ of the data is used as the test samples for auditing.
Figure~\ref{fig:reg_figure} demonstrates the relation between the accuracy and the degree of fairness with respect to the regularization parameter $\lambda$. Strong regularization penalty (high values of $\lambda$) results in small values of the test statistic, but the classifier has low test accuracy. On the contrary, weak penalization leads to undesirable fairness level but higher prediction accuracy.
The pink dashed line in Figure~\ref{fig:reg_figure} shows the rejection threshold of the Wasserstein projection test at significance level $\alpha = 0.05$ for varying value of the regularization parameter $\lambda$. We can observe that the Wasserstein projection test recommends a rejection of the null hypothesis $\mathcal H_0^\mathrm{opp}$ for a wide range of $\lambda$. Only at $\lambda$ sufficiently large that the test fails to reject the null hypothesis.
\section{Concluding Remarks and Broader Impact}
\label{sec:conclusion}
In this paper, we propose a statistical hypothesis test for group fairness of classification algorithms based on the theory of optimal transport. Our test statistic relies on computing the projection distance from the empirical distribution supported on the test samples to the manifold of distributions that renders the classifier fair. When the notion of fairness is chosen to be either the probabilistic equal opportunity or the probabilistic equalized odds, we show that the projection can be computed efficiently. We provide the limiting distribution of the test statistic and show that our Wasserstein projection test is asymptotically correct. Our proposed test also offers the flexibility to incorporate the geometric information of the feature space into testing procedure. Finally, analyzing the most favorable distribution can help interpreting the reasons behind the outcome of the test.
The Wasserstein projection hypothesis test is the culmination of a benevolent motivation and effort, and it aims to furnish the developers, the regulators and the general public a quantitative method to verify certain notions of fairness in the classification setting. At the same time, we acknowledge the risks and limitations of the results presented in this paper.
First, it is essential to keep in mind that this paper focuses on \textit{probabilistic} notions of fairness, in particular, we provide the Wasserstein statistical test for probabilistic equality of opportunity and probabilistic equalized odds. Probabilistic notions are only approximations of the original definitions, and the employment of probabilistic notions are solely for the technical purposes. Due to the sensitivity of the test result on the choice of fairness notions, a test that is designed for probabilistic notions may not be applicable to test for original notions of fairness due to the interplay with the threshold $\tau$ and the radical difference of both the test statistic and the limiting distribution. If a logistic classifier $h_\beta$ is rejected using our framework for probabilistic equal opportunity, it does \textit{not} necessarily imply that the classifier $h_\beta$ fails to satisfy the equal opportunity criterion, and vice versa. The same argument holds when we test for probabilistic equalized odds.
Second, the outcome of the Wasserstein projection test is dependent on the choice of the underlying metric on the feature, the sensitive attribute and the label spaces. Indeed, the test outcome can change if we switch the metric of the feature space, for example, from the Euclidean norm to a 1-norm. In the scope of this paper, we do not study how sensitive the test outcome is with respect to the choice of the metric, nor can we make any recommendation on the optimal choice of the metric. Nevertheless, it is reasonable to recommend that the metric should be chosen judiciously, and the action of tuning the metric in order to obtain favorable test outcome should be prohibited.
Third, to simplify the computation, we have assumed absolute trust on the sensitive attributes and the label. The users of our test should be mindful if there is potential corruption to these values. Moreover, our test is constructed under the assumption that there is no missing values in the test data. This assumption, unfortunately, may not hold in real-world implementations. Constructing statistical test which is robust to adversarial attacks and missing data using the Wasserstein projection framework is an interesting research direction.
Fourth, the statistical test in this paper is for a simple null hypothesis. In practice, the regulators may be interested in a relaxed fairness test in which the difference of the conditional expectations is upper bounded by a fixed positive constant $\epsilon$. The extension of the Wasserstein hypothesis testing framework for a composite null hypothesis is non-trivial, thus we leave this idea for future study.
Finally, any auditing process for algorithmic fairness can become a dangerous tool if it falls into the hand of unqualified or vicious inspectors. The results in this paper are developed to broaden our scientific understanding, and we recommend that the test and its outcomes should be used as an informative reference, but \textit{not} as an absolute certification to promote any particular classifier or as a justification for any particular classification decision.
We thus sincerely recommend that the tools proposed in this paper be exercised with utmost consideration.
\textbf{Acknowledgments.}
Research supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF40$\_$180545. Material in this paper is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0397. Additional support is gratefully acknowledged from NSF grants 1915967, 1820942, 1838676, and also from the China Merchant Bank.
Finally, we would like to thank Nian Si and Michael Sklar for helpful comments and discussions.
|
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Hack A Slot Machine With Iphone
Hack A Slot Machine With Iphone – R. Paul Wilson On: Why the hole is a game of cat and mouse for Cheaters and The House
In fact, it is a difficult question to answer since, in my experience, any game is vulnerable to attacks by hackers or opportunistic players and the more confident someone is with regards to game security, the more concerned I am that they don't watch those games as closely as they should.
Slot machines, for example, occupy the majority of floor space in modern high-end resorts and are carefully monitored; not only from the ground or the sky but inside with unusual activity reported automatically.
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However, these devices have been successfully targeted and the more protections they have, the greater the chance that something may have been missed or not thought of.
Since their introduction, holes have been the site for many games of cat and mouse between predators and home.
Early machines have many defects that are gradually abused and appear over time. My favorite of these is the ability to place a bandit arm behind a winning barrel.
The arm is pulled down, cocking the gear to turn the wheels. But before the tension is released, which will determine the result, the arm is forced, passing it through the gear, locking the wheels in their previous position so that when the arm returns to its initial position, the cycles are registered another (symbol ) win!
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The poor and easily accessible locks are quickly taken and give access to the internal mechanism where you can add weights, filed gears and any possible way to affect the resulting effort.
As devices become more secure and device flaws are addressed, hackers find more clever ways to beat them.
A feature designed to control the amount of coins issued is a light sensor that registers every passing coin, blocking the light for a split second. This is an old-fashioned way of becoming familiar, but it leads to another clever solution.
A small bulb is connected to the end of a piece of thin plastic, designed to slide into the hopper and contact a sensor that reads the light and records how many times a coin has passed into the tray.
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The billing software works like this: every time the electricity breaks down, it registers a bill as paid out. In production, it will keep pulling coins from inside and releasing them until the correct number of coins are out.
See also Rocky Slot Machine Free
To beat this, a pot bulb is positioned in front of the sensor and turns on whenever the machine is charged. This means that the coins are now falling from the bank and into the tray, but after the fire since the intruder is lighting his own device in the sensor!
Expert slot thieves quickly measure how long to hold the fire so that the machine will extract a few more coins each time the machine pays or is paid out.
This led to the devices being rebuilt with a plastic guard attached that was placed in front of the sensor from the bottom so that any attempt to use "fire" blocked the sensor completely – making this scam impossible.
Find Out If Gas Station Slots Are Fair Or If You're Getting Cheated
The solution s 'secure devices from fraudsters operating the light, but also offers an easy way to manipulate payments.
While slot machine developers are happily putting a hidden gem into every new machine on the market, this new cheating system is emptying the machines on the floors!
Paw Monkey is very easy compared to "light", which needs to be sold, connected to a battery and looks like what it is – a compact game machine.
The "Papa" is a piece of thick, hard wire bent to fit the inner shape of the machine so that the end of the coin can be quickly married up to the bottom of the conductive plastic guard that prevents the flame from being used.
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Now some attackers have pushed up the guard in order to block the sensor as the coins are dropped.
Since the machine is a rotating machine that ejects coins back and forth, past the sensor to be counted, it spits out coins until the required number is paid. As with fire, the paw has to be played in and out with care so as not to press the device and introduce a problem to safety.
Essentially, cheaters make each coin longer in terms of how long it takes to fire from the sensor's perspective. It seems as if a coin falls but in reality many are allowed to pass each time the fire breaks.
See also Slot Machine Triple Chance
Clearly, the method we invented to protect slot machines from a secret light source has proven to be effective even if used to block the sensor for a long time!
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As soon as the slot machines are working, the place becomes quieter (although I occasionally miss the cacophony of tumbling silver dollars) and the slot thieves are no longer able to milk the machines for excess coins.
With these new cashless machines, players insert paper money and receive a bar code with recorded profits for the seller or a separate machine to pay through a network system.
A few ballsy football players will play tricks on customers while pretending to help them cash out, secretly changing a big win or cash-out for a smaller amount on an otherwise paper slip. resemble.
These guys were quickly aware and took since s is all beautiful Islam to avoid anyone else taking advantage of their customers.
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The electronic bill readers have a defect that is recognized and used thanks to a simple device, made to open a pocket garage door.
This device has a thin, flat wire projecting that is slid under a bill as it is inserted into the reader. Each click of the button sends a token to the reader that lets you register a dollar bill for every click of the button!
Cheaters will put in one dollar, hit the buttons until the machine registers nine dollars (more than this will shake the management) and walk away while the shills step up and take over the machines. They will play for an hour or more until they burn through some of that fairy money or get a few hidden prizes.
The shill will come out, take the book to the cashier and leave with hundreds of dollars. Thieves will hit many places and machines in one night, securing thousands of dollars for a "night on the town."
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This evolution of cheating describes the constant cat and mouse game played between the s and anyone who is ready to beat them by any and all means possible.
Complete reliance on any security mechanism invites risk because those who break through, under or over the walls feel different from those who build them.
For example, when microchips were inserted into the machine slot, a resourceful employee reprogrammed a few and put them in the live machine during routine maintenance.
See also Reel Em In Slot Machine Locations
Look at this: at a conference several years ago, I watched a company tell their customers that the bill reader scam described above was "impossible" – when I had seen it happen with my own eyes. just a few hours before!
How Addictive Mobile Games Hack Your Brain To Drain Your Wallet
There is always a way to defeat a strategy. All it takes is someone with the right perspective (and motivation).
Poker Basics: How s Make Money On Poker It's no secret that s are in the business of making money. From roulette to slots, every single game & hellip;
Top 10 Dos And Don'ts Of Etiquette I have spent more than 30 years on earth, and one of the things I wish I could…
How to Bet Late in a Horse Race Holding your nerve and being able to bet late in a horse race takes skill and practice. You can read and …"If you are a tool, how do you make people hug? Turn yourself into a slot machine." – Tristan Harris, co-founder of Time Well Used and former Google.
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As part of the series How design can hack your brain – and after the Infinite Scroll and Hook Design concept made by Nir Eyal – today's turn is for the Personal Slot Machine – Are you ready for it?
If you've lost your phone for a while, you may be in a state of mild panic until you find it. According to a study from Harvard University, about 73% of people claim to experience the unique taste of anxiety, because most adults spend more than 3 hours per day tapping, clicking and dragging on their devices.
Most of us have become so in tune with our digital lives that we don't even feel our phones alerting in our pockets when they aren't even there.
New forms of addiction and addictive behaviors have emerged in society with the rapid development of technology. Today I will focus on the third step of the Hook Model – variable reward – which acts as a bridge between modern digital services and Gaming.
Ways Criminals 'hack' Into Atms
While digital services are only a product of the internet, gaming has been here almost since the beginning of civilization, considering that the sport of wagering has evolved seamlessly with recent technology. With the internet
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 637
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Kohli on India's biggest Test win: Kohli threw accolades on Prithvi Shaw, Ravindra Jadeja and the team for a gargantuan victory.
In which ended up being a cakewalk for India, they literally conquered the first Test against West Indies. A record-breaking victory (by an innings and 272 runs) came India's way in under three complete days.
Speaking during the post-match presentation, captain Virat Kohli was contended with the way his team played in these three days. Having lost India's last series 1-4 against and in England, Kohli was asked about the difference in conditions of England and India.
"I don't think you can compare the two conditions [England and India]. That was a bigger challenge. We understand that with the ability we have, we will dominate in these conditions. We were very clinical," he was quoted.
Kohli didn't feel it to be correct to speak on what went wrong for the inexperienced opposition. "It's not something for to speak on [West Indies' performance] I'm sure they'll sort out the errors. We don't want to focus on what the opposition does," he said.
Apart from praising the team's effort as a whole, Kohli was delighted on the way debutant Prithvi Shaw and all-rounder Ravindra Jadeja batted during India's innings. "Delighted for Prithvi and Jaddu. Playing his first game, seeing him dominate – the guy showed he is different quality. That's why he's been pushed to the Test team. It is exciting to see from the captain's perspective.
"Jaddu as well – he has got important runs for us before and we wanted him to get three figures. We believe he can turn matches for us," said the Indian captain.
Although the Indian fast bowlers picked only three (out of 20) wickets in two innings, Kohli was satisfied to see them bowled in the way they did. "If you see the first innings, the way Umesh and Shami ran in to bowl. Few wickets with the new ball and you can put the opposition under pressure. Shami took wickets on a pitch that was offering nothing. Didn't want to play four bowlers with the heat factor," Kohli added.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 3,860
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Fulham boss Marco Silva keeping fingers crossed that Sheffield United clash goes ahead amid rising Covid-19 cases
Sheffield Wednesday promotion rivals bring in former Tottenham Hotspur and Manchester United man in League One promotion race twist
Keiren Westwood training with Sheffield Wednesday's League One rivals as he plots comeback
Sheffield Wednesday modern icon Keiren Westwood is plotting a comeback to professional football and is spending time training with one of their League One rivals, he has revealed.
Thursday, 4th November 2021, 5:00 am
The 37-year-old, who has maintained his determination to find a new club since he left the Owls in the summer, made a heroes return to Hillsborough on Tuesday evening while enjoying a co-commentary stint during their 3-0 win over Sunderland.
Westwood had a handful of irons in the fire but none came to fruition as he continues to look for the right option.
The wait for the right opportunity goes on and revealed he has been keeping in shape with the help of Wednesday's League One rivals Crewe Alexandra.
"I'm really thankful to Crewe, the manager and all the staff there," he said. "I've known the goalkeeper coach since I was 19 when I was at Carlisle, Fred Barber.
"A club was interested in me in the summer and unfortunately it didn't really work out so Fred said to go and train at Crewe.
"I said I wasn't sure because of Covid and stuff. He said they were sorted for all that and that I'd be great for the young ones and that. Their number one is Will Jääskeläinen.
"He told me to come in and help out as much as I can, help out and show the standards of what he needs to do.
Former Sheffield Wednesday goalkeeper Keiren Westwood is training with Crewe Alexandra.
"It's been brilliant and I've really, really enjoyed it."
Westwood is based in the North West with his young family and though he appreciates the Football League has a different financial landscape post-coronavirus, is confident he'll find a new club as the season goes on.
The two-time Owls player of the year and former Republic of Ireland keeper is now looking towards winter and the January transfer window when clubs will have the option to move existing goalkeepers on.
"I do feel like I can still do the job," he said. "I'm really, really fit training with Crewe. If anybody knows Fred Barber and the way he works they know I'm fit because he works ridiculously hard – it's very intense and very rewarding. It keeps me feeling young!
"Would I like to be at a club? Yes."
Looking back on his seven-year stint with Wednesday, Westwood expressed regret at how it ended – with relegation from the Championship – but detailed his pride and continuing support for the club.
"I've got nothing but good memories of Wednesday to be honest," he said. "There were a few little bumps in the road but I had a fantastic time here.
"I wish I was playing tonight to be honest! I'm looking down at the pitch ahead of such a huge game. I'd be thriving for a game like today.
"I don't think the club can look back now, they've got to move forward and look at the positives.
"It's still a huge football club, the numbers are still coming to the stadium, they still want to support the lads and the club wants to be promoted.
"The thing is, so does everybody else and it's all been geared up towards that. They've got a really good squad, a squad that should be competing right up the top end of the table and hopefully they can."
League OneKeiren WestwoodSheffieldHillsborough
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,237
|
using Machine.Specifications;
using PlainElastic.Net.IndexSettings;
using PlainElastic.Net.Utils;
namespace PlainElastic.Net.Tests.Builders.IndexSettings
{
[Subject(typeof(WordDelimiterTokenFilter))]
class When_complete_WordDelimiterTokenFilter_built
{
Because of = () => result = new WordDelimiterTokenFilter()
.Name("name")
.Version("3.6")
.GenerateWordParts(false)
.GenerateNumberParts(false)
.CatenateWords(true)
.CatenateNumbers(true)
.CatenateAll(true)
.SplitOnCaseChange(false)
.PreserveOriginal(true)
.SplitOnNumerics(false)
.StemEnglishPossessive(false)
.ProtectedWords("2", "3")
.ProtectedWordsPath("4")
.TypeTable("$=>DIGIT", "%=>DIGIT")
.TypeTablePath("5")
.CustomPart("{ Custom }")
.ToString();
It should_start_with_name = () => result.ShouldStartWith("'name': {".AltQuote());
It should_contain_type_part = () => result.ShouldContain("'type': 'word_delimiter'".AltQuote());
It should_contain_version_part = () => result.ShouldContain("'version': '3.6'".AltQuote());
It should_contain_generate_word_parts_part = () => result.ShouldContain("'generate_word_parts': false".AltQuote());
It should_contain_generate_number_parts_part = () => result.ShouldContain("'generate_number_parts': false".AltQuote());
It should_contain_catenate_words_part = () => result.ShouldContain("'catenate_words': true".AltQuote());
It should_contain_catenate_numbers_part = () => result.ShouldContain("'catenate_numbers': true".AltQuote());
It should_contain_catenate_all_part = () => result.ShouldContain("'catenate_all': true".AltQuote());
It should_contain_split_on_case_change_part = () => result.ShouldContain("'split_on_case_change': false".AltQuote());
It should_contain_preserve_original_part = () => result.ShouldContain("'preserve_original': true".AltQuote());
It should_contain_split_on_numerics_part = () => result.ShouldContain("'split_on_numerics': false".AltQuote());
It should_contain_stem_english_possessive_part = () => result.ShouldContain("'stem_english_possessive': false".AltQuote());
It should_contain_protected_words_part = () => result.ShouldContain("'protected_words': [ '2','3' ]".AltQuote());
It should_contain_protected_words_path_part = () => result.ShouldContain("'protected_words_path': '4'".AltQuote());
It should_contain_type_table_part = () => result.ShouldContain("'type_table': [ '$=>DIGIT','%=>DIGIT' ]".AltQuote());
It should_contain_type_table_path_part = () => result.ShouldContain("'type_table_path': '5'".AltQuote());
It should_contain_custom_part = () => result.ShouldContain("{ Custom }".AltQuote());
It should_return_correct_result = () => result.ShouldEqual(("'name': { " +
"'type': 'word_delimiter'," +
"'version': '3.6'," +
"'generate_word_parts': false," +
"'generate_number_parts': false," +
"'catenate_words': true," +
"'catenate_numbers': true," +
"'catenate_all': true," +
"'split_on_case_change': false," +
"'preserve_original': true," +
"'split_on_numerics': false," +
"'stem_english_possessive': false," +
"'protected_words': [ '2','3' ]," +
"'protected_words_path': '4'," +
"'type_table': [ '$=>DIGIT','%=>DIGIT' ]," +
"'type_table_path': '5'," +
"{ Custom } }").AltQuote());
private static string result;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,165
|
class DefinitionsError < StandardError; end
class ChekkuError < StandardError; end
class AppNameNotSaneError < ChekkuError; end
class AppNameNotStringError < ChekkuError; end
class DefinitionNotFoundError < ChekkuError; end
class DefinitionValidationError < DefinitionsError; end
class NotInstalledError < StandardError; end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,652
|
namespace Microsoft.Azure.Management.LabServices.Models
{
using Newtonsoft.Json;
using System.Linq;
/// <summary>
/// Details of the status of an operation.
/// </summary>
public partial class LatestOperationResult
{
/// <summary>
/// Initializes a new instance of the LatestOperationResult class.
/// </summary>
public LatestOperationResult()
{
CustomInit();
}
/// <summary>
/// Initializes a new instance of the LatestOperationResult class.
/// </summary>
/// <param name="status">The current status of the operation.</param>
/// <param name="errorCode">Error code on failure.</param>
/// <param name="errorMessage">The error message.</param>
/// <param name="requestUri">Request URI of the operation.</param>
/// <param name="httpMethod">The HttpMethod - PUT/POST/DELETE for the
/// operation.</param>
/// <param name="operationUrl">The URL to use to check long-running
/// operation status</param>
public LatestOperationResult(string status = default(string), string errorCode = default(string), string errorMessage = default(string), string requestUri = default(string), string httpMethod = default(string), string operationUrl = default(string))
{
Status = status;
ErrorCode = errorCode;
ErrorMessage = errorMessage;
RequestUri = requestUri;
HttpMethod = httpMethod;
OperationUrl = operationUrl;
CustomInit();
}
/// <summary>
/// An initialization method that performs custom operations like setting defaults
/// </summary>
partial void CustomInit();
/// <summary>
/// Gets the current status of the operation.
/// </summary>
[JsonProperty(PropertyName = "status")]
public string Status { get; private set; }
/// <summary>
/// Gets error code on failure.
/// </summary>
[JsonProperty(PropertyName = "errorCode")]
public string ErrorCode { get; private set; }
/// <summary>
/// Gets the error message.
/// </summary>
[JsonProperty(PropertyName = "errorMessage")]
public string ErrorMessage { get; private set; }
/// <summary>
/// Gets request URI of the operation.
/// </summary>
[JsonProperty(PropertyName = "requestUri")]
public string RequestUri { get; private set; }
/// <summary>
/// Gets the HttpMethod - PUT/POST/DELETE for the operation.
/// </summary>
[JsonProperty(PropertyName = "httpMethod")]
public string HttpMethod { get; private set; }
/// <summary>
/// Gets the URL to use to check long-running operation status
/// </summary>
[JsonProperty(PropertyName = "operationUrl")]
public string OperationUrl { get; private set; }
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,873
|
\section{Introduction}
Let $H$ be an $h$-vertex graph and $G$ be an $n$-vertex graph. An $H$-\emph{tiling} in $G$ is a collection of vertex-disjoint subgraphs of $G$ isomorphic to $H$. An $H$-\emph{factor} is an $H$-tiling which covers all the vertices of $G$. Determining sufficient conditions for the existence of an $H$-factor is one of the fundamental lines of research in extremal graph theory. One important reason is due to a result of Hell and Kirkpatrick \cite{1983Hell} which shows that the decision problem for $H$-factors is \emph{NP}-complete, given that $H$ has a connected component of size at least 3.
The first result in this line is due to Dirac \cite{Dirac1952}, who proved that every $n$-vertex graph $G$ with $\delta(G)\geq\frac{n}{2}$ contains a Hamiltonian cycle, in particular if $n$ is even then $G$ has a perfect matching. The celebrated Hajnal--Szemer\'{e}di theorem \cite{HajnalSze1970} states that for all integers $n, r$ with $r\geq 2$ and $r|n$, any $n$-vertex graph $G$ with $\delta(G)\geq \left(1-\frac{1}{r}\right)n$ contains a $K_{r}$-factor. The $K_{3}$-factor case was previously proved by Corr\'{a}di and Hajnal \cite{MR200185}. The balanced complete $r$-partite graph witnesses the tightness of the minimum degree condition.
For non-cliques $H$, Alon and Yuster \cite{MR1376050} proved that for any constant $\varepsilon>0$ and large $n\in\mathbb{N}$ divisible by $|H|$, $\delta(G)\geq \left(1-\frac{1}{\chi(H)}+\varepsilon\right)n$ guarantees an $H$-factor, namely, the relevant parameter in the degree condition is $\chi(H)$, the \emph{chromatic number of $H$}.
However, as given in \cite{cooley2007perfect,Kawa}, there are graphs $H$ for which the term $1-1/\chi(H)$ in the minimum degree condition can be improved significantly and determining the best possible bound on $\delta(G)$ for an arbitrary graph $H$ has been an intriguing problem.
There had been many excellent results (see e.g. \cite{MR1767021, MR1829855, MR1995690} and survey \cite{MR2588541}) until it was finally settled by K\"{u}hn and Osthus \cite{DKDO2009} in 2009. There are also several significant generalisations of the Hajnal--Szemer\'{e}di theorem in the setting of partite graphs \cite{MR3354296, MR1910115, MR2433861}, directed graphs \cite{MR3406450} and hypergraphs \cite{MR2500161}.
\subsection{Main result}
Note that the extremal example that achieves the optimality of the bound on $\delta(G)$ in the Hajnal--Szemer\'{e}di theorem contains a large independent set, which is ``regular'' and rather rare among all graphs. Following the spirit of the well-known Ramsey--Tur\'{a}n theory (see \cite{MR0299512,MR1476449, MR716422, MR2500161}), a natural question on the Hajnal--Szemer\'{e}di theorem is to determine the minimum degree condition forcing a clique factor when the host graph has sublinear independence number. The following Ramsey--Tur\'{a}n type problem was proposed by Balogh, Molla and Sharifzadeh \cite{MR3570984}.
\begin{problem}[\cite{MR3570984}]\label{pro1.1}
Let $r\geq 3$ be an integer and $G$ be an $n$-vertex graph with $\alpha(G)=o(n)$. What is the minimum degree condition on $G$ that guarantees a $K_{r}$-factor?
\end{problem}
Balogh, Molla and Sharifzadeh \cite{MR3570984} studied the $K_{3}$-factors and showed that the minimum degree condition $\delta(G)\ge\frac{n}{2}+\mu n$ guarantees a triangle factor, for any $\mu>0$ and large $n\in 3\mathbb N$.
Later, Nenadov and Pehova \cite{MR4080942} generalised Problem \ref{pro1.1} to $K_{\ell}$-independence numbers for $\ell\geq 2$: is it true that for every $r, \ell\in \mathbb{N}$ with $r\geq \ell\geq 2$ and $\mu>0$ there exist a constant $\alpha$ and $n_{0}\in \mathbb{N}$ such that every graph $G$ on $n\geq n_{0}$ vertices where $r$ divides $n$ with $\delta(G)\geq \max\left\{\frac{1}{2}+\mu, \frac{r-\ell}{r}+\mu\right\}n$ and $\alpha_{\ell}(G)\leq \alpha n$ has a $K_{r}$-factor? Nenadov and Pehova \cite{MR4080942} also proved a general upper bound for the minimum degree conditions, which is asymptotically optimal when $\ell=r-1$. Chang, Han, Kim, Wang and Yang \cite{CHKWY} provided a negative answer to this problem. They \cite{CHKWY} also determined an asymptotically optimal degree condition for $\ell\geq \frac{3}{4}r$.
Recently, Knierim and Su \cite{MR4193066} proved the following result, which solves Problem \ref{pro1.1} asymptotically.
\begin{theorem}[\cite{MR4193066}]\label{thm1.31}
Given constant $\mu >0$, $r\in \mathbb{N}$ with $r\geq 4$, there exists $\alpha >0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $r$ dividing $n$, $\delta(G)\geq \left(1-\frac{2}{r}+\mu\right)n$ and $\alpha(G)\leq \alpha n$. Then $G$ contains a $K_{r}$-factor.
\end{theorem}
It is natural to study Problem~\ref{pro1.1} for arbitrary $H$ other than cliques.
The first attempt would be to prove a result that matches the Alon--Yuster theorem \cite{MR1376050}, which we shall formulate below.
For our problem, we find that the relevant parameter for the minimum degree condition is the vertex arboricity of $H$.
\begin{defn}[]\label{def1.41}
The \emph{vertex arboricity $ar(H)$ of a graph $H$} is the minimum integer $r$ such that the vertices of $H$ can be partitioned into $r$ subsets each of which induces a forest. The corresponding partition is called an \emph{acyclic partition} of $H$. We say that an acyclic partition of $H$ is \emph{optimal} if it has exactly $ar(H)$ parts.
\end{defn}
Note that this definition extends that of the chromatic number of $H$, where each color class is required to be an independent set.
Our first result provides an analogue of the Alon--Yuster theorem \cite{MR1376050} in host graphs with sublinear independence number.
\begin{theorem}\label{coro1.311}
Given $\mu >0$, $h\in \mathbb{N}$ with $h\geq 3$ and an $h$-vertex graph $H$, there exists $\alpha >0$ such that the following holds for sufficiently large $n\in h\mathbb{N}$. Let $G$ be an $n$-vertex graph such that $\delta(G)\geq \max\left\{\left(1-\frac{1}{ar(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ contains an $H$-factor.
\end{theorem}
Similar to the Alon--Yuster theorem \cite{MR1376050}, the minimum degree condition in Theorem~\ref{coro1.311} is asymptotically tight for infinitely many graphs $H$ which we define now.
Let $H$ be a graph and $\mathscr P$ be a family of vertex partitions of $H$ where each partition has $\ell$ parts.
For $\mathcal P\in \mathscr P$, let $x_{1}\leq x_{2}\leq\cdots \leq x_{\ell}$ denote the sizes of the parts of $\mathcal P$.
Put $\mathcal{D}(\mathcal P):=\{x_{i+1}-x_{i}\mid i=1, \dots, \ell-1\}$.
Let $\mathcal{D}(H, \mathscr P)$ denote the union of all the sets $\mathcal{D}(\mathcal P)$ taken over all $\mathcal P\in \mathscr P$.
Let $\mathrm{hcf}_{1}(H, \mathscr P)$ be the highest common factor of all integers in $\mathcal{D}(H, \mathscr P)$.
(If $\mathcal{D}(H, \mathscr P)=\emptyset$, then we set $\mathrm{hcf}_{1}(H, \mathscr P):=\infty$.)
Let $\mathrm{hcf}_{2}(H, \mathscr P)$ be the highest common factor of all the orders of components of $H$.
If $\ell=1$ and $\mathrm{hcf}_{2}(H, \mathscr P)=1$, then $\mathrm{hcf}(H, \mathscr P)=1$.
If $\ell=2$, $\mathrm{hcf}_{2}(H, \mathscr P)=1$ and $\mathrm{hcf}_{1}(H, \mathscr P)\leq 2$, then $\mathrm{hcf}(H, \mathscr P)=1$.
If $\ell \geq3$ and $\mathrm{hcf}_{1}(H, \mathscr P)=1$, then $\mathrm{hcf}(H, \mathscr P)=1$.
Given a graph $H$, let $\text{PC}$ be the family of all proper colorings of $H$ with $\chi(H)$ colors, and let $\text{AP}$ be the family of all acyclic partitions of $H$ with $ar(H)$ parts.
Then for $\ell\geq 2$, whether $\mathrm{hcf}(H, \text{PC})=1$ or not is the dichotomy found by K\"{u}hn and Osthus \cite{DKDO2009} for the minimum degree conditions for $H$-factors in general graphs.
That is, the term $1-\frac{1}{\chi(H)}$ in the Alon--Yuster theorem is asymptotically tight if and only if $\mathrm{hcf}(H, \text{PC})\neq 1$. In the problem of K\"{u}hn and Osthus \cite{DKDO2009}, $\ell=1$ implies that $H$ is an independent set, which is a trivial case. However, for our problem, $\ell=1$ means that $H$ is a forest, in which case we have the following result.
\begin{theorem}\label{thm1.2345}
Given $\mu >0$, $h\in \mathbb{N}$ with $h\geq 3$ and an $h$-vertex forest $H$ with $\mathrm{hcf}_{2}(H, \text{AP})=1$, there exists $\alpha >0$ such that the following holds for sufficiently large $n\in h\mathbb{N}$. Let $G$ be an $n$-vertex graph such that $\delta(G)\geq \mu n$ and $\alpha(G)\leq \alpha n$. Then $G$ contains an $H$-factor.
\end{theorem}
Note that the degree condition in Theorem \ref{thm1.2345} is quite small. Its proof follows the lattice-based absorbing method \cite{MR3632565} used in previous works as well as in this paper. The proof is omitted here and will appear in the first author's doctoral thesis.
For our problem, we show that the minimum degree condition in Theorem~\ref{coro1.311} is asymptotically tight for $H$ with $\mathrm{hcf}(H, \text{AP})\neq 1$ and conjecture that the relation is actually ``if and only if'' (see Conjecture~\ref{conj} below). The proof of the following Proposition \ref{prop5.111} is presented in full details in Section \ref{sec555}.
\begin{prop}[]\label{prop5.111}
Given $\alpha>0$ and $h\in \mathbb{N}$, let $H$ be an $h$-vertex graph with $\mathrm{hcf}(H, \text{AP})\neq 1$.
Then the following holds for sufficiently large $n$.
There exists an $n$-vertex graph $G$ with $\delta(G)\geq \max\{(1-\frac{1}{ar(H)})n-2, \frac{n}{2}-2\}$ and $\alpha(G)\leq \alpha n$ such that $G$ has no $H$-factor.
\end{prop}
Comparing with Theorem \ref{coro1.311}, we actually prove a slightly stronger result. To state it, we first introduce some notation.
\begin{defn}[]\label{def1.42}
Let $\widetilde{\mathcal{H}}$ be a family of graphs such that every element $H\in \widetilde{\mathcal{H}}$ has an acyclic partition $\mathcal{P}=\{T_{1},\ldots, T_{r}\}$ with $r:=ar(H)$ such that i) $H[T_{1}]$ is an independent set and ii) $2|T_{1}|=|T_{i}|$ for each $i\in [2, r]$.
Given a graph $H$, let $f(H)$ be an integer such that
\[
f(H) := \begin{cases} 2ar(H)-1 \quad \text{ if } H\in \widetilde{\mathcal{H}}; \\
2ar(H) \hfill \text{ otherwise}.
\end{cases}
\]
\end{defn}
Note that all cliques of odd order belong to $\widetilde{\mathcal{H}}$ and $f(K_{r})=r$. Our main result reads as follows.
\begin{theorem}\label{thm1.3}
Given $\mu >0$, $h\in \mathbb{N}$ and an $h$-vertex graph $H$, there exists $\alpha >0$ such that the following holds for sufficiently large $n\in h\mathbb{N}$. Let $G$ be an $n$-vertex graph such that $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ contains an $H$-factor.
\end{theorem}
Since $f(H)\le 2ar(H)$, Theorem~\ref{coro1.311} follows from Theorem \ref{thm1.3}.
Theorem \ref{thm1.3} unifies and generalises the results of Balogh--Molla--Sharifzadeh~\cite{MR3570984} and Knierm--Su~\cite{MR4193066} on clique factors.
We remark that the minimum degree condition in Theorem \ref{thm1.3} is asymptotically sharp for (all graphs $H$ with ${\mathrm{hcf}}(H, \text{AP})\neq 1$ and) infinitely many graphs $H$ with ${\mathrm{hcf}}(H, \text{AP})=1$.
To illustrate this we introduce the following notation.
\begin{defn}[]\label{def1.31}
The \emph{critical arboricity} $ar_{cr}(H)$ of a graph $H$ is defined as $\frac{\left(ar(H)-1\right)|V(H)|}{|V(H)|-\sigma(H)}$, where $\sigma(H)$ denotes the minimum size of a part taken over all optimal acyclic partitions of $H$.
\end{defn}
Note that $ar(H)-1 < ar_{cr}(H)\le ar(H)$.
We supply the following general lower bound serving as a ``space barrier'' for this problem.
\begin{prop}\label{prop1.8}
Let $h\in \mathbb{N}$ and $H$ be an $h$-vertex graph. Then for any $\alpha>0$, the following holds for sufficiently large $n\in h\mathbb{N}$. There exists an $n$-vertex graph $G$ with $\delta(G)\geq\left(1-\frac{1}{ar_{cr}(H)}\right)n-1$ and $\alpha(G)\leq \alpha n$ which does not have an $H$-factor.
\end{prop}
In particular, Proposition \ref{prop1.8} implies that the degree condition in Theorem \ref{thm1.3} is tight for the graphs $H$ with $f(H)=2ar_{cr}(H)$.
In particular, when $ar_{cr}(H)<ar(H)$, $f(H)=2ar_{cr}(H)$ implies that $H\in \widetilde{\mathcal{H}}$ and thus $\mathrm{hcf}(H, \text{AP})=1$.
The following is an example for such $H$.
Let $r\in \mathbb{N}$ and $H$ be an $(r+1)$-partite graph $K_{2r+1, \dots, 2r+1}$.
By definition we have $K_{2r+1, \dots, 2r+1}\in \widetilde{\mathcal{H}}, f(K_{2r+1, \dots, 2r+1})=2r+1$ and $ar_{cr}(K_{2r+1, \dots, 2r+1})=\frac{2r+1}{2}$. The proof of Proposition \ref{prop1.8} is presented in full details in Section \ref{sec555}.
In summary, the degree condition in Theorem \ref{thm1.3} is tight for the graphs $H$ with i) ${\mathrm{hcf}}(H, \text{AP})\\ \neq 1$ and ii) $H\in \widetilde{\mathcal{H}}$ and $ar_{cr}(H) = ar(H)-1/2$.
Nevertheless, we put forward the following conjecture for the $H$-factor problem.
\begin{conjecture}\label{conj}
Given $\mu >0$, $h\in \mathbb{N}$ and an $h$-vertex graph $H$ with ${\mathrm{hcf}}(H, \text{AP})= 1$, there exists $\alpha >0$ such that the following holds for sufficiently large $n\in h\mathbb{N}$. Let $G$ be an $n$-vertex graph such that $\delta(G)\geq \max\left\{\left(1-\frac{1}{ar_{cr}(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ contains an $H$-factor.
\end{conjecture}
For complete multi-partite graphs $H$, the smallest open case is $H=K_{3,3,4}$ where $f(H)=5$, $ar_{cr}(H)=\frac{20}{9}$ and ${\mathrm{hcf}}(H, \text{AP})=1$.
\subsection{Proof strategy}
Our proof makes use of the absorption method and builds on the techniques developed in \cite{MR3529107, MR3632565, MR3290271}. The absorption method was introduced by R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di about a decade ago in \cite{MR2500161}. Since then, it has turned out to be an important tool for studying the existence of spanning structures in graphs, digraphs and hypergraphs.
Now we sketch the proof idea. The main tasks are to (i) build an absorbing set and (ii) cover almost all of the remaining vertices with an $H$-tiling. For (i), we need the following notation of absorbers and absorbing sets in \cite{MR4080942}.
\begin{defn}
Let $G$ be an $n$-vertex graph and $H$ be an $h$-vertex graph. Then
\begin{itemize}
\item[(1)] a subset $A\subseteq V(G)$ is a $\xi$-\emph{absorbing set} in $G$ for some constant $\xi$ if for any subset $U\subseteq V(G)\backslash A$ of size at most $\xi n$ and $|A\cup U|\in h\mathbb{N}$, $G[A\cup U]$ contains an $H$-factor.
\item[(2)] for any subset $S\in V(G)$ of size $h$ and an integer $t$, we call a subset $A_{S}\subseteq V(G)\backslash S$ an $(H, t)$-\emph{absorber} for $S$ if $|A_{S}|=ht$ and both $G[A_{S}]$ and $G[A_{S}\cup S]$ contain an $H$-factor.
\end{itemize}
\end{defn}
Widely used constructions of absorbing set by R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di \cite{MR2500161} and H\`{a}n, Person and Schacht \cite{MR2496914} rely on the property that
\begin{quote}
\emph{every $h$-subset $S$ has $\Omega(n^{ht})$ $(H, t)$-absorbers for $S$.}
\end{quote}
However, as pointed out in \cite{MR3570984}, in our setting this is usually impossible because when we construct the absorbers using the independence number condition, it does not give such a strong counting. Instead, a new approach due to Nenadov and Pehova \cite{MR4080942} guarantees an absorbing set provided that
\begin{quote}
\emph{every $h$-set $S$ has $\Omega(n)$ vertex-disjoint $(H, t)$-absorbers for $S$.}
\end{quote}
However, it is still unclear how to verify this for our problem and the bulk of the work on building absorbing sets is to handle this. Here we instead build a partition $V(G)=B\cup U$ satisfying that
\begin{quote}
\emph{$|B|=o(n)$ and every $h$-set in $U$ has $\Omega(n)$ vertex-disjoint absorbers.}
\end{quote}
Similar ideas already appeared in our recent work, e.g.~\cite{CHKWY}. Then the arguments reduce to finding an absorbing set $A$ in $G[U]$ and then covering vertices of $B$ by a small $H$-tiling vertex disjoint from $A$ so as to yield a desired absorbing set.
Our proof for this is also significantly more involved than that in \cite{MR4193066} and uses the lattice-based absorption method by the second author \cite{MR3632565}.
Our second task is to establish an almost perfect $H$-tiling. Following a standard application of the regularity lemma, we obtain an $\varepsilon$-regular partition and then build a reduced multigraph where two vertices are connected by a double-edge if the corresponding clusters form a regular pair of density larger than $\frac{1}{2}$.
Our proof uses a crucial notion of $K_{r}$-embeddable structures (see Definition~\ref{def2.7}) from the work of Knierim and Su \cite{MR4193066}, which was used to model all possible ways how $K_r$ is embedded into a collection of clusters.
To be more precise, a $K_{r}$-embeddable structure, say $\mathcal{K}$, is a clique in the reduced multigraph such that for $a,b\in \mathbb{N}$ with $a+b=|V(\mathcal{K})|$ and $a+2b=r$, $\mathcal{K}$ has $b$ vertices that are pairwise connected by multiple edges (if $b>1$).
Another key ingredient in their approach is that they build an almost perfect fractional tiling with $K_{r}$-embeddable structures via a novel idea (see Definition~\ref{def2.8} and Lemma~\ref{lem4.4}).
We adapt this approach to $H$-tilings which roughly boils down to embedding vertex-disjoint copies of $H$ in different $K_{r}$-embeddable structures where $r=f(H)$.
To apply Lemma~\ref{lem4.4} in our proof, it suffices to show that for every $K_{r}$-embeddable structure $\mathcal{K}$ with $a,b$ given as above and $V(\mathcal{K})=\{V_1,V_2,\ldots,V_{a+b}\}$, we can find \emph{an almost perfect $H$-tiling in an arbitrary collection of subclusters $V'_i\subseteq V_i$, where $2|V_i'|=|V_j'|$ for every $i\in [a],j\in[a+1,a+b]$.}
Unlike in \cite{MR4193066} where they embed a copy of $K_{r}$ each containing exactly one vertex in $V_i$ and one edge in $V_{j}$ for every $i\in [a],j\in[a+1,a+b]$, we instead need to embed an independent set in each $V_i$ and a forest in $V_j$ for every $i\in [a]$ and $j\in [a+1, a+b]$.
For this we shall use a result of Erd\H{o}s, Hajnal, S\'{o}s and Szemer\'{e}di \cite{{MR716422}} which allows us to embed two forests in a regular pair with density above $\frac{1}{2}$, one in each side, such that they are complete to each other.
Note that the $K_{r}$-embeddable structures may have different orders ranging from $\lceil\frac{r}{2}\rceil$ to $r$. For every $K_{r}$-embeddable structure $\mathcal{K}$ with integers $a,b$ as above, we define a suitably-chosen auxiliary graph $Q(a,b)$ which admits an acyclic partition $\{T_1,T_2,\ldots, T_{a+b}\}$ such that $T_i$ induces an independent set and $2|T_i|=|T_j|$\footnote{This is the origin of the family $\widetilde{H}$.} for every $i\in [a],j\in[a+1,a+b]$. In particular, $Q(a,b)$ has an $H$-factor, and thus it suffices to find an almost perfect $Q(a,b)$-tilings in the same context (see Lemma~\ref{lem2.16} and Corollary~\ref{coro2.17}).
\subsection{Basic notation and organization}
We first introduce some notation throughout the paper. For a graph $G:= G(V, E)$, we write $e(G)=|E(G)|$. We say an independent set $I$ is an $\ell$-\emph{independent set} if $|I|=\ell$. Similarly, we can define an $\ell$-\emph{tree} and an $\ell$-\emph{forest}. The \emph{girth} $g(G)$ of a graph $G$ is the length of a shortest cycle. For $U\subseteq V$, let $G[U]$ be the induced subgraph of $G$ on $U$. Let $G-U:=G[V\backslash U]$. For two subsets $A, B\subseteq V(G)$, we use $E(A, B)$ to denote the set of edges joining $A$ and $B$. We write $N_{G}(v)$ for the set of neighbors of $v$ in $G$. For convenience, we use $d_{G}(v)$ to denote the number of edges which contain $v$ in $G$. We omit the subscript $G$ if the graph is clear from the context. For $m\in \mathbb{N}$, we write $[V]^{\geq m}$ for the family of all subsets $U\subseteq V$ with $|U|\geq m$. For a vertex set $W$ and a positive integer $\ell$, we write $\tbinom{W}{\ell}$ to denote the set of all $\ell$-\emph{subsets} of distinct elements of $W$. We write $V(G)=V_{1}\cup V_{2}$ for a \emph{bipartition} of $G$ if $G[V_{i}]$ is an independent set for each $i\in [2]$. For any integers $a\leq b$, let $[a, b]:=\{i\in \mathbb{Z}: a\leq i\leq b\}$ and $[a]:= [1, a]$.
When we write $\beta\ll \gamma$, we always mean that $\beta, \gamma$ are constants in $(0, 1)$, and $\beta\ll \gamma$ means that there exists $\beta_{0}=\beta_{0}(\gamma)$ such that the subsequent arguments hold for all $0<\beta\leq \beta_{0}$. Hierarchies of other lengths are defined analogously.
The rest of the paper is organized as follows. In Section \ref{sec555}, we will prove Proposition \ref{prop5.111} and Proposition \ref{prop1.8}. In Section \ref{sec2}, we will give a proof of our main result and introduce the regularity lemma. In Section \ref{sec3}, our main work is to find a fractional tiling and establish an embedding lemma. We will prove Lemma \ref{lem3.1} in Section \ref{sec4}. In Section \ref{sec5}, we present some necessary results and tools to introduce the latticed-based absorbing method and prove Lemma \ref{lem3.3}.
\section{Proofs of Proposition \ref{prop5.111} and Proposition \ref{prop1.8}}\label{sec555}
We first give a result which is commonly used in the proof of Proposition \ref{prop5.111} and Proposition \ref{prop1.8}.
\begin{lemma}[\cite{ER1959}]\label{lem6.1}
For any $\alpha>0$, $k\in \mathbb{N}$ with $k\geq 3$, there exists an $n$-vertex graph $G$ for sufficiently large $n$ such that $\alpha(G)<\alpha n$ and $g(G)>k$.
\end{lemma}
Now we give the proof of Proposition \ref{prop5.111}.
\begin{proof} [Proof of Proposition \ref{prop5.111}]
Given $\alpha>0$ and $h\in \mathbb{N}$, we shall choose $\frac{1}{n}\ll \alpha$.
Let $\ell:=ar(H)$. We divide the proof into following three cases: $\ell=1$, $\ell=2$ and $\ell\geq 3$.
We first give a construction for every graph $H$ with $\mathrm{hcf}_{2}(H, \text{AP})\geq 2$, which will commonly be used when $\ell=1, 2$. Note that there exists $p\in \{\lfloor\frac{n}{2}\rfloor, \lfloor\frac{n}{2}\rfloor+1\}$ that is non-divisible by $\mathrm{hcf}_{2}(H, \text{AP})$.
Let $G_{0}$ be an $n$-vertex graph with $V(G_{0})=V_{1}\cup V_{2}$ such that $|V_{1}|=p$ and $|V_{2}|=n-p$, and $G_{0}[V_{i}]$ be a complete graph for each $i\in[2]$.
It holds that $\delta(G_{0})\geq \frac{n}{2}-2$ and $\alpha(G_{0})=2$.
Let $H'$ be a copy of $H$ in $G_{0}$.
Since $G_{0}$ is disconnected, $|V(H')\cap V_{1}| \equiv 0 \pmod{\mathrm{hcf}_{2}(H, \text{AP})}$.
Thus, for every $H$-tiling $\mathcal{H}$ in $G_0$, $|\bigcup_{H'\in \mathcal{H}}(V(H')\cap V_{1})| \equiv 0 \pmod{\mathrm{hcf}_{2}(H, \text{AP})}$.
Note that $|V_{1}|=p$ is non-divisible by $\mathrm{hcf}_{2}(H, \text{AP})$.
Hence, $\mathcal{H}$ is not an $H$-factor in $G_0$.
If $\ell=1$, then by the assumption that $\mathrm{hcf}(H, \text{AP})\neq 1$, it holds that $\mathrm{hcf}_{2}(H, \text{AP})\geq 2$. We are done by the construction $G_0$ as above.
If $\ell=2$, then by the assumption that $\mathrm{hcf}(H, \text{AP})\neq 1$, we may assume that $\mathrm{hcf}_{2}(H, \text{AP})=1$ and $\mathrm{hcf}_{1}(H, \text{AP})\geq 3$ as the case $\mathrm{hcf}_{2}(H, \text{AP})\geq 2$ would be handled by $G_0$.
Let $G$ be an $n$-vertex graph with $V(G)=V_{1}\cup V_{2}$, $|V_{1}|=\lfloor\frac{n}{2}\rfloor+1$, $|V_{2}|=\lceil\frac{n}{2}\rceil-1$ and $G[V_{1}, V_{2}]$ be a complete bipartite graph.
Let $G[V_{i}]$ be a subgraph with $\alpha(G[V_{i}])\leq \alpha n$ and $g(G[V_{i}])\geq h+1$ given by Lemma \ref{lem6.1} for each $i\in [2]$.
It holds that $\delta(G)\geq \frac{n}{2}-1$ and $\alpha(G)\leq \alpha n$.
Let $H'$ be a copy of $H$ in $G$.
Then $g(G[V_{i}])\geq h+1$ implies that $V(H')\cap V_{i}$ induces a forest in $H'$ for every $i\in[2]$.
Hence, $\mathcal{P}:=\{V(H')\cap V_{1}, V(H')\cap V_{2}\}$ is an acyclic partition of $H'$ and thus $|(V(H')\cap V_{1})|-|(V(H')\cap V_{2})| \equiv 0 \pmod{\mathrm{hcf}_{1}(H, \text{AP})}$.
For every $H$-tiling $\mathcal{H}$ in $G$, it holds that $|\bigcup_{H'\in \mathcal{H}}(V(H')\cap V_{1})|-|\bigcup_{H'\in \mathcal{H}}(V(H')\cap V_{2})| \equiv 0 \pmod{\mathrm{hcf}_{1}(H, \text{AP})}$.
Note that $\mathrm{hcf}_{1}(H, \text{AP})\geq 3$ and $|V_{1}|-|V_{2}|\in \{1, 2\}$.
Hence, $\mathcal{H}$ is not an $H$-factor in $G$.
If $\ell\geq 3$, then by the assumption that $\mathrm{hcf}(H, \text{AP})\neq 1$, it holds that $\mathrm{hcf}_{1}(H, \text{AP})\neq1$. Let $G$ be an $n$-vertex graph with $V(G)=V_{1}\cup \cdots\cup V_{\ell}$, $|V_{1}|=\lfloor\frac{n}{\ell}\rfloor+1$, $|V_{2}|=\lfloor\frac{n}{\ell}\rfloor$, $\lfloor\frac{n}{\ell}\rfloor-1\leq |V_{i}|\leq \lceil\frac{n}{\ell}\rceil$ for each $i\in [3, \ell]$ and $G[V_{i}, V_{j}]$ be a complete bipartite graph for distinct $i, j\in [\ell]$.
Let $G[V_{i}]$ be a subgraph with $\alpha(G[V_{i}])\leq \alpha n$ and $g(G[V_{i}])\geq h+1$ given by Lemma \ref{lem6.1} for each $i\in [\ell]$.
It holds that $\delta(G)\geq n-(\lfloor\frac{n}{\ell}\rfloor+1)\geq (1-\frac{1}{\ell})n-1$ and $\alpha(G)\leq \alpha n$.
Let $H'$ be a copy of $H$ in $G$.
Then by the same arguments in the case above, it holds that for every $H$-tiling $\mathcal{H}$ in $G$, $|\bigcup_{H'\in \mathcal{H}}(V(H')\cap V_{1})|-|\bigcup_{H'\in \mathcal{H}}(V(H')\cap V_{2})| \equiv 0 \pmod{\mathrm{hcf}_{1}(H, \text{AP})}$.
Note that $\mathrm{hcf}_{1}(H, \text{AP})\neq1$ and $|V_{1}|-|V_{2}|=1$.
Hence, $\mathcal{H}$ is not an $H$-factor in $G$.
\end{proof}
Next we prove Proposition \ref{prop1.8}.
\begin{proof} [Proof of Proposition \ref{prop1.8}]
Given $\alpha>0$ and $h\in \mathbb{N}$, we shall choose $\frac{1}{n}\ll \alpha$.
Let $H$ be an $h$-vertex graph and $\ell:=ar(H)$, $G$ be an $n$-vertex graph with $V(G)=V_{1}\cup \cdots \cup V_{\ell}$, $|V_{1}|=\frac{\sigma(H)}{h}n-1$, $|V_{2}|=\big\lceil\frac{h-\sigma(H)}{(\ell-1)h}n\big\rceil+1$, $\big\lfloor\frac{h-\sigma(H)}{(\ell-1)h}n\big\rfloor\leq |V_{i}|\leq \big\lceil\frac{h-\sigma(H)}{(\ell-1)h}n\big\rceil$ for each $i\in [3, \ell]$ and $G[V_{i}, V_{j}]$ be a complete bipartite graph for distinct $i, j\in [\ell]$.
Let $G[V_{i}]$ be a subgraph with $\alpha(G[V_{i}])\leq \alpha n$ and $g(G[V_{i}])\geq h+1$ given by Lemma \ref{lem6.1} for each $i\in [\ell]$. It holds that $\delta(G)\geq n-\left(\big\lceil\frac{h-\sigma(H)}{(\ell-1)h}n\big\rceil+1\right)\geq\left(1-\frac{1}{ar_{cr}(H)}\right)n-1$ and $\alpha(G)\leq \alpha n$. Let $H'$ be a copy of $H$ in $G$. Then $g(G[V_{i}])\geq h+1$ implies that $H'[V(H')\cap V_{i}]$ is a forest for each $i\in[\ell]$, and $\mathcal{P}:=\{V(H')\cap V_{1}, \dots, V(H')\cap V_{\ell}\}$ is an acyclic partition of $H'$. By the definition of $\sigma(H)$, $|V(H')\cap V_{1}|\geq \sigma(H)$. Since $|V_{1}|=\frac{\sigma(H)}{h}n-1$, there are at most $\big\lfloor\frac{|V_{1}|}{\sigma(H)}\big\rfloor\leq\frac{n}{h}-1$ vertex-disjoint copies of $H$ in $G$. Hence, $G$ has no $H$-factor.
\end{proof}
In the next section, we prove Theorem \ref{thm1.3}.
\section{Proof of Theorem \ref{thm1.3}}\label{sec2}
\subsection{Proof of the main result}\label{sec3}
In this subsection, we introduce the central lemmas that are needed for the proof of our main theorem. This subsection is devoted to explaining how they work together to give the proof of Theorem \ref{thm1.3}. The proofs of these lemmas are then presented in full details in Section \ref{sec3} and Section \ref{sec5} respectively.
A crucial and necessary step in our proof is to find an $H$-tiling in the graph $G$ which covers all but a small set of vertices. The following result guarantees the existence of such an $H$-tiling. The proof of Lemma \ref{lem3.1} will be presented in Section \ref{sec3}.
\begin{lemma}[]\label{lem3.1}
Given $\mu, \delta>0$, an $h$-vertex graph $H$ with $h\in \mathbb{N}$ and $h\geq 3$, there exists $\alpha>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ contains an $H$-tiling which covers all but at most $\delta n$ vertices.
\end{lemma}
Lemma \ref{lem3.3} provides an absorbing set in the graph $G$, whose proof can be found in Section \ref{sec5}.
\begin{lemma}[]\label{lem3.3}
Given $\mu, \gamma$ with $0<\gamma\leq \frac{\mu}{2}$, an $h$-vertex graph $H$ with $h\in \mathbb{N}$ and $h\geq 3$, there exist $\alpha, \xi>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ contains a $\xi$-absorbing set $A$ of size at most $\gamma n$.
\end{lemma}
Now we give the proof of Theorem \ref{thm1.3} using Lemma \ref{lem3.1} and Lemma \ref{lem3.3}.
\begin{proof} [Proof of Theorem \ref{thm1.3}] Given $\mu>0$, $h\in\mathbb{N}$ with $h\geq 3$ and an $h$-vertex graph $H$, we shall choose
\begin{center}
$\frac{1}{n}\ll \alpha\ll \delta\ll \xi\ll\gamma\ll \mu$.
\end{center}
Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. By Lemma \ref{lem3.3} with $\gamma\leq \frac{\mu}{2}$, we find a $\xi$-absorbing set $A\subseteq V(G)$ of size at most $\gamma n$ for some $\xi>0$. Let $G_{1}:= G-A$. Then we have
\begin{center}
$\delta(G_{1})\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}-\gamma n\geq \max\left\{\left(1-\frac{2}{f(H)}+\frac{\mu}{2}\right)n, \left(\frac{1}{2}+\frac{\mu}{2}\right)n\right\}.$
\end{center}
Therefore by applying Lemma \ref{lem3.1} on $G_{1}$ with $\delta$, we obtain an $H$-tiling $\mathcal{H}$ that covers all but a set $L$ of at most $\delta n$ vertices in $G_{1}$. Since $\delta\ll \xi$, the absorbing property of $A$ implies that $G[A\cup L]$ contains an $H$-factor, which together with $\mathcal{H}$ forms an $H$-factor in $G$.
\end{proof}
\subsection{Regularity}
To find an almost perfect tiling, an important ingredient in our proof is Szemer\'{e}di's Regularity Lemma. In this paper, we make use of a degree form of the regularity lemma \cite{MR1395865}.
We shall first introduce some notation.
Given a graph $G$ and a pair $(V_{1}, V_{2})$ of vertex-disjoint subsets in $V(G)$, the \emph{density} of $(V_{1}, V_{2})$ is defined as
\begin{center}
$d(V_{1}, V_{2})=\frac{e(V_{1}, V_{2})}{|V_{1}||V_{2}|}$.
\end{center}
\begin{defn}[]\label{def2.1}
Given $\varepsilon>0$, a graph $G$ and a pair $(V_{1}, V_{2})$ of vertex-disjoint subsets in $V(G)$, we say that the pair $(V_1, V_2)$ is $\varepsilon $-\emph{regular} if for all $X\subseteq V_{1}$ and $Y\subseteq V_{2}$ satisfying
\[
X \subseteq V_{1}, |X| \ge \varepsilon |V_{1}| ~\text{and}~ Y \subseteq V_{2}, |Y| \ge \varepsilon |V_{2}|,
\]
we have
\[
|d(X,Y) - d(V_1,V_2)| \le \varepsilon.
\]
\end{defn}
\begin{lemma}[\cite{MR1395865}, Slicing Lemma]\label{lem2.2}
Assume $(V_{1}, V_{2})$ is $\varepsilon$-regular with density $d$. For some $\alpha\geq \varepsilon$, let $V_{1}'\subseteq V_{1}$ with $|V_{1}'|\geq \alpha|V_{1}|$ and $V_{2}'\subseteq V_{2}$ with $|V_{2}'|\geq \alpha|V_{2}|$. Then $(V_{1}', V_{2}')$ is $\varepsilon'$-regular with $\varepsilon':=\max\{2\varepsilon, \varepsilon/\alpha\}$ and for its density $d'$ we have $|d'-d|<\varepsilon$.
\end{lemma}
\begin{lemma}[\cite{MR1395865}, Degree form of the Regularity Lemma]\label{lem2.3}
For every $\varepsilon > 0$ there is an $N = N(\varepsilon )$ such that the following holds for any real number $\beta\in [0, 1]$ and $n\in \mathbb{N}$. Let $G$ be a graph with $n$ vertices.
Then there exist an $(\varepsilon,\beta)$-regular partition $V(G)=V_{0}\cup \cdots \cup V_{k} $ and a spanning subgraph $G' \subseteq G$ with the following properties:
\item $({\rm 1})$ $ \frac{1}{\varepsilon}\leq k \le N $;
\item $({\rm 2})$ $|V_{i}| \le \varepsilon n$ for $i\in [0, k]$ and $|V_{1}|=|V_{2}|=\cdots=|V_{k}| =m$ for some $m\in \mathbb{N}$;
\item $({\rm 3})$ $d_{G'}(v) > d_{G}(v) - (\beta + \varepsilon )n$ for all $v \in V(G)$;
\item $({\rm 4})$ each $V_{i}$ is an independent set in $G' $ for $i\in [k]$;
\item $({\rm 5})$ all pairs $(V_{i}, V_{j})$ are $\varepsilon $-regular (in $G'$) with density 0 or at least $\beta$ for distinct $i, j\neq0$.
\end{lemma}
A widely-used auxiliary graph accompanied with the regular partition is the reduced graph. To differentiate between dense and very dense pairs of partitions, we employ the following definitions of reduced multigraph.
\begin{defn}[Reduced graph]\label{def2.4}
Let $k\in \mathbb{N}$, $\beta, \varepsilon>0$, $G$ be a graph with a vertex partition $V(G)=V_0\cup \cdots \cup V_k$ and $G'\subseteq G$ be a subgraph fulfilling the properties of Lemma \ref{lem2.3}. We denote by $R_{\beta, \varepsilon}$ the \emph{reduced graph} for the $(\varepsilon,\beta)$-partition, which is defined as follows. Let $V(R_{\beta, \varepsilon})=\{V_{1}, \ldots, V_{k}\}$ and for two distinct clusters $V_{i}$ and $V_{j}$ we draw a double-edge between $V_{i}$ and $V_{j}$ if $d_{G'}(V_i, V_j)\geq \frac{1}{2}+\beta$, a single-edge if $\beta\leq d_{G'}(V_i, V_j)<\frac{1}{2}+\beta$ and no edge otherwise.
\end{defn}
The following fact presents a minimum degree of the reduced graph provided the minimum degree of $G$, where a double-edge is counted as two edges.
\begin{fac}[]\label{fact2.5}
Let $n, h\in \mathbb{N}$, $\mu>0$, $0<\varepsilon, \beta\leq \frac{\mu}{10}$ with $\beta\in [0, 1]$, $H$ be an $h$-vertex graph and $G$ be an $n$-vertex graph with $\delta(G)\geq\left(1-\frac{2}{f(H)}+\mu\right)n$. Let $V(G)=V_{0}\cup \cdots \cup V_{k}$ be a vertex partition of $V(G)$ satisfying Lemma \ref{lem2.3} $(1)$-$(5)$. We denote the reduced graph as $R_{\beta, \varepsilon}$. Then for every $V_{i}\in V(R_{\beta, \varepsilon})$ we have
\begin{center}
$d_{R_{\beta, \varepsilon}}(V_{i})\geq2\left(1-\frac{2}{f(H)}+\frac{\mu}{2}\right)k$.
\end{center}
\end{fac}
\begin{proof} Note that $|V_{0}|\leq \varepsilon n$ and $|V_{i}|=m$ for each $i\in [k]$. Every edge in $R_{\beta, \varepsilon}$ represents less than $\left(\frac{1}{2}+\beta\right) m^{2}$ edges in $G'-V_{0}$. Thus we have
\begin{align*}
d_{R_{\beta, \varepsilon}}(V_{i}) & \geq \frac{|V_{i}|\left(\delta(G)-(\beta+\varepsilon)n-\varepsilon n\right)}{\left(\frac{1}{2}+\beta\right) m^{2}} \\
& \geq \frac{\left(1-\frac{2}{f(H)}+\mu-2\varepsilon-\beta\right)mn}{\left(\frac{1}{2}+\beta\right) m^{2}} \\
& \geq 2\left(1-\frac{2}{f(H)}+\mu-2\varepsilon-\beta\right)(1-2\beta)k\\
& > 2\left(1-\frac{2}{f(H)}+\frac{\mu}{2}\right)k,
\end{align*}
since $0<\varepsilon, \beta\leq \frac{\mu}{10}$ and $\left(\frac{1}{2}+\beta\right)^{-1}\geq 2(1-2\beta)$.
\end{proof}
\begin{rmk}\label{remk2.6}
Let $R$ be a multigraph with multiplicity 2. Note that $|N_{R}(i)|\geq \frac{1}{2}d_{R}(i)$ for each $i\in V(R)$. The \emph{double-edge neighborhood} of $i\in V(R)$ is a set of vertices in $V(R)$ which are connected to $i$ through double-edges. Similarly, we define the \emph{single-edge neighborhood}.
\end{rmk}
\section{Almost perfect tilings}\label{sec3}
To obtain an almost perfect $H$-tiling in the graph $G$, we first define a family $\mathcal{Q}$ of suitably-chosen auxiliary graphs $Q(a,b)$ (to be defined later) for $a, b\in \mathbb{N}$ such that $a+2b=f(H)$ and $Q(a,b)$ contains an $H$-factor. This roughly reduces the problem to finding in $G$ a collection of vertex-disjoint copies of members from $\mathcal{Q}$ which altogether cover almost all vertices.
Here our proof adopts a standard application the regularity lemma on $G$ to get a reduced graph $R$. A key step in it is to construct certain structures in $R$ for embedding $Q(a,b)$. In this case, we use an idea from the work of Knierim and Su \cite{MR4193066} to find a fractional tiling with $K_{f(H)}$-embeddable structures (see Definition~\ref{def2.7}); and then develop a tool (see Corollary~\ref{coro2.17}) for embedding $Q(a,b)$ under certain pseudorandomness conditions.
\subsection{Fractional tilings}
The main result in this subsection is Lemma \ref{lem4.4} which provides us a fractional tiling with some special structures in the reduced graph.
Here, we first present some related notation about these special structures. They are formalised as follows.
\begin{defn}[\cite{MR4193066}, Definition 2.6]\label{def2.7}
Let $R$ be a multigraph with multiplicity 2. Then a $K_{r}$-\emph{multi-embedding} to $R$ is a mapping $\phi: V(K_{r})\rightarrow V(R)$ with the following properties:
\begin{itemize}
\item for any $i\in V(R)$ the induced subgraph on the vertex set $\phi^{-1}(i)$ (if not empty) in $K_{r}$ is either an isolated vertex or an edge (in particular, $|\phi^{-1}(i)|\le 2$);
\item if $uv\in E(K_{r})$, then $\phi(u)$ and $\phi(v)$ are connected by at least one edge in $R$ (as long as $\phi(u)$ and $\phi(v)$ differ);
\item if $|\phi^{-1}(i)|=|\phi^{-1}(j)|=2$ for distinct $i, j\in V(R)$, then $i$ and $j$ are connected by a double-edge.
\end{itemize}
\end{defn}
We also need some definitions which are related to the $K_{r}$-multi-embedding. If $\phi$ is a $K_{r}$-multi-embedding to $R$, then the corresponding subgraph $R[\phi(K_{r})]=:\mathcal{K}$ is a \emph{$K_{r}$-embeddable structure} in $R$. We write $i_{\mathcal{K}}(v)=|\phi^{-1}(v)|$ for every $v\in V(R)$ and by Definition~\ref{def2.7} we know that $i_{\mathcal{K}}(v)\in\{0,1,2\}$.
We use $\mathcal{F}(R, r):=\{\mathcal{K}_{1},\dots, \mathcal{K}_{\ell}\}$ to denote the family of all $K_{r}$-embeddable structures in $R$.
\begin{defn}[]\label{def2.8}
Let $R$ be a $k$-vertex multigraph with multiplicity 2 and $\mathcal{F}(R, r)$ be given as above. Then a fractional $\mathcal{F}(R, r)$-tiling $\omega$ in $R$ is a weight function from the members of $\mathcal{F}(R, r)$ to the interval $[0, 1]$ such that for every vertex $v\in V(R)$ it holds that
\[
\omega(v) :=\sum\limits_{\mathcal{K}\in\mathcal{F}(R, r)}\omega(\mathcal{K})i_{\mathcal{K}}(v)\leq 1.
\]
\end{defn}
We call $\omega(R):=\sum_{v\in V(G)}\omega(v)$ the \emph{total weight} of the fractional $\mathcal{F}(R, r)$-tiling $\omega$ and it is a \emph{perfect} fractional tiling for $R$ if $\omega(R)=|V(R)|$.
We shall use the following result to find a fractional $\mathcal{F}(R, r)$-tiling with a large total weight, whose proof will be given in Section \ref{sec3.3}.
\begin{lemma}[]\label{lem4.4}
For $h\in \mathbb{N}$, $h\geq3$, an $h$-vertex graph $H$ and positive constants $\mu, \eta$, there exist $\beta, \varepsilon, \alpha>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$, $\alpha(G)\leq \alpha n$ and $R:=R_{\beta, \varepsilon}$ be a reduced graph for an $(\varepsilon, \beta)$-regular partition of $G$. Then $R$ contains a fractional $\mathcal{F}(R, f(H))$-tiling $\omega$ such that $\omega(R)\geq (1-\eta)|V(R)|$.
\end{lemma}
\subsection{An embedding Lemma}\label{sec2.2}
The main goal of this subsection is to prove an embedding lemma for our purpose. Recall that Lemma~\ref{lem4.4} gives us a large fractional tiling with $K_{f(H)}$-embeddable structures.
Let $\{\mathcal{K}_{1}, \dots, \mathcal{K}_{\ell}\}$ be all the $K_{f(H)}$-embeddable structures in $R$. Roughly speaking, Lemma \ref{lem4.4} tells us that one can correspond the proportion of weight occupied by $\mathcal{K}_{i}$, e.g. $\omega(\mathcal{K}_{i})$ into disjoint vertex sets $W_{i}$ in $G$, in each of which we shall find an almost perfect $H$-tiling. To achieve this, e.g. for $\mathcal{K}_{1}$, we have unique non-negative integers $a, b$ such that $a+b=|V(\mathcal{K}_{1})|$ and $a+2b=f(H)$, and we build an intermediate auxiliary graph $Q(a, b)$ such that we can find an almost perfect $Q(a, b)$-tiling in $W_{1}$ and $Q(a, b)$ itself has an $H$-factor.
To elaborate on this, we first define a graph $Q(a, b, s, F_{1},\ldots, F_{b})$ with given forests $F_{1},\ldots, F_{b}$.
\begin{defn}[]\label{def2.15}
Let $a, b, s\in \mathbb{N}$ and $F_{i}$ be any $2s$-forest for each $i\in [b]$. Then we construct a graph $Q:=Q(a, b, s, F_{1}, \dots, F_{b})$ with $V(Q)=U_{1}\cup\cdots\cup U_{a+b}$ which satisfies following conditions:
\begin{itemize}
\item $Q[U_{i}, U_{j}]$ is a complete bipartite graph for distinct $i, j\in [a+b]$;
\item $Q[U_{i}]$ is an $s$-independent set for $i\in [a]$;
\item $Q[U_{a+j}]$ is a $2s$-forest $F_{j}$ for $j\in [b]$.
\end{itemize}
\end{defn}
We omit the index $(a, b, s, F_{1}, \dots, F_{b})$ if it is clear from context. By the definition of $Q$, we have $2|U_{i}|=|U_{j}|$ for each $i\in [a]$ and $j\in [a+1, a+b]$.
The following lemma is an essential gadget which allows us to embed an auxiliary graph in $G[V_{1}\cup \cdots\cup V_{a+b}]$ with $a, b$ given as above.
\begin{lemma}[Embedding lemma]\label{lem2.17}
Let $a, b, s$ be positive integers and $\beta>0$. Then there exist $N_0\in \mathbb{N}$ and positive constants $\alpha, \varepsilon$ such that the following holds for any $N\ge N_0$. Let $G$ be a graph with $V(G)=V_{1}\cup \cdots \cup V_{a+b}$, $\alpha(G)\leq \alpha |V(G)|$, $|V_{i}|\geq N$ for each $i\in [a+b]$ such that $(V_{i}, V_{j})$ is $\varepsilon$-regular, $d(V_{i}, V_{j})\geq \beta$ for distinct $i\in [a]$, $j\in [a+b]$ and $d(V_{i}, V_{j})\geq \frac{1}{2}+\beta$ for distinct $i, j\in [a+1, a+b]$. Then for any given $2s$-forests $\{F_{1}, \dots, F_{b}\}$ there exists a copy of $Q(a, b, s, F_{1}, \dots, F_{b})$ in $G$ whose vertex set, say $U_{1}\cup \cdots\cup U_{a+b}$, satisfies $U_{i}\subseteq V_{i}$ for each $i\in [a+b]$.
\end{lemma}
To make use of Lemma \ref{lem2.17}, we shall need the following lemma which guarantees the existence of $Q(a,b)$ as aforementioned
\begin{lemma}\label{lem2.16}
Let $h\in\mathbb{N}$, $H$ be an $h$-vertex graph and $a, b\in \mathbb{N}$ with $a+2b=f(H)$. Then there exist $s\in \mathbb{N}$ with $s\le h$ and a family of forests $\{F_{1}, \dots, F_{b}\}$ such that $Q(a, b, s, F_{1}, \dots, F_{b})=:Q(a,b)$ contains an $H$-factor.
\end{lemma}
Note that the $Q(a, b)$ is not necessary unique. In the rest of the proof, we fix an instance of $Q(a, b)$ as returned by Lemma \ref{lem2.16}, which serves as building blocks in our proof of Lemma \ref{lem3.1}. For convenience, we formulate this in the following corollary.
\begin{coro}\label{coro2.17}
For any constant $\beta>0$, positive integers $a,b, h$ and an $h$-vertex graph $H$ with $a+2b=f(H)$, there exist $\alpha, \varepsilon>0$ and an integer $s$ with $s\leq h$ such that the following holds for sufficiently large $N$. Let $G$ be a graph with $\alpha(G)\leq \alpha |V(G)|$, $V(G)=V_{1}\cup\cdots \cup V_{a+b}$, $|V_{i}|\ge N$ for each $i\in [a+b]$, $(V_{i}, V_{j})$ be $\varepsilon$-regular with $d(V_{i}, V_{j})\geq \beta$ for distinct $i\in [a]$, $j\in [a+b]$ and $d(V_{i}, V_{j})\geq \frac{1}{2}+\beta$ for distinct $i, j\in [a+1, a+b]$. Then there exists a copy of $Q(a,b)$ in $G$ whose vertex set, say $U_{1}\cup \cdots \cup U_{a+b}$, satisfies $2|U_{i}|=|U_{j}|=2s$ for $i\in [a]$ and $j\in [a+1, a+b]$ and $U_{i}\subseteq V_{i}$ for every $i\in [a+b]$.
\end{coro}
\begin{proof} Given $a, b, h\in \mathbb{N}$, we choose $\frac{1}{N}\ll \alpha\ll\varepsilon\ll \beta$.
Applying Lemma \ref{lem2.17} with $F_1,F_2,\ldots,F_b$ obtained from Lemma~\ref{lem2.16}, we can embed a copy of $Q(a,b)$ into $G$ as desired.
\end{proof}
\subsection{Proof of Lemma \ref{lem3.1}}\label{sec4}
Equipped with a fractional tiling (Lemma \ref{lem4.4}) in the reduced graph and an embedding lemma (Corollary \ref{coro2.17}), we are able to find an almost perfect $H$-tiling in the original graph $G$.
\begin{proof} [Proof of Lemma \ref{lem3.1}] Given $h\in \mathbb{N}$, an $h$-vertex graph and positive constants $\delta, \mu$, we shall choose
\begin{center}
$\frac{1}{n}\ll \alpha\ll \frac{1}{k}\ll \varepsilon\ll \beta, \eta\ll \delta, \mu, \frac{1}{h}$.
\end{center}
Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$.
Applying Lemma \ref{lem2.3} with $\varepsilon, \beta>0$, we obtain an $\varepsilon$-regular partition $\mathcal{P}=\{V_{0}, V_{1}, \dots, V_{k}\}$ of $V(G)$.
Let $m:=|V_{i}|$ for each $i\in [k]$ and $R:=R_{\beta, \varepsilon}$ be a reduced multigraph of the partition $\mathcal{P}$ with multiplicity $2$ and $V(R)=\{V_{1}, \dots, V_{k}\}$.
Write $\mathcal{F}:=\mathcal{F}(R, f(H))=\{\mathcal{K}_{1},\dots, \mathcal{K}_{\ell}\}$.
By applying Lemma \ref{lem4.4} on $R$ with $\eta$, we obtain a fractional $\mathcal{F}$-tiling $\omega$ such that
\begin{center}
$\omega(R)\geq (1-\eta)k$.
\end{center}
Then we construct $\omega'$ from the fractional tiling $\omega$ by scaling the weight function $\omega$ of every $K_{f(H)}$-embeddable-structure with a factor of $(1-\eta)$ i.e. for every $K_{f(H)}$-embeddable-structure $\mathcal{K}$ we have $\omega'(\mathcal{K})=(1-\eta)\omega(\mathcal{K})$. Thus $\omega'$ has total weight at least $(1-2\eta)k$.
For each $K_{f(H)}$-embeddable-structure $\mathcal{K}$ with $\omega'(\mathcal{K})>0$, we have a unique pair of integers $a, b\in \mathbb{N}$ such that $a+b=|V(\mathcal{K})|$ and $a+2b=f(H)$. Write $C_{\mathcal{K}}=\omega'(\mathcal{K})m$. Now we construct a $Q(a,b)$-tiling $\mathcal{Q}_{\mathcal{K}}$ by greedily picking vertex-disjoint copies of $Q(a,b)$ in $G$ such that $\mathcal{Q}_{\mathcal{K}}$ is maximal subject to the fact that it contains at most $i_{\mathcal{K}}(V_i)C_{\mathcal{K}}$ vertices from each $V_i, i\in [k]$. We repeat the process for every $K_{f(H)}$-embeddable-structure $\mathcal{K}$ with positive weight, such that the corresponding $Q(a,b)$-tilings $\mathcal{Q}_{\mathcal{K}}$ are pariwise vertex-disjoint. Note that at the end of this process the set of uncovered vertices in each $V_i$, denoted by $V_i'$, has size \[|V_i'|\ge|V_i|-\sum_{\mathcal{K}\in\mathcal{F}}i_{\mathcal{K}}(V_i)C_{\mathcal{K}}=m-\omega'(V_i)m\ge m-(1-2\eta)m\ge 2\eta m.\]
Now, we claim that every $\mathcal{Q}_{\mathcal{K}}$ covers at least $i_{\mathcal{K}}(V_i)(C_{\mathcal{K}}-h)$ vertices from each $V_i, i\in[k]$. Otherwise, by assuming that $V(\mathcal{K})=\{V_1,V_2,\ldots,V_{a+b}\}$ and applying Lemma~\ref{lem2.2} and Corollary~\ref{coro2.17}, we can pick one more copy of $Q(a,b)$ in $G[V_1'\cup\cdots\cup V_{a+b}']$ which contains at most $i_{\mathcal{K}}(V_i)h$ vertices in each $V_i'$ for $i\in[a+b]$. This contradicts the maximality of $\mathcal{Q}_{\mathcal{K}}$.
Therefore, the total number of vertices covered as above is at least \[\sum_{i\in[k]}\sum_{\mathcal{K}\in\mathcal{F}}(C_{\mathcal{K}}-h)i_{\mathcal{K}}(V_i)=m\sum_{i\in[k]}\sum_{\mathcal{K}\in\mathcal{F}}\omega'(\mathcal{K})i_{\mathcal{K}}(V_i)
-h\sum_{i\in[k]}\sum_{\mathcal{K}\in\mathcal{F}}i_{\mathcal{K}}(V_i)\ge (1-3\eta)mk,\]
where the last inequality follows as $\omega'(R)=\sum_{i\in[k]}\sum_{\mathcal{K}\in\mathcal{F}}\omega'(\mathcal{K})i_{\mathcal{K}}(V_i)\ge(1-2\eta)k$ and $\frac{1}{n}\ll \frac{1}{k}\ll \varepsilon\ll \eta$. As each $Q(a,b)$ contains an $H$-factor and $\eta\ll \delta$, the union of these $\mathcal{Q}_{\mathcal{K}}$ provides an $H$-tiling which covers all but at most $n-(1-3\eta)mk\le \delta n$ vertices in $G$ and this completes the proof.
\end{proof}
In the next subsection, we prove Lemma \ref{lem4.4}, Lemma \ref{lem2.17} and Lemma \ref{lem2.16}.
\subsection{Proof of related lemmas}\label{sec3.3}
\subsubsection{Proof of Lemma \ref{lem4.4}}
The proof of Lemma \ref{lem4.4} relies on the following two results Lemma \ref{lem4.1} \cite{MR4193066} and Lemma \ref{lem4.2} \cite{MR4193066}.
\begin{lemma}[\cite{MR4193066}, Lemma 4.4]\label{lem4.1}
For every $r\in \mathbb{N}$ with $r\ge 4$ and $\eta, \mu>0$, there exist $\alpha>0$ and $n_{0}\in \mathbb{N}$ such that every graph $G$ on $n\geq n_{0}$ vertices with $\delta(G)\geq \left(1-\frac{2}{r}+\mu\right)n$ and $\alpha(G)< \alpha n$ has a fractional $K_{r}$-tiling $\omega$ such that
\begin{center}
$|\{v\in G: \omega(v)<1-\eta\}|\leq \eta n$.
\end{center}
\end{lemma}
\begin{lemma}[\cite{MR4193066}, Lemma 4.7]\label{lem4.2}
For every $r\in \mathbb{N}$ with $r\geq 4$ and $\mu, \eta>0$, there exist $\beta, \varepsilon, \gamma>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \left(1-\frac{2}{r}+\mu\right)n$, $\alpha(G)\leq \gamma n$ and $R:=R_{\beta, \varepsilon}$ be a reduced multigraph with multiplicity $2$, $k:=|V(R)|$. There is a graph $\Gamma$ with $\delta(\Gamma)\geq \left(1-\frac{2}{r}+\frac{\mu}{4}\right)|V(\Gamma)|$ and $\alpha(\Gamma)\leq \gamma|V(\Gamma)|$ such that the following holds.
If $\Gamma$ has a fractional $K_{r}$-tiling with total wight at least $(1-\eta)|V(\Gamma)|$, then $G$ contains a $K_{r}$-tiling covering at least $(1-2\eta)n$ vertices. Moreover, $R$ contains a fractional $\mathcal{F}(R,r)$-tiling $\omega$ such that $\omega(R)\geq (1-\eta)k$.
\end{lemma}
The ``moreover'' part of the statement is not a part of the original statement of the Lemma \ref{lem4.2} in \cite{MR4193066} but is stated explicitly in the proof.
For convenience, we need the following corollary.
\begin{coro}\label{coro3.3}
For every $r\in \mathbb{N}$ with $r\geq 4$ and $\mu, \eta>0$, there exist $\beta, \varepsilon, \gamma>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \left(1-\frac{2}{r}+\mu\right)n$, $\alpha(G)\leq \gamma n$ and $R:=R_{\beta, \varepsilon}$ be a reduced multigraph with multiplicity $2$. Then $R$ contains a fractional $\mathcal{F}(R,r)$-tiling $\omega$ such that $\omega(R)\geq (1-\eta)|V(R)|$.
\end{coro}
Corollary \ref{coro3.3} comes directly from Lemma \ref{lem4.1} and Lemma \ref{lem4.2} by applying Lemma \ref{lem4.1} on the graph $\Gamma$ in Lemma \ref{lem4.2} where $\frac{\mu}{4}$ plays the role of $\mu$.
Next, we prove Lemma \ref{lem4.4}.
\begin{proof} [Proof of Lemma \ref{lem4.4}] We write $k:=|V(R)|$. If $f(H)\geq 4$, then applying Corollary \ref{coro3.3} with $r=f(H)$ we obtain that there exists a fractional $\mathcal{F}\left(R,f(H)\right)$-tiling $\omega$ and
\begin{center}
$\omega(R)=\sum_{V_{i}\in V(R)}\omega(V_{i})\geq (1-\eta)k$.
\end{center}
If $f(H)=2, 3$, then $\delta(G)\geq \left(\frac{1}{2}+\mu\right)n$. Applying Corollary \ref{coro3.3} with $r=4$, we obtain that there exists a fractional $\mathcal{F}(R,4)$-tiling $\omega$. Next, we construct from $\omega$ a fractional $\mathcal{F}(R,3)$-tiling $\omega_{1}$ and a fractional $\mathcal{F}(R,2)$-tiling $\omega_{2}$ such that $\omega_{1}(V_i)=\omega_{2}(V_i)=\omega(V_i)$ for every vertex $V_i$ in $R$. Note that $\mathcal{F}(R, 4)$ is the family of all $K_{4}$-embeddable structures in $R$. For every $\mathcal{K}\in \mathcal{F}(R, 4)$, if $\mathcal{K}$ is a copy of $K_{4}$ in $R$ and the triangles in the $\mathcal{K}$ are denoted as $\{\mathcal{K}^{1}, \mathcal{K}^{2}, \mathcal{K}^{3}, \mathcal{K}^{4}\}$, then we define $\omega_{1}(\mathcal{K}^{i})=\frac{1}{3}\omega(\mathcal{K})$ for each $i\in [4]$. If $\mathcal{K}$ is a triangle in $R$, say $V_1V_2V_3$ such that $2i_{\mathcal{K}}(V_i)=i_{\mathcal{K}}(V_3)=2, i\in[2]$, then we have three $K_3$-embeddable structures $\mathcal{K}^1,\mathcal{K}^2,\mathcal{K}^3$ defined as follows: $\mathcal{K}^1=V_1V_3$ with $i_{\mathcal{K}^1}(V_1)=1$ and $i_{\mathcal{K}^1}(V_3)=2$; $\mathcal{K}^2=V_2V_3$ with $i_{\mathcal{K}^2}(V_2)=1$ and $i_{\mathcal{K}^2}(V_3)=2$; $\mathcal{K}^3=V_1V_2V_3$ with $i_{\mathcal{K}^3}(V_i)=1$ for every $i\in[3]$. In this case, we define $\omega_{1}(\mathcal{K}^{3})=2\omega_{1}(\mathcal{K}^{i})=\frac{2}{3}\omega(\mathcal{K})$ for each $i\in[2]$. If $\mathcal{K}$ is a double-edge in $R$, say $V_1V_2\in E(R)$ and the multiple edges in the double-edge are denoted as $\{\mathcal{K}^{1}, \mathcal{K}^{2}\}$, then $\mathcal{K}^{1}$ (or $\mathcal{K}^{2}$) can be simply regarded as a $K_{3}$-embeddable structure with $i_{\mathcal{K}^1}(V_1)=1$ and $i_{\mathcal{K}^1}(V_2)=2$ (resp. $i_{\mathcal{K}^2}(V_1)=2$ and $i_{\mathcal{K}^2}(V_2)=1$). Here we define $\omega_{1}(\mathcal{K}^{i})=\frac{2}{3}\omega(\mathcal{K})$ for each $i\in [2]$.
In all cases, it is easy to see that $\omega_{1}(V_{i})=\omega(V_{i})$ for every $i\in [k]$ and thus $\omega_{1}(R)=\sum_{V_{i}\in V(R)}\omega_{1}(V_{i})=\omega(R)$.
Next we shall construct a fractional $\mathcal{F}(R,2)$-tiling $\omega_{2}$ from $\omega$ such that $\sum_{V_{i}\in V(R)}\omega_{2}(V_{i})\geq (1-\eta)k$. By Definition \ref{def2.7}, every vertex $V_i$ in $V(R)$ is a $K_{2}$-embeddable structure, and we define $\omega_{2}(V_{i})=\omega(V_{i})$ for every $i\in [k]$. Then $\omega_{2}(R)=\sum_{V_{i}\in V(R)}\omega_{2}(V_{i})=\omega(R)$.
\end{proof}
\subsubsection{Proof of Lemma \ref{lem2.17}}
Before the proof of Lemma \ref{lem2.17}, we need several results as follows. The first one is due to Gy\'{a}rf\'{a}s, Szemer\'{e}di and Tuza \cite{MR1395865} and independently, Sumner \cite{MR634555}.
\begin{lemma}[\cite{MR1395865}, \cite{MR634555}]\label{lem2.9}
A k-chromatic graph contains every tree on $k$ vertices as a subgraph.
\end{lemma}
In our context, we shall use the following corollary of Lemma \ref{lem2.9}.
\begin{coro}\label{coro2.10}
Let $n\geq k$ be any integers and $G$ be an $n$-vertex graph with $\alpha(G)\leq \frac{n}{k}$. Then every $k$-tree is contained in $G$.
\end{coro}
The next gadget in our proof is Lemma \ref{lem2.14} proved by Erd\H{o}s, Hajnal, S\'{o}s and Szemer\'{e}di~\cite{MR716422}, which enables us to embed one tree inside the common neighborhood of other trees under
certain density conditions.
Before the statement of Lemma \ref{lem2.14}, we first need the following notation of $(r_{1}, r_{2})$-graphs~\cite{MR716422}.
\begin{defn}[\cite{MR716422}, Definition 2.2]\label{def2.13}
For $r_{1}, r_{2}\in \mathbb{N}$, a graph $G(V, E)$ is said to be an $(r_{1}, r_{2})$-$graph$ with root $v\in V(G)$ if $|V(G)|\leq r_{1}^{r_{2}}+1$ and each $u\in V(G)$ with distance at most $r_{1}$ from $v$ has degree at least $r_{2}$.
\end{defn}
Obviously, for $r\geq 1$ an arbitrary tree of $r+1$ vertices is a subgraph of any $(r, r)$-graph.
\begin{lemma}[\cite{MR716422}, Lemma 2.4]\label{lem2.14}
Given $r_{1}, r_{2}, p\in \mathbb{N}$ and $c>0$, there exist positive constants $c'$ and $s$ such that the following holds for sufficiently large $n$. Let $V_{0}, V_{1}, \dots, V_{p}$ be vertex sets each of size $n$ and $G$ be a graph defined on $V_{0}$ with $|E(G)|\geq sn$. Then for all given mappings $f_{i}: E(G)\rightarrow [V_{i}]^{\geq cn}$ with $i\in[p]$, there exists an $(r_{1}, r_{2})$-graph $H_{1}\subseteq G$ with
\begin{center}
$\big|\bigcap_{e\in E(H_{1})} f_{i}(e)\big|\geq c' n$ for every $i\in [p]$.
\end{center}
\end{lemma}
Note that to invoke Lemma \ref{lem2.14} in the proof of Lemma \ref{lem2.17}, we need Proposition \ref{prop2.12} which gives a lower bound on the number of edges in the setting of small independence number.
\begin{prop}[]\label{prop2.12}
Let $G$ be an $n$-vertex graph with $\alpha(G)\leq \alpha n$. Then $e(G)\geq \frac{(1-\alpha)n}{2\alpha}$.
\end{prop}
\begin{proof}
Note that
\begin{align*}
1+\frac{2e(G)}{n} & =\frac{\sum_{v \in V(G)}\left(d(v)+1\right)}{n} \geq \frac{n}{\sum_{v \in V(G)}\frac{1}{d(v)+1}}\ge\frac{1}{\alpha},
\end{align*}
where the first inequality follows from the fact that the arithmetic mean is at least the harmonic mean, and the last inequality follows from a well-known result in \cite{MR1090733} stating that $\alpha(G) \geq\sum_{v\in V(G)}\frac{1}{d(v)+1}$.
Thus $e(G)\geq \frac{(1-\alpha)n}{2\alpha}$.
\end{proof}
Next we prove Lemma~\ref{lem2.17}.
\begin{proof}[Proof of Lemma \ref{lem2.17}] Given $a, b, s\in \mathbb{N}$ and $\beta>0$, we shall choose
\[\frac{1}{N}\ll \alpha\ll\varepsilon\ll \varepsilon_{a+b-1}\ll \cdots\ll \varepsilon_{1}\ll c'\ll \beta,\frac{1}{s}\] and let $G, F_1,\ldots,F_b$ be given in Lemma~\ref{lem2.17}.
The proof will proceed by induction on $a+b$. The base case $a+b=1$ is clear as either $Q$ is an $s$-independent set or a $2s$-forest $F_{1}$: If $a=1$ and $b=0$, then we only need to choose a vertex set $U_{1}\subseteq V_{1}$ with $|U_{1}|=s$ which is easily derived since $|V_{1}|\geq N \geq s$. If $a=0$ and $b=1$, then we only need to embed a $2s$-forest $F_{1}$ in $G[V_{1}]$. By Corollary \ref{coro2.10} with $\alpha\leq \frac{1}{2s}$, $G[V_{1}]$ contains every $2s$-forest.
Next we show that our statement holds for $a+b=k$ assuming it holds for $a+b=\ell<k$.
First assume $a\geq 1$. Since $(V_{i}, V_{j})$ is $\varepsilon$-regular in $G$ for distinct $i, j\in [1, a+b]$, there exists a subset $V_{1}'\subseteq V_{1}$ such that every vertex in $V_{1}'$ has at least $\left(d(V_{1}, V_{j})-\varepsilon\right)|V_{j}|$ neighbors in $V_{j}$ for each $j\in [2, a+b]$ and $|V_{1}'|\geq \left(1-\varepsilon(a+b-1)\right)|V_{1}|$. In $V'_{1}$, we choose $s$ vertices $\{v_{1}, \dots, v_{s}\}$ with $S_{j}^{1}:=N(v_{1})\cap V_{j}$, $S_{j}^{i}:= N(v_{i})\cap S_{j}^{i-1}$ and $|N(v_{i})\cap S_{j}^{i-1}|\geq (\beta-\varepsilon)|S_{j}^{i-1}|\geq \frac{\beta}{2}|S_{j}^{i-1}|$ for each $i\in [2, s]$ and each $j\in [2, a+b]$. Then the $s$ vertices in $V'_{1}$ have a common neighborhood $S_{j}:=S_{j}^{s}$ in $V_{j}$ for each $j\in [2, a+b]$ with $|S_{j}|\geq (\frac{\beta}{2})^{s}|V_{j}|$. Applying Lemma \ref{lem2.2} to $G[\bigcup_{i=2}^{a+b} S_{i}]$, we obtain that $(S_{i}, S_{j})$ is $\varepsilon'$-regular with $\varepsilon':=\max\left\{2\varepsilon, \frac{\varepsilon2^{s}}{\beta^{s}}\right\}=\frac{\varepsilon2^{s}}{\beta^{s}}$ and $d(S_{i}, S_{j})\geq d(V_{i}, V_{j})-\varepsilon\geq d(V_{i}, V_{j})-\frac{\beta}{2}$ for distinct $i, j\in [2, a+b]$. Since $\varepsilon\ll \varepsilon_{a+b-1}$, it holds that $(S_{i}, S_{j})$ is $\varepsilon_{a+b-1}$-regular for distinct $i, j\in [2, a+b]$.
Note that $|S_{j}|\geq (\frac{\beta}{2})^{s} |V_{j}|\geq (\frac{\beta}{2})^{s} N$. We apply the induction hypothesis to find in $G[\bigcup_{i=2}^{a+b} S_{i}]$ a copy of $Q(a-1, b, s, F_{1}, \dots, F_{b})$ which together with $\{v_{1}, \dots, v_{s}\}$ forms a copy of $Q(a, b, s, F_{1}, \dots, F_{b})$ as desired.
Now assume $a=0$ and $b\geq 2$. Recall that $d(V_{i}, V_{j})\geq \frac{1}{2}+\beta$ for distinct $i, j\in [b]$. Inside $V_{1}$, there exists a subset $V_{1}'$ such that every vertex in $V_{1}'$ has at least $(d(V_{1}, V_{j})-\varepsilon)|V_{j}|\geq \left(\frac{1}{2}+\frac{\beta}{2}\right)|V_{j}|$ neighbors in $V_{j}$ for every $j\in [2, b]$ and
\[
|V'_{1}|\geq |V_{1}|-\varepsilon |V_{1}|b=(1-\varepsilon b)|V_{1}|\geq (1-\varepsilon b)N.
\]
Our next step is to embed $Q(a, b, s, F_{1}, \dots, F_{b})$ into $G[V'_{1}\cup V_{2} \cup \cdots \cup V_{b}]$.
Note that for $u, v\in V'_{1}$ and $i\in [2, b]$, we have
\begin{equation}\label{eq1}
|N(u)\cap N(v)\cap V_{i}|\geq \beta|V_{i}|.
\end{equation}
By Proposition \ref{prop2.12}, we have
\[e(V'_{1})\geq \frac{(1-\alpha)}{2\alpha}|V'_{1}|\geq \frac{(1-\alpha)}{2\alpha}(1-\varepsilon b)N.\]
To apply Lemma \ref{lem2.14}, we define for every $i\in [2, b]$ a mapping
\begin{center}
$f_{i}: E(G[V'_{1}])\rightarrow [V_{i}]^{\geq \beta |V_{i}|}$,
\end{center}
by letting $f_{i}(uv):= N(u)\cap N(v)\cap V_{i}$ for every $uv\in E(G[V'_{1}])$. From (\ref{eq1}), it holds that $|f_{i}(uv)|\geq \beta |V_{i}|$ for every $uv\in E(G[V'_{1}])$. Lemma \ref{lem2.14} applied with $r_{1}=r_{2}=2s$ and $c=\beta$ implies that there exist a constant $c'$ and a $(2s, 2s)$-subgraph $H_{1}\subseteq G[V'_{1}]$ such that for $i\in[2, b]$, $|\bigcap_{e\in E(H_{1})} f_{i}(e)|\geq c' |V_{i}|$. By definition, $H_{1}$ contains a subgraph isomorphic to $F_{1}$. Write $S_{i}:= \bigcap_{e\in E(F_{1})} f_{i}(e)$ for each $i\in [2, b]$. Then for every $i\in [2, b]$, we have $|S_{i}| \geq c'|V_{i}|\geq c'N$. By Lemma \ref{lem2.2}, we have that $(S_{i}, S_{j})$ is $\varepsilon'$-regular where $\varepsilon':=\max\left\{2\varepsilon, \frac{\varepsilon}{c'}\right\}\leq \varepsilon_{a+b-1}$ since $\varepsilon\ll \varepsilon_{a+b-1}, c'$ and $d(S_{i}, S_{j})\geq \frac{1}{2}+\frac{\beta}{2}$. By induction hypothesis, $G[\bigcup_{i=2}^{b} S_{i}]$ contains a copy of $Q(a, b-1, s, F_{2}, \dots, F_{b})$ which together with $F_{1}$ forms a copy of $Q(a, b, s, F_{1}, \dots, F_{b})$ as desired.
\end{proof}
\subsubsection{Proof of Lemma \ref{lem2.16}}
\begin{proof}[Proof of Lemma \ref{lem2.16}] Let $r:=ar(H)$ and $\mathcal{T}=\{T_{0}, T_{1}, \dots, T_{r-1}\}$ be an acyclic partition of $H$.
For every $T_{i}\in \mathcal{T}$, $\mathcal{P}_{i}=\{V_{i,1}, V_{i,2}\}$ is a bipartition of $T_{i}$, and $T_{i, j}:=V_{i,j}$ is an independent set in $H$ for $j\in[2]$.
We divide the proof into the following two cases.
If $f(H)=2r$, then we take $s=h$ and construct $F_{1}= F_2=\cdots=F_{b}$ such that $F_{1}$ is a $2h$-forest which consists of two vertex-disjoint copies of $H[T_{j}]$ for each $j\in [0, r-1]$. Recall that $\frac{a}{2}+b=r$. To show that $Q(a, b, h, F_{1}, \dots, F_{b})$ contains an $H$-factor, we build an auxiliary matrix \[A=\{a_{ij}\}_{r\times r}=\begin{pmatrix}
T_{1} & T_{2} & \dots & T_{0}\\
T_{2} & T_{3} & \dots & T_{1} \\
\vdots & \vdots& \ddots & \vdots\\
T_{0} & T_{1} &\dots & T_{r-1}
\end{pmatrix}\]
where $a_{ij}:=T_{i+j-1\pmod{r}}$ for $i, j\in [r]$. Note that every row (or column) of $A$ corresponds to a permutation of $\mathcal{T}=\{T_{0}, T_{1}, \dots, T_{r-1}\}$. Now we construct two matrices $A_{1}$ and $A_{2}$ as follows by ``cutting'' each of the first $\frac{a}{2}$ columns in half and then swapping columns two by two: \[A_{1}=\{a^{1}_{i, j}\}_{r\times (a+b)}=\begin{pmatrix}
T_{1,1} & T_{1,2}& \dots & T_{\frac{a}{2},1} & T_{\frac{a}{2},2} & T_{\frac{a}{2}+1} & \dots & T_{0}\\
T_{2,1} & T_{2,2}& \dots & T_{\frac{a}{2}+1,1} & T_{\frac{a}{2}+1,2} & T_{\frac{a}{2}+2} & \dots & T_{1}\\
\vdots &\vdots & \ddots &\vdots & \vdots & \vdots & \ddots & \vdots\\
T_{0,1} & T_{0,2}& \dots & T_{\frac{a}{2}-1,1} & T_{\frac{a}{2}-1,2} & T_{\frac{a}{2}} & \dots & T_{r-1}\\
\end{pmatrix}\]
where $a^{1}_{i, 2j-1}:=T_{i+j-1\pmod{r}, 1}$ and $a^{1}_{i, 2j}:=T_{i+j-1\pmod{r}, 2}$ for each $i\in [r]$ and $j\in[\frac{a}{2}]$; \[A_{2}=\{a^{2}_{i, j}\}_{r\times (a+b)}=\begin{pmatrix}
T_{1,2} & T_{1,1}& \dots &T_{\frac{a}{2},2} & T_{\frac{a}{2},1} & T_{\frac{a}{2}+1} & \dots & T_{0}\\
T_{2,2} & T_{2,1}& \dots &T_{\frac{a}{2}+1,2} & T_{\frac{a}{2}+1,1} & T_{\frac{a}{2}+2} & \dots & T_{1}\\
\vdots &\vdots & \ddots &\vdots & \vdots & \vdots & \ddots & \vdots\\
T_{0,2} & T_{0,1}& \dots &T_{\frac{a}{2}-1,2} & T_{\frac{a}{2}-1,1} & T_{\frac{a}{2}} & \dots & T_{r-1}\\
\end{pmatrix}\]
where $a^{2}_{i, 2j-1}:=T_{i+j-1\pmod{r}, 2}$ and $a^{2}_{i, 2j}:=T_{i+j-1\pmod{r}, 1}$ for each $i\in [r]$ and $j\in[\frac{a}{2}]$.
Then $A^{\ast}: =\begin{pmatrix}
A_{1}\\
A_{2}
\end{pmatrix}$ is a $2r\times (a+b)$ matrix and observe that for each $i\in[a]$, disjoint union of the elements taken over the $i$th column gives us an $h$-independent set, whilst each of the last $b$ columns of $A^{\ast}$ provides a copy of $F_1$. Clearly, $Q(a, b, h, F_{1}, \dots, F_{b})$ has an $H$-factor with $2r$ vertex-disjoint copies of $H$.
If $f(H)=2r-1$, then we take $s=\frac{2h}{2r-1}$. By the definition of $\widetilde{\mathcal{H}}$, we know that in $\mathcal{T}=\{T_{0}, T_{1}, \dots, T_{r-1}\}$, $T_0$ is an independent set in $H$ and $|T_i|=2|T_0|, i\in[r-1]$. Thus $|T_0|=\frac{h}{2r-1}$ and recall that $\frac{a+1}{2}+b=r$. Now we construct $F_{1},F_2,\ldots ,F_{b}$ such that each $F_{i}$ is a $4|T_0|$-forest which consists of two vertex-disjoint copies of $H[T_{r-b+i-1}]$.
To show that $Q(a, b, \frac{2h}{2r-1}, F_{1}, \dots, F_{b})$ contains an $H$-factor, we build an auxiliary matrix $A=\{a_{1j}\}_{1\times r}=\begin{pmatrix}
T_{0}&\dots& T_{r-1}
\end{pmatrix}$
where $a_{1j}:=T_{j}$ for each $j\in [0, r-1]$. It holds that $A$ corresponds to an acyclic partition of $H$. Similarly, we construct two matrices $A_{1}$ and $A_{2}$ as follows: \[A_{1}=\{a^{1}_{i, j}\}_{1\times (a+b)}=\begin{pmatrix}
T_{0} & T_{1,1} & T_{1,2} & \dots & T_{\frac{a-1}{2},1} & T_{\frac{a-1}{2},2} & T_{r-b} & \dots & T_{r-1}
\end{pmatrix},\] \[A_{2}=\{a^{2}_{i, j}\}_{1\times (a+b)}=\begin{pmatrix}
T_{0} & T_{1,2} & T_{1,1} & \dots & T_{\frac{a-1}{2},2} & T_{\frac{a-1}{2},1} & T_{r-b} & \dots & T_{r-1}
\end{pmatrix}.\]
Let $A^{\ast}: =\begin{pmatrix}
A_{1}\\
A_{2}
\end{pmatrix}$ be a $2\times (a+b)$ matrix. By taking a disjoint union of the elements in the columns as above, each of the first $a$ columns of $A^{\ast}$ provides a $\frac{2h}{2r-1}$-independent set while the last $b$ columns respectively provides copies of $F_i, i\in [b]$. Clearly, $Q(a, b, \frac{2h}{2r-1}, F_{1}, \dots, F_{b})$ has an $H$-factor containing two vertex-disjoint copies of $H$.
\end{proof}
\section{Absorbing}\label{sec5}
In this section, we give a proof of Lemma \ref{lem3.3}. We shall use a result of Nenadov and Pehova \cite{MR4080942} which gives a sufficient condition on the existence of an absorbing set.
\begin{lemma}[\cite{MR4080942}, Lemma 2.2]\label{lem5.1}
Let $H$ be a graph with $h$ vertices, $\gamma>0$ and $t\in \mathbb{N}$ be constants. Then there exists $\xi:=\xi(h, t, \gamma)$ such that the following holds for sufficiently large $n$. Suppose that $G$ is a graph with $n$ vertices such that every $S\in \binom{V(G)}{h}$ has a family of at least $\gamma n$ vertex-disjoint $(H, t)$-absorbers. Then $G$ contains a $\xi$-absorbing set of size at most $\gamma n$.
\end{lemma}
So the key point in the proof of Lemma \ref{lem3.3} is to build linearly many vertex-disjoint absorbers for every $S\in \binom{V(G)}{h}$.
To achieve this, we employ the latticed-based absorbing method \cite{HMWY2021} and we first need the notion of $H$-reachability from \cite{HMWY2021} which originates in \cite{MR3338027}.
\begin{defn}
Let $G, H$ be given as aforementioned and $m, t\in \mathbb{N}$. Then we say that two vertices $u, v\in V(G)$ are \emph{$(H, m, t)$-reachable} (in $G$) if for any vertex set $W$ of $m$ vertices, there is a set $S\subseteq V(G)\backslash W$ of size at most $ht-1$ such that both $G[S\cup \{u\}]$ and $G[S\cup \{v\}]$ have $H$-factors, where we call such $S$ an \emph{$H$-connector} for $u, v$. Moreover, a set $U\subseteq V(G)$ is \emph{$(H, m, t)$-closed} if every two vertices $u, v\in U$ are $(H, m, t)$-reachable, where the corresponding $H$-connector for $u, v$ may not be contained in $U$. If two vertices $u, v\in V(G)$ are $(H, m, 1)$-reachable, then we say $u$ is \emph{1-reachable} to $v$. If $u, v\in U$ are $(H, m, t)$-reachable, and the corresponding $H$-connector for $u, v$ is contained in $U$, then we say that $u, v\in U$ are \emph{$(H, m, t)$-inner-reachable}. Similarly, we can define \emph{$(H, m, t)$-inner-closed} and \emph{1-inner-reachable}.
\end{defn}
The following result from \cite{HMWY2021} builds a sufficient condition to ensure that every subset $S\subseteq V(G)$ with $|S|=h$ has linearly many vertex-disjoint absorbers.
\begin{lemma}[\cite{HMWY2021}, Lemma 3.9]\label{lem5.2}
Given $\beta>0$, $t, h\in \mathbb{N}$ with $h\geq 3$ and an $h$-vertex graph $H$, the following holds for sufficiently large $n\in \mathbb{N}$. Let $G$ be an $n$-vertex graph such that $V(G)$ is $(H, \beta n, t)$-closed. Then every $S\in \tbinom{V(G)}{h}$ has a family of at least $\frac{\beta}{h^{2}t}n$ vertex-disjoint $(H, t)$-absorbers.
\end{lemma}
Based on this lemma, it suffices to show that $V(G)$ is closed. However, we are only able to prove a slightly weaker result which states that the graph $G$ admits a vertex partition $V(G)=B\cup U$ where $B$ is a small vertex set and $U$ is inner-closed.
\begin{lemma}[]\label{lem7.1}
Given $h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constants $\tau, \mu$ with $0<\tau<\mu$, there exist positive constants $\alpha, \beta$ and $t\in \mathbb{N}$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ admits a partition $V(G)=B\cup U$ with $|B|\leq \tau n$ and $U$ is $(H, \beta n, t)$-inner-closed.
\end{lemma}
Clearly, we shall focus on the subgraph $G[U]$ and obtain an absorbing set by applying Lemma~\ref{lem5.2} and Lemma~\ref{lem5.1} on $G[U]$.
The next step is to deal with the vertex set $B$. We shall pick mutually vertex-disjoint copies of $H$ each covering a vertex in $B$. To achieve this, we use the following result which enables us to find linearly many copies of $H$ covering any given vertex.
\begin{lemma}[]\label{lem5.7}
Given $h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and a constant $\mu>0$, there exists $\alpha>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. If $W$ is a subset of $V(G)$ with $|W|\leq \frac{\mu}{2}n$, then there exists at least one copy of $H$ covering $v$ in $G-W$ for each $v\in V(G)\backslash W$.
\end{lemma}
Now we give the proof of Lemma \ref{lem3.3} by using Lemma \ref{lem5.1}, Lemma \ref{lem5.2}, Lemma \ref{lem7.1} and Lemma \ref{lem5.7}.
\begin{proof} [Proof of Lemma \ref{lem3.3}]
Let $h\in \mathbb{N}$, $H$ be an $h$-vertex graph and constants $\gamma, \mu$ with $0<\gamma\leq \frac{\mu}{2}$. Then we shall choose $\tau=\frac{\gamma}{2h}$ and
\begin{center}
$\frac{1}{n}\ll \alpha\ll \xi\ll \frac{1}{t}, \beta\ll \gamma, \mu$.
\end{center}
Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$.
Applying Lemma \ref{lem7.1} on $G$, we obtain that $G$ admits a vertex partition $V(G)=B\cup U$ such that $|B|\leq \tau n$ and $U$ is $(H, \beta n, t)$-inner-closed.
Applying Lemma \ref{lem5.2} on $G[U]$, it holds that every $S\in \tbinom{U}{h}$ has a family of at least $\frac{\beta}{h^{2}t}|U|\geq \frac{\beta}{2h^{2}t}n$ vertex-disjoint $(H, t)$-absorbers. Applying Lemma \ref{lem5.1} on $G[U]$ where $\frac{\gamma}{2}$ plays the role of $\gamma$, we obtain a $\xi$-absorbing set $A_{1}$ in $G[U]$ of size at most $\frac{\gamma}{2}n$.
Now we shall iteratively pick vertex-disjoint copies of $H$ each covering one vertex in $B$ while avoiding using any vertex in $A_{1}$, and we claim that every vertex in $B$ can be covered in this way. Let $G_{2}:= G-A_{1}$. For each $v\in B$, we apply Lemma \ref{lem5.7} iteratively to find a copy of $H$ covering $v$ in $G_{2}$, while avoiding $A_{1}$ and all copies of $H$ found so far. This is possible as during the process the number of vertices that we need to avoid is at most
\begin{center}
$h|B|+|A_{1}|\leq h\tau n+ \frac{\gamma}{2}n=\gamma n\leq \frac{\mu}{2}n$.
\end{center}
Let $W$ be the union of the vertex sets over all the $|B|$ vertex-disjoint copies of $H$ as above and $A:= A_{1}\cup W$. Recall that $A_{1}$ is a $\xi$-absorbing set for $G[U]$, and $G[W]$ has an $H$-factor. Thus $A$ is a $\xi$-absorbing set for $G$ with $|A|\leq \gamma n$.
\end{proof}
Now it remains to prove Lemma \ref{lem7.1} and Lemma \ref{lem5.7} whose proofs will be given in next two subsections respectively.
\subsection{Proof of Lemma \ref{lem7.1}}
To prove Lemma \ref{lem7.1}, we divide the proof into two steps (i): $G$ admits a partition $V(G)=B\cup U$ where $B$ is a small vertex set and every vertex in $U$ is $1$-inner reachable to linearly many vertices; (ii): $U$ is inner-closed.
The following result is the first step.
\begin{lemma}[]\label{lem5.5}
Given $h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constants $\tau, \mu$ with $0<\tau<\mu$, there exist positive constants $\alpha, \beta_{1}, \gamma_{1}$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$. Then $G$ admits a vertex partition $V(G)=B\cup U$ such that $|B|\leq \tau n$ and every vertex in $U$ is $(H, \beta_{1}n, 1)$-inner-reachable to at least $\gamma_{1}n$ other vertices in $G[U]$.
\end{lemma}
In the second step, we only need to apply the following result on $G[U]$.
\begin{lemma}[]\label{lem5.4}
Given $h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constants $\mu, \beta_{1}, \gamma_{1}$ with $0<\mu, \beta_{1}, \gamma_{1}<1$, there exist positive constants $\alpha, \beta$ and $t\in \mathbb{N}$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$ such that every vertex in $V(G)$ is $(H, \beta_{1}n, 1)$-reachable to at least $\gamma_{1}n$ other vertices. Then $V(G)$ is $(H, \beta n, t)$-closed.
\end{lemma}
Obviously, Lemma \ref{lem7.1} is an immediate corollary of the above-mentioned two lemmas. In the following, we will give the proofs of Lemma \ref{lem5.5} and Lemma \ref{lem5.4}.
\subsubsection{Proof of Lemma \ref{lem5.5}: $1$-reachability}
The proof of Lemma \ref{lem5.5} goes roughly as follows. We first apply the regularity lemma on $G$ to obtain a partition and a reduced graph $R$ with multiplicity 2. A result of Knierim and Su \cite{MR4193066} guarantees that every cluster $V_i$ is covered by a $K_{f(H)+1}$-embeddable structure, say $\mathcal{K}_i$ in the reduced graph $R$ (see Lemma~\ref{lem5.11}). In this case, for each $V_i$, by using Corollary~\ref{coro2.17} on $\mathcal{K}_i$, we are able to show that almost all vertices in $V_i$ are $1$-reachable to linearly many other vertices from $V_i$, where the bad vertices would be given iteratively at each stage of the process.
As sketched above, we need the following result which investigates the structure around every cluster in the reduced graph.
\begin{lemma}[\cite{MR4193066}, Lemma 3.7]\label{lem5.11}
For $r\geq 4$ and $k\in \mathbb{N}$, let $R$ be a multigraph with multiplicity 2 on $k$ vertices and $\delta(R)> \left(1-\frac{2}{r}\right)2k$. Then any double-edge $ij\in E(R)$ is contained in some $K_{r+1}$-embeddable structure.
\end{lemma}
\begin{proof} [Proof of Lemma \ref{lem5.5}]Given $h\in \mathbb{N}$, an $h$-vertex graph $H$ and constants $\tau, \mu$ with $0<\tau< \mu$, we shall choose
\begin{center}
$\frac{1}{n}\ll\frac{1}{N}\ll\alpha\ll \beta_{1}, \gamma_{1}\ll\frac{1}{k}\ll \varepsilon \ll \tau, \mu$.
\end{center}
Let $\beta=\frac{\mu}{10}$, $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$ and $\alpha(G)\leq \alpha n$.
Applying Lemma \ref{lem2.3} on $G$ with $\varepsilon, \beta>0$, we obtain an $\varepsilon$-regular partition $\mathcal{P}=\{V_{0}, V_{1}, \dots, V_{k}\}$ of $G$.
Let $m:=|V_{i}|$ for each $i\in [k]$ and $R:=R_{\beta, \varepsilon}$ be a reduced multigraph with multiplicity $2$ of the $\varepsilon$-regular partition $\mathcal{P}$. By Fact \ref{fact2.5}, we have $\delta(R)\geq 2\left(1-\frac{2}{f(H)}+\frac{\mu}{2}\right)k$ when $f(H)\geq 4$ and $\delta(R)\geq (1+\mu)k$ when $f(H)=2, 3$. Clearly, every cluster is in a double-edge. Applying Lemma \ref{lem5.11} with $r=\max\{f(H), 4\}$, we have that every cluster is in a $K_{f(H)+1}$-embeddable structure when $f(H)\geq 4$ and every cluster is in a $K_{5}$-embeddable structure when $f(H)=2, 3$. Therefore every cluster is covered by a $K_{f(H)+1}$-embeddable structure.
Write $\mathcal{F}:=\mathcal{F}(R, f(H)+1)$ for the family of all $K_{f(H)+1}$-embeddable structures in $R$. Since every cluster is in some $K_{f(H)+1}$-embeddable structure, there exists a minimal subfamily $\{\mathcal{K}_{1}, \dots, \mathcal{K}_{\ell}\}$ such that $V(R)=\bigcup_{i=1}^{\ell}$ $V(\mathcal{K}_{i})$. Now we first define the vertex set $B$ as follows:
\begin{itemize}
\item [(1)]
Let $S_{i}=V(\mathcal{K}_{i})\backslash \bigcup_{p=1}^{i-1} V(\mathcal{K}_{p})$ for $i\in [\ell]$, where $S_{1}=V(\mathcal{K}_{1})$. Then it follows from the minimality that $S_{i}\neq \emptyset$ and we write $S_{i}=\{V_{i_{1}}, \dots, V_{i_{s_{i}}}\}$ for some integer $s_{i}:=|S_{i}|$;
\item [(2)]
For $i\in [\ell]$ and $j\in [s_{i}]$, $B_{i_{j}}:=\Big\{v\in V_{i_{j}}\Big| |N(v)\cap V_{s}|\leq (d(V_{i_{j}}, V_{s})-\varepsilon)|V_{s}|$ for some $V_{s}\in V(\mathcal{K}_{i})\, \text{with}\, s\neq i_{j}\Big\}$, and $B_{i}:=\bigcup_{j=1}^{s_{i}} B_{i_{j}}$;
\item [(3)]
$B:=\bigcup_{i=1}^{\ell} B_{i}$.
\end{itemize}
Observe that $\{S_{1}, S_{2}, \dots, S_{\ell}\}$ is a partition of $V(R)$ and $|S_{i}|\le 2r+1$, $i\in [\ell]$. Moreover, for every $i\in [\ell]$ and $j\in [s_{i}]$, we have $|B_{i_{j}}|\le \varepsilon m|\mathcal{K}_{i}|$. Thus $|B|\leq \varepsilon m(2r+1)k\leq \tau n$ since $\varepsilon \ll \tau$.
Let $U:=V(G)\backslash B=\bigcup_{i=1}^{k} U_{i}$ where $U_{i}:= V_{i}\backslash B$. Then $|U_{i}|\geq m-\varepsilon m(2r+1)$. By Lemma \ref{lem2.2}, $d(U_{i}, U_{j})\geq d(V_{i}, V_{j})-\varepsilon$ for distinct $i, j\in [k]$. Next, we shall prove that every $v\in U$ is 1-inner-reachable to linearly many vertices in $U$.
For each $p\in [k]$ and any vertex $v\in U_{p}$, we choose the minimum $q\in [\ell]$ such that $V_{p}\in V(\mathcal{K}_{q})$. Since $\mathcal{K}_{q}$ is a $K_{f(H)+1}$-embeddable structure, there exists a subgraph of $\mathcal{K}_{q}$, say $\mathcal{K}'_{q}$, which is a $K_{f(H)}$-embeddable structure such that $i_{\mathcal{K}'_{q}}(V_{p})=1$. Without loss of generality, we may assume $p=1$ and write $V(\mathcal{K}'_{q})=\{V_{1}, \dots, V_{a}, \dots, V_{a+b}\}$ for some integers $a, b\in \mathbb{N}$ and $a+2b=f(H)$. Thus for distinct $i, j\in [a+b]$, it follows from (2) and the fact $\varepsilon\ll \beta, \frac{1}{r}$ that every vertex $u\in U_{i}$ has at least $|N(u)\cap V_{j}|-|B\cap V_{j}|> d(V_{i}, V_{j})m-\varepsilon m-\varepsilon m(2r+1)\geq \frac{\beta}{2}m$ neighbors in $U_{j}$.
Recall that $v\in U_{1}$. We denote by $U_{1}^{\ast}$ the set of vertices $u\in U_{1}$ such that for every $j\in [2, a+b]$, $|N(u)\cap N(v)\cap U_{j}|\geq (\frac{\beta}{2})^{2}m$. The following claim would complete our proof because $|U_{1}^{\ast}|\geq(1-\varepsilon(2r+1))|U_{1}|\geq (1-\varepsilon(2r+1))^{2}m\geq \gamma_{1}n$, where $\gamma_{1}\ll \frac{1}{k}\ll \varepsilon$.
\begin{claim}
The vertex $v$ is $(H, \beta_{1}n, 1)$-reachable to every $u\in U^{\ast}_{1}$.
\end{claim}
To prove this, for every $u\in U_{1}^{\ast}$, we arbitrarily choose $N_{1}\subseteq U_{1}\backslash \{u, v\}$ with $|N_{1}|\geq (\frac{\beta}{2})^{2}m$ and write $N_j:=N(u)\cap N(v)\cap U_{j}$ for each $j\in [2, a+b]$. Then $|N_j|\geq (\frac{\beta}{2})^{2}m$ for each $j\in [a+b]$. For each $j\in [a+b]$, $N'_{j}$ comes from $N_j$ by deleting any $\beta_{1}n$ vertices.
Hence, $|N_j'|\geq |N_j|-\beta_{1}n\geq (\frac{\beta}{2})^{2}m-\beta_{1}n\geq \frac{\beta^{2}}{8}m$ since $\frac{1}{n}\ll \frac{1}{N}\ll\beta_{1}\ll\frac{1}{k}, \mu$ and $\beta=\frac{\mu}{10}$.
By Lemma \ref{lem2.2}, $(N'_{i}, N'_{j})$ is $\varepsilon'$-regular with $\varepsilon':=\max\left\{2\varepsilon, \frac{8\varepsilon}{\beta^{2}}\right\}=\frac{8\varepsilon}{\beta^{2}}$ and $d(N'_{i}, N'_{j})\geq d(V_{i}, V_{j})-\varepsilon$ for distinct $i, j\in [a+b]$.
Applying Corollary \ref{coro2.17} on $G[N'_{1}\cup \cdots\cup N'_{a+b}]$, we obtain a copy of $Q(a,b)$ which induces an independent set inside $N_1'$. Hence, by definition $v$ is $(H, \beta_{1}n, 1)$-reachable to $u$.
\end{proof}
\subsubsection{Proof of Lemma \ref{lem5.4}}
In the following, we shall use the latticed-based absorbing method developed by Han \cite{MR3632565} and begin with the following notion introduced by Keevash and Mycroft \cite{MR3290271}.
Let $G$ be an $n$-vertex graph. We will often work with a vertex partition $\mathcal{P}=\{V_{1}, \dots, V_{r}\}$ of $V(G)$ for some integer $r\geq 1$. For any subset $S\subseteq V(G)$, the \emph{index vector} of $S$ with respect to $\mathcal{P}$, denoted by $i_{\mathcal{P}}(S)$, is the vector in $\mathbb{Z}^{r}$ whose $i$th coordinate is the size of the intersections of $S$ with $V_{i}$ for each $i\in [r]$. For each $j\in [r]$, let $\textbf{u}_{j}\in \mathbb{Z}^{r}$ be the $j$th unit vector, i.e. $\textbf{u}_{j}$ has 1 on the $j$th coordinate and 0 on the other coordinates. A \emph{transferral} is a vector of the form $\textbf{u}_{i}-\textbf{u}_{j}$ for some distinct $i, j\in [r]$. A vector $\textbf{v}\in \mathbb{Z}^{r}$ is an \emph{$s$-vector} if all its coordinates are non-negative and their sum is $s$. Given $\mu>0$ and an $h$-vertex graph $H$, we say that an $h$-vector \textbf{v} is \emph{$(H, \mu)$-robust} if for any set $W$ of at most $\mu n$ vertices, there is a copy of $H$ in $G-W$ whose vertex set has an index vector equal to $\textbf{v}$. Let $I^{\mu}(\mathcal{P})$ be the set of all $(H, \mu)$-robust $h$-vectors and $L^{\mu}(\mathcal{P})$ be the lattice (i.e. the additive subgroup) generated by $I^{\mu}(\mathcal{P})$.
Here is a brief proof outline for Lemma~\ref{lem5.4}. In order to prove that $V(G)$ is closed, we adopt a less direct approach and build on the merging techniques developed in \cite{HMWY2021}.
We first partition $V(G)$ into a constant number of parts each of which is closed (see Lemma~\ref{lem5.8}).
Then we try to merge some of them into a larger (still closed) part by analyzing the graph structures.
Lemma~\ref{lem5.9} allows us to iteratively merge two distinct parts into a closed one, given the existence of a transferral. Therefore, the key step is to find a transferral (see Lemma~\ref{lem5.10}), where we shall use the regularity method and Corollary~\ref{coro2.17}.
The following lemma can be used to construct a partition such that each part is closed.
\begin{lemma}[\cite{HMWY2021}, Lemma 3.10]\label{lem5.8}
For any positive constants $\gamma_{1}, \beta_{1}$, $h\in \mathbb{N}$ with $h\geq 3$ and an $h$-vertex graph $H$, there exist $\beta_{2}=\beta_{2}(\gamma_{1}, \beta_{1}, h)>0$ and $t_{2}\in \mathbb{N}$ such that the following holds for sufficiently large n. Let $G$ be an $n$-vertex graph such that every vertex in $V(G)$ is $(H, \beta_{1} n, 1)$-reachable to at least $\gamma_{1} n$ other vertices. Then there is a partition $\mathcal{P}=\{V_{1}, \dots, V_{p}\}$ of $V(G)$ with $p\leq \lceil\frac{1}{\gamma_{1}}\rceil$ such that for each $i\in [p]$, $V_{i}$ is $(H, \beta_{2} n, t_{2})$-closed and $|V_{i}|\geq \frac{\gamma_{1}}{2}n$.
\end{lemma}
\begin{lemma}[\cite{HMWY2021}, Lemma 4.4]\label{lem5.9}
Given any positive integers $h, t\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constant $\beta>0$, the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with a partition $\mathcal{P}=\{V_{1}, \dots, V_{p}\}$ of $V(G)$ such that each $V_{i}$ is $(H, \beta n, t)$-closed. For distinct $i, j\in [p]$, if there exist two $h$-vectors $\textbf{s}, \textbf{t}\in I^{\beta}(\mathcal{P})$ such that $\textbf{s} - \textbf{t}=\textbf{u}_{i} - \textbf{u}_{j}$, then $V_{i}\cup V_{j}$ is $\left(H, \frac{\beta n}{2}, 2ht\right)$-closed.
\end{lemma}
Note that to invoke Lemma \ref{lem5.9}, we need the following result which provides a sufficient condition for the existence of a transferral.
\begin{lemma}[]\label{lem5.10}
Given $p, h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constants $\mu, \delta_{1}>0$, there exist $\alpha, \beta'>0$ such that the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$, $\alpha(G)\leq \alpha n$, $\mathcal{P}=\{V_{1}, \dots, V_{p}\}$ be a partition of $V(G)$ with $|V_{i}|\geq \delta_{1} n$ for each $i\in [p]$. If $p\geq 2$, then there exist two $h$-vectors $\textbf{s}, \textbf{t}\in I^{\beta'}(\mathcal{P})$ such that $\textbf{s} - \textbf{t}=\textbf{u}_{i} - \textbf{u}_{j}$ for some distinct $i, j\in [p]$.
\end{lemma}
Now, we have collected all the tools needed for the proof of Lemma \ref{lem5.4}.
\begin{proof} [Proof of Lemma \ref{lem5.4}] Given $h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constants $\beta_{1}, \gamma_{1}, \mu>0$, we shall choose
\begin{center}
$\frac{1}{n}\ll\alpha\ll\beta, \frac{1}{t}\ll \beta_{2}, \frac{1}{t_{2}}\ll \beta_{1}, \gamma_{1}, \mu$.
\end{center}
Let $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$, $\alpha(G)\leq \alpha n$ and every vertex in $V(G)$ is $(H, \beta_{1}n, 1)$-reachable to at least $\gamma_{1}n$ other vertices. Applying Lemma \ref{lem5.8} on $G$, we obtain a partition $\mathcal{P}_{0}=\{V_{1}, \dots, V_{p}\}$ for some $p\leq \lceil\frac{1}{\gamma_{1}}\rceil$, where each $V_{i}$ is $(H, \beta_{2}n, t_{2})$-closed and $|V_{i}|\geq \frac{\gamma_{1}n}{2}$.
Let $\mathcal{P}'=\{U_{1}, \dots, U_{p'}\}$ be a vertex partition of $G$ with minimum $|\mathcal{P}'|$ such that $|U_{i}|\geq \frac{\gamma_{1}n}{2}$ and $U_{i}$ is $(H, \beta n, t)$-closed. We claim that $p'=1$. If $p'\geq 2$, then by Lemma \ref{lem5.9} and Lemma \ref{lem5.10}, there exist two distinct vertex parts $U_{i}$ and $U_{j}$ for distinct $i, j\in [p']$ such that $U_{i}\cup U_{j}$ is $(H, \beta'n, t')$-closed for some $\beta'$ and $t'$. By taking $U_{i}\cup U_{j}$ as a new part in partition and renaming all the parts if necessary, we get a partition $\mathcal{P}''$ with $|\mathcal{P}''|<|\mathcal{P}'|$, which contradicts the minimality of $|\mathcal{P}'|$. Hence, $V(G)$ is $(H, \beta n, t)$-closed.
\end{proof}
Next, we give a proof of Lemma \ref{lem5.10}. In order to prove Lemma \ref{lem5.10}, we use the regularity lemma (Lemma \ref{lem2.3}) and an embedding result (Claim \ref{claim5.13}). In particular, such an embedding result allows us to construct vertex-disjoint copies of $H$ with different index vectors, which can be used to show the existence of a transferral. This roughly reduces the problem to finding in the reduced graph a crossing $K_{f(H)+1}$-embeddable structure, which will be made precise later.
\begin{proof}[Proof of Lemma \ref{lem5.10}] Given $p, h, t\in \mathbb{N}$, an $h$-vertex graph $H$ and positive constants $\mu, \delta_{1}$, we shall choose
\begin{center}
$\frac{1}{n}\ll\alpha\ll\frac{1}{N}\ll\beta'\ll\frac{1}{k}\ll\varepsilon\ll\mu, \delta_{1}$.
\end{center}
Let $\beta=\frac{\mu}{10}$, $r:=ar(H)$, $\mathcal{T}=\{T_{1}, \dots, T_{r}\}$ be an acyclic partition of $H$, $\mathcal{P}=\{V_{1}, \dots, V_{p}\}$ be a vertex partition of $V(G)$ with $|V_{i}|\geq \delta_{1} n$ for each $i\in [p]$. Anchoring at the current vertex partition of $V(G)$, we apply Lemma \ref{lem2.3} with $\varepsilon, \beta>0$ and refine the current partition.
After refinement, we denote the $\varepsilon$-regular partition by $\mathcal{P}'=\{V_{0}, V_{1, 1}, \dots, V_{1, s_{1}}, \dots, V_{p, 1}, \dots, V_{p, s_{p}}\}$ where $V_{i, j}\subseteq V_{i}$ and $s_{i}\in \mathbb{N}$ for each $i\in [p]$, $j\in [s_{i}]$. Let $R:=R_{\beta, \varepsilon}$ be the reduced graph with $|V(R)|=k$ and $\mathcal{V}_{i}:=\{V_{i, 1}, \dots, V_{i, s_{i}}\}$ be a vertex subset of $V(R)$ for each $i\in[p]$.
For every cluster $V_{i, j}$, $D_{i, j}$ denotes the double-edge neighborhood of $V_{i, j}$.
It holds that $2|D_{i, j}|+(k-|D_{i, j}|) \geq \delta(R)$ which means that
\begin{equation}\label{eq2}
|D_{i, j}|\geq \delta(R)-k,
\end{equation}
for each $V_{i, j}\in V(R)$.
We call a subgraph $\mathcal{K}\subseteq R$ \emph{crossing} with respect to the partition $\mathcal{P}$ if $V(\mathcal{K})\cap \mathcal{V}_{i}\neq \emptyset$ and $V(\mathcal{K})\cap \mathcal{V}_{j}\neq \emptyset$ for some distinct $i, j\in [p]$. A \emph{double-edged} $\mathcal{K}$ has every two adjacent vertices connected by a double-edge. We use $K^{=}_{3}$ to denote the triangle which contains exactly one double-edge and write $f:=f(H)$ throughout this proof.
The following claim provides a sufficient condition for the existence of a transferral. Its proof is postponed to the end of this subsection.
\begin{claim}\label{claim5.13}
If there is a crossing $K_{f+1}$-embeddable structure $\mathcal{K}$ in $R$, then there exist two $h$-vectors $\textbf{s}, \textbf{t}\in I^{\beta'}(\mathcal{P})$ such that $\textbf{s} - \textbf{t}=\textbf{u}_{i} - \textbf{u}_{j}$ for distinct integers $i$ and $j$.
\end{claim}
Thus, we may assume that there is no crossing $K_{f+1}$-embeddable structure. Note that if there exists a crossing double-edge between $\mathcal{V}_{i}$ and $\mathcal{V}_{j}$ for some distinct $i, j\in [p]$, then by Lemma \ref{lem5.11} the double-edge is contained in a $K_{f+1}$-embeddable structure which is crossing, a contradiction. So we may further assume that there is no crossing double-edge in $R$. In this case, we shall find a crossing $K_{f+1}$-embeddable structure and this gives a final contradiction.
In the following, we assume $|\mathcal{V}_{i}|\leq |\mathcal{V}_{i+1}|$ for each $i\in [p-1]$ and $x:=|\mathcal{V}_{1}|$ for some integer $x\leq \frac{k}{2}$. To find a crossing $K_{f+1}$-embeddable structure without any crossing double-edge, we divide the proof into the following four cases.
Assume $f\geq 8$. By Fact \ref{fact2.5}, we have $\delta(R)\geq \left(\frac{3}{2}+\mu\right)k$. Hence for $V_{1, 1}\in \mathcal{V}_{1}$, by (\ref{eq2}) it holds that
\begin{center}
$\frac{k}{2}\geq x=|\mathcal{V}_{1}|\geq |D_{1, 1}|\geq \delta(R)-k\geq \left(\frac{1}{2}+\mu\right)k,$
\end{center}
a contradiction.
Assume $f=7, 6$. By Fact \ref{fact2.5}, we have $\delta(R)\geq 2\left(1-\frac{2}{f}+\frac{\mu}{2}\right)k$. Let $V_{1, i}V_{1, j}$ be a double-edge in $R[\mathcal{V}_{1}]$ for distinct $i, j\in [s_{1}]$, $\mathcal{V}':=(N_{R}(V_{1, i})\cap N_{R}(V_{1, j}))\backslash\mathcal{V}_{1}$ and $y:=|\mathcal{V}'|$. Since there is no crossing double-edge, it holds that
\begin{align}\label{eq3}
y & \geq 2\left[2\left(1-\frac{2}{f}+\frac{\mu}{2}\right)k-2x\right]-(k-x)\nonumber\\
& =\left(3-\frac{8}{f}+2\mu\right)k-3x\nonumber\\
& \geq \left(2-\frac{8}{f}+2\mu\right)k-x.
\end{align}
For any $V_{w, \ell}\in \mathcal{V}'$, it holds that \begin{align*}
|N_{R[\mathcal{V}']}(V_{w, \ell})|& \geq \nonumber \frac{1}{2}\left[\delta(R)-2|V(R)\backslash(\mathcal{V}_{1}\cup\mathcal{V}')|-|\mathcal{V}_{1}|\right]\\ \nonumber
& =\frac{1}{2}\left[2\left(1-\frac{2}{f}+\frac{\mu}{2}\right)k-2(k-x-y)-x\right] \\ \nonumber
& =\frac{1}{2}\left[\left(-\frac{4}{f}+\mu\right)k+x+2y\right]. \nonumber
\end{align*}
If there is a copy of $K_{f-3}$ in $R[\mathcal{V}']$, then this copy combined with $V_{1, i}V_{1, j}$ forms a crossing copy of $K_{f+1}$-embeddable structure. So by Tur\'{a}n's theorem, we must have
\begin{align*}
& \frac{1}{2}\left[\left(-\frac{4}{f}+\mu\right)k+x+2y\right] <\frac{f-5}{f-4}y; \\
& \Longleftrightarrow \frac{1}{f-4}y <\frac{1}{2}\left(\frac{4}{f}-\mu\right)k-\frac{1}{2}x;\\
& \Longleftrightarrow y<\left(2-\frac{8}{f}-\frac{f-4}{2}\mu\right)k-\frac{f-4}{2}x.
\end{align*}
However, this contradicts \eqref{eq3} and $f\geq 6$.
\begin{figure}[htbp]\label{Figure1}
\centering
\includegraphics[scale=0.8]{1.pdf}
\caption{Illustration of a graph ordering}
\label{L1}
\end{figure}
Assume $f=5$. For any cluster $V_{1, i}\in \mathcal{V}_{1}$ for $i\in [s_{1}]$, it holds that $|N_{R[\mathcal{V}_{1}]}(V_{1, i})|\geq \frac{\frac{6}{5}k-(k-x)}{2}=\frac{x}{2}+\frac{k}{10}$. Thus every double-edge $V_{1, i}V_{1, j}$ is contained in a copy of $K^{=}_{3}$ in $R[\mathcal{V}_{1}]$. Based on this, we shall first show that there is no double-edged $K_{5}$ in $R[\mathcal{V}_{1}]$. Suppose for a contradiction that there exists a double-edged $K_{5}$ in $R[\mathcal{V}_{1}]$, whose vertex set is denoted by $\{V_{1,1}, V_{1,2}, \dots, V_{1,5}\}$. Then we claim that there exists a crossing $K_{7}$-embeddable structure which consists one cluster outside $\mathcal{V}_{1}$ and three clusters in the double-edged $K_{5}$. Indeed, we calculate the edges between $\{V_{1,1}, V_{1,2}, \dots, V_{1,5}\}$ and $V(R)\backslash \mathcal{V}_{1}$. We aim to show $e(\{V_{1,1}, V_{1,2}, \dots, V_{1,5}\}, V(R)\backslash \mathcal{V}_{1})>2(k-x)$, which would imply the existence of one cluster in $V(R)\backslash \mathcal{V}_{1}$ with at least three neighbors in the double-edged $K_{5}$. Note that this gives a $K_{7}$-embeddable structure. It suffices to show
\begin{align*}
& \sum_{i=1}^{5}d(V_{1, i}) >\sum_{i=1}^{5}d_{R[\mathcal{V}_{1}]}(V_{1, i})+2(k-x);\\
&\Longleftarrow 5\times\left(\frac{6}{5}+\mu\right)k>5\times 2x+2(k-x);\\
&\Longleftrightarrow x<\frac{4+5\mu}{8}k.
\end{align*}
As $x=|\mathcal{V}_{1}|\leq \frac{k}{2}$, we are done. In the following, we may further assume that there is no double-edged $K_{5}$ in $R[\mathcal{V}_{1}]$.
Now we shall define an ordering of the graphs in $\{K^{=}_{3}, \mbox{double-edged}\:K_{3}, \mbox{double-edged}\:K_{4}, \\\mbox{double-edged}\:K_{5}\}$ as illustrated in Figure 1. Let $S$ be a maximal element in the chain which is a subgraph of $R[\mathcal{V}_{1}]$. Without loss of generality, let $V(S)=\{V_{1, 1}, \dots, V_{1, a}\}$. Then $a\in \{3, 4\}$. Observe that $\bigcap_{j\in [a]}D_{1,j}=\emptyset$. Then we have $e(V(S), V(R)\backslash \mathcal{V}_{1})\geq \sum_{i=1}^{a}d(V_{1, i})-(2a-1)(x-a)-2a^{2}=\sum_{i=1}^{a}d(V_{1, i})-(2a-1)x-a$. Since $x\leq \frac{k}{2}$, no matter $a=3$ or $4$, we always have
\begin{center}
$x <\frac{(6a-10)k-5a}{10a-15}$\text{, that is, }$\frac{6}{5}ak-(2a-1)x-a>2(k-x)$,
\end{center}
which implies that $e(V(S), V(R)\backslash \mathcal{V}_{1})\geq \sum_{i=1}^{a}d(V_{1, i})-(2a-1)x-a >2(k-x).$ This means that there is a cluster in $V(R)\setminus \mathcal{V}_{1}$ which has at least three neighbors in $V(S)$, giving a crossing $K_{6}$-embeddable structure consisting of one cluster in $V(R)\backslash \mathcal{V}_{1}$ and three clusters in $V(S)$. So we are done.
Finally we have $f=4, 3, 2$. We assume that there is no crossing $K_{3}^{=}$ as otherwise we are done. By Fact \ref{fact2.5}, we have $\delta(R)\geq (1+\mu)k$ and without loss of generality, we may assume $V_{1,1}V_{2,1}$ is a crossing single-edge. Then $N(V_{1, 1})\cap D_{2, 1}=\emptyset$ and $N(V_{2, 1})\cap D_{1, 1}=\emptyset$. We have $d(V_{1, 1})\leq k-|D_{1, 1}|-|D_{2, 1}|+2|D_{1, 1}|$ and $d(V_{2, 1})\leq k-|D_{1, 1}|-|D_{2, 1}|+2|D_{2, 1}|$ which yields $(2+2\mu)k\leq2\delta(R)\leq d(V_{1, 1})+d(V_{2, 1})\leq2k$, a contradiction.
\end{proof}
Now we prove Claim \ref{claim5.13}.
\begin{proof} [Proof of Claim \ref{claim5.13}]
Recall that $V(R)=\{V_{1, 1}, \dots, V_{1, s_{1}}, \dots, V_{p, 1}, \dots, V_{p, s_{p}}\}$ with $|V(R)|=k$ and $\mathcal{V}_{i}=\{V_{i, 1}, \dots, V_{i, s_{i}}\}$, for each $i\in[p]$.
Without loss of generality, we assume that $\mathcal{K}$ is a crossing $K_{f+1}$-embeddable structure in $R$ such that $V(\mathcal{K})=\{U_1, U_2, \dots, U_{a+b}\}$ for distinct clusters $U_1, U_2, \dots, U_{a+b}\in V(R)$ such that $U_i\in\mathcal{V}_i$ for $i\in[2]$, where $a+2b=f+1$.
If there exists a cluster in $V(\mathcal{K})$, say $U_{q}\in \mathcal{V}_{\ell}$ for some $q\in [a+b]$ and $\ell\in[p]$, such that $i_{\mathcal{K}}(U_q)=1$, then $\mathcal{K}-\{U_{q}\}=:\mathcal{K}'$ is a $K_{f}$-embeddable structure. Now we shall find two $h$-vectors $\textbf{s}, \textbf{t}\in I^{\beta'}(\mathcal{P})$ as required. Let $U'_{i}$ be a subset of $U_{i}$ for every $i\in[a+b]$, by deleting any $\beta'n$ vertices. Since $(U_{q}, U_{i})$ is an $(\varepsilon,\beta)$-regular pair for each $i\in[a+b]\setminus \{q\}$ and $\frac{1}{n}\ll\frac{1}{N}\ll \beta'\ll\frac{1}{k}\ll\beta$, we pick a vertex $v\in U'_{q}$ such that $|N(v)\cap U'_{i}|\geq (\beta-\varepsilon)m-\beta'n\ge\frac{\beta}{2} m\ge N$. Write $Y_i=N(v)\cap U'_{i}$ for each $i\in[a+b]\setminus \{q\}$. By Lemma \ref{lem2.2}, $(Y_{i}, Y_{j})$ is $\varepsilon'$-regular with $\varepsilon':=\max\left\{2\varepsilon, \frac{2\varepsilon}{\beta}\right\}=\frac{2\varepsilon}{\beta}$ for distinct $i, j\in[a+b]\setminus \{q\}$ and $d(Y_{i}, Y_{j})\geq d(U_{i}, U_{j})-\varepsilon$. Applying Corollary \ref{coro2.17} on $G[\cup_{i\neq q}Y_{i}]$, there exists a copy of $Q(a-1,b)$. Recall that $U_i\in\mathcal{V}_i$ for $i\in[2]$. Thus there exists $i_0\in[2]$ such that $\ell\neq i_0$. Let $H_{1}$ be a copy of $H$ in $Q(a-1,b)$ such that $V(H_{1})\cap U_{i_{0}}\neq\emptyset$. Then we replace any vertex in $V(H_{1})\cap U_{i_0}$ with $v$ and get another copy of $H$, say $H_2$. The two copies $H_1,H_2$ respectively give two $h$-vectors $\textbf{s}, \textbf{t}\in I^{\beta'}(\mathcal{P})$ such that $\textbf{s} - \textbf{t}=\textbf{u}_{\ell} - \textbf{u}_{i_0}$.
Now it remains to deal with the case that every $U_{i}\in V(\mathcal{K})$ satisfies $i_{\mathcal{K}}(U_i)=2$, $i\in [a+b]$. Then $a=0, f=2b-1$. Here we obtain that $H\in\widetilde{\mathcal{H}}$ and $b=ar(H)$.
Let $\mathcal{T}=\{T_{1}, \dots, T_{b}\}$ be an acyclic partition of $H$ such that $H[T_1]$ is an $s$-independent set for some $s\in\mathbb{N}$ and $|T_i|=2s,i\in[2,b]$.
Let $F_{1}=K_{1, 2s-1}$ and $F_{i}=2H[T_{i}]$ for each $i\in [2, b]$.
For each $U_{i}$, let $U'_{i}$ be obtained from $U_{i}$ by deleting any $\beta'n$ vertices. Since $\frac{1}{n}\ll\frac{1}{N}\ll \beta'\ll\frac{1}{k}$, $|U'_{i}|\geq m-\beta'n\geq \frac{m}{2}\ge N$ for each $i\in [b]$. By Lemma \ref{lem2.2}, $d(U'_{i}, U'_{j})\geq d(U_{i}, U_{j})-\varepsilon\ge\frac{1}{2}+\frac{\beta}{2}$ and $(U'_{i}, U'_{j})$ is $\varepsilon'$-regular with $\varepsilon':=\max\{2\varepsilon, \frac{\varepsilon|U_{i}|}{|U'_{i}|}\}=2\varepsilon$ for distinct $i,j\in [b]$. Applying Lemma \ref{lem2.17} on $G[U'_{1}\cup \cdots\cup U'_{b}]$, we obtain a copy of $Q(0, b, 2s, F_{1}, \dots, F_b)$, say $Q$, such that $V(Q)=X_1\cup X_2\cup\cdots\cup X_b$ and $X_i\subseteq U_i', i\in[b]$. Note that $Q[X_1]$ induces a copy of $K_{1,2s-1}$ whose center is denoted by $v^*$.
It is easy to derive that $Q$ contains a copy of $H$, say $H_{1}$, such that $V(H_{1})\cap X_{1}$ are leaves of the $K_{1,2s-1}$, $|V(H_{1})\cap U_{i}|=2|V(H_{1})\cap U_{1}|=2s$ for every $i\in [2, b]$. By removing from $H_{1}$ any vertex in $V(H_{1})\cap U_{2}$ and adding the center $v^*$, we obtain another copy of $H$, say $H_{2}$ such that $|V(H_2)\cap U_{1}|=s+1,|V(H_2)\cap U_{2}|=2s-1$ and $|V(H_2)\cap U_{i}|=2s, i\in[3,b]$. So $H_{1}$ and $H_{2}$ provide two $h$-vectors $\textbf{s}, \textbf{t}\in I^{\beta'}(\mathcal{P})$ such that $\textbf{s} - \textbf{t}=\textbf{u}_{2} - \textbf{u}_{1}$.
\end{proof}
\subsection{Proof of Lemma \ref{lem5.7}}
This subsection is devoted to the proof of Lemma \ref{lem5.7} which states that the given minimum degree and independence number suffice to guarantee that every vertex is in a copy of $H$ while excluding any vertex set $W$ of size $o(n)$. To achieve this goal, we need the following result which investigates the structure around every vertex in the original graph $G$.
\begin{lemma}[\cite{MR4193066}, Lemma 3.11]\label{lem5.12}
Given $r\in \mathbb{N}$ with $r\geq 4$ and constants $\varepsilon, \beta, \mu$ with $0<\varepsilon, \beta\leq \frac{\mu}{10}$, the following holds for sufficiently large $n$. Let $G$ be an $n$-vertex graph with $\delta(G)\geq \left(1-\frac{2}{r}+\mu\right)n$, $\mathcal{P}=\{V_{0}, \dots, V_{k}\}$ be an $\varepsilon$-regular partition for some integer $k$ and $R$ be a reduced multigraph with multiplicity $2$. Fix a vertex $v$ in $G$, and let $Q_{v}$ be the set of clusters $V_{i}\in V(R)$ such that $|N_{V_{i}}(v)|\geq \beta|V_{i}|$. Then there exists a multi-embedding of $K_{r}$ into $R$ embedding at most one vertex in $V(R)\backslash Q_{v}$.
\end{lemma}
\begin{proof} [Proof of Lemma \ref{lem5.7}] Given $h\in \mathbb{N}$ with $h\geq 3$, an $h$-vertex graph $H$ and constant $\mu$, we choose
\begin{center}
$\frac{1}{n}\ll \frac{1}{N}\ll \alpha\ll\frac{1}{k}\ll \varepsilon\ll \mu$.
\end{center}
Let $\beta=\frac{\mu}{10}$, $G$ be an $n$-vertex graph with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}$, $\alpha(G)\leq \alpha n$, $W\subseteq V(G)$ with $|W|\leq \frac{\mu}{2}n$ and $G_{1}:= G-W$. Then we have
\begin{center}
$\delta(G_{1})\geq \max\left\{\left(1-\frac{2}{f(H)}+\mu\right)n, \left(\frac{1}{2}+\mu\right)n\right\}-|W|\geq \max\left\{\left(1-\frac{2}{f(H)}+\frac{\mu}{2}\right)n, \left(\frac{1}{2}+\frac{\mu}{2}\right)n\right\}$.
\end{center}
Applying Lemma \ref{lem2.3} to $G_{1}$ with $\varepsilon, \beta>0$, we obtain an $(\varepsilon,\beta)$-regular partition $\mathcal{P}=\{V_{0}, V_{1}, \dots, V_{k}\}$ of $G_{1}$.
Let $R:=R_{\beta, \varepsilon}$ be a reduced multigraph for the partition $\mathcal{P}$ with multiplicity $2$, $|V(R)|=k$ and $m:=|V_{i}|$ for each $i\in [k]$. By Fact \ref{fact2.5}, we have $\delta(R)\geq 2\left(1-\frac{2}{f(H)}+\frac{\mu}{2}\right)k$ when $f(H)\geq 4$ and $\delta(R)\geq 2\left(\frac{1}{2}+\frac{\mu}{2}\right)k$ when $f(H)=2, 3$. Applying Lemma \ref{lem5.12} with $r=\max\{f(H), 4\}$ where $\frac{\mu}{2}$ plays the role of $\mu$, for any vertex $v\in V(G_{1})$, we obtain a $K_{f(H)}$-embeddable structure $\mathcal{K}$ such that $|V(\mathcal{K})\backslash Q_{v}|\le1$ where $Q_{v}$ is as defined in Lemma \ref{lem5.12}. Assume $V(\mathcal{K})=\{V_{1}, \dots, V_{\ell}\}$.
If $V(\mathcal{K})\backslash Q_{v}=\emptyset$, then $\mathcal{K}$ is a subgraph of $R[Q_{v}]$. Let $S_{i}:= N(v)\cap V_{i}$ for each $i\in [\ell]$. Then by the definition of $Q_{v}$, we have $|S_{i}|\geq \beta m\geq N$ for each $i\in [\ell]$ since $\frac{1}{n}\ll\frac{1}{N}\ll\frac{1}{k}\ll\beta$. By Lemma \ref{lem2.2}, $(S_{i}, S_{j})$ is $\varepsilon'$-regular with $\varepsilon':=\max\left\{2\varepsilon, \frac{\varepsilon}{\beta}\right\}=\frac{\varepsilon}{\beta}$ and $d(S_{i}, S_{j})\geq d(V_{i}, V_{j})-\varepsilon$ for distinct $i, j\in [\ell]$. Corollary \ref{coro2.17} applied on $G[S_{1}\cup \cdots\cup S_{\ell}]$ with $a=2\ell-f(H)$ and $b=f(H)-\ell$ gives a copy $Q$ of $Q(a, b)$. Note that $V(Q)\subseteq N(v)$. Hence, $v$ is in a copy of $H$.
Now assume $V(\mathcal{K})\backslash Q_{v}=\{V_{\ell}\}$, and thus $i_{\mathcal{K}}(V_{\ell})=1$. Similarly, we can choose subsets $S_{i}\subseteq V_{i}\cap N(v)$ as above for each $i\in [\ell-1]$. Applying Corollary \ref{coro2.17} on $G[S_{1}\cup S_{2}\cup \cdots\cup S_{\ell-1}\cup V_{\ell}]$ with $a=2\ell-f(H)$, $b=f(H)-\ell$ and the fact that $i_{\mathcal{K}}(V_{\ell})=1$, we obtain a copy $Q$ of $Q(a, b)$ such that $Q[V_{\ell}]$ is an independent set. Replacing any vertex in the independent set with $v$, we conclude that $v$ is in a copy of $H$.
\end{proof}
\bibliographystyle{abbrv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,673
|
Q: Does initializing a variable with true or false establish the direction of the content in Javascript? <script>
var switchDirection = false;
function doAnimation() {
var divAdvert = document.getElementById("divAdvert");
var currentLeft = divAdvert.offsetLeft;
var newLocation;
if (!switchDirection) {
newLocation = currentLeft + 2;
if (currentLeft >= 400) {
switchDirection = true;
}
} else {
newLocation = currentLeft - 2;
if (currentLeft <= 0) {
switchDirection = false;
}
}
divAdvert.style.left = newLocation + "px";
}
setInterval(doAnimation, 10);
</script>
I want to know why a new variable with the value of false has been declared in the beginning of the function.
A: No, nothing like this. May be developer using this variable later in page as this is global variable. He is calling function from setInterval hence its going to called repeatedly. So, he is keeping that value in some global variable as it will be lost if kept inside function.
Name is like that just to know what was the intention of variable.Its simple variable.
This line is changing the location :-
divAdvert.style.left = newLocation + "px";
When switchDirection is false , auther is adding value:-
if (!switchDirection) {
newLocation = currentLeft + 2;
Once it is more that 400 , author changing the variable value
if (currentLeft >= 400) {
switchDirection = true;
}
Then he is moving it in other direction and again vice versa,
newLocation = currentLeft - 2;
if (currentLeft <= 0) {
switchDirection = false;
}
Instead of giving value true false , s/he could have used values left and right or ltr and rtl but as he has only 2 values, he used true and false.
Its simple variable , you can give any name like dir or xyz it will work in same way.
And if you change all the false with true and true with false code will work same.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,523
|
Americans paid an average price of $33,560, or 2.6% more than a year ago, for new cars and trucks in April, according to data from auto researcher Kelley Blue Book. The rise was fueled by automakers' new models, and a return to big, expensive SUVs and other trucks.
The estimated average transaction price of a new car or truck sold in the U.S. in April was $33,560 — 2.6% higher than in the month a year ago, according to data from auto researcher Kelley Blue Book.
One factor fueling the rise was a flood of redesigned or refreshed models from automakers — such as the aluminum-body Ford F-150 pickup, Kia Sedona minivan, Chrysler 300 big sedan and other's fresh in showrooms. Those tend to sell at full price, and early buyers of a new model also tend to be people who check all the options boxes.
The transaction price boost also came partly from the shift in sales mix in the month in favor of pricier trucks and SUVs, including even hot sales for expensive full-size SUVs, such as General Motors' full-size Chevrolet Suburban, GMC Yukon and Cadillac Escalade.
"Prices were up across most of the industry, but we are seeing some of the greatest increases in the truck segments," said Alec Gutierrez, senior analyst for Kelley Blue Book. "Full-size trucks were particularly strong, up 4.5%, while mid-size trucks were up 3.5%. Incentive spending on trucks also has been lighter this year, indicating a great market for these units right now."
In addition to market factors, prices are moving generally up to cover the additional costs of more fuel-efficient drivetrains required by tightening federal fuel economy standards, and the costs of sophisticated safety gear, such as automatic braking to prevent a collision, both required for top safety ratings and becoming expected by many buyers.
Note: The KBB estimate does not include the variety of consumer incentives that may be applied, some of which may be regional and not apply for all buyers.
• Volkswagen Group (Audi, Volkswagen, Porsche), $39,203, down 0.9%.
• General Motors (Buick, Cadillac, Chevrolet, GMC), $38,632, up 2.9%.
• Fiat Chrysler Automobiles, (Alfa Romeo, Chrysler, Dodge, Fiat, Jeep, Ram), $33,901, up 3.3%.
• Toyota Motor (Lexus, Scion, Toyota), $30,463, up 1.1%.
• Nissan North America (Nissan, Infiniti), $27,767, up 2.4%.
• American Honda (Acura, Honda), $27,564, up 1.9%.
•Hyundai Motor Group (Hyundai, Kia), $24,980, up 4.7%. "The redesigned Sonata, up 7.9%, helped the Hyundai brand to a 4.2% gain, while the Kia Sedona, up 16.5%, lifted the Kia brand 5.2% in April," said Tim Fleming, analyst for Kelley Blue Book.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,128
|
<?php
namespace FlamingSms\Controller;
use Zend\Mvc\Controller\AbstractActionController;
use FlamingSms\Service\SmsService;
/**
* NotificationController
*
* @author Flemming Andersen <flemming@flamingcode.com>
* @copyright (c) 2013, Flaming Code
* @link http://github.com/FlamingCode/FlamingSms for the canonical source repository
* @license http://opensource.org/licenses/MIT MIT
*/
class NotificationController extends AbstractActionController
{
/**
*
* @var SmsService
*/
protected $smsService;
public function coolsmsAction()
{
$request = $this->getRequest();
$response = $this->getResponse();
$response->setContent('');
$headers = $response->getHeaders();
$headers->addHeaderLine('Content-Type', 'text/plain');
$data = $request->getQuery();
$status = isset($data['status']) ? $data['status'] : null;
$smsId = isset($data['msgid']) ? $data['msgid'] : null;
$statuscode = isset($data['statuscode']) ? $data['statuscode'] : null;
if(null != $status && null != $smsId) {
if (null != $statuscode)
$status .= '-' . $statuscode;
if ($sms = $this->getSmsService()->findOutgoingSmsByGatewayId($smsId)) {
$this->getSmsService()->handleStatusNotification($sms, $status);
$response->setStatusCode(200);
$response->setContent('OK');
} else {
$response->setStatusCode(404);
}
} else {
$response->setStatusCode(400);
}
return $response;
}
/**
*
* @return SmsService
*/
public function getSmsService()
{
if (!$this->smsService) {
$this->smsService = $this->getServiceLocator()->get('FlamingSms\Service\SmsService');
}
return $this->smsService;
}
/**
*
* @param SmsService $smsService
* @return IncommingController
*/
public function setSmsService(SmsService $smsService)
{
$this->smsService = $smsService;
return $this;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,633
|
\section{Introduction}\label{intro}
Notation follows \cite{bang2009} so we only repeat a few definitions here (see also Section
\ref{sec:prelim}).
Let $D=(V,A)$ be a digraph. An {\bf out-tree} ({\bf in-tree}) is an oriented tree in which every vertex except one, called the {\bf root}, has in-degree (out-degree) one. An {\bf out-branching} ({\bf in-branching}) of $D$ is a spanning out-tree (in-tree) in $D$. For a subdigraph $H$ of $D$ and a vertex $s$ of $H$ we denote by $B_{s,H}^+$, (resp., $B_{s,H}^-$) an arbitrary out-branching (resp., in-branching) rooted at $s$ in $H$. To simplify the notation, we set $B_s^+=B_{s,D}^+$ and $B_s^-=B_{s,D}^-$.
A digraph $D$ is {\bf strong} if there exists a path from $x$ to $y$ in $D$ for every ordered pair of distinct vertices $x$, $y$ of $D$ and $D$ is {\bf $k$-arc-strong} if $D\setminus{}A'$ is strong for every subset $A' \subseteq A$ of size at most $k - 1$. For a subset $X$ of $V$, we denote by $D\left\langle X \right\rangle$ the subdigraph of $D$ induced by $X$.
The following well-known theorem, due to Edmonds, provides a characterization for the existence of $k$ arc-disjoint out-branchings rooted at the same vertex.
\begin{thm}\label{edmonds1973} {\bf(Edmonds' Branching Theorem)}
A directed multigraph $D = (V,A)$ with a special vertex $s$ has $k$ arc-disjoint out-branchings rooted at $s$ if and only if
\begin{equation}
\label{kbranchcond}
d^-(X) \geq k,\;\; \forall \;\; \emptyset\neq X \subseteq V - s.
\end{equation}
\end{thm}
Note that, by Menger's Theorem, (\ref{kbranchcond}) is equivalent to the existence of $k$ arc-disjoint $(s,v)$-paths for every $v\in V-s$. Hence (\ref{kbranchcond}) can be checked in polynomial time via maximum flow calculations, see e.g.,
\cite[Section 5.4]{bang2009}.
Lov\'asz \cite{lovaszJCT21} gave a constructive proof of Theorem \ref{edmonds1973} which implies the existence of a polynomial algorithm for constructing a set of $k$ arc-disjoint branchings from a given root when (\ref{kbranchcond}) is satisfied.
A natural related problem is to ask for a characterization of digraphs having an out-branching and an in-branching which are arc-disjoint. Such pair will be called \textbf{a good pair} in this paper and more precisely we call it a \textbf{good $(u,v)$-pair} if the roots $u$ and $v$ are specified. Thomassen showed (see \cite{bangJCT51} and \cite{bangJGT100}) that for general digraphs it is NP-complete to decide if a given digraph has a good pair. This makes it interesting to study classes of digraphs for which we can find a good pair or decide that none exists in polynomial time.
For acyclic digraphs there can only be one choice for the vertices $u,v$ as $u$ must be able to reach all other vertices by a directed path and $v$ must be reachable by all other vertices by a directed paths. A characterization of acyclic digraphs with a good pair and a polynomial algorithm for finding a good pair when it exists was given in \cite{bangJGT42}. A polynomial algorithm was also given in
\cite{bercziIPL109}. In \cite{bangJCT51}
the first author gave a complete characterization of tournaments with no good $(u,v)$-pair \JBJ{and gave a polynomial algorithm for either producing a good $(u,v)$-pair for a given tournament $T$ and two vertices $u,v$ of $T$ or providing a certificate for the non-existence of such a pair in $T$. } Bang-Jensen and Huang characterized quasi-transitive digraphs with a good $(u,u)$-pair \cite{bangJGT20b}. A digraph is {\bf quasi-transitive} if the presence of arcs $uv$ and $vw$ implies an arc between $u$ and $w$. It is easy to see that every semicomplete digraph is quasi-transitive. Gutin and Sun \cite{gutinDM343} generalized this result to digraphs of the form $D=T[H_1,H_2,\ldots{},H_{|V(T)|}]$, where $T$ is a semicomplete digraph. Such a digraph is called a {\bf composition of $T$} and the precise definition is not important here (see e.g. \cite[Page 9]{bang2009}).\\
\JBJ{The following conjecture, due to Thomassen, is wide open and it is not even known whether already $K=3$ suffices for all digraphs.}
\begin{conj}\cite{thomassen1989}\label{conj:CT}
There exists an integer $K$ such that every $K$-arc-strong digraph $D$ has a good $(u,v)$-pair for every choice of vertices $u,v$ of $D$.
\end{conj}
Bang-Jensen, Bessy, Havet and Yeo \cite{bangJGT100} showed that every digraph of independence number at most 2 and arc-connectivity at least 2 has a good $(u,v)$-pair \JBJ{for at least one choice of vertices $u,v$} and they showed that the same condition is not sufficient to guarantee a good $(u,v)$-pair for every choice of $u$ and $v$. Hence $K$ in Conjecture \ref{conj:CT} must be at least 3. To the best of our knowledge it is open whether $K=3$ would suffice for all digraphs.
The following conjecture due to Bang-Jensen and Yeo would imply Conjecture \ref{conj:CT}.
\begin{conj}\cite{bangC24}
\label{conj:BJAY}
There exists an integer $K$ such that every $K$-arc-strong digraph $D=(V,A)$ has an arc-partition $A=A_1\cup A_2$ such that each of the subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$ are spanning and strong.
\end{conj}
The next result implies that the Conjecture \ref{conj:BJAY} holds with $K=3$ for the case of semicomplete digraphs.
\begin{thm}\cite{bangC24}
\label{thm:SD2arcstrong}
Let $D=(V,A)$ be a 2-arc-strong semicomplete digraph. Then $D$ has an arc-partition $A=A_1\cup A_2$ such that each of the subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$ are spanning and strong except if $D$ is isomorphic to the digraph $S_4$ on vertices $\{v_1,v_2,v_3,v_4\}$ and
arc set \\
$\{v_1v_2,v_2v_3,v_3v_4,v_4v_1,v_2v_4,v_4v_2,v_1v_3,v_3v_1\}$.
\end{thm}
It is not difficult to check that $S_4$ has a good $(u,v)$-pair for all possible choices of $u,v\in V(S_4)$. Hence every 2-arc-strong semicomplete digraph has a good pair for every possible choice of $u,v$. \JBJ{So Conjecture \ref{conj:CT} holds for semicomplete digraphs with $K=2$.} In \cite{bangJGT95} Theorem \ref{thm:SD2arcstrong} was generalized to semicomplete compositions, that is, digraphs of the form $S[H_1,\ldots{},H_{|V(S)|}]$. From that result a complete characterization for the existence of good $(u,v)$-pairs in 2-arc-strong semicomplete compositions can be obtained.
Bang-Jensen and Huang considered the case that $u=v$ for strong semicomplete digraphs by proving Theorem \ref{sABC}.
\begin{thm}\cite{bangJGT20b} \label{sABC}
Let $D$ be a strong semicomplete digraph and let $u\in V(D)$ be arbitrary vertex. Suppose that $D$ does not contain a good pair with the same root $u$. Then the following holds, where $A, B, C$ form a partition of
$V(D)-u$ such that $N^+_D(u) = A\cup C$ and $N^-_D(u)= B\cup C$.
There is precisely one arc $e$ leaving the terminal component of $D\left\langle A \right\rangle$ and precisely one arc $e^{\prime}$ entering the initial component of $D\left\langle B\right\rangle$ and $e=e^{\prime}$.
\end{thm}
\JBJ{For two vertices $x,y$ of a digraph $D$ we} use $P_{x,y}$ to denote a path from $x$ to $y$. Such a path is also called an $(x,y)$-path.
Bang-Jensen's characterization of tournaments with good $(u,v)$-pairs in \cite{bangJCT51} is quite complicated and does not extend to semicomplete digraphs \JBJ{so we will not describe it here}.
In this paper, we prove the following a surprisingly simple characterization for the existence of a good $(u,v)$-pair in semicomplete digraphs.
\begin{figure}[H]
\centering
\subfigure{\begin{minipage}[t]{0.15\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\filldraw[black](0,9) circle (3pt)node[label=left:$u$](u){};
\filldraw[black](0,8) circle (3pt)node[label=left:$v$](v){};
\path[draw, line width=0.8pt] (u) edge (v);
\end{tikzpicture}\caption*{(a)}\end{minipage}}
\subfigure{\begin{minipage}[t]{0.15\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\filldraw[black](0,10) circle (3pt)node[label=left:$u$](u){};
\filldraw[black](0,9) circle (3pt)node[](w){};
\filldraw[black](0,8) circle (3pt)node[label=left:$v$](v){};
\path[draw, line width=0.8pt] (u) edge[bend left=30] (v);
\path[draw, line width=0.8pt] (u) edge (w) edge (v);
\end{tikzpicture}\caption*{(b)}\end{minipage}}
\subfigure{\begin{minipage}[t]{0.15\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\filldraw[black](0,10) circle (3pt)node[label=left:$u$](u){};
\filldraw[black](0,9) circle (3pt)node[](w){};
\filldraw[black](0,8) circle (3pt)node[label=left:$v$](v){};
\path[draw, line width=0.8pt] (u) edge[bend left=30] (v);
\path[draw, line width=0.8pt] (v) edge[bend left=30] (u);
\path[draw, line width=0.8pt] (u) edge (w) edge (v);
\end{tikzpicture}\caption*{(c)}\end{minipage}}
\subfigure{\begin{minipage}[t]{0.15\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\filldraw[black](0,10) circle (3pt)node[label=left:$u$](a){};
\filldraw[black](0,9) circle (3pt)node[](b){};
\filldraw[black](0,8) circle (3pt)node[](c){};
\filldraw[black](0,7) circle (3pt)node[label=left:$v$](d){};
\path[draw, line width=0.8pt] (a) edge[bend left=30] (c);
\path[draw, line width=0.8pt] (b) edge[bend left=30] (d);
\path[draw, line width=0.8pt] (d) edge[bend right=50] (a);
\path[draw, line width=0.8pt] (a) edge (b) edge (c) edge (d);
\end{tikzpicture}\caption*{(d)}\end{minipage}}
\subfigure{\begin{minipage}[t]{0.15\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\filldraw[black](0,10) circle (3pt)node[label=left:$u$](a){};
\filldraw[black](0,9) circle (3pt)node[](b){};
\filldraw[black](0,8) circle (3pt)node[](c){};
\filldraw[black](0,7) circle (3pt)node[label=left:$v$](d){};
\path[draw, line width=0.8pt] (a) edge[bend left=30] (c);
\path[draw, line width=0.8pt] (b) edge[bend left=30] (d);
\path[draw, line width=0.8pt] (d) edge[bend left=30] (b);
\path[draw, line width=0.8pt] (d) edge[bend right=50] (a);
\path[draw, line width=0.8pt] (a) edge (b) edge (c) edge (d);
\end{tikzpicture}\caption*{(e)}\end{minipage}}
\subfigure{\begin{minipage}[t]{0.15\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\filldraw[black](0,10) circle (3pt)node[label=left:$u$](a){};
\filldraw[black](0,9) circle (3pt)node[](b){};
\filldraw[black](0,8) circle (3pt)node[](c){};
\filldraw[black](0,7) circle (3pt)node[label=left:$v$](d){};
\path[draw, line width=0.8pt] (a) edge[bend left=30] (c);
\path[draw, line width=0.8pt] (b) edge[bend left=30] (d);
\path[draw, line width=0.8pt] (c) edge[bend left=30] (a);
\path[draw, line width=0.8pt] (d) edge[bend right=50] (a);
\path[draw, line width=0.8pt] (a) edge (b) edge (c) edge (d);
\end{tikzpicture}\caption*{(f)}\end{minipage}}
\caption{Semicomplete digraphs that have no good $(u,v)$-pair. The digraph in (e) is isomorphic to the digraph in (f).}
\label{fig4}
\end{figure}
\iffalse
The characterizations in Theorems \ref{Tuv} and \ref{SDgoodpair} are both quite complicated and not very intuitive. In fact Theorem \ref{SDgoodpair} implies the following much more intuitive characterization. The only if direction follows from the definition of in- and out-branchings. We will show that other direction follows from Theorem \ref{SDgoodpair} and results on arc-disjoint paths with prescribed end-vertices in semicomplete digraphs from \cite{bangJCT51}. \fi
\begin{thm}
\label{thm:SDbranchchar}
Let $D=(V,A)$ be a semicomplete digraph with $u,v\in V$ (possibly $u=v$). Then $D$ has a good $(u,v)$-pair if and only if it satisfies (i) and (ii) below.
\begin{itemize}
\item[(i)] For every choice of $z,w\in V$ there are arc-disjoint paths $P_{u,z},P_{w,v}$ in $D$
\item[(ii)] $D$ is not one of the digraphs in Figure \ref{fig4}(b)-(f).
\end{itemize}
\end{thm}
It is easy to see that (i) must hold if $D$ has a good $(u,v)$-pair
and when $D$ has at least 5 vertices the theorem says that (i) is also sufficient.
{\bf The rest of the paper is organized as follows:} We start out with Section \ref{sec:prelim} which contains some extra definitions and results that will be used in the paper. In Section \ref{sec:paths} we unify results from \cite{bangJCT51} on arc-disjoint paths in semicomplete digraphs in order to use these when we assume condition (i) of Theorem \ref{thm:SDbranchchar} holds. In Section \ref{sec:extend} we prove some important lemmas on how to extend special arc-disjoint in- and out-trees to larger ones. Then in Section \ref{sec:B+P} we study the problem of finding an out-branching rooted at a specific vertex which is arc-disjoint from a path with prescribed end vertices. Finally we prove Theorem \ref{thm:SDbranchchar} in Section \ref{sec:proof}. We also give an alternative, semingly more involved characterization of semicomplete digraphs without a good $(u,v)$-pair for specified vertices $u,v$. This characterization is used heavily in \cite{bangQTDbranchpaper}.
\section{Further terminology and Preliminaries}\label{sec:prelim}
If a digraph is not strong, then we can label its strong components $D_1,\ldots{},D_p$ such that there is no arc from $D_j$ to $D_i$ when $j>i$. We call such an ordering an \textbf{acyclic ordering} of the strong components of $D$. For a non-strong semicomplete digraph $D$ it is easy to see that the ordering $D_1,\ldots{},D_p$ is unique and we call $D_1$ (resp., $D_p$) the {\bf initial (resp., terminal)} strong component of $D$.
The set of vertices of a digraph $D=(V,A)$ which can reach (resp., can be reached from) every other vertex in $V$ by a directed path is denoted by $Out(D)$ (resp., $In(D)$).
Note that a vertex $v$ belongs to $Out(D)$ (resp., $In(D)$) if and only if it is the root of some out-branching (resp., in-branching) of $D$. Hence $Out(D)=In(D)$ if and only if $D$ is strong.
\begin{lem}\label{outinstrong}\cite{bangJGT100}
Let $D$ be a digraph. Then the induced subdigraphs $D\left\langle Out(D)\right\rangle$ and
$D\left\langle In(D)\right\rangle$ are strong.
\end{lem}
We will use the following classical result by Camion. It was originally proved only for tournaments but almost the same proof works for semicomplete digraphs (one can also use the easy fact that every strong semicomplete digraph $D=(V,A)$ on at least 3 vertices contains a spanning strong subtournament $T=(V,A')$ where $A'\subseteq A$).
\begin{thm}\cite{camionCRASP249}\label{thm:camion}
Every strong semicomplete digraph of order at least 2 has a hamiltonian cycle.
\end{thm}
The following extension of Redei's Theorem \cite{redeiALS7} is easy to prove from Theorem \ref{thm:camion} and the fact that if $D$ is not strong and $D_1,\ldots{},D_p$, $p\geq 2$ is the unique ordering of the strong components of $D$ then every vertex of $V(D_i)$ dominates every vertex of $V(D_j)$ when $1\leq i<j\leq p$.
\begin{lem}
Every strong semicomplete digraph $D$ has a hamiltonian path starting at any prescribed vertex $x$.
\JBJ{If $D$ is a non-strong semicomplete digraph and $D_1,\ldots{},D_p$, $p\geq 2$ is the unique ordering of its strong components}, then $D$ has an $(x,y)$-hamiltonian path for every choice of vertices $x\in V(D_1)$ and $y\in V(D_p)$.
\end{lem}
\begin{thm}\cite{bangJGT100}
Let $D$ be a semicomplete digraph of order at least 4 and let $v$ be arbitrary vertex of $In(D)$. There is a pair of arc-disjoint out- and in-branchings in $D$ such that the in-branching is rooted at $v$ if and only if $D$ is not a digraph such that both $v$ and its in-neighbor has in-degree one.
\end{thm}
\begin{lem}\cite{bangJGT100}\label{nonstrong-OI}
Every non-strong semicomplete digraph of order at least 4 has a good $(u, v)$-pair for every choice of $u \in Out(D)$ and $v \in In(D)$.
\end{lem}
\iffalse
The case that $D$ has arc-connectivity 2 or it is non-strong also has been verified.
\begin{thm}\cite{bangJCT102}\label{bang2012}
Every 2-arc-strong semicomplete digraph $D$ contains a good $(u,v)$-pair for every choice of $u, v \in V(D)$.
\end{thm}\fi
\begin{lem}\label{OutbranPath}
Let $D$ be a digraph and let $u,v$ be two vertices of $D$ such that \JBJ{$u\in Out(D)$.} Suppose that $D$ has no out-branching $B_u^+$ which is arc-disjoint from some $(u,v)$-path. Then there exists a partition $V_1,V_2$ of $V(D)$ such that $v\in V_1,u\in V_2$ and $d_D^+(V_2)=1$.
\end{lem}
\begin{proof}
By the assumption, there are no two arc-disjoint out-branchings rooted at $u$ in $D$. It follows by Edmonds' branching theorem (Theorem \ref{edmonds1973}) that there exists a partition $U_1,U_2$ of $V(D)$ such that $u\in U_2$ and $d_D^-(U_1)=d_D^+(U_2)\leq 1$. Since $D$ contains an out-branching with root $u$ and $u\in U_2$, we have $d_D^+(U_2)=1$, moreover, let $xy$ be the only arc leaving $U_2$, then $y$ dominates all vertices of $U_2-x$ and it can reach all vertices of $U_1$
If $v\in U_1$, then $U_1,U_2$ is the desired partition. So we may assume that $v$ belongs to $U_2$. By the assumption of the lemma, there can be no pair of arc-disjoint $(u,x)$- and $(u,v)$-paths in $D\left\langle U_2\right\rangle$. Otherwise, using that $y$ dominates all of $U_2-x$, \JBJ{it is easily seen that $D$ has an out-branching $B_u^+$ which is arc-disjoint from some $(u,v)$-path}, contradiction. Thus by Menger's theorem, there exists a proper subset $U$ of $U_2$ such that $u\in U,\{v,x\} \subseteq U_2- U$ and $d_D^+(U)=1$. Then $V_2=U$ and $V_1=V(D)-U$ is the desired partition.
\end{proof}
\section{Arc-disjoint paths in semicomplete digraphs}\label{sec:paths}
In this section we unify and slightly generalize results on arc-disjoint paths from \cite{bangJCT51} in order to use these in the next sections.
\begin{definition}\label{def1}
Let $D=(V,A)$ be a semicomplete digraph and let $u,w,v$ be three vertices of $D$. The 4-tuple $(D,u,w,v)$ is said to be of
{\bf type $A$}, if there exists a partition $V_1, V_2$ of $V$ such that $v\in V_1$, $u,w\in V_2$ and there is exactly one arc from $V_2$ to $V_1$. (Note that in this type $D$ may be non-strong.)
{\bf type $B$}, if there exists a partition $V_1, V_2, V_3$ of $V$ such that $u,v\in V_2$, $w\in V_3$ and all arcs between $V_i$ and $V_j$ with $i<j$ \JBJ{go} from $V_i$ to $V_j$ \JBJ{except for precisely one arc} which goes from the terminal component of $D\left\langle V_3\right\rangle$ to the initial component of $D\left\langle V_1\right\rangle$.
{\bf type $2\alpha+2$}, for some $\alpha\geq 1$ if there exists a partition $V_1,\ldots,V_{2\alpha+2}$ of $V$ such that $v\in V_2, w\in V_{2\alpha+1}, u\in V_{2\alpha+2}$ and all arcs between $V_i$ and $V_j$ with $i<j$ \JBJ{go} from $V_i$ to $V_j$ with the following exceptions. There exists precisely one arc from $V_{i+2}$ to $V_{i}$ for all $i\in[2\alpha]$ and it goes from the terminal component of $D\left\langle V_{i+2}\right\rangle$ to the initial component of $D\left\langle V_i\right\rangle$.
{\bf type $2\alpha+3$}, for some $\alpha\geq 1$ if there exists a partition $V_1,\ldots,V_{2\alpha+3}$ of $V$ such that $v\in V_2, u\in V_{2\alpha+2}, w\in V_{2\alpha+3}$ and all arcs between $V_i$ and $V_j$ with $i<j$ \JBJ{go} from $V_i$ to $V_j$ with the following exceptions. There exists precisely one arc from $V_{i+2}$ to $V_{i}$ for all $i\in[2\alpha+1]$ and it goes from the terminal component of $D\left\langle V_{i+2}\right\rangle$ to the initial component of $D\left\langle V_i\right\rangle$.
\end{definition}
\begin{figure}[H]
\centering
\subfigure{
\begin{minipage}[t]{0.2\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\draw[line width=1pt] (0,9) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,7.3) ellipse [x radius=20pt, y radius=10pt];
\draw[-stealth,line width=1.5pt] (-1.1,8.5) -- (-1.1,7.5);
\filldraw[white](0.5,7.3) circle (3pt)node[](a){}; \filldraw[white](0.5,9) circle (3pt)node[](b){};
\path[draw, line width=1pt] (a) edge[bend right=40] (b);
\coordinate [label=center:$u\;w$] () at (0,7.3);
\coordinate [label=center:$v$] () at (0,9);
\end{tikzpicture}\caption*{Type $A$}
\end{minipage}}
\subfigure{
\begin{minipage}[t]{0.2\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\draw[line width=1pt] (0,10) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,8.6) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,7.3) ellipse [x radius=20pt, y radius=10pt];
\draw[-stealth,line width=1.5pt] (-1.1,9.5) -- (-1.1,7.5);
\filldraw[white](0.5,7.3) circle (3pt)node[](a){}; \filldraw[white](0.5,10) circle (3pt)node[](b){};
\path[draw, line width=1pt] (a) edge[bend right=40] (b);
\coordinate [label=center:$u\;v$] () at (0,8.6);
\coordinate [label=center:$w$] () at (0,7.3);
\end{tikzpicture}\caption*{Type $B$}
\end{minipage}}
\subfigure{\begin{minipage}[t]{0.2\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\draw[line width=1pt] (0,10) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,9) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,8) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,7) ellipse [x radius=20pt, y radius=10pt];
\draw[-stealth,line width=1.5pt] (-1.1,9.5) -- (-1.1,7.5);
\filldraw[white](0.5,7.3) circle (0.1pt)node[](b){};
\filldraw[white](0.5,8.3) circle (0.1pt)node[](c){};
\filldraw[white](0.5,9.3) circle (0.1pt)node[](d){};
\filldraw[white](0.5,10.3) circle (0.1pt)node[](e){};
\filldraw[white](0.5,6.7) circle (0.1pt)node[](b1){};
\filldraw[white](0.5,7.7) circle (0.1pt)node[](c1){};
\filldraw[white](0.5,8.7) circle (0.1pt)node[](d1){};
\path[draw, line width=0.8pt] (b1) edge[bend right=60] (d);
\path[draw, line width=0.8pt] (c1) edge[bend right=60] (e);
\coordinate [label=center:$u$] () at (0,7);
\coordinate [label=center:$v$] () at (0,9);
\coordinate [label=center:$w$] () at (0,8);
\end{tikzpicture}\caption*{Type $2\alpha+2$}
\end{minipage}}
\subfigure{\begin{minipage}[t]{0.2\linewidth}
\centering\begin{tikzpicture}[scale=0.8]
\draw[line width=1pt] (0,10) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,9) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,8) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,7) ellipse [x radius=20pt, y radius=10pt];
\draw[line width=1pt] (0,6) ellipse [x radius=20pt, y radius=10pt];
\draw[-stealth,line width=1.5pt] (-1.1,9.5) -- (-1.1,6.5);
\filldraw[white](0.5,7.3) circle (0.1pt)node[](b){};
\filldraw[white](0.5,8.3) circle (0.1pt)node[](c){};
\filldraw[white](0.5,9.3) circle (0.1pt)node[](d){};
\filldraw[white](0.5,10.3) circle (0.1pt)node[](e){};
\filldraw[white](0.5,5.7) circle (0.1pt)node[](a1){};
\filldraw[white](0.5,6.7) circle (0.1pt)node[](b1){};
\filldraw[white](0.5,7.7) circle (0.1pt)node[](c1){};
\filldraw[white](0.5,8.7) circle (0.1pt)node[](d1){};
\path[draw, line width=0.8pt] (a1) edge[bend right=60] (c);
\path[draw, line width=0.8pt] (b1) edge[bend right=60] (d);
\path[draw, line width=0.8pt] (c1) edge[bend right=60] (e);
\coordinate [label=center:$u$] () at (0,7);
\coordinate [label=center:$v$] () at (0,9);
\coordinate [label=center:$w$] () at (0,6);
\end{tikzpicture}\caption*{Type $2\alpha+3$}
\end{minipage}}
\caption{\JBJ{Illustration of Definition \ref{def1}}. The vertex sets \JBJ{$V_1,V_2\ldots{}$} are labeled from top to bottom. The bold arcs indicate that all arcs not shown going up in the figure are present in the shown direction. The third and fourth digraphs are of type $2\alpha+2$ and $2\alpha+3$ with $\alpha=1$, respectively.}
\label{fig1}
\end{figure}
\begin{lem}\label{type--nopaths}
Let $D$ be a semicomplete digraph let $u,w,v$ be three vertices of $D$. Suppose that $(D,u,w,v)$ is one of the types in Definition \ref{def1}, then for any vertex $z\in V_1$, there is no pair of arc-disjoint $(u,z)$- and $(w,v)$-paths in $D$.
\end{lem}
\begin{proof}
It is not difficult to check that if $(D,u,w,v)$ is of the type $A$ or $B$, then $D$ cannot contain arc-disjoint $(u,z)$- and $(w,v)$-paths as both of the paths must use the arc entering $V_1$.
For the case that $(D,u,w,v)$ is of type $2\alpha+2$ for some $\alpha\geq 1$, we use $x_{i+2}y_i$ to denote the arc from the terminal component of $D\left\langle V_{i+2}\right\rangle$ to the initial component of $D\left\langle V_i\right\rangle$. Let $P$ be an arbitrary $(u,z)$-path. Note that the path $P$ must use the arc $x_{2\alpha+2}y_{2\alpha}$ and at least one of the arcs of kind $x_{2k+1}y_{2k-1},~k\in[\alpha]$. Let $x_{2j+1}y_{2j-1},~j\in[\alpha]$ be the first arc of the kind $x_{2k+1}y_{2k-1}$ as we go along $P$ from $u$. In $D-A(P)$ there is no path from $w$ to $\bigcup_{i<2j+1}V_i$, because there are only two arcs, i.e., $x_{2j+2}y_{2j}$ and $x_{2j+1}y_{2j-1}$, entering $\bigcup_{i<2j+1}V_i$ and these two arcs are in $A(P)$. Note that $v\in V_2 \subseteq \bigcup_{i<2j+1}V_i$ as $j\in[\alpha]$, so there is no $(w,v)$-path in $D-A(P)$ and then $D$ has no arc-disjoint $(u,z)$- and $(w,v)$-paths.
Similarly, it is not difficult to prove that if $(D,u,w,v)$ is of type $2\alpha+3$ for some $\alpha\geq 1$, there cannot exist arc-disjoint $(u,z)$- and $(w,v)$-paths in $D$.
\end{proof}
The next two results from \cite{bangJCT51} were only stated and proved for tournaments but it is easy to check that the proofs are also valid for semicomplete digraphs.
\begin{thm}\label{jbjuzvw}\cite{bangJCT51}
Let $D$ be a semicomplete digraph and let $x_1,y_1,x_2,y_2$ be distinct vertices such that $D$ contains an $(x_i,y_i)$-path for all $i\in[2]$. Then $D$ has a pair of arc-disjoint $(x_1,y_1)$- and $(x_2,y_2)$-paths unless $x_1,y_1,x_2,y_2$ all belong to the same strong component $D_j$ of $D$ and for some $i\in[2]$, $(D_j,x_i,x_{3-i},y_{3-i})$ is one of the types in Definition \ref{def1} and the vertex $y_i$ belongs to $V_1$.
\end{thm}
\begin{thm}\label{jbjuvw}\cite{bangJCT51}
Let $D$ be a semicomplete digraph and $x,y,z\in V(D)$ three distinct vertices such that there exist an $(x,y)$- and a $(y,z)$-path in $D$. There exists a pair of arc-disjoint $(x,y)$- and $(y,z)$-paths in $D$ if and only if for every arc $e$ there exists either an $(x,y)$-path or a $(y,z)$-path in $D-e$.
\end{thm}
Now we \JBJ{prove} the following common generalization of Theorems \ref{jbjuzvw} and \ref{jbjuvw}.
\begin{thm}\label{goodpair-paths
Let $D$ be a semicomplete digraph and let $x_1,y_2,x_2,y_2$ be four vertices (not necessarily distinct) such that $D$ contains an $(x_i,y_i)$-path for all $i\in[2]$. Then $D$ has a pair of arc-disjoint $(x_1,y_1)$- and $(x_2,y_2)$-paths unless one of the following statements holds.
(i) $D$ is non-strong, $x_1=x_2,y_1=y_2$ and $\{x_1\}, \{y_1\}$ are two consecutive components in the acyclic ordering of the strong components of $D$.
(ii) The four vertices all belong to the same strong component $D_j$ and for some $i$, $(D_j,x_i,x_{3-i},y_{3-i})$ is one of the types in Definition \ref{def1} and the vertex $y_i$ belongs to $V_1$.
\end{thm}
\begin{proof}
There is clearly no arc-disjoint $(x_1,y_1)$- and $(x_2,y_2)$-paths in $D$ when (i) holds. If (ii) holds, by Lemma \ref{type--nopaths}, there is no arc-disjoint $(x_1,y_1)$- and $(x_2,y_2)$-paths in $D_j$ and consequently no such pair in $D$.
Next we suppose that none of (i) and (ii) holds and show that $D$ has a pair of arc-disjoint $(x_1,y_1)$- and $(x_2,y_2)$-paths. Suppose first that $x_1,y_2,x_2,y_2$ do not all belong to the same strong component. In particular $D$ is non-strong. Let $D_1,\ldots,D_l~(l\geq 2)$ be the unique acyclic ordering of the strong components of $D$. If $x_i$ and $y_i$ belong to the same component of $D$ for all $i\in[2]$, then $D$ clearly has the desired paths. So we may assume that $x_1$ and $y_1$ belong to components $D_p$ and $D_q$ with $q>p$, respectively. Then $x_1y_1\in A(D)$ and $D$ has the desired paths when $x_2\notin D_p$ or $y_2\notin D_q$. Therefore we may assume that $x_1,x_2\in D_p$ and $y_1,y_2\in D_q$. Let $D^{\prime}=D\left\langle V(D_p\cup\cdots\cup D_q)\right\rangle$. If there is a good $(x_1,y_2)$-pair in $D^{\prime}$, then $D^{\prime}$ (and consequently, $D$) has the desired paths. We may assume that $|D^{\prime}|\leq 3$ by Lemma \ref{nonstrong-OI}. It is not difficult to check that $D$ has the desired paths if $|D^{\prime}|= 3$. Hence, $|D^{\prime}|=2$, which implies that (i) holds, a contradiction with our assumption.
Therefore we may assume that $x_1,y_2,x_2,y_2$ all belong to the same strong component $D_j$. By Theorem \ref{jbjuzvw}, we may assume that $|\{x_1\}\cup\{y_1\}\cup\{x_2\}\cup\{y_2\}|<4$. By the assumption in the lemma $D_j$ has an $(x_i,y_i)$-path for all $i\in[2]$, so we can assume that $x_i\neq y_i$ for all $i\in[2]$. If $y_1=y_2$, then it follows from by Menger's theorem and the fact that $(D_j,x_1,x_2,y_2)$ is not of the type $A$ that the desired paths exist. So we may assume that $y_1\neq y_2$ and by symmetry we have $x_1\neq x_2$.
The only remaining case is $x_i=y_{3-i}$ for some $i\in [2]$. Assume without loss of generality that
$x_1=y_2$. If $D_j$ has a good $(x_1,x_1)$-pair, then $D_j$ clearly has the desired paths. So we may assume that $D_j$ has the structure given in Theorem \ref{sABC}. Let $e=e^{\prime}=ab$ and let $A_1,\ldots, A_l$ be the acyclic ordering of the strong components of $D\left\langle A\right\rangle$. Clearly, $\{x_1r:r\in A\cup C\}\cup \{ab\}\cup B_{b,B}^+$ is an out-branching $B^+_{x_1}$ with root $x_1$ in $D_j$, where $B_{b,B}^+$ exists as $b\in Out(B)$. If $D_j-A(B^+_{x_1})$ has an $(x_2,x_1)$-path, then we are done so we may assume that $x_2$ can not reach any vertex of $\{x_1\}\cup B\cup C$ in $D_j-A(B^+_{x_1})$. This means that $x_2\in A_t$ for some $t$. Let $V_3$ be the set of vertices which $x_2$ can reach in $D_j-A(B^+_{x_1})$. Clearly, $x_2\in\bigcup_{i=t}^l A_i\subseteq V_3\subseteq A$ and there is only one arc $ab$ leaving $V_3$. By symmetry, one can construct an in-branching $B^-_{x_1}$ rooted at $x_1$ in $D_j$ and let $V_1$ be the set of vertices which can reach $y_1$ in $D_j-A(B^-_{x_1})$. Then $y_1\in V_1\subseteq B$ and there is only one arc $ab$ entering $V_1$. Set $V_2=V-V_1-V_3$. Then $(D_j,x_1,x_2,y_2)$ is of type $B$ with partition $V_1,V_2,V_3$ and $y_1\in V_1$, which contradicts our assumption.
This completes the proof.
\end{proof}
\section{Extending arc-disjoint in- and out-trees in semicomplete digraphs}\label{sec:extend}
\begin{lem}\label{adjacentsomevex}
Let $D$ be a semicomplete digraph and let $H\subseteq D$ be a subdigraph. For any oriented tree $T$ in $D$, if all arcs in $H$ not in $A(T)$ are adjacent to some (fixed) vertex $h$ of $H$, then the digraph $H-h$ is either a single vertex or two vertices joined by one arc. In particular, $|V(H)|\leq 3$.
\end{lem}
\begin{proof}
The lemma follows by the fact that $H-h$ is semicomplete and all arcs in $H-h$ are used in $T$.
Hence $H-h$ has at most two vertices and if it has such vertices $u,v$ then there is only one arc between these.
\end{proof}
Let $D$ be a semicomplete digraph and let $X,Y$ be two disjoint subsets of $V(D)$ such that all vertices in $X$ (resp., in $Y$) are covered by an out-tree $T^+_u$ rooted at $u$ (resp., an in-tree $T^-_v$ rooted at $v$) in $D$ and assume that $T^+_u$ and $T^-_v$ are arc-disjoint. Let $X^{\prime}\subseteq X$ (resp., $Y^{\prime}\subseteq Y$) be the set of vertices covered by $T^-_v$ (resp., $T^+_u$) (possibly $X^{\prime}$ or $Y^{\prime}$ is empty).
We say that the pair $(T^+_u,T^-_v)$ is \textbf{extendable} (on $X\cup Y$) if $D$ has an out-tree $\hat{T}^+_u$ and an in-tree $\hat{T}^-_v$ which are arc-disjoint and such that
each of them covers all vertices in $X\cup Y$. \JBJ{Note that it is not required that all arcs of $T^+_u$ ($T^-_v$) are arcs of $\hat{T}^+_u$ ($\hat{T}^-_v$).} The following lemmas are used to describe non-extendable pairs $(T^+_u,T^-_v)$. \JBJ{Note that $X$ and $Y$ are always the same below.}
\begin{lem}\label{XYnoarc}
Suppose that $X$ dominates $Y$ and no arc between $X$ and $Y$ is used in $T^+_u$ or $T^-_v$. If $(T^+_u,T^-_v)$ is non-extendable, then $X^{\prime}=Y^{\prime}=\emptyset$, $|X\cup Y|\leq 3$ and $D\left\langle X\right\rangle$ (resp., $D\left\langle Y\right\rangle$) is a single vertex or an arc covered by $T^+_u$ (resp., by $T^-_v$).
\end{lem}
\begin{proof}
Suppose that $X^{\prime}\neq \emptyset$ or there is an arc $ab$ in $D\left\langle X\right\rangle$ not used in $T^+_u$, let $x\in X^{\prime}\cup \{a\}$, then $(T^+_u,T^-_v)$ can be extended as follows: $T^+_u\cup \{xr:r\in Y-Y^{\prime}\}$, $T^-_v\cup \{ry:r\in X-X^{\prime}\}$ or $T^-_v\cup \{ab\}\cup \{ry:r\in X-a\}$, where $y$ is any vertex of $Y$. By our assumption we may assume that $X^{\prime}=\emptyset$ and all arcs in $D\left\langle X\right\rangle$ are used in $T^+_u$. By symmetry, we have $Y^{\prime}=\emptyset$ and all arcs in $D\left\langle Y\right\rangle$ are used in $T^-_v$. This means that $|X|\leq 2$ and $|Y|\leq 2$. If $|X|=|Y|=2$, say $X=\{x_1,x_2\},Y=\{y_1,y_2\}$, then $(T^+_u,T^-_v)$ can be extended by adding arcs $x_iy_i,i\in[2]$ to $T^+_u$ and arcs $x_iy_{3-i},i\in[2]$ to $T^-_v$. So the lemma holds by assumption.
\end{proof}
\begin{lem}\label{XYblue}
Let $a\in Y, b\in X$ be two vertices. Suppose that the arc between $a$ and $b$ belongs to $T^-_v$ and all other arcs between $X$ and $Y$ go from $X$ to $Y$ and none of these arcs are used in $T^+_u$ and $T^-_v$. If $(T^+_u,T^-_v)$ is non-extendable, then $X^{\prime}=\{b\}$ and one of the following statements holds:
(i) $X=X^{\prime}$, $a\notin Y^{\prime}$ and all in-arcs of $a$ in $D\left\langle X\cup Y\right\rangle$ are used in $T^-_v$.
(ii) $|(X-b)\cup Y|\leq 3$ and $D\left\langle X-b\right\rangle$ (resp., $D\left\langle Y\right\rangle$) is a single vertex or an arc covered by $T^+_u$ (resp., by $T^-_v$). Moreover, all arcs in $D\left\langle X\right\rangle$ which not covered by $T^+_u$ are out-arcs of $b$.
\end{lem}
\begin{proof}
As the arc between $a$ and $b$ is used in $T^-_v$, the vertex $b$ clearly belongs to $X^{\prime}$. If there exists an $x\in X^{\prime}-b$, then $(T^+_u,T^-_v)$ can be extended as follows: $T^+_u\cup \{xr:r\in Y-Y^{\prime}\}$ and $T^-_v\cup \{ra:r\in X-X^{\prime}\}$. It follows by the assumption that $x$ does not exist and then $X^{\prime}=\{b\}$.
First we consider the case $X=X^{\prime}=\{b\}$. Note that in this case, $T^-_v$ covers all vertices of $X\cup Y$. If $a\in Y^{\prime}$, that is, $a\in T^+_u$, or there is an in-arc $a_Ia$ of $a$ in $D\left\langle X\cup Y\right\rangle$ not used in $T^-_v$, then $T^+_u\cup \{br:r\in Y-Y^{\prime}\}$ or $T^+_u\cup \{a_Ia\}\cup\{br:r\in Y-Y^{\prime}-a\}$ extends $T^+_u$, a contradiction. So (i) follows by the assumption.
For the case that $X-b\neq \emptyset$, the first statement of (ii) follows from Lemma \ref{XYnoarc} when we consider $X-b$ and $Y$. Suppose that there is an arc $wz$ with $w\neq b$ in $D\left\langle X\right\rangle$ which is not in $T^+_u$. Then $(T^+_u,T^-_v)$ can be extended in the following way: $T^+_u\cup \{wr:r\in Y-Y^{\prime}\}$ and $T^-_v\cup \{wz\}\cup\{ra:r\in X-b-w\}$, which contradicts our assumption. So all arcs in $D\left\langle X\right\rangle$ which not covered by $T^+_u$ are out-arcs of $b$ and then (ii) holds.
\end{proof}
\begin{lem}\label{XYarc}
Let $ab$ be an arc from $Y$ to $X$. Suppose that all arcs between $X$ and $Y$ go from $X$ to $Y$ except for the arc $ab$. If $X^{\prime}=Y^{\prime}=\emptyset$ and $(T^+_u,T^-_v)$ is non-extendable, then one of the following statements holds:
(i) $X=\{b,x\}, Y=\{a,y\}$, $A(D\left\langle X\right\rangle)=\{bx\}$ and
$A(D\left\langle Y\right\rangle)=\{ya\}$ and either $bx\in T^+_u$ or $ya\in T^-_v$;
(ii) $Y=\{a\}$ and either all out-arcs of $b$ in $D\left\langle X\right\rangle$ are used in $T^+_u$ or all arcs not covered by $T^+_u$ in $D\left\langle X\right\rangle$ are out-arcs of $b$.
(iii) $X=\{b\}$ and either all in-arcs of $a$ in $D\left\langle Y\right\rangle$ are used in $T^-_v$ or all arcs not covered by $T^-_v$ in $D\left\langle Y\right\rangle$ are in-arcs of $a$.
\end{lem}
\begin{proof}
First observe that no arc between $X$ and $Y$ is used in $T^+_u$ and $T^-_v$ as $X^{\prime}=Y^{\prime}=\emptyset$. Suppose that $|X|\geq 3$ and $|Y|\geq 2$, say $x_1,x_2,b\in X$ and $y,a\in Y$. Then $T^+_u\cup \{x_1y,x_2a\}\cup\{br:r\in Y-a-y\}$ extends $T^+_u$ and $T^-_v\cup \{x_1a\}\cup\{ry:r\in X-x_1\}$ extends $T^-_v$, a contradiction. So we may assume that either $|X|\leq 2$ or $|Y|= 1$. By symmetry we have that either $|Y|\leq 2$ or $|X|=1$.
Next we show that if $|Y|\geq 2$ and $D\left\langle X\right\rangle$ has an arc $wz$ with $w\neq b$ not in $T^+_u$, then $(T^+_u,T^-_v)$ is extendable. Let $y$ be a vertex in $Y-a$. Then $(T^+_u,T^-_v)$ can be extended as follows: $T^+_u\cup \{wr:r\in Y\}$ and $T^-_v\cup \{wz\}\cup\{ry:r\in X-w\}$. By assumption, we may assume that either $|Y|= 1$ or all arcs in $D\left\langle X\right\rangle$ which not in $T^+_u$ are out-arcs of $b$. By symmetry, we have that either $|X|= 1$ or all arcs in $D\left\langle Y\right\rangle$ which not in $T^-_v$ are in-arcs of $a$.
Suppose that $|X|=|Y|=2$, say $X=\{x,b\}$ and $Y=\{y,a\}$. If $x$ dominates $b$, then $xb$ belongs to $T^+_u$ by the argument above and hence $xab\cup xy$ and $T^-_v\cup xby$ extend $T^+_u$ and $T^-_v$, respectively\footnote{Note that in this case the new out-tree does not use all arcs of the old out-tree $T^+_u$.}. So we assume that $D\left\langle X\right\rangle$ consists of arc $bx$. By symmetry, we have $D\left\langle X\right\rangle=ya$. Moreover, either $bx\in T^+_u$ or $ya\in T^-_v$, otherwise, $(bya,bxa)$ extends $(T^+_u,T^-_v)$. This implies that (i) holds.
Now it suffices to consider the case that $|X|=1$ or $|Y|=1$. Suppose that $|Y|=1$. If $D\left\langle X\right\rangle$ has an out-arc $bb_o$ of $b$ and an arc $wz$ with $w\neq b$ such that none of the arcs $bb_0$ and $wz$ are used in $T^+_u$, then $(T^+_u,T^-_v)$ can be extended in the following way: $T^+_u\cup \{wa\}$ and $T^-_v\cup \{wz,bb_o\}\cup\{ra:r\in X-b-w\}$, which contradicts our assumption. So (ii) holds and by symmetry (iii) holds if $|X|=1$.
\end{proof}
\section{Arc-disjoint branchings and paths in semicomplete digraphs}\label{sec:B+P}
If $D$ has a good $(u,v)$-pair, then for every vertex $w$ of $D$ it is the case that $D$ has a $(w,v)$-path which is arc-disjoint from some out-branching $B^+_u$ rooted at $u$. It turns out that this partial problem also has a simple and very natural characterization for semicomplete digraphs.
\begin{thm}\label{Buvwiff}
Let $D=(V,A)$ be a semicomplete digraph and let $u,w,v$ be three vertices (not necessarily distinct) such that $D$ has an out-branching rooted at $u$ and a $(w,v)$-path. Then $D$ has an out-branching rooted at $u$ which is arc-disjoint from some $(w,v)$-path if and only if $D$ has a pair of arc-disjoint $(u,z)$- and $(w,v)$-paths for every vertex $z\in V$.
\end{thm}
\begin{proof}
The necessity is trivial. To see the sufficiency, observe that for the case $u=w$, by Lemma \ref{OutbranPath}, we may assume that there exists a partition $V_1,V_2$ of $V$ such that $v\in V_1,u\in V_2$ and $d_D^+(V_2)=1$. Then for any vertex $z\in V_1$, $D$ has no pair of arc-disjoint $(u,z)$- and $(w,v)$-paths, contradicting our assumption. So it suffices to consider the case $u\neq w$.
Let $D^{\prime}$ be an auxiliary digraph obtained from $D$ by adding a new vertex $s$ together with arcs $su,sw$. If there are two arc-disjoint out-branchings with root $s$ in $D^{\prime}$, then it is clear that $D$ has the desired branching and path. Hence by Theorem \ref{edmonds1973}, we may assume that there is a subset $V_1\subseteq V(D^{\prime}) - \{s\}$ such that $d_{D^{\prime}}^-(V_1) \leq 1$. Let $V_2=V-V_1$. Clearly, $s\in V_2$. By the construction and the fact that $d_{D^{\prime}}^-(V_1) \leq1$, we have either $u\notin V_1$ or $w\notin V_1$. Furthermore, since $D$ has an out-branching rooted at $u$, we have either $u\in V_1,w\in V_2$ or $u,w\in V_2$.
If $u\in V_1,w\in V_2$, then the arc $su$ is the only arc entering $V_1$ in $D^{\prime}$ and hence $V_1$ has in-degree zero in $D$. Since there is a $(w,v)$-path in $D$, the vertex $v$ belongs to $V_2$ and $D\left\langle{}V_2\right\rangle$ contains a
$(w,v)$-path $P$. Now the desired pair can be obtained by taking $P$ and an out-branching in $D$ consisting of an out-branching with root $u$ in $D^{\prime}\left\langle V_1\right\rangle$ and all arcs $\{ur:r\in V_2-s\}$. So we may assume that both $u$ and $w$ belong to $V_2$ and moreover, $d_{D^{\prime}}^-(V_1) =d_{D}^-(V_1)=1$ as $D$ has an out-branching rooted at $u$. Let $xy$ be the arc entering $D\left\langle V_1\right\rangle$. By the assumption, there is a pair of arc-disjoint $(u,y)$- and $(w,v)$-paths $P_{u,y}$ and $P_{w,v}$
in $D$. Thus we must have $v\in V_2$. Now we construct an out-branching $B_u^+$ in $D$ which is arc-disjoint with $P_{w,v}$ as follows: $B_{u}^+=B_{y,D\left\langle V_1\right\rangle}^+\cup P_{u,y}\cup\{yr:r\in V_2-\{s\}-V(P_{u,y})\}$, where $B_{y,D\left\langle V_1\right\rangle}^+$ exists as $D$ has an out-branching rooted at $u$. This completes the proof.
\end{proof}
\newpage
\begin{thm}\label{outbranpath}
Let $D$ be a semicomplete digraph and let $u,w,v$ be three vertices (not necessarily distinct) such that $D$ contains an out-branching rooted at $u$ and a $(w,v)$-path. Then $D$ has an out-branching with root $u$ which is arc-disjoint from some $(w,v)$-path unless one of the following statements holds.
(i) $Out(D)=\{u\}=\{w\}$ and \JBJ{$u$ is the only in-neighbour of }$v$ in $D$. In particular, $(D,u,w,v)$ is of type $A$ with $V_1=\{v\}$ in Definition \ref{def1}.
(ii) The vertices $u,w,v$ belong to the same component of $D$, i.e., $D\left\langle Out(D)\right\rangle$, and $(D\left\langle Out(D)\right\rangle,u,w,v)$ is one of the types in Definition \ref{def1}.
\end{thm}
\begin{proof}
Observe that $u\in Out(D)$ as there is an out-branching rooted at $u$ in $D$. Then the theorem follows by Theorem \ref{Buvwiff} and Theorem \ref{goodpair-paths}. It should be noted that if $(D\left\langle Out(D)\right\rangle,w,u,z)$ is one of the types in Definition \ref{def1} and $v\in V_1$, then $(D\left\langle Out(D)\right\rangle,u,w,v)$ is of type $A$.
\end{proof}
\section{Arc-disjoint in- and out-branchings in semicomplete digraphs}\label{sec:proof}
\noindent{}Now we are ready to prove Theorem \ref{thm:SDbranchchar}.
We first state the following result that will be used in the proof. An arc $xy$ of a strong digraph $D$ is a {\bf cut-arc} if $D\setminus xy$ is not strong.
\begin{thm}\label{thmOI}
Let $D$ be a strong semicomplete digraph of order at least 4 with a cut-arc $xy$ and let $u \in Out(D-xy),v \in In(D-xy)$ be two vertices. Suppose that $D$ contains no good $(u,v)$-pair. Then either $D$ is isomorphic to one of the digraphs shown in Figure \ref{fig4} (d)-(f) or $D-xy$ has exactly two strong components and one of the following statements holds.
(i) $In(D-xy)=\{v\}=\{x\}$ and $d_{D}^+(y)=1$. Say $N_D^{+}(y)=\{z\}$. There is no $(u,z)$-path in $D-yz$.
(ii) $Out(D-xy)=\{u\}=\{y\}$ and $d_{D}^-(x)=1$. Say $N_D^{-}(x)=\{z\}$. There is no $(z,v)$-path in $D-zx$.
\end{thm}
\noindent{} Note that if (i) holds, then we have $z\in V_1$, $V_{l-1}=\{x\}=\{v\},V_l=\{y\}$ and $u\notin V_1\cup V_{l-1}$ (possibly $u=y$) where $D_1,\ldots,D_l~(l\geq 3)$ is acyclic ordering of the strong components of $D-yz$ and $V_i=V(D_i)$.\\
\begin{proof} Let $Y=In(D-xy)$ and $X=V(D)-Y$. As $u$ belongs to $Out(D-xy)=Out(D\left\langle X\right\rangle)$ and $v\in Y= In(D-xy)$, $D$ has an out-branching rooted at $u$ in $D\left\langle X\right\rangle$ and an in-branching rooted at $v$ in $D\left\langle Y\right\rangle$. Let $T^+_u$ and $T^-_v$ be any such out- and in-branchings, respectively. As $D\left\langle Y\right\rangle$ is strong, by symmetry, we may assume that the second statement of Lemma \ref{XYarc} holds. That is, $Y=\{x\}=\{v\}$ and either all out-arcs of $y$ in $D\left\langle X\right\rangle$ are used in $T^+_u$ or all arcs of $D\left\langle X\right\rangle$ which are not used by $T^+_u$ are adjacent to $y$.
Suppose that $d_D^+(y)\geq 2$. Let $T^+_u$ be a hamitonian path starting at $u$ in $D\left\langle X\right\rangle$. Then there is an out-arc of $y$ not used in $T^+_u$ and hence, by the remark above, all arcs not covered by $T^+_u$ in $D\left\langle X\right\rangle$ are adjacent to $y$. It follows by Lemma \ref{adjacentsomevex} and the fact $|V(D)|\geq 4$ that $|X|=3$. Suppose that $T^+_u=u_1u_2u_3$ with $u_1=u$ is the hamiltonian path in $D\left\langle X\right\rangle$. Then $y\in\{u_1,u_3\}$. First consider the case $y=u_1=u$. If $u_1u_3\notin A(D)$, then $y$ dominates $x$ as $d_D^+(y)\geq 2$. Then $T^+_u\cup \{u_3v\}$ and $u_3yv\cup \{u_2v\}$ is a good $(u,v)$-pair, which contradicts our assumption. So $u_1u_3\in A(D)$. If $u_2$ dominates $u_1$, then $T^+_u\cup \{u_2v\}$ and $u_2u_1u_3v$ form a good $(u,v)$-pair, a contradiction again. Then $D$ is isomorphic to the digraph shown in Figure \ref{fig4} (d) if $u_3u_1\notin A(D)$ (resp., in Figure \ref{fig4} (f) if $u_3u_1\in A(D)$. For the case that $y=u_3$, the vertex $u_3$ dominates $u_1$ as $d_D^+(y)\geq 2$ and then $u_1u_2vu_3$ and $u_2u_3u_1v$ form a good $(u,v)$-pair, a contradiction again.
\JBJ{It remains }to consider the case $d_D^+(y)=1$. Say $N_D^+(y)=\{z\}$. This means that $yz$ is a cut-arc of $D$. Let $V_1,\ldots,V_l$ be the acyclic ordering of the strong components of $D-yz$. It follows by $d_D^+(y)=1$ that $|V_l|=|\{y\}|=1$. Since there \JBJ{is} only one $(x,y)$-path in $D$ (recall that $xy$ is a cut-arc), we have $V_{l-1}=\{x\}=\{v\}$. As $u \in Out(D-xy)$ and $v \in In(D-xy)$, we have $u\neq v$ and then $u\notin V_{l-1}$.
Note that if $u\notin V_1$, then the statement (i) holds so we may assume that $u\in V_1$. Next we claim that there is a good $(u,v)$-pair, which contradicts our assumption. Recall that $X=V_{1}\cup\cdots\cup V_{l-2}\cup \{y\}$. As $|V(D)|\geq 4$ we have that $|V_{1}\cup\cdots\cup V_{l-2}|\geq 2$. Moreover, there is a spanning out-tree (that is, an out-branching) $T^+_u$ in $D\left\langle X\right\rangle$ such that $yz$ does not belong to $T^+_u$. By the argument in the first paragraph of the proof, all arcs not covered by $T^+_u$ in $D\left\langle X\right\rangle$ are adjacent to $y$. Thus all arcs in $D\left\langle V_{1}\cup\cdots\cup V_{l-2}\right\rangle$ are used in $T^+_u$, which means that $|V_{1}\cup\cdots\cup V_{l-2}|\leq 2$. It follows by $|V(D)|\geq 4$ that equality holds, say $D\left\langle V_{1}\cup\cdots\cup V_{l-2}\right\rangle=uw$, then $uwvy$ and $wyuv$ form the desired pair, which completes the proof.
\end{proof}
\begin{lem}\label{counterexample}
Let $D$ be a semicomplete digraph and $u,v$ two distinct vertices. If $D$ is isomorphic to one of digraphs shown in Figure \ref{fig4}, then there is no good $(u,v)$-pair in $D$.
\end{lem}
\begin{proof}
If $D$ has a good $(u,v)$-pair, then the size of $D$ must be at least $2(|V(D)|-1)$, moreover, if $v$ dominates $u$, then $|A(D)|\geq 2|V(D)|-1$ since the arc $vu$ cannot be used in any $(u,v)$-pair. This shows that $D$ has no good $(u,v)$-pair if it is isomorphic to one of digraphs in Figure \ref{fig4} (a)-(d).
For the case that $D$ is isomorphic to the digraph in Figure \ref{fig4} (e), suppose that $D$ has a good $(u,v)$-pair ($B_u^+,B_v^-$). Then every arc except for $vu$ either belongs to $B_u^+$ or $B_v^-$ as $|A(D-vu)|=2|V(D)|-2$. Let $V(D)-\{u,v\}=\{w,z\}$ such that $w$ dominates $z$. The only out-arc $zv$ of $z$ must belong to $B_{v}^-$ and $vw$ must belong to $B_{u}^+$. By the definitions of out- and in-branchings, we have $wv,uw\in A(B_v^-)$ and then $B_u^+=uz\cup vwz$. However, $B_u^+$ is not an out-branching rooted at $u$, a contradiction. By a similar argument, $D$ has no good $(u,v)$-pair when it is not isomorphic to the digraph shown in Figure \ref{fig4} (f).
\end{proof}
\noindent{}{\bf Proof of Theorem \ref{thm:SDbranchchar}:}
\begin{proof}
Observe that if there is a good $(u,v)$-pair, then clearly there are arc-disjoint $(u,z)$- and $(w,v)$-paths for every choice of vertices $z$ and $w$. Hence the necessity follows by Lemma \ref{counterexample}. Next we show the sufficiency. If $u=v$, then it follows from Theorem \ref{sABC} that the desired branchings exist since if there was an arc $e=pq$ as in the theorem, then $D$ would have no pair of arc-disjoint $(u,q)$- and $(p,x)$-paths. Hence we can assume that $u\neq v$.
Suppose first that $D$ is non-strong. Since there is a $(u,z)$-path and a $(w,v)$-path for any $z,w$, we have that $u$ belongs to $Out(D)$ and $v$ belongs to $In(D)$. By Lemma \ref{nonstrong-OI}, we may assume that $|V(D)|\leq 3$. Since $D$ is not isomorphic to one of digraphs shown in Figure \ref{fig4} (a)-(b), either $|Out(D)|=2$ or $|In(D)|=2$ and then there is a good $(u,v)$-pair in $D$.
It remains to consider the case that $D$ is strong. Further, by Theorem \ref{thm:SD2arcstrong}, we may assume that $D$ is not 2-arc-strong, implying that it has a cut-arc. Let $xy$ be a cut-arc of $D$ and let $U_1,\ldots,U_l$ $(l\geq 2)$ be the acyclic ordering of the strong components of $D-xy$. Note that $y\in U_1$ and $x\in U_l$. Suppose that $u$ belongs to $U_i$ and $v$ belongs to $U_j$. Since there exist arc-disjoint $(u,y)$- and $(x,v)$-paths in $D$, either $u$ belongs to $U_1$ or $v$ belongs to $U_l$ (or both), that is, either $i=1$ or $j=l$.
Note that $|V(D)|\geq 3$ as it has two arc-disjoint $(u,v)$-paths. If $|V(D)|=3$, say $V(D)=\{u,v,w\}$, then $u$ dominates $w,v$ and $w$ dominates $v$. It can be checked easily that there is a good $(u,v)$-pair if $v$ dominates $w$ or $w$ dominates $u$. Hence we can assume that $v$ dominates $u$ and then $D$ is isomorphic to the digraph shown in Figure \ref{fig4} (c), contradicting our assumption. Therefore, we may assume that $|V(D)|\geq 4$.
Suppose first that $u\in U_1$ and $v\in U_l$, that is, $i=1,j=l$. By Theorem \ref{thmOI} we are done, unless either (i) or (ii) in the theorem holds. By symmetry, we may assume that the statement (i) of Theorem \ref{thmOI} holds. However, then there is no pair of arc-disjoint $(u,z)$- and $(y,v)$-paths in $D$, which contradicts our assumption.
Consider next the case $i=1$ and $j\neq l$. Let $X=U_1\cup\cdots\cup U_j$ and $Y=V(D)-X$.
If $D\left\langle X\right\rangle$ does not have an out-branching rooted at $u$ which is arc-disjoint from some $(y,v)$-path, then by Lemma \ref{outbranpath}, $(D,u,y,v)$ is one of the types in Definition \ref{def1}. It follows by Lemma \ref{type--nopaths} that $D$ has no arc-disjoint $(u,z)$- and $(y,v)$-paths for any vertex $z\in V_1$, contradicting our assumption. So we may assume that $D\left\langle X\right\rangle$ has an out-branching $T^+_u$ rooted at $u$ and a $(y,v)$-path $P_{y,v}$ which are arc-disjoint.
Let $P_{x}$ be a hamiltonian path ending in $x$ in $D\left\langle Y\right\rangle$. Such path $P_{x}$ exists as $x$ belongs to $In(D-xy)$. Let $T^-_v=P_x\cup\{xy\}\cup P_{y,v}$. Clearly, all vertices of $P_{y,v}$ are covered by $T^-_v$, that is, $X^{\prime}=V(P_{y,v})$. Now Lemma \ref{XYblue} implies that we are done unless $|X^{\prime}|=|V(P_{y,v})|=1$. This means that $v=y\in Out(D-xy)$ and then $X=Out(D-xy)$. Moreover, $|X|\geq 2$ as $u\neq v$. Hence, we may assume that the second statement of Lemma \ref{XYblue} holds. That is, $|(X-y)\cup Y|\leq 3$ and $D\left\langle X-y\right\rangle$ (resp., $D\left\langle Y\right\rangle$) is a single vertex or an arc covered by $T^+_u$ (resp., by $T^-_v$). Moreover, all arcs in $D\left\langle X\right\rangle$ which not covered by $T^+_u$ are out-arcs of $y$.
On the other hand, as $|V(D)|\geq 4$, we have $|(X-y)\cup Y|\geq 3$ and then the equality holds. Suppose that $|Y|=2$, say $Y=\{x,x^{\prime}\}$, then $|X|=2$ and $X=\{u,v\}$ with $v=y$ as $u\neq v$. In this case, $uvx^{\prime}\cup ux$ and $ux^{\prime}xv$ form a good $(u,v)$-pair. Thus it remains to consider the case that $|Y|=1$ and $|X|=3$, say $X=\{u,v,w\}$ with $v=y$. Recall that $D\left\langle X\right\rangle$ is strong as $X=Out(D-xy)$. Since $D\left\langle X-y\right\rangle$ is an arc covered by $T^+_u$ (as we are in Case (ii) of Lemma \ref{XYblue}) and all arcs in $D\left\langle X\right\rangle$ which are not used by $T^+_u$ are out-arcs of $y$ (i.e., $v$), we may assume that $uwvu$ is a hamitonian cycle in $D\left\langle X\right\rangle$ and $T^+_u=uwv$. Then $D$ is isomorphic to the digraph in Figure \ref{fig4} (d) or (e),
contradicting our assumption.
Finally assume that that $i\neq 1$ and $j=l$. By applying similar arguments as above to the digraph obtained from $D$ by reversing all arcs, we either find that $D$ has the desired branchings or it would be isomorphic to one of the digraphs in Figure \ref{fig4} (d) or (f), a contradiction again. This completes the proof of Theorem \ref{thm:SDbranchchar}. \end{proof}
All our arguments leading to the proof of Theorem \ref{thm:SDbranchchar} are constructive and can be converted to polynomial algorithms. Hence we have the following corollary.
\begin{coro}
There exists a polynomial algorithm which given a semicomplete digraph $D=(V,A)$ and vertices $u,v$ of $D$ either constructs a good $(u,v)$-pair or produces a certificate that $D$ has no such pair.
\end{coro}
The following equivalent structural characterization of semicomplete digraphs with good $(u,v)$-pairs is often easier to use when one wishes to study digraphs that are more general than semicomplete digraphs.
In particular we use it in \cite{bangQTDbranchpaper} to study good $(u,v)$-pairs in so-called semicomplete compositions.
\begin{thm}\label{SDgoodpair}
Let $D$ be a semicomplete digraph and $u,v$ be arbitrary chosen vertices (possibly $u=v$). Then $D$ has a good $(u,v)$-pair if and only if $(D,u,v)$ satisfies none of the following conditions.
(i)
$D$ is isomorphic to one of the digraphs in Figure \ref{fig4}
(ii) $D$ is non-strong and either $u$ is not in the initial component of $D$ or $v$ is not in the terminal component of $D$.
(iii) $D$ is strong and there exists an arc $e\in A(D)$ such that $u$ is not in the initial component of $D-e$ and $v$ is not in the terminal component of $D-e$.
(iv) $D$ is strong and there exists a partition $V_1,\ldots,V_{2\alpha+3}$ of $V(D)$ for some $\alpha\geq 1$ such that $v\in V_2, u\in V_{2\alpha+2}$ and all arcs between $V_i$ and $V_j$ with $i<j$ from $V_i$ to $V_j$ with the following exceptions. There exists precisely one arc from $V_{i+2}$ to $V_{i}$ for all $i\in[2\alpha+1]$ and it goes from the terminal component of $D\left\langle V_{i+2}\right\rangle$ to the initial component of $D\left\langle V_i\right\rangle$.
\end{thm}
\begin{proof}
We first prove the necessity. Lemma \ref{counterexample} shows that $D$ has no such pair if (i) holds. It is not difficult to check that when (ii) or (iii) holds, there is no good $(u,v)$-pair in $D$ as no such pair can cover the vertices in the initial and terminal components in the same time. If (iv) holds, let $z$ and $w$ be two vertices of $V_1$ and $V_{2\alpha+3}$, respectively. Then there is no pair of arc-disjoint $(u,z)$- and $(w,v)$-paths (due to Lemma \ref{type--nopaths}). By Theorem \ref{thm:SDbranchchar}, $D$ has no good $(u,v)$-pair.
Now suppose that $(D,u,v)$ satisfies none of the conditions (i)-(iv). We prove that then there exists a good $(u,v)$-pair. If $D$ is non-strong, since $(D,u,v)$ does not satisfy condition (ii), we have $u\in Out(D)$ and $v\in In(D)$. From Lemma \ref{nonstrong-OI}, we may assume that $D$ has order at most three. Since $D$ is not isomorphic to one of digraphs shown in Figure \ref{fig4} (a)-(b), either $|Out(D)|=2$ or $|In(D)|=2$ and then there is a good $(u,v)$-pair in $D$.
For the case that $D$ is strong, suppose for contradiction that there is no good $(u,v)$-pair. Then by Theorem \ref{thm:SDbranchchar}, there exist $z,w$ such that there is no pair of arc-disjoint $(u,z)$- and $(w,v)$-paths in $D$. Clearly, $D$ has a $(u,z)$-path and a $(w,v)$-path as $D$ is strong. By Theorem \ref{goodpair-paths}, either $(D,u,w,v)$ is one of the types in Definition \ref{def1}, or $(D,w,u,z)$ is one of the types in Definition \ref{def1} and the vertex $v$ belongs to $V_1$. It is not difficult to check that $(D,u,v)$ satisfies condition (iii) or (iv), which contradicts our assumption. This completes the proof.
\end{proof}
\noindent{}{\bf Conflict of interest statement}\\
There are no sources of conflict of interest regarding this paper.\\
\noindent{}{\bf Data availability statement}\\
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
\noindent{}{\bf Acknowledgement:} Financial support from the Independent Research Fund Denmark under grant DFF-7014-00037B is gratefully acknowledged. The second author was supported by China Scholarship Council (CSC) No. 202106220108.
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"redpajama_set_name": "RedPajamaArXiv"
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La Sportiva mountain runner Giulia Amadori brings us in Liguria on her favorite training tracks
In the Alps may be still winter but not in Liguria, where the spring has already blossomed: together with La Sportiva ambassador Giulia Amadori, let's go to the discover of the perfect five training tracks around the amazing scenery of this Italian region.
Waiting for Sciacchetrail, the iconic race of Cinque Terre that will take place next March 24th, we start warming up together with the home athlete and La Sportiva ambassador Giulia Amadori.
Giulia, born in 1981 and 100% Ligurian, takes us on her favourite training tracks, which are mainly based in the nearby of her town, Rapallo, and which differ both in terms of length and technicality .
5 Campanili
Starting from the small square of Zoagli, this track touches five small villages scattered along the Semorile valley, a difficult and harsh territory that offers exceptional views of Rapallo, Portofino, and the entire Golfo di Tigullio. From the centre of the village, the asphalt quickly gives way to the typical Ligurian creuze (mule track), giving life to a tough track characterized by a succession of steep climbs and as many descents, obviously relatively short. For those who don't dare to do it running, this route is extremely beautiful to do even as a trek, as it is little known and crowded. Thanks to its ring structure, is possible to return to the start point without ever having to retrace your steps.
distance: 11 km
difference in height +/-: 340 meters
> DISCOVER THE TRACK
Marina trail
With start and arrival points in the centre of Lavagna, this track has become in the last three years a proper Trail Race, which in the month of March (just a few days from the SciaccheTrail) brings in the area many fans and newcomers of the discipline. On the 21 km of development, in fact, there are not many technical passages but mainly narrow paths that climb along the Ligurian hills, offering from the first to the last meter unique landscapes, between glimpses of sea and Mediterranean Macchia. After the first ascent of about 650 meters, you stay on the mountain's ridge before starting the descent to Sant 'Anna. After crossing Cavi Borgo you go up for the last time before the final descent to Lavagna, from where you started. Even if the different in height is not too tough, the steep slopes and the intense climbs, make sure that this path never leaves room for boredom.
Sestri Lavante - Moneglia
A beautiful route that allows, starting from the town of Sestri Levante, to skirt the Ligurian Sea up to Moneglia for 15, easy-to-run kilometers. The first part of the route leads to Punta Manara (140 meters), a promontory that stretches out in the Gulf of Tigulio, before descending again to the sea in the center of Riva Trigoso. Another ascent leads to Monte Moneglia, at an altitude of 521m, from where you start the final descent to the village along steep and narrow paths.
I always suggest avoiding this track in the warmer months of the year especially in the central hours. Do this at night with the fronts, instead, is simply wonderful! Once in Moneglia you can take the train to get back to Sestri Levante in a very short time.
distance: 15,5 km
For me this is the classic ride I do during the week to keep me always active, because I can do it easily starting from home. With a total length of just over ten kilometers, the Montallegro tour is a ring that starts from Rapallo with a steep and pretty hard climb due to the numerous stairways, which leads to the beautiful Nostra Signora di Montallegro sanctuary, where the view on the gulf is definitely beautiful. From here you can descend on an alternative path among the cobblestones of the Via dei pellegrini, getting back to the start.
Mount Zatta ring
This route, certainly less known than the previous ones, differs mainly due to the difference in altitude, which is about 800mt, and because it is the only one to wind its way entirely in the Ligurian hinterland, starting from the town of Arzeno. From here, go up towards Mount Biscia where it begins a beautiful plateau that leads to the nearby Chiappozzo mountain. From here, the first descent starts along a small valet that ends in a shaded forest, and ideal to refresh before the climb to Mount Zatta, which with its 1,404 meters is an Apennine watershed between the Valle del Taro, the Valle Graveglia and the Val di Vara. From here, proceeding along the ridge you will reach a little chapel, after which will start the descent, backing to the starting point. The track does not present any technical difficulty and it is all easy-to-run.
Join the top climbers at Milano Climbing Expo
Epic Ski Tour starts again from Davos on December 20th
Highlight Products
Bushido Woman
Trail Vest
Auster Short M
Joy Tank W
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,688
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package com.evolveum.midpoint.web.page.admin.users;
import java.util.List;
import javax.xml.namespace.QName;
import org.apache.wicket.ajax.AjaxRequestTarget;
import org.apache.wicket.model.Model;
import org.apache.wicket.request.mapper.parameter.PageParameters;
import com.evolveum.midpoint.gui.api.util.WebComponentUtil;
import com.evolveum.midpoint.prism.PrismObject;
import com.evolveum.midpoint.security.api.AuthorizationConstants;
import com.evolveum.midpoint.util.logging.Trace;
import com.evolveum.midpoint.util.logging.TraceManager;
import com.evolveum.midpoint.web.application.AuthorizationAction;
import com.evolveum.midpoint.web.application.PageDescriptor;
import com.evolveum.midpoint.web.component.FocusSummaryPanel;
import com.evolveum.midpoint.web.component.objectdetails.AbstractObjectMainPanel;
import com.evolveum.midpoint.web.component.objectdetails.AbstractRoleMainPanel;
import com.evolveum.midpoint.web.component.progress.ProgressReportingAwarePage;
import com.evolveum.midpoint.web.page.admin.PageAdminAbstractRole;
import com.evolveum.midpoint.web.page.admin.roles.AbstractRoleMemberPanel;
import com.evolveum.midpoint.web.page.admin.users.component.OrgMemberPanel;
import com.evolveum.midpoint.web.page.admin.users.component.OrgSummaryPanel;
import com.evolveum.midpoint.web.security.GuiAuthorizationConstants;
import com.evolveum.midpoint.web.session.UserProfileStorage.TableId;
import com.evolveum.midpoint.web.util.OnePageParameterEncoder;
import com.evolveum.midpoint.xml.ns._public.common.common_3.AreaCategoryType;
import com.evolveum.midpoint.xml.ns._public.common.common_3.OrgType;
/**
* @author lazyman
*/
@PageDescriptor(url = "/admin/org/unit", encoder = OnePageParameterEncoder.class, action = {
@AuthorizationAction(actionUri = PageAdminUsers.AUTH_ORG_ALL, label = PageAdminUsers.AUTH_ORG_ALL_LABEL, description = PageAdminUsers.AUTH_ORG_ALL_DESCRIPTION),
@AuthorizationAction(actionUri = AuthorizationConstants.AUTZ_UI_ORG_UNIT_URL, label = "PageOrgUnit.auth.orgUnit.label", description = "PageOrgUnit.auth.orgUnit.description") })
public class PageOrgUnit extends PageAdminAbstractRole<OrgType> implements ProgressReportingAwarePage {
private static final Trace LOGGER = TraceManager.getTrace(PageOrgUnit.class);
public PageOrgUnit() {
initialize(null);
}
public PageOrgUnit(final PrismObject<OrgType> unitToEdit) {
initialize(unitToEdit);
}
public PageOrgUnit(final PrismObject<OrgType> unitToEdit, boolean isNewObject) {
initialize(unitToEdit, isNewObject);
}
public PageOrgUnit(final PrismObject<OrgType> unitToEdit, boolean isNewObject, boolean isReadonly) {
initialize(unitToEdit, isNewObject, isReadonly);
}
public PageOrgUnit(PageParameters parameters) {
getPageParameters().overwriteWith(parameters);
initialize(null);
}
@Override
protected OrgType createNewObject() {
return new OrgType();
}
@Override
public Class<OrgType> getCompileTimeClass() {
return OrgType.class;
}
@Override
protected Class getRestartResponsePage() {
return PageOrgTree.class;
}
@Override
protected FocusSummaryPanel<OrgType> createSummaryPanel() {
return new OrgSummaryPanel(ID_SUMMARY_PANEL, getObjectModel(), this);
}
@Override
protected AbstractObjectMainPanel<OrgType> createMainPanel(String id) {
return new AbstractRoleMainPanel<OrgType>(id, getObjectModel(),
getProjectionModel(), this) {
private static final long serialVersionUID = 1L;
@Override
protected boolean isFocusHistoryPage(){
return PageOrgUnit.this.isFocusHistoryPage();
}
@Override
protected void viewObjectHistoricalDataPerformed(AjaxRequestTarget target, PrismObject<OrgType> object, String date){
PageOrgUnit.this.navigateToNext(new PageOrgUnitHistory(object, date));
}
};
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 291
|
La stagione dei Dallas Cowboys è stata la 50ª della franchigia nella National Football League e la prima giocata al nuovo Cowboys Stadium. La squadra vinse il titolo di division e nel primo turno di playoff batté i Philadelphia Eagles. Fu eliminata nel successivo dai Minnesota Vikings.
Scelte nel Draft 2009
Roster
Calendario
Stagione regolare
Playoff
Note
Altri progetti
Collegamenti esterni
2009
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,330
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{"url":"https:\/\/sources.debian.org\/src\/hmisc\/4.2-0-1\/man\/xYplot.Rd\/","text":"## File: xYplot.Rd\n\npackage info (click to toggle)\nhmisc 4.2-0-1\n 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545 \\name{xYplot} \\alias{xYplot} \\alias{panel.xYplot} \\alias{prepanel.xYplot} \\alias{Dotplot} \\alias{panel.Dotplot} \\alias{prepanel.Dotplot} \\alias{Cbind} \\alias{[.Cbind} \\alias{setTrellis} \\alias{numericScale} \\title{xyplot and dotplot with Matrix Variables to Plot Error Bars and Bands} \\description{ A utility function \\code{Cbind} returns the first argument as a vector and combines all other arguments into a matrix stored as an attribute called \\code{\"other\"}. The arguments can be named (e.g., \\code{Cbind(pressure=y,ylow,yhigh)}) or a \\code{label} attribute may be pre-attached to the first argument. In either case, the name or label of the first argument is stored as an attribute \\code{\"label\"} of the object returned by \\code{Cbind}. Storing other vectors as a matrix attribute facilitates plotting error bars, etc., as \\code{trellis} really wants the x- and y-variables to be vectors, not matrices. If a single argument is given to \\code{Cbind} and that argument is a matrix with column dimnames, the first column is taken as the main vector and remaining columns are taken as \\code{\"other\"}. A subscript method for \\code{Cbind} objects subscripts the \\code{other} matrix along with the main \\code{y} vector. The \\code{xYplot} function is a substitute for \\code{xyplot} that allows for simulated multi-column \\code{y}. It uses by default the \\code{panel.xYplot} and \\code{prepanel.xYplot} functions to do the actual work. The \\code{method} argument passed to \\code{panel.xYplot} from \\code{xYplot} allows you to make error bars, the upper-only or lower-only portions of error bars, alternating lower-only and upper-only bars, bands, or filled bands. \\code{panel.xYplot} decides how to alternate upper and lower bars according to whether the median \\code{y} value of the current main data line is above the median \\code{y} for all \\code{groups} of lines or not. If the median is above the overall median, only the upper bar is drawn. For \\code{bands} (but not 'filled bands'), any number of other columns of \\code{y} will be drawn as lines having the same thickness, color, and type as the main data line. If plotting bars, bands, or filled bands and only one additional column is specified for the response variable, that column is taken as the half width of a precision interval for \\code{y}, and the lower and upper values are computed automatically as \\code{y} plus or minus the value of the additional column variable. When a \\code{groups} variable is present, \\code{panel.xYplot} will create a function in frame 0 (\\code{.GlobalEnv} in \\R) called \\code{Key} that when invoked will draw a key describing the \\code{groups} labels, point symbols, and colors. By default, the key is outside the graph. For S-Plus, if \\code{Key(locator(1))} is specified, the key will appear so that its upper left corner is at the coordinates of the mouse click. For R\/Lattice the first two arguments of \\code{Key} (\\code{x} and \\code{y}) are fractions of the page, measured from the lower left corner, and the default placement is at \\code{x=0.05, y=0.95}. For \\R, an optional argument to \\code{sKey}, \\code{other}, may contain a list of arguments to pass to \\code{draw.key} (see \\code{\\link[lattice]{xyplot}} for a list of possible arguments, under the \\code{key} option). When \\code{method=\"quantile\"} is specified, \\code{xYplot} automatically groups the \\code{x} variable into intervals containing a target of \\code{nx} observations each, and within each \\code{x} group computes three quantiles of \\code{y} and plots these as three lines. The mean \\code{x} within each \\code{x} group is taken as the \\code{x}-coordinate. This will make a useful empirical display for large datasets in which scatterdiagrams are too busy to see patterns of central tendency and variability. You can also specify a general function of a data vector that returns a matrix of statistics for the \\code{method} argument. Arguments can be passed to that function via a list \\code{methodArgs}. The statistic in the first column should be the measure of central tendency. Examples of useful \\code{method} functions are those listed under the help file for \\code{summary.formula} such as \\code{smean.cl.normal}. \\code{xYplot} can also produce bubble plots. This is done when \\code{size} is specified to \\code{xYplot}. When \\code{size} is used, a function \\code{sKey} is generated for drawing a key to the character sizes. See the bubble plot example. \\code{size} can also specify a vector where the first character of each observation is used as the plotting symbol, if \\code{rangeCex} is set to a single \\code{cex} value. An optional argument to \\code{sKey}, \\code{other}, may contain a list of arguments to pass to \\code{draw.key} (see \\code{\\link[lattice]{xyplot}} for a list of possible arguments, under the \\code{key} option). See the bubble plot example. \\code{Dotplot} is a substitute for \\code{dotplot} allowing for a matrix x-variable, automatic superpositioning when \\code{groups} is present, and creation of a \\code{Key} function. When the x-variable (created by \\code{Cbind} to simulate a matrix) contains a total of 3 columns, the first column specifies where the dot is positioned, and the last 2 columns specify starting and ending points for intervals. The intervals are shown using line type, width, and color from the trellis \\code{plot.line} list. By default, you will usually see a darker line segment for the low and high values, with the dotted reference line elsewhere. A good choice of the \\code{pch} argument for such plots is \\code{3} (plus sign) if you want to emphasize the interval more than the point estimate. When the x-variable contains a total of 5 columns, the 2nd and 5th columns are treated as the 2nd and 3rd are treated above, and the 3rd and 4th columns define an inner line segment that will have twice the thickness of the outer segments. In addition, tick marks separate the outer and inner segments. This type of display (an example of which appeared in \\emph{The Elements of Graphing Data} by Cleveland) is very suitable for displaying two confidence levels (e.g., 0.9 and 0.99) or the 0.05, 0.25, 0.75, 0.95 sample quantiles, for example. For this display, the central point displays well with a default circle symbol. \\code{setTrellis} sets nice defaults for Trellis graphics, assuming that the graphics device has already been opened if using postscript, etc. By default, it sets panel strips to blank and reference dot lines to thickness 1 instead of the Trellis default of 2. \\code{numericScale} is a utility function that facilitates using \\code{xYplot} to plot variables that are not considered to be numeric but which can readily be converted to numeric using \\code{as.numeric()}. \\code{numericScale} by default will keep the name of the input variable as a \\code{label} attribute for the new numeric variable. } \\usage{ Cbind(\\dots) xYplot(formula, data = sys.frame(sys.parent()), groups, subset, xlab=NULL, ylab=NULL, ylim=NULL, panel=panel.xYplot, prepanel=prepanel.xYplot, scales=NULL, minor.ticks=NULL, sub=NULL, \\dots) panel.xYplot(x, y, subscripts, groups=NULL, type=if(is.function(method) || method=='quantiles') 'b' else 'p', method=c(\"bars\", \"bands\", \"upper bars\", \"lower bars\", \"alt bars\", \"quantiles\", \"filled bands\"), methodArgs=NULL, label.curves=TRUE, abline, probs=c(.5,.25,.75), nx=NULL, cap=0.015, lty.bar=1, lwd=plot.line$lwd, lty=plot.line$lty, pch=plot.symbol$pch, cex=plot.symbol$cex, font=plot.symbol$font, col=NULL, lwd.bands=NULL, lty.bands=NULL, col.bands=NULL, minor.ticks=NULL, col.fill=NULL, size=NULL, rangeCex=c(.5,3), \\dots) prepanel.xYplot(x, y, \\dots) Dotplot(formula, data = sys.frame(sys.parent()), groups, subset, xlab = NULL, ylab = NULL, ylim = NULL, panel=panel.Dotplot, prepanel=prepanel.Dotplot, scales=NULL, xscale=NULL, \\dots) prepanel.Dotplot(x, y, \\dots) panel.Dotplot(x, y, groups = NULL, pch = dot.symbol$pch, col = dot.symbol$col, cex = dot.symbol$cex, font = dot.symbol$font, abline, \\dots) setTrellis(strip.blank=TRUE, lty.dot.line=2, lwd.dot.line=1) numericScale(x, label=NULL, \\dots) } \\arguments{ \\item{\\dots}{ for \\code{Cbind} \\code{\\dots} is any number of additional numeric vectors. Unless you are using \\code{Dotplot} (which allows for either 2 or 4 \"other\" variables) or \\code{xYplot} with \\code{method=\"bands\"}, vectors after the first two are ignored. If drawing bars and only one extra variable is given in \\code{\\dots}, upper and lower values are computed as described above. If the second argument to \\code{Cbind} is a matrix, that matrix is stored in the \\code{\"other\"} attribute and arguments after the second are ignored. For bubble plots, name an argument \\code{cex}. Also can be other arguments to pass to \\code{labcurve}. } \\item{formula}{ a \\code{trellis} formula consistent with \\code{xyplot} or \\code{dotplot} } \\item{x}{ \\code{x}-axis variable. For \\code{numericScale} \\code{x} is any vector such as \\code{as.numeric(x)} returns a numeric vector suitable for x- or y-coordinates. } \\item{y}{ a vector, or an object created by \\code{Cbind} for \\code{xYplot}. \\code{y} represents the main variable to plot, i.e., the variable used to draw the main lines. For \\code{Dotplot} the first argument to \\code{Cbind} will be the main \\code{x}-axis variable. } \\item{data,subset,ylim,subscripts,groups,type,scales,panel,prepanel,xlab,ylab}{ see \\code{trellis.args}. \\code{xlab} and \\code{ylab} get default values from \\code{\"label\"} attributes. } \\item{xscale}{allows one to use the default \\code{scales} but specify only the \\code{x} component of it for \\code{Dotplot}} \\item{method}{ defaults to \\code{\"bars\"} to draw error-bar type plots. See meaning of other values above. \\code{method} can be a function. Specifying \\code{method=quantile}, \\code{methodArgs=list(probs=c(.5,.25,.75))} is the same as specifying \\code{method=\"quantile\"} without specifying \\code{probs}. } \\item{methodArgs}{ a list containing optional arguments to be passed to the function specified in \\code{method} } \\item{label.curves}{ set to \\code{FALSE} to suppress invocation of \\code{labcurve} to label primary curves where they are most separated or to draw a legend in an empty spot on the panel. You can also set \\code{label.curves} to a list of options to pass to \\code{labcurve}. These options can also be passed as \\code{\\dots} to \\code{xYplot}. See the examples below. } \\item{abline}{ a list of arguments to pass to \\code{panel.abline} for each panel, e.g. \\code{list(a=0, b=1, col=3)} to draw the line of identity using color 3. To make multiple calls to \\code{panel.abline}, pass a list of unnamed lists as \\code{abline}, e.g., \\code{abline=list(list(h=0),list(v=1))}. } \\item{probs}{ a vector of three quantiles with the quantile corresponding to the central line listed first. By default \\code{probs=c(.5, .25, .75)}. You can also specify \\code{probs} through \\code{methodArgs=list(probs=\\dots)}. } \\item{nx}{ number of target observations for each \\code{x} group (see \\code{cut2} \\code{m} argument). \\code{nx} defaults to the minimum of 40 and the number of points in the current stratum divided by 4. Set \\code{nx=FALSE} or \\code{nx=0} if \\code{x} is already discrete and requires no grouping. } \\item{cap}{ the half-width of horizontal end pieces for error bars, as a fraction of the length of the \\code{x}-axis } \\item{lty.bar}{ line type for bars } \\item{lwd, lty, pch, cex, font, col}{ see \\code{trellis.args}. These are vectors when \\code{groups} is present, and the order of their elements corresponds to the different \\code{groups}, regardless of how many bands or bars are drawn. If you don't specify \\code{lty.bands}, for example, all band lines within each group will have the same \\code{lty}. } \\item{lty.bands, lwd.bands, col.bands}{ used to allow \\code{lty}, \\code{lwd}, \\code{col} to vary across the different band lines for different \\code{groups}. These parameters are vectors or lists whose elements correspond to the added band lines (i.e., they ignore the central line, whose line characteristics are defined by \\code{lty}, \\code{lwd}, \\code{col}). For example, suppose that 4 lines are drawn in addition to the central line. Specifying \\code{lwd.bands=1:4} will cause line widths of 1:4 to be used for every group, regardless of the value of \\code{lwd}. To vary characteristics over the \\code{groups} use e.g. \\code{lwd.bands=list(rep(1,4), rep(2,4))} or \\code{list(c(1,2,1,2), c(3,4,3,4))}. } \\item{minor.ticks}{ a list with elements \\code{at} and \\code{labels} specifying positions and labels for minor tick marks to be used on the x-axis of each panel, if any. } \\item{sub}{an optional subtitle} \\item{col.fill}{ used to override default colors used for the bands in method='filled bands'. This is a vector when \\code{groups} is present, and the order of the elements corresponds to the different \\code{groups}, regardless of how many bands are drawn. The default colors for 'filled bands' are pastel colors matching the default colors superpose.line$col (plot.line$col) } \\item{size}{ a vector the same length as \\code{x} giving a variable whose values are a linear function of the size of the symbol drawn. This is used for example for bubble plots. } \\item{rangeCex}{ a vector of two values specifying the range in character sizes to use for the \\code{size} variable (lowest first, highest second). \\code{size} values are linearly translated to this range, based on the observed range of \\code{size} when \\code{x} and \\code{y} coordinates are not missing. Specify a single numeric \\code{cex} value for \\code{rangeCex} to use the first character of each observations's \\code{size} as the plotting symbol. } \\item{strip.blank}{ set to \\code{FALSE} to not make the panel strip backgrounds blank } \\item{lty.dot.line}{ line type for dot plot reference lines (default = 1 for dotted; use 2 for dotted) } \\item{lwd.dot.line}{ line thickness for reference lines for dot plots (default = 1) } \\item{label}{ a scalar character string to be used as a variable label after \\code{numericScale} converts the variable to numeric form } } \\value{ \\code{Cbind} returns a matrix with attributes. Other functions return standard \\code{trellis} results. } \\section{Side Effects}{ plots, and \\code{panel.xYplot} may create temporary \\code{Key} and \\code{sKey} functions in the session frame. } \\details{ Unlike \\code{xyplot}, \\code{xYplot} senses the presence of a \\code{groups} variable and automatically invokes \\code{panel.superpose} instead of \\code{panel.xyplot}. The same is true for \\code{Dotplot} vs. \\code{dotplot}. } \\author{ Frank Harrell \\cr Department of Biostatistics \\cr Vanderbilt University \\cr \\email{f.harrell@vanderbilt.edu} \\cr Madeline Bauer \\cr Department of Infectious Diseases \\cr University of Southern California School of Medicine \\cr \\email{mbauer@usc.edu} } \\seealso{ \\code{\\link[lattice]{xyplot}}, \\code{\\link[lattice]{panel.xyplot}}, \\code{\\link{summarize}}, \\code{\\link{label}}, \\code{\\link{labcurve}}, \\code{\\link{errbar}}, \\code{\\link[lattice:xyplot]{dotplot}}, \\code{\\link{reShape}}, \\code{\\link{cut2}}, \\code{\\link[lattice:panel.functions]{panel.abline}} } \\examples{ # Plot 6 smooth functions. Superpose 3, panel 2. # Label curves with p=1,2,3 where most separated d <- expand.grid(x=seq(0,2*pi,length=150), p=1:3, shift=c(0,pi)) xYplot(sin(x+shift)^p ~ x | shift, groups=p, data=d, type='l') # Use a key instead, use 3 line widths instead of 3 colors # Put key in most empty portion of each panel xYplot(sin(x+shift)^p ~ x | shift, groups=p, data=d, type='l', keys='lines', lwd=1:3, col=1) # Instead of implicitly using labcurve(), put a # single key outside of panels at lower left corner xYplot(sin(x+shift)^p ~ x | shift, groups=p, data=d, type='l', label.curves=FALSE, lwd=1:3, col=1, lty=1:3) Key() # Bubble plots x <- y <- 1:8 x[2] <- NA units(x) <- 'cm^2' z <- 101:108 p <- factor(rep(c('a','b'),4)) g <- c(rep(1,7),2) data.frame(p, x, y, z, g) xYplot(y ~ x | p, groups=g, size=z) Key(other=list(title='g', cex.title=1.2)) # draw key for colors sKey(.2,.85,other=list(title='Z Values', cex.title=1.2)) # draw key for character sizes # Show the median and quartiles of height given age, stratified # by sex and race. Draws 2 sets (male, female) of 3 lines per panel. # xYplot(height ~ age | race, groups=sex, method='quantiles') # Examples of plotting raw data dfr <- expand.grid(month=1:12, continent=c('Europe','USA'), sex=c('female','male')) set.seed(1) dfr <- upData(dfr, y=month\/10 + 1*(sex=='female') + 2*(continent=='Europe') + runif(48,-.15,.15), lower=y - runif(48,.05,.15), upper=y + runif(48,.05,.15)) xYplot(Cbind(y,lower,upper) ~ month,subset=sex=='male' & continent=='USA', data=dfr) xYplot(Cbind(y,lower,upper) ~ month|continent, subset=sex=='male',data=dfr) xYplot(Cbind(y,lower,upper) ~ month|continent, groups=sex, data=dfr); Key() # add ,label.curves=FALSE to suppress use of labcurve to label curves where # farthest apart xYplot(Cbind(y,lower,upper) ~ month,groups=sex, subset=continent=='Europe', data=dfr) xYplot(Cbind(y,lower,upper) ~ month,groups=sex, type='b', subset=continent=='Europe', keys='lines', data=dfr) # keys='lines' causes labcurve to draw a legend where the panel is most empty xYplot(Cbind(y,lower,upper) ~ month,groups=sex, type='b', data=dfr, subset=continent=='Europe',method='bands') xYplot(Cbind(y,lower,upper) ~ month,groups=sex, type='b', data=dfr, subset=continent=='Europe',method='upper') label(dfr$y) <- 'Quality of Life Score' # label is in Hmisc library = attr(y,'label') <- 'Quality\\dots'; will be # y-axis label # can also specify Cbind('Quality of Life Score'=y,lower,upper) xYplot(Cbind(y,lower,upper) ~ month, groups=sex, subset=continent=='Europe', method='alt bars', offset=unit(.1,'inches'), type='b', data=dfr) # offset passed to labcurve to label .4 y units away from curve # for R (using grid\/lattice), offset is specified using the grid # unit function, e.g., offset=unit(.4,'native') or # offset=unit(.1,'inches') or unit(.05,'npc') # The following example uses the summarize function in Hmisc to # compute the median and outer quartiles. The outer quartiles are # displayed using \"error bars\" set.seed(111) dfr <- expand.grid(month=1:12, year=c(1997,1998), reps=1:100) month <- dfr$month; year <- dfr$year y <- abs(month-6.5) + 2*runif(length(month)) + year-1997 s <- summarize(y, llist(month,year), smedian.hilow, conf.int=.5) xYplot(Cbind(y,Lower,Upper) ~ month, groups=year, data=s, keys='lines', method='alt', type='b') # Can also do: s <- summarize(y, llist(month,year), quantile, probs=c(.5,.25,.75), stat.name=c('y','Q1','Q3')) xYplot(Cbind(y, Q1, Q3) ~ month, groups=year, data=s, type='b', keys='lines') # Or: xYplot(y ~ month, groups=year, keys='lines', nx=FALSE, method='quantile', type='b') # nx=FALSE means to treat month as a discrete variable # To display means and bootstrapped nonparametric confidence intervals # use: s <- summarize(y, llist(month,year), smean.cl.boot) s xYplot(Cbind(y, Lower, Upper) ~ month | year, data=s, type='b') # Can also use Y <- cbind(y, Lower, Upper); xYplot(Cbind(Y) ~ ...) # Or: xYplot(y ~ month | year, nx=FALSE, method=smean.cl.boot, type='b') # This example uses the summarize function in Hmisc to # compute the median and outer quartiles. The outer quartiles are # displayed using \"filled bands\" s <- summarize(y, llist(month,year), smedian.hilow, conf.int=.5) # filled bands: default fill = pastel colors matching solid colors # in superpose.line (this works differently in R) xYplot ( Cbind ( y, Lower, Upper ) ~ month, groups=year, method=\"filled bands\" , data=s, type=\"l\") # note colors based on levels of selected subgroups, not first two colors xYplot ( Cbind ( y, Lower, Upper ) ~ month, groups=year, method=\"filled bands\" , data=s, type=\"l\", subset=(year == 1998 | year == 2000), label.curves=FALSE ) # filled bands using black lines with selected solid colors for fill xYplot ( Cbind ( y, Lower, Upper ) ~ month, groups=year, method=\"filled bands\" , data=s, label.curves=FALSE, type=\"l\", col=1, col.fill = 2:3) Key(.5,.8,col = 2:3) #use fill colors in key # A good way to check for stable variance of residuals from ols # xYplot(resid(fit) ~ fitted(fit), method=smean.sdl) # smean.sdl is defined with summary.formula in Hmisc # Plot y vs. a special variable x # xYplot(y ~ numericScale(x, label='Label for X') | country) # For this example could omit label= and specify # y ~ numericScale(x) | country, xlab='Label for X' # Here is an example of using xYplot with several options # to change various Trellis parameters, # xYplot(y ~ x | z, groups=v, pch=c('1','2','3'), # layout=c(3,1), # 3 panels side by side # ylab='Y Label', xlab='X Label', # main=list('Main Title', cex=1.5), # par.strip.text=list(cex=1.2), # strip=function(\\dots) strip.default(\\dots, style=1), # scales=list(alternating=FALSE)) # # Dotplot examples # s <- summarize(y, llist(month,year), smedian.hilow, conf.int=.5) setTrellis() # blank conditioning panel backgrounds Dotplot(month ~ Cbind(y, Lower, Upper) | year, data=s) # or Cbind(\\dots), groups=year, data=s # Display a 5-number (5-quantile) summary (2 intervals, dot=median) # Note that summarize produces a matrix for y, and Cbind(y) trusts the # first column to be the point estimate (here the median) s <- summarize(y, llist(month,year), quantile, probs=c(.5,.05,.25,.75,.95), type='matrix') Dotplot(month ~ Cbind(y) | year, data=s) # Use factor(year) to make actual years appear in conditioning title strips # Plot proportions and their Wilson confidence limits set.seed(3) d <- expand.grid(continent=c('USA','Europe'), year=1999:2001, reps=1:100) # Generate binary events from a population probability of 0.2 # of the event, same for all years and continents d\\$y <- ifelse(runif(6*100) <= .2, 1, 0) s <- with(d, summarize(y, llist(continent,year), function(y) { n <- sum(!is.na(y)) s <- sum(y, na.rm=TRUE) binconf(s, n) }, type='matrix') ) Dotplot(year ~ Cbind(y) | continent, data=s, ylab='Year', xlab='Probability') # Dotplot(z ~ x | g1*g2) # 2-way conditioning # Dotplot(z ~ x | g1, groups=g2); Key() # Key defines symbols for g2 # If the data are organized so that the mean, lower, and upper # confidence limits are in separate records, the Hmisc reShape # function is useful for assembling these 3 values as 3 variables # a single observation, e.g., assuming type has values such as # c('Mean','Lower','Upper'): # a <- reShape(y, id=month, colvar=type) # This will make a matrix with 3 columns named Mean Lower Upper # and with 1\/3 as many rows as the original data } \\keyword{hplot} \\concept{trellis} \\concept{lattice}","date":"2019-08-25 13:47:49","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\": 0.5786404609680176, \"perplexity\": 8645.910904241146}, \"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\/1566027330233.1\/warc\/CC-MAIN-20190825130849-20190825152849-00026.warc.gz\"}"}
| null | null |
Q: Is there a way for changing text file names in a folder using C++ I am working with a bunch of txt files(thousands) on my project. Each txt file has 'csv' information on it. The problem is that each txt file has a random name and I cannot create a code for loading them in my project due to it. So, I want to rename them in a particular pattern to make easier the loading of the files in my work. I will use C++ for accomplish this task.
I put all the txt files in a folder but I cannot see a way of renaming them using C++. How can I do this? is there a way to do it? Can someone help me?
A: You can use std::filesystem::directory_iterator and std::filesystem::rename (c++17), as documented here.
A: Disclaimer
This answer validity is based on a comment where the author precised they were not bound to the C++ language (it may be worth editing the question, the C++ tag, and the OS). This solution may work for UNIX systems supporting bash, that is most Linux distributions and all releases of Apple's macOS prior to macOS Catalina (correct me if I'm wrong).
Bash command line
Using the following bash command should rename all the files in a folder with increasing numbers, that is:
*
*toto.csv -> 1.csv
*titi.csv -> 2.csv etc
It assumes the ordering is not important.
a=1; for i in *; do mv -n "$i" "$a.csv" ; let "a +=1"; done
To test it, you can prepare a test folder by opening a terminal and typing:
mkdir test
cd test
touch toto.csv titi.csv tata.csv
ls
Output:
tata.csv titi.csv toto.csv
Then you can run the following command:
a=1; for i in *; do mv -n "$i" "$a.csv" ; let "a +=1"; done
ls
Output:
1.csv 2.csv 3.csv
Explication:
*
*a=1 declare a variable
*for i in *; begin to iterate over all files in the folder
*do mv will move (rename) a file of the list (that is, the variable $i) to a new name called a.csv
*and we increment the counter a, and close the loop.
*the option -n will make sure no file gets overwritten by the command mv
I assumed there was no specific criterion to rename the files. If there is a specific structure (pattern) in the renaming, the bash command can probably accommodate it, but the question should then give more details about these requirements :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 668
|
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,931
|
package com.obdobion.funnel.provider;
import java.io.File;
import java.io.IOException;
import java.text.ParseException;
/**
* <p>
* InputReader interface.
* </p>
*
* @author Chris DeGreef fedupforone@gmail.com
*/
public interface InputReader
{
/**
* <p>
* close.
* </p>
*
* @throws java.io.IOException if any.
* @throws java.text.ParseException if any.
*/
void close() throws IOException, ParseException;
/**
* <p>
* length.
* </p>
*
* @return a long.
* @throws java.io.IOException if any.
*/
long length() throws IOException;
/**
* <p>
* open.
* </p>
*
* @param _inputFile a {@link java.io.File} object.
* @throws java.io.IOException if any.
* @throws java.text.ParseException if any.
*/
void open(File _inputFile) throws IOException, ParseException;
/**
* <p>
* position.
* </p>
*
* @return a long.
* @throws java.io.IOException if any.
*/
long position() throws IOException;
/**
* <p>
* read.
* </p>
*
* @param row an array of byte.
* @return a int.
* @throws java.io.IOException if any.
*/
int read(final byte[] row) throws IOException;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,914
|
Back to Journals » International Journal of Chronic Obstructive Pulmonary Disease » Volume 9 » Issue 1
Clinical issues of mucus accumulation in COPD
Authors Ramos F, Krahnke J, Kim V
Received 18 September 2013
Accepted for publication 2 December 2013
Published 24 January 2014 Volume 2014:9(1) Pages 139—150
DOI https://doi.org/10.2147/COPD.S38938
Frederick L Ramos, Jason S Krahnke, Victor Kim
Division of Pulmonary and Critical Care Medicine, Department of Medicine, Temple University School of Medicine, Philadelphia, PA, USA
Abstract: Airway mucus is part of the lung's native immune function that traps particulates and microorganisms, enabling their clearance from the lung by ciliary transport and cough. Mucus hypersecretion and chronic productive cough are the features of the chronic bronchitis and chronic obstructive pulmonary disease (COPD). Overproduction and hypersecretion by goblet cells and the decreased elimination of mucus are the primary mechanisms responsible for excessive mucus in chronic bronchitis. Mucus accumulation in COPD patients affects several important outcomes such as lung function, health-related quality of life, COPD exacerbations, hospitalizations, and mortality. Nonpharmacologic options for the treatment of mucus accumulation in COPD are smoking cessation and physical measures used to promote mucus clearance. Pharmacologic therapies include expectorants, mucolytics, methylxanthines, beta-adrenergic receptor agonists, anticholinergics, glucocorticoids, phosphodiesterase-4 inhibitors, antioxidants, and antibiotics.
Keywords: chronic obstructive pulmonary disease, chronic bronchitis, mucus, sputum
A Letter to the Editor has been received and published for this article.
Chronic obstructive pulmonary disease (COPD) is characterized by a persistent airflow limitation that is associated with an enhanced chronic inflammatory response to noxious particles or gases.1 The World Health Organization estimates that over 200 million people have COPD worldwide, and it also predicts that COPD will be the third leading cause of death in the world by 2030, which is higher than its place in 2004 as the fourth leading cause of death.2 The increased output from goblet cells and mucous glands in COPD patients is variably described as "chronic mucus hypersecretion", "chronic sputum production", or "chronic bronchitis" (CB). Sputum and mucus are commonly used interchangeably, but these are distinct substances. While mucus is generally cleared by cilia, the ciliated epithelium becomes damaged with chronic inflammation and the increased volume of secretions often requires clearance by cough. Sputum refers to the expectorated secretions.3 CB is commonly defined as the presence of a chronic, productive cough and sputum production for at least 3 consecutive months in 2 consecutive years. This review presents the normal anatomy and physiology related to airway mucus and the pathophysiology of increased mucus production in COPD. Clinical consequences of mucus overproduction, as well as its therapeutic options, are also discussed.
Normal anatomy and physiology related to airway mucus
Airway mucus is secreted by goblet cells found in the superficial mucosa and the mucous glands in the submucosa. Goblet cells decrease in number further into the airways, and they eventually disappear at the level of the terminal bronchioles. The quantity of mucous glands, which produce the majority of the airway mucus, decrease distally as they approach the respiratory bronchioles. The mucosa is a surface layer of pseudostratified columnar epithelial cells that have cilia on their luminal surfaces. The rhythmic beating of the cilia enables the "mucociliary elevator" that transports mucus and liquid, as well as inhaled particles, pathogens, and dissolved chemicals, from distal to proximal airways. After the mucus ascends the trachea, it is driven through the vocal cords by the larynx ciliary epithelium. The mucus is then swallowed after the pharynx is entered. The vocal cords are covered by squamous epithelium, so they do not have cilia, but they promote cough clearance by closing, while expiratory pressure builds; they then open suddenly so airflow is forceful.4,5 The secreted mucins – in particular, the polymeric mucins MUC5AC and MUC5B – serve as the organizing framework of the mucus gel in the airways. The mucins also prevent barrier dehydration, present carbohydrate ligands to sequester the pathogens, and via binding to other components of the secretion, they have the potential to act as sinks for host-protective proteins and peptides.6
Airway mucus is part of the lung's innate immune function that traps particulates and microorganisms, facilitating their clearance from the lung by means of ciliary transport or cough.3,6 In normal conditions, mucin production efficiently defends the airways. However, in mucin secretory cell hyperplasia and metaplasia, there is overproduction, with pathological consequences.6,7
Mechanism of mucus accumulation in COPD
Mucus hypersecretion and chronic productive cough is a feature of CB.1 The primary mechanisms responsible for excessive mucus production in CB in COPD are the overproduction and hypersecretion by goblet cells, and the decreased elimination of mucus.7 There is also hypertrophy of the submucosal glands that Reid8 described with a ratio of the thickness of the submucosal glands and the thickness between the epithelium and cartilage that covers the bronchi. The size of the submucosal glands correlates with the degree of airway inflammation (Figure 1).9
Mucus hypersecretion in COPD is a consequence of cigarette smoke exposure,10,11 acute and chronic viral infection,12 bacterial infection,13 or inflammatory cell activation of mucin gene transcription.13 This leads to the overproduction of mucus and to hypersecretion from increased degranulation, primarily by neutrophil elastase. This is compounded by a difficulty in clearing secretions because of poor ciliary function, distal airway occlusion, and an ineffective cough that is secondary to respiratory muscle weakness and reduced peak expiratory flow.13–15
Figure 1 Causes of excessive mucus in COPD.
Notes: Reprinted with permission of the American Thoracic Society. Copyright © 2013 American Thoracic Society. Kim V, Criner GJ, 2013, Chronic bronchitis and chronic obstructive pulmonary disease. Am J Respir Crit Care Med. 187(3):228–237.7 Official journal of the American Thoracic Society.
Abbreviations: PEF, peak expiratory flow; COPD, chronic obstructive pulmonary disease.
Under increased airway inflammation, the airway epithelium remodels and undergoes metaplasia, implying that there is a phenotypic change that occurs within an adult cell type, and that hyperplasia also occurs, denoting an increase in the total cell number within a given tissue type.16 Saetta et al's17 study of surgical specimens shows that smokers with both CB and airflow limitation have an increased number of goblet cells and inflammatory cells in peripheral airway epithelium. Innes et al18 revealed that goblet cell hypertrophy and hyperplasia occur in the large airways of habitual cigarette smokers, and this hypertrophy results in epithelial mucin stores that are significantly higher than normal; Figures 2 and 3 show examples of airway epithelium remodeling.
Figure 2 MM and smooth muscle hypertrophy in a small airway from a COPD patient.
Note: Hematoxylin and eosin stain.
Abbreviations: MM, mucous metaplasia; SMH, smooth muscle hypertrophy; COPD, chronic pulmonary disease.
Figure 3 Goblet cell hyperplasia.
Notes: Periodic acid Schiff–Alcian Blue stain. Higher magnification of a small airway quadrant is shown here. Goblet cells appear in an intense blue–purple color with periodic acid Schiff–Alcian Blue stain.
Cellular and molecular mechanisms in the pathogenesis of mucus hypersecretion in CB include acquired cystic fibrosis transmembrane conductance regulator dysfunction19 and activation of the epidermal growth factor receptor.20 Smokers with and without COPD have reduced chloride conductance in the lower airway, and this ion transport abnormality is associated with the presence of CB and dyspnea.19 Cigarette smoke also increases mucin MUC5AC synthesis via epidermal growth factor receptor activation in the airway epithelial cells.20 Upregulation of the basic fibroblast growth factor21 and transforming growth factor-β,22 as well as a higher frequency of the tumor necrosis factor-α polymorphism23 have also been implicated in the pathogenesis of CB.
From a review of population-based studies, Kim and Criner7 estimated the prevalence of CB to be between 3.4%–22% among adults. The wide range of prevalence estimates may be due to varying definitions of CB (ie, chronic phlegm versus chronic cough and phlegm), as well as the possible inclusion of subjects with bronchiectasis.
The prevalence of CB is higher in COPD patients.13,24,25 In the Evaluation of COPD Longitudinally to Identify Predictive Surrogate Endpoints (ECLIPSE) cohort,25 34.6% of the 2,161 subjects reported the presence of CB, as defined by the American Thoracic Society–Diffuse Lung Disease (ATS-DLD) questionnaire ("phlegm on most days for 3 or more consecutive months during the year and trouble with phlegm for 2 or more years"). Of the 1,061 Global initiative for chronic Obstructive Lung Disease (Gold) 2–4 subjects in the COPDGene study,24 CB, by the same definition, was found in 27.3% of patients. The sex predilection of CB is unclear, with some studies finding males more affected,24,26,27 while others report a female predominance.28–30
Smoking is the primary risk factor for CB. In one study,31 the 30-year cumulative incidence of CB was 42% in continuous smokers, 26% in ex-smokers, and 22% in never-smokers. Occupational exposure to biological dust32 or to combustion byproducts, inorganic dusts, or fumes and organic dusts33 may also be risk factors. Gastroesophageal reflux disease is another possible risk factor for CB.34
Mucus accumulation and outcomes
CB affects important outcomes in COPD, including declines in lung function, health-related quality of life, as well as COPD exacerbations and hospitalizations, and mortality. Table 1 summarizes the findings of select studies.
Table 1 Selected studies on chronic bronchitis and outcomes
Notes: *Statistically significant. Data are presented as mean ± SD or number (percentage), except as indicated. IRR, OR, RR, and HR are all from multivariate analyses with adjustments for covariates.
Abbreviations: FEV1, forced expiratory volume in 1 second; SE, standard error; CI, confidence interval; FVC, forced vital capacity; OR, odds ratio; IRR, incidence rate ratio; HR, hazard ratio; CB+, group with chronic bronchitis; CB−, group without chronic bronchitis; GOLD, Global initiative for chronic Obstructive Lung Disease; SGRQ, St George's Respiratory Questionnaire; mMRC, modified Medical Research Council; RR, relative risk; SD, standard deviation; COPD, chronic obstructive pulmonary disease.
Several studies demonstrated that chronic mucus production is related to a progressive decline in lung function.35,36 In a longitudinal study that followed 1,757 men and 2,191 women for 12 years, the presence of chronic sputum production was associated with an accelerated loss in forced expiratory volume in one second (FEV1) in men of 4.5±2 mL/year after adjusting for height, age, and cigarette smoking, but a statistically insignificant decline in FEV1 in women of 1.7±1.5 mL/year.35 Using data from the Copenhagen City Heart Study,36 which was conducted with 5,354 females and 4,081 males, comparing two spirometry results 5 years apart, Vestbo et al found an excessive FEV1 decline of 22.8 mL/year (95% confidence interval [CI]: 8.2–37.4), after adjusting for age, height, weight change, and smoking, in males with chronic mucus hypersecretion compared with males without mucus hypersecretion; in women, the adjusted excess decline was not statistically significant at 12.6 mL/year (95% CI: 0.7–24.6).
The presence of CB may also predict the development of airflow obstruction.37–39 In an international population-based cohort study of young adults with normal lung function, chronic cough and phlegm predicts the development of COPD (defined as a ratio of FEV1 to forced vital capacity [FVC] <0.7) with an incidence rate ratio of 1.85 (95% CI: 1.17–2.93) after adjusting for smoking.37 Subjects who reported chronic cough and phlegm have a nearly threefold increased risk of developing COPD (FEV1/FVC <0.7) with respect to asymptomatic subjects (incidence rate ratio: 2.88; 95% CI: 1.44–5.79). Among the 1,412 participants in the Tucson Epidemiological Study of Airway Obstructive Disease,38 the presence of CB is an independent risk factor for incident airflow limitation among subjects <50 years old (adjusted hazard ratio [HR]: 2.2; 95% CI: 1.3–3.8), but not among subjects ≥50 years old (adjusted HR: 0.9; 95% CI: 0.6–1.4).
Several analyses show that chronic sputum production correlates with a worse quality of life and more limitations due to physical health.24–26 Our analysis of the COPDGene study reports higher modified Medical Research Council (mMRC) dyspnea scores and St George's Respiratory Questionnaire (SGRQ) scores in COPD subjects with CB symptoms.24 In the ECLIPSE cohort, the presence of CB is related to worse total scores on the mMRC and SGRQ across all COPD disease severities (GOLD II to IV).25 Among subjects with COPD in the Proyecto Latinoamericano de Investigacio´n en Obstruccio´n Pulmonar (PLATINO) study, those with CB have a worse general health status and more physical activity limitations.26
COPD exacerbations and hospitalization
Vestbo et al36 showed that hospitalization due to COPD is associated with chronic mucus hypersecretion with a relative risk of 2.4 (95% CI: 1.3–4.5) for males and 2.6 (95% CI: 1.2–5.3) for females. A cross-sectional multicenter analysis of 433 COPD subjects found that chronic cough and sputum were associated with frequent COPD exacerbations during the previous year (adjusted odds ratio [OR]: 4.15 [95% CI: 2.43–7.08]; P<0.0001), including severe exacerbations requiring hospitalizations (adjusted OR: 4.08 [95% CI: 1.18–14.09]; P=0.03).40 Examination of the COPDGene cohort concludes the fact that a history of exacerbations in the previous year is higher in the CB group (1.21±1.62 versus 0.63±1.12 per patient; P=0.027), and more subjects in that group reported a history of severe exacerbations (26.6% versus 20%; P=0.024).24
However, not all studies link CB with exacerbations and hospitalizations. In ECLIPSE,25 exacerbation frequency in the year before recruitment in those with and without CB was the same. In addition, an analysis of the PLATINO study26 showed a nonsignificant difference in exacerbation frequency between COPD subjects with and without CB.
The differing results could be explained by the numerous differences in the study population and in the design of the comparison of the selected studies that report COPD exacerbations or hospitalizations. Two studies are population-based and involve patients from Denmark36 or Latin America.26 The other three are cross-sectional studies involving patients from France,40 the United States,24 or from multiple nations.25 Vestbo et al36 used a definition of chronic mucus hypersecretion, while the other studies use the more classic definition of CB.24–26,40 The prevalence of CB in COPD patients using the classic definition is 14%,26 27%,24 35%,25 and 74%.40 Across the studies, there were also small differences in the percentage of current smokers and the severity of disease, as measured by airflow obstruction. In two studies with similar demographics and the number of current smokers, the study that showed a positive association between CB and exacerbations had more patients with frequent exacerbations,24 while the other did not show an association and it had fewer patients with frequent exacerbations.25 Therefore, prospective studies are needed to clarify the association between CB and COPD exacerbations and hospitalizations.
CB is a risk factor for respiratory-related27,31,41 and all-cause 31,38,42,43 mortality. In a Finnish study of 1,711 middle-aged males with 40-year mortality data, persistent CB predicts risk of respiratory-related deaths (adjusted HR: 2.54; 95% CI: 1–6.46; P=0.049) and all-cause mortality (adjusted HR: 1.64; 95% CI: 1.23–2.19; P=0.001), after adjusting for pulmonary function.31 Prescott et al,41 who followed 14,223 participants for 10–12 years, reported that chronic mucus hypersecretion is a significant predictor of death from pulmonary infection, with a multivariate relative risk (RR) of 3.5 (95% CI: 1.8–7.1), but not of death without pulmonary infection. In a 9- to 12-year mortality follow-up of 8,427 Caucasian adults, Speizer et al27 reported the association of cough or phlegm and an increase in COPD mortality with an adjusted OR of 3.75 (95% CI: 1.28–11) in men and 11.04 (95% CI: 2.52–48.5) in women. In the Tuscon Epidemiologic Survey of Airway Obstructive Disease,38 the risk of all-cause mortality was higher in patients younger than 50 years of age with CB (HR: 2.2; 95% CI: 1.3–3.8), but not among subjects 50 years of age or older (HR: 1; 95% CI: 0.7–1.3). In an analysis of the 13,756 subjects in the Copenhagen City Heart Study,42 the presence of chronic mucus hypersecretion increased mortality from all causes with an adjusted RR of 1.3 (95% CI: 1.1–1.4) in males and 1.1 (95% CI: 0.9–1.3) in females. An investigation of 1,061 French males showed a multivariate RR of 1.35 (P<0.01) for the relationship between chronic mucus hypersecretion and mortality during a 22-year follow-up.43 The mechanism behind this association in unclear, but a possible cause could be the increased inflammatory state seen in those with CB, leading to increased cardiovascular events.38
Other studies, however, have not shown a statistically significant relationship between chronic mucus production and mortality. There is a statistically insignificant trend towards death in Tockman and Comstock's study44 of 10-year mortality in 884 males with chronic phlegm production (RR: 1.65; 95% CI: 0.95–2.89). Mannino et al45 reported similar findings for risk of death in the presence of respiratory symptoms (cough, sputum, or wheeze) without obstructive lung disease.
Numerous differences are apparent after comparing the different studies that report mortality. These differences could account for the inconsistent relationship between CB and mortality. First, as mentioned previously, the definition used of mucus accumulation with or without spirometry-defined airflow obstruction varies. The negative studies use chronic phlegm production44 or a combination of respiratory symptoms (cough, sputum, or wheeze).45 The positive studies use the classic CB definition,31,38 or symptoms of chronic mucus hypersecretion,41,42 or chronic phlegm27,43. All of the symptoms are self-reported, and so responses may be varied depending on how the question is phrased. Second, standard of care likely varies in the populations being studied, as the research on this subject matter includes patients from all areas of the globe at different points in time. Third, the association of mortality and chronic sputum, phlegm production, or CB may be influenced by factors such as the presence of COPD or smoking history. Most of the studies report results that have adjusted for these covariates, but unmeasured differences associated with the presence of these factors may influence the findings. Fourth, some authors suggest that the relationship between mucus accumulation and mortality is further affected by severity of airflow obstruction,27 infection,41 and age.38
The main goals of therapy should target the different pathophysiologic mechanisms of CB by reducing mucus overproduction, decreasing mucus hypersecretion by controlling inflammation, facilitating mucus elimination by increasing ciliary transport, reducing mucus tenacity, increasing shear stress to augment mucus detachment, and modifying cough (Table 2).
Table 2 Summary of therapeutic interventions for chronic bronchitis
Abbreviations: PT, physiotherapy; HFCWO, high frequency chest wall oscillation; DNA, deoxyribonucleic acid; SABA, short-acting beta agonist; LABA, long-acting beta agonist; PEF, peak expiratory flow; PDE-4, phosphodiesterase-4.
Nonpharmacological therapy
Smoking cessation can improve cough in many patients with CB by improving mucociliary function and by decreasing goblet cell hyperplasia.46 Smoking cessation has also been shown to decrease airway injury and lower levels of mucus in exfoliated sputum tracheobronchial cells when compared to those that continued to smoke.47 A large longitudinal follow-up study found that the incidence rates of CB were much higher in current smokers compared to ex-smokers (42% versus 26%, respectively).31 Unfortunately, there is a paucity of data regarding the effects of smoking cessation on sputum symptomatology.
Physical measures
Mucus clearance is aided by maneuvers that promote coughing and increase minute ventilation. This augments shear stresses on mucosal surfaces generated by increased airflow. It also increases humidification of the airway and regulates mucus hydration. Thus, methods such as the application of positive expiratory pressure and use of flutter valves or high-frequency chest compression vests may be of value, but they have not been studied for use in COPD in large clinical trials. Although cystic fibrosis studies have demonstrated that chest percussion and postural drainage improve mucociliary clearance,48 these methods have not been well studied in the COPD patient population. There are a few trials that have studied chest physiotherapy or directed coughing techniques in COPD.49 These trials have shown some improvements in mucus clearance, but no changes in lung function.
Pharmacological therapy
Expectorants and mucolytics
Guaifenesin works by promoting vagally-mediated increases in airway secretions.50 Long-term use of guaifenesin has not been shown to be of benefit in COPD or CB.51 Inhaled hypertonic saline works by rehydrating mucus by drawing water from the epithelial cells and by promoting cough.52,53 While this method has shown improvement in lung function in cystic fibrosis, it has only been shown in one study in COPD to improve dyspnea and exercise capacity.54 Inhaled dornase alfa hydrolyzes deoxyribonucleic acid (DNA), thereby improving lung function and decreasing exacerbation frequency in cystic fibrosis patients, in whom airway mucus concentrations of DNA are high. However, the concentration of DNA in the sputum of COPD patients is much lower,55 and studies have shown that dornase alfa is not beneficial and, in fact, it may be harmful.56
Methylxanthines and beta-adrenergic receptor agonists
Both methylxanthines and short-acting beta-adrenergic receptor agonists promote mucus clearance by increasing airway luminal diameter, increasing ciliary beat frequency via an increase in intracellular cyclic adenosine monophosphate levels, and increasing mucus hydration by stimulating airway Cl− secretion via activation of the cystic fibrosis transmembrane regulator. This decreases mucus viscosity, allowing for easier transport by airway cilia.57–59
In animal models, short-term administration of beta-agonists is associated with upregulation of mucociliary clearance.60,61 Similarly, methylxanthines improve mucociliary clearance not only via their bronchodilatory properties, but also by stimulating ciliary beat frequency, augmenting airway epithelial ion transport to increase mucus hydration, and promoting mucus secretion in the lower airways.62 Clinical studies of theophylline in CB have shown improved lung function, but no consistent change in cough and sputum production.63,64
The effects of long-acting beta-adrenergic receptor agonists on mucociliary function have been attributed to their beneficial effects on lung function.64–67 Long-acting beta-adrenergic receptor agonists also reduce hyperinflation and increase peak expiratory flow, which are essential components of effective cough.68 In vitro evidence has shown that salmeterol can stimulate ciliary beat frequency.58 Similarly, formoterol significantly improves mucociliary clearance when compared with placebo in patients with bronchitis.69
Anticholinergics, by their action on the muscarinic receptor, are believed to help mucus clearance by increasing luminal diameter and by decreasing surface and submucosal gland mucin secretion.70–72 They are also thought to facilitate cough-induced mucus clearance. However, anticholinergics may desiccate airway secretions by depleting airway surface liquid, thereby making secretions more difficult to expectorate. In vivo, the literature does not support the use of anticholinergics for the treatment of CB. Ipratropium bromide has been shown to reduce the quantity and severity of coughs in CB,72 but it is not effective in improving the mucociliary clearance in COPD.73 Tiotropium has been shown to improve lung function74 and reduce cough, but mucociliary clearance was not improved.75
There is in vitro evidence that glucocorticoids reduce inflammation and mucus production.76,77 In a murine model of asthma, inhaled corticosteroids decrease goblet cell hyperplasia.78 Dexamethasone has also been shown to decrease epithelial mucin gene, MUC5AC, expression in human bronchial epithelial cells;79 glucocorticoids may also hasten mucociliary clearance.80 Inhaled corticosteroids reduce exacerbation frequency and improve quality of life scores in COPD.81–83 Whether inhaled corticosteroids are more beneficial in COPD patients with CB or in airway-predominant phenotypes remains to be determined.
Phosphodiesterase-4 inhibitors
Phosphodiesterase-4 (PDE-4) inhibition decreases inflammation and promotes smooth-muscle relaxation in the airways by preventing the hydrolysis of cyclic adenosine monophosphate to its inactive metabolite. Cilomilast and roflumilast are highly specific second-generation oral PDE-4 inhibitors. A meta-analysis of 23 randomized trials of roflumilast or cilomilast compared with placebo found that treatment with a PDE-4 inhibitor only modestly increased FEV1 (45.59 mL; 95% CI: 39.1–52.03), but it reduced the likelihood of an exacerbation (OR: 0.78; 95% CI: 0.72–0.85).84 Roflumilast has been shown to significantly improve prebronchodilator FEV1, decrease the rate of moderate to severe exacerbations, and decrease the total number of exacerbations by 17% (95% CI: 8–25).85 Two trials have evaluated the use of roflumilast in moderate to severe COPD patients.86 The majority of patients (78%–100%) enrolled had chronic cough and sputum production at baseline. One study randomized patients to roflumilast plus salmeterol or salmeterol alone, and the second study randomized patients to roflumilast plus tiotropium or tiotropium alone.87 In both trials, roflumilast significantly improved the primary endpoint, prebronchodilator FEV1, as well as the exacerbation rate. Thus, as CB increases the risk for exacerbation, PDE-4 inhibitors may play a preferential role in preventing the development of exacerbation in patients with CB and COPD.
Given that oxidative stress is crucial to the pathogenesis of COPD,87 antioxidant therapy may be of benefit in COPD treatment. Thiol compounds are powerful antioxidants and include N-acetylcysteine (NAC), N-acestyln, carbocysteine, erdosteine, and fudosteine. The two most extensively studied antioxidant medications for COPD are NAC and carbocysteine. NAC is a precursor of L-cysteine and reduced glutathione, which reduces the cellular levels of oxidative stress and the production of reactive oxygen species. NAC also reduces disulfide bonds and sulfhydryl bonds that link together mucin polymers, thereby reducing sputum viscosity. Carbocysteine is a blocked thiol derivative of L-cysteine with in vitro free radical scavenging and anti-inflammatory properties, and it may work on the fucose and sialic acid content in mucus.88
The Bronchitis Randomized On NAC Cost-Utility Study89 (BRONCHUS) is the largest trial of N-acetylcysteine use in COPD to date. In this multicenter study, 523 patients with a mean predicted FEV1 of 57% were randomized to NAC 600 mg daily or placebo, and they were followed for 3 years. The mean exacerbation rates of the subjects were 2.4–2.5 exacerbations/year. There were no differences in FEV1 decline in terms of time or health-related quality of life between the two groups. There was also no overall difference in the number of exacerbations. However, in a post hoc analysis, those without inhaled corticosteroids (about 30% of the entire group) had a significant reduction in exacerbations with NAC when compared to placebo.
A more recent study, the High-Dose N-Acetylcysteine in Stable COPD (HIACE) study,90 enrolled 120 subjects with stable COPD who were randomized to receive NAC 600 mg twice daily or placebo daily for 1 year. The primary outcomes were change in small airway function, as assessed by forced expiratory flow at 25% to 75% (FEF 25%–75%) and forced oscillation technique parameters, which were measured by applying external oscillatory pressure during tidal breathing. The secondary outcomes measurements were change in symptoms, as assessed by the mMRC dyspnea scale and SGRQ scores, exacerbation frequency, and hospitalizations. The patients were predominantly male (93.2%) with a mean age of 70.8±0.74 years and a predicted FEV1% of 53.9%±2.0%. There were no differences in baseline FEF25–75, mMRC dyspnea score, and exacerbation frequency within the previous year prior to enrollment between the two groups. Patients in the NAC group had exhibited decreased small airway resistance over the duration of the study when compared to the placebo group. The NAC group had a statistically significant increase (P=0.037) in FEF25–75, (0.72±0.07 L/second versus 0.80 ±0.07 L/second) compared to placebo (0.679±0.07 L/second to 0.677±0.07 L/second), and the reactance at 6 Hz improved in the NAC group by 23%, which was compared to a decrease in the placebo group by 10.7% (P=0.04). Additionally, the mean exacerbation frequency in the NAC group was lower (0.96/year) compared to the placebo group (1.71/year) (P=0.19).
The Effect of Carbocysteine on Acute Exacerbation of Chronic Obstructive Pulmonary Disease (PEACE) study randomized 709 patients with at least two exacerbations within the 2 years prior to enrollment to carbocysteine 500 mg three times daily or placebo, with the primary endpoint of exacerbation rate over 1 year.91 Numbers of exacerbations per patient per year declined significantly in the carbocysteine group when compared with the placebo group (1.01 [standard error: 0.06] versus 1.35 [standard error: 0.06]); RR: 0.75 (95% CI: 0.62–0.92; P=0.004). There were no significant interactions between COPD severity, smoking, and use of inhaled corticosteroids, and the primary endpoint. These three studies provided conflicting results on the efficacy of antioxidants on exacerbation frequency. A 600 mg daily dose of NAC may be too low to see a clinical effect when compared to a 600 mg twice daily dose, which was used in the HIACE study,90 and a 600 mg three times daily dose used in patients with idiopathic pulmonary fibrosis. Additionally, patients in the PEACE trial and HIACE trial had a lower mean predicted FEV1 (44% and 53.9%, respectively) compared to 57% in the BRONCHUS trial.89,90,91 Therefore, it is possible that antioxidant therapy is more efficacious in those with lower lung function.
Poole et al92 performed a meta-analysis of mucolytic agents for CB or COPD. They included randomized, placebo-controlled studies of at least 2 months' duration. They found 30 trials involving 7,436 participants to be methodologically acceptable for further analysis. The majority of the studies involved the use of NAC (n=15) or carbocysteine (n=4). Compared to placebo, there was a 17% reduction in the number of exacerbations per patient with oral mucolytics (a reduction of 0.04 exacerbations per participant per month; 95% CI: –0.04 to –0.03). This may not be clinically relevant since there was very high heterogeneity (I2=87%) in assessing the exacerbation frequency outcome. There was no overall effect on lung function or increase in adverse effects from the medications.
Chronic antibiotic therapy is generally not indicated for patients with emphysema or CB. Macrolide therapy, however, has been shown to have anti-inflammatory properties and may play a role in the treatment of those with CB. These therapies have been shown to inhibit proinflammatory cytokines, decrease neutrophil burst, inhibit migration and increase apoptosis, decrease eosinophilic inflammation, increase mucociliary transport, reduce goblet cell secretion, and decrease bronchoconstriction.93 The effect of chronic macrolide therapy on COPD exacerbations was assessed in 109 patients with COPD who were randomly assigned to receive erythromycin 250 mg or placebo twice daily for 1 year.94 The erythromycin group had significantly fewer exacerbations than the placebo group. A recent, large, prospective, placebo-controlled, randomized trial on the use of azithromycin (250 mg daily for 1 year) to prevent acute exacerbations of COPD showed that azithromycin was associated with a significant decrease in exacerbation frequency and an improvement in health-related quality of life.95 There was, however, no significant additional benefit of azithromycin in those with CB at baseline.
Future therapy
There is a novel inhaled therapy, BIO-11006, that is currently undergoing Phase II testing (BREATH 1 trial96) in patients with CB. This drug inhibits the function of the myristoylated alanine-rich C kinase substrate protein (MARCKS), which has been shown to be a vital component for the secretion of mucus and inflammatory mediators. Preliminary results indicated that patients have improvements in lung function and reductions in both cough and sputum production.96
There is no available evidence of the benefit of performing routine microbiologic cultures of mucus from COPD patients. Sputum production is one of the characteristic symptoms of COPD, and a change in amount or quality of sputum beyond a day-to-day variation may indicate an exacerbation.1 The presence of purulent sputum is 94.4% sensitive and 77% specific for the yield of a high bacterial load, and it indicates a clear subset of patient episodes identified at presentation that is likely to benefit most from antibiotic therapy.97 Thus, the 2013 GOLD guidelines1 recommend that the presence of purulent sputum during an exacerbation can be a sufficient indication for starting empirical antibiotics, and that a sputum culture with antibiotic sensitivity testing should be performed when there is lack of response to the initial antibiotic treatment.
We believe more studies are needed on why some smokers develop CB and others do not, and on how smoking cessation affects the natural history of CB. In addition, more research on the pathophysiology of this disease process will help in the development of better therapies that directly target CB in order to improve symptoms, while decreasing exacerbations and mortality. The effects of the presence of radiology-confirmed bronchiectasis, a clinically similar phenotype, on the symptoms and outcomes of CB deserve further study.
We believe that the additional study of higher doses of antioxidants such as NAC, and more in-depth studies of selective PDE-4 inhibitors like roflumilast are needed. The identification of more therapeutic targets is necessary for drug development in order to improve outcomes specifically related to CB.
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Rubin BK. Mucolytics, expectorants, and mucokinetic medications. Respir Care. 2007;52(7):859–865.
Thomson ML, Pavia D, McNicol MW. A preliminary study of the effect of guaiphenesin on mucociliary clearance from the human lung. Thorax. 1973;28(6):742–747.
Levin MH, Sullivan S, Nielson D, Yang B, Finkbeiner WE, Verkman AS. Hypertonic saline therapy in cystic fibrosis: Evidence against the proposed mechanism involving aquaporins. J Biol Chem. 2006;281(35):25803–25812.
Boucher RC. Cystic fibrosis: a disease of vulnerability to airway surface dehydration. Trends Mol Med. 2007;13(6):231–240.
Valderramas SR, Atallah AN. Effectiveness and safety of hypertonic saline inhalation combined with exercise training in patients with chronic obstructive pulmonary disease: a randomized trial. Respir Care. 2009;54(3):327–333.
Fahy JV, Steiger DJ, Liu J, Basbaum CB, Finkbeiner WE, Boushey HA. Markers of mucus secretion and DNA levels in induced sputum from asthmatic and from healthy subjects. Am Rev Respir Dis. 1993;147(5):1132–1137.
O'Donnell AE, Barker AF, Ilowite JS, Fick RB. Treatment of idiopathic bronchiectasis with aerosolized recombinant human DNase I. rhDNase Study Group. Chest. 1998;113(5):1329–1334.
Tamaoki J, Kondo M, Takizawa T. Effect of cAMP on ciliary function in rabbit tracheal epithelial cells. J Appl Physiol (1985). 1989;66(3):1035–1039.
Devalia JL, Sapsford RJ, Rusznak C, Toumbis MJ, Davies RJ. The effects of salmeterol and salbutamol on ciliary beat frequency of cultured human bronchial epithelial cells, in vitro. Pulm Pharmacol. 1992;5(4):257–263.
Salathe M. Effects of beta-agonists on airway epithelial cells. J Allergy Clin Immunol. 2002;110(Suppl 6):S275–S281.
Nguyen LP, Omoluabi O, Parra S, et al. Chronic exposure to beta-blockers attenuates inflammation and mucin content in a murine asthma model. Am J Respir Cell Mol Biol. 2008;38(3):256–262.
Nguyen LP, Lin R, Parra S, et al. Beta2-adrenoceptor signaling is required for the development of an asthma phenotype in a murine model. Proc Natl Acad Sci U S A. 2009;106(7):2435–2440.
Wanner A. Effects of methylxanthines on airway mucociliary function. Am J Med. 1985;79(6A):16–21.
Anderson G, Peel ET, Pardoe T, Jones R. Sustained-release theophylline in chronic bronchitis. Br J Dis Chest. 1982;76(3):261–265.
Taylor DR, Buick B, Kinney C, Lowry RC, McDevitt DG. The efficacy of orally administered theophylline, inhaled salbutamol, and a combination of the two as chronic therapy in the management of chronic bronchitis with reversible air-flow obstruction. Am Rev Respir Dis. 1985;131(5):747–751.
Foster WM, Langenback EG, Bergofsky EH. Lung mucociliary function in man: interdependence of bronchial and tracheal mucus transport velocities with lung clearance in bronchial asthma and healthy subjects. Ann Occup Hyg. 1982;26(1–4):227–244.
Jones PW, Bosh TK. Quality of life changes in COPD patients treated with salmeterol. Am J Respir Crit Care Med. 1997;155(4):1283–1289.
Mahler DA, Donohue JF, Barbee RA, et al. Efficacy of salmeterol xinafoate in the treatment of COPD. Chest. 1999;115(4):957–965.
van Noord JA, Aumann JL, Janssens E, et al. Effects of tiotropium with and without formoterol on airflow obstruction and resting hyperinflation in patients with COPD. Chest. 2006;129(3):509–517.
Melloni B, Germouty J. [The influence of a new beta agonist: formoterol on mucociliary function]. Rev Mal Respir. 1992;9(5):503–507. French.
Bateman ED, Rennard S, Barnes PJ, et al. Alternative mechanisms for tiotropium. Pulm Pharmacol Ther. 2009;22(6):533–542.
Wine JJ, Joo NS. Submucosal glands and airway defense. Proc Am Thorac Soc. 2004;1(1):47–53.
Ghafouri MA, Patil KD, Kass I. Sputum changes associated with the use of ipratropium bromide. Chest. 1984;86(3):387–393.
Bennett WD, Chapman WF, Mascarella JM. The acute effect of ipratropium bromide bronchodilator therapy on cough clearance in COPD. Chest. 1993;103(2):488–495.
Casaburi R, Briggs DD Jr, Donohue JF, Serby CW, Menjoge SS, Witek TJ Jr. The spirometric efficacy of once-daily dosing with tiotropium in stable COPD: a 13-week multicenter trial. The US Tiotropium Study Group. Chest. 2000;118(5):1294–1302.
Hasani A, Toms N, Agnew JE, Sarno M, Harrison AJ, Dilworth P. The effect of inhaled tiotropium bromide on lung mucociliary clearance in patients with COPD. Chest. 2004;125(5):1726–1734.
Hattotuwa KL, Gizycki MJ, Ansari TW, Jeffery PK, Barnes NC. The effects of inhaled fluticasone on airway inflammation in chronic obstructive pulmonary disease: a double-blind, placebo-controlled biopsy study. Am J Respir Crit Care Med. 2002;165(12):1592–1596.
Innes AL, Carrington SD, Thornton DJ, et al. Ex vivo sputum analysis reveals impairment of protease-dependent mucus degradation by plasma proteins in acute asthma. Am J Respir Crit Care Med. 2009;180(3):203–210.
Leung SY, Eynott P, Nath P, Chung KF. Effects of ciclesonide and fluticasone propionate on allergen-induced airway inflammation and remodeling features. J Allergy Clin Immunol. 2005;115(5):989–996.
Chen Y, Nickola TJ, DiFronzo NL, Colberg-Poley AM, Rose MC. Dexamethasone-mediated repression of MUC5AC gene expression in human lung epithelial cells. Am J Respir Cell Mol Biol. 2006;34(3):338–347.
O'Riordan TG, Mao Y, Otero R, Lopez J, Sabater JR, Abraham WM. Budesonide affects allergic mucociliary dysfunction. J Appl Physiol (1985). 1998;85(3):1086–1091.
Burge PS, Calverley PM, Jones PW, Spencer S, Anderson JA, Maslen TK. Randomised, double blind, placebo controlled study of fluticasone propionate in patients with moderate to severe chronic obstructive pulmonary disease: the ISOLDE trial. BMJ. 2000; 320(7245):1297–1303.
Calverley P, Pauwels R, Vestbo J, et al; TRial of Inhaled STeroids ANd long-acting beta2 agonists study group. Combined salmeterol and fluticasone in the treatment of chronic obstructive pulmonary disease: a randomised controlled trial. Lancet. 2003;361(9356):449–456.
Szafranski W, Cukier A, Ramirez A, et al. Efficacy and safety of budesonide/formoterol in the management of chronic obstructive pulmonary disease. Eur Respir J. 2003;21(1):74–81.
Chong J, Poole P, Leung B, Black PN. Phosphodiesterase 4 inhibitors for chronic obstructive pulmonary disease. Cochrane Database Syst Rev. 2011CD002309.
Calverley PM, Rabe KF, Goehring UM, Kristiansen S, Fabbri LM, Martinez FJ; M2-124 and M2-125 study groups. Roflumilast in symptomatic chronic obstructive pulmonary disease: two randomised clinical trials. Lancet. 2009;374(9691):685–694.
Fabbri LM, Calverley PM, Izquierdo-Alonso JL, et al; M2-127 and M2-128 study groups. Roflumilast in moderate-to-severe chronic obstructive pulmonary disease treated with longacting bronchodilators: two randomised clinical trials. Lancet. 2009;374(9691):695–703.
MacNee W. Oxidants/antioxidants and COPD. Chest. 2000; 117(5 Suppl 1):303S–317S.
Ishibashi Y, Takayama G, Inouye Y, Taniguchi A. Carbocisteine normalizes the viscous property of mucus through regulation of fucosylated and sialylated sugar chain on airway mucins. Eur J Pharmacol. 2010;641(2–3):226–228.
Decramer M, Rutten-van Mölken M, Dekhuijzen PN, et al. Effects of N-acetylcysteine on outcomes in chronic obstructive pulmonary disease (Bronchitis Randomized on NAC Cost-Utility Study, BRONCUS):a randomised placebo-controlled trial. Lancet. 2005;365(9470):1552–1560.
Tse HN, Raiteri L, Wong KY, et al. High-dose N-acetylcysteine in stable COPD: the 1-year, double-blind, randomized, placebo-controlled HIACE study. Chest. 2013;144(1):106–118.
Zheng JP, Kang J, Huang SG, et al. Effect of carbocisteine on acute exacerbation of chronic obstructive pulmonary disease (PEACE Study):a randomised placebo-controlled study. Lancet. 2008;371(9629):2013–2018.
Poole P, Black PN, Cates CJ. Mucolytic agents for chronic bronchitis or chronic obstructive pulmonary disease. Cochrane Database Syst Rev. 2012;8:CD001287.
Gotfried MH. Macrolides for the treatment of chronic sinusitis, asthma, and COPD. Chest. 2004;125(Suppl 2):52S–60S; quiz 60S–61S.
Seemungal TA, Wilkinson TM, Hurst JR, Perera WR, Sapsford RJ, Wedzicha JA. Long-term erythromycin therapy is associated with decreased chronic obstructive pulmonary disease exacerbations. Am J Respir Crit Care Med. 2008;178(11):1139–1147.
Albert RK, Connett J, Bailey WC, et al; COPD Clinical Research Network. Azithromycin for prevention of exacerbations of COPD. N Engl J Med. 2011;365(8):689–698.
BioMarck Pharmaceuticals [webpage on the Internet]. BioMarck Pharmaceutical Reports Phase II Study Results for COPD Inhalation Solution: BREATH 1 study assessed effectiveness and safety of BIO-11006 inhalation solution. Raleigh, NC: BioMarck Pharmaceuticals; 2011. Available from: http://www.biomarck.com/downloads/biomarck-pharmaceuticals-reports-phase-ii.swf. Accessed October 21, 2013.
Stockley RA, O'Brien C, Pye A, Hill SL. Relationship of sputum color to nature and outpatient management of acute exacerbations of COPD. Chest. 2000;117(6):1638–1645.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,689
|
Worlds within stone: the inner and outer rock-art landscapes of northern Australia and southern Africa
Paul S.C. Tacon, Sven Ouzman
Research output: Chapter in Book/Conference paper › Chapter
The hunter-gatherer rock-arts of northern Australia and southern Africa have much in common: an abundance of engraved and painted rock-art which occurs at topographically comparable locales. Though northern Australian and southern African hunter-gatherers had no contact, their world-understandings have tended to find expression in similar ways – ways which often involve rock-art imagery. In the case of northern Australia and southern Africa, rock-art points the way, often literally, to multiple landscapes that co-exist but which do not seem to have been equally accessible to all hunter-gatherers.
In both regions the notion is pervasive that inner worlds of extra-ordinary experience simultaneously and immanently exist alongside and intertwined with the outer world of 'ordinary' existence. Rock-art sites represent places where these worlds connected. As important as the rock-art imagery in this respect is the rock itself; by no means a neutral support for imagery, it was and it is an active, a living and sometimes a dangerous entity. Ethnography, rock-art imagery and a consideration of rock and place, taken together, allow exploration of the nature of landscape perception and use among the hunter-gatherers of northern Australia and southern Africa.
The figured landscapes of rock-art
Looking at pictures in place
Christopher Chippindale, George Nash
Cambridge University Ppress
Tacon, P. S. C., & Ouzman, S. (2004). Worlds within stone: the inner and outer rock-art landscapes of northern Australia and southern Africa. In C. Chippindale, & G. N. (Eds.), The figured landscapes of rock-art: Looking at pictures in place (pp. 39-68). Cambridge: Cambridge University Ppress.
Tacon, Paul S.C. ; Ouzman, Sven. / Worlds within stone : the inner and outer rock-art landscapes of northern Australia and southern Africa. The figured landscapes of rock-art: Looking at pictures in place. editor / Christopher Chippindale ; George Nash. Cambridge : Cambridge University Ppress, 2004. pp. 39-68
@inbook{108c386cd0ff4c318b27b879b5dfb804,
title = "Worlds within stone: the inner and outer rock-art landscapes of northern Australia and southern Africa",
abstract = "The hunter-gatherer rock-arts of northern Australia and southern Africa have much in common: an abundance of engraved and painted rock-art which occurs at topographically comparable locales. Though northern Australian and southern African hunter-gatherers had no contact, their world-understandings have tended to find expression in similar ways – ways which often involve rock-art imagery. In the case of northern Australia and southern Africa, rock-art points the way, often literally, to multiple landscapes that co-exist but which do not seem to have been equally accessible to all hunter-gatherers.In both regions the notion is pervasive that inner worlds of extra-ordinary experience simultaneously and immanently exist alongside and intertwined with the outer world of 'ordinary' existence. Rock-art sites represent places where these worlds connected. As important as the rock-art imagery in this respect is the rock itself; by no means a neutral support for imagery, it was and it is an active, a living and sometimes a dangerous entity. Ethnography, rock-art imagery and a consideration of rock and place, taken together, allow exploration of the nature of landscape perception and use among the hunter-gatherers of northern Australia and southern Africa.",
keywords = "Archaeology, Rock art, Australia, Southern Africa, Stone, San, Aboriginal, Bushman",
author = "Tacon, {Paul S.C.} and Sven Ouzman",
editor = "Christopher Chippindale and {George Nash}",
booktitle = "The figured landscapes of rock-art",
publisher = "Cambridge University Ppress",
Tacon, PSC & Ouzman, S 2004, Worlds within stone: the inner and outer rock-art landscapes of northern Australia and southern Africa. in C Chippindale & GN (eds), The figured landscapes of rock-art: Looking at pictures in place. Cambridge University Ppress, Cambridge, pp. 39-68.
Worlds within stone : the inner and outer rock-art landscapes of northern Australia and southern Africa. / Tacon, Paul S.C.; Ouzman, Sven.
The figured landscapes of rock-art: Looking at pictures in place. ed. / Christopher Chippindale; George Nash. Cambridge : Cambridge University Ppress, 2004. p. 39-68.
T1 - Worlds within stone
T2 - the inner and outer rock-art landscapes of northern Australia and southern Africa
AU - Tacon, Paul S.C.
AU - Ouzman, Sven
N2 - The hunter-gatherer rock-arts of northern Australia and southern Africa have much in common: an abundance of engraved and painted rock-art which occurs at topographically comparable locales. Though northern Australian and southern African hunter-gatherers had no contact, their world-understandings have tended to find expression in similar ways – ways which often involve rock-art imagery. In the case of northern Australia and southern Africa, rock-art points the way, often literally, to multiple landscapes that co-exist but which do not seem to have been equally accessible to all hunter-gatherers.In both regions the notion is pervasive that inner worlds of extra-ordinary experience simultaneously and immanently exist alongside and intertwined with the outer world of 'ordinary' existence. Rock-art sites represent places where these worlds connected. As important as the rock-art imagery in this respect is the rock itself; by no means a neutral support for imagery, it was and it is an active, a living and sometimes a dangerous entity. Ethnography, rock-art imagery and a consideration of rock and place, taken together, allow exploration of the nature of landscape perception and use among the hunter-gatherers of northern Australia and southern Africa.
AB - The hunter-gatherer rock-arts of northern Australia and southern Africa have much in common: an abundance of engraved and painted rock-art which occurs at topographically comparable locales. Though northern Australian and southern African hunter-gatherers had no contact, their world-understandings have tended to find expression in similar ways – ways which often involve rock-art imagery. In the case of northern Australia and southern Africa, rock-art points the way, often literally, to multiple landscapes that co-exist but which do not seem to have been equally accessible to all hunter-gatherers.In both regions the notion is pervasive that inner worlds of extra-ordinary experience simultaneously and immanently exist alongside and intertwined with the outer world of 'ordinary' existence. Rock-art sites represent places where these worlds connected. As important as the rock-art imagery in this respect is the rock itself; by no means a neutral support for imagery, it was and it is an active, a living and sometimes a dangerous entity. Ethnography, rock-art imagery and a consideration of rock and place, taken together, allow exploration of the nature of landscape perception and use among the hunter-gatherers of northern Australia and southern Africa.
KW - Archaeology
KW - Rock art
KW - Australia
KW - Southern Africa
KW - Stone
KW - San
KW - Aboriginal
KW - Bushman
BT - The figured landscapes of rock-art
A2 - Chippindale, Christopher
A2 - null, George Nash
PB - Cambridge University Ppress
Tacon PSC, Ouzman S. Worlds within stone: the inner and outer rock-art landscapes of northern Australia and southern Africa. In Chippindale C, GN, editors, The figured landscapes of rock-art: Looking at pictures in place. Cambridge: Cambridge University Ppress. 2004. p. 39-68
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,216
|
export interface TwoFactorAuthSettings {
maxVerificationFailuresBeforeUserLockout: number;
providers: Array<TwoFactorAuthProviderConfig>;
totalAllowedTimeForVerification: number;
useSystemTwoFactorAuthSettings: boolean;
verificationCodeCheckRateLimit: string;
minVerificationCodeSendPeriod: number;
}
export interface TwoFactorAuthSettingsForm extends TwoFactorAuthSettings{
providers: Array<TwoFactorAuthProviderConfigForm>;
verificationCodeCheckRateLimitEnable: boolean;
verificationCodeCheckRateLimitNumber: number;
verificationCodeCheckRateLimitTime: number;
}
export type TwoFactorAuthProviderConfig = Partial<TotpTwoFactorAuthProviderConfig | SmsTwoFactorAuthProviderConfig |
EmailTwoFactorAuthProviderConfig>;
export type TwoFactorAuthProviderConfigForm = Partial<TotpTwoFactorAuthProviderConfig | SmsTwoFactorAuthProviderConfig |
EmailTwoFactorAuthProviderConfig> & TwoFactorAuthProviderFormConfig;
export interface TotpTwoFactorAuthProviderConfig {
providerType: TwoFactorAuthProviderType;
issuerName: string;
}
export interface SmsTwoFactorAuthProviderConfig {
providerType: TwoFactorAuthProviderType;
smsVerificationMessageTemplate: string;
verificationCodeLifetime: number;
}
export interface EmailTwoFactorAuthProviderConfig {
providerType: TwoFactorAuthProviderType;
verificationCodeLifetime: number;
}
export interface TwoFactorAuthProviderFormConfig {
enable: boolean;
}
export enum TwoFactorAuthProviderType{
TOTP = 'TOTP',
SMS = 'SMS',
EMAIL = 'EMAIL',
BACKUP_CODE = 'BACKUP_CODE'
}
interface GeneralTwoFactorAuthAccountConfig {
providerType: TwoFactorAuthProviderType;
useByDefault: boolean;
}
export interface TotpTwoFactorAuthAccountConfig extends GeneralTwoFactorAuthAccountConfig {
authUrl: string;
}
export interface SmsTwoFactorAuthAccountConfig extends GeneralTwoFactorAuthAccountConfig {
phoneNumber: string;
}
export interface EmailTwoFactorAuthAccountConfig extends GeneralTwoFactorAuthAccountConfig {
email: string;
}
export interface BackupCodeTwoFactorAuthAccountConfig extends GeneralTwoFactorAuthAccountConfig {
codesLeft: number;
codes?: Array<string>;
}
export type TwoFactorAuthAccountConfig = TotpTwoFactorAuthAccountConfig | SmsTwoFactorAuthAccountConfig |
EmailTwoFactorAuthAccountConfig | BackupCodeTwoFactorAuthAccountConfig;
export interface AccountTwoFaSettings {
configs: AccountTwoFaSettingProviders;
}
export type AccountTwoFaSettingProviders = {
[key in TwoFactorAuthProviderType]?: TwoFactorAuthAccountConfig;
};
export interface TwoFaProviderInfo {
type: TwoFactorAuthProviderType;
default: boolean;
contact?: string;
minVerificationCodeSendPeriod?: number;
}
export interface TwoFactorAuthProviderData {
name: string;
description: string;
activatedHint: string;
}
export interface TwoFactorAuthProviderLoginData extends Omit<TwoFactorAuthProviderData, 'activatedHint'> {
icon: string;
placeholder: string;
}
export const twoFactorAuthProvidersData = new Map<TwoFactorAuthProviderType, TwoFactorAuthProviderData>(
[
[
TwoFactorAuthProviderType.TOTP, {
name: 'security.2fa.provider.totp',
description: 'security.2fa.provider.totp-description',
activatedHint: 'security.2fa.provider.totp-hint'
}
],
[
TwoFactorAuthProviderType.SMS, {
name: 'security.2fa.provider.sms',
description: 'security.2fa.provider.sms-description',
activatedHint: 'security.2fa.provider.sms-hint'
}
],
[
TwoFactorAuthProviderType.EMAIL, {
name: 'security.2fa.provider.email',
description: 'security.2fa.provider.email-description',
activatedHint: 'security.2fa.provider.email-hint'
}
],
[
TwoFactorAuthProviderType.BACKUP_CODE, {
name: 'security.2fa.provider.backup_code',
description: 'security.2fa.provider.backup-code-description',
activatedHint: 'security.2fa.provider.backup-code-hint'
}
]
]
);
export const twoFactorAuthProvidersLoginData = new Map<TwoFactorAuthProviderType, TwoFactorAuthProviderLoginData>(
[
[
TwoFactorAuthProviderType.TOTP, {
name: 'security.2fa.provider.totp',
description: 'login.totp-auth-description',
placeholder: 'login.totp-auth-placeholder',
icon: 'mdi:cellphone-key'
}
],
[
TwoFactorAuthProviderType.SMS, {
name: 'security.2fa.provider.sms',
description: 'login.sms-auth-description',
placeholder: 'login.sms-auth-placeholder',
icon: 'mdi:message-reply-text-outline'
}
],
[
TwoFactorAuthProviderType.EMAIL, {
name: 'security.2fa.provider.email',
description: 'login.email-auth-description',
placeholder: 'login.email-auth-placeholder',
icon: 'mdi:email-outline'
}
],
[
TwoFactorAuthProviderType.BACKUP_CODE, {
name: 'security.2fa.provider.backup_code',
description: 'login.backup-code-auth-description',
placeholder: 'login.backup-code-auth-placeholder',
icon: 'mdi:lock-outline'
}
]
]
);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,763
|
Rejestrator rozmów – urządzenie umożliwiające nagrywanie treści audio rozmów telefonicznych lub łączności radiowej. Współczesny rejestrator rozmów udostępnia nagrania w postaci cyfrowej, wraz z informacjami dodatkowymi takimi jak data i godzina rozpoczęcia rozmowy, czas jej trwania oraz numery telefoniczne rozmówców (w przypadku rozmowy telefonicznej).
Producenci
CyberTech Recording Solution
Compol II
DGT
LogiREC
Metasoft
NiceLog
Prioriti Voice
SIM
TRX
Vidicode
Zobacz też
rejestrator dźwięku
Urządzenia telekomunikacyjne
Rozliczalność
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,768
|
Q: Connecting to People Search through Tableau? We are creating a report that would need employee information. We have the employee ID, but we are also looking for the name and office location. Is it possible to have tableau hit people search to pull this information?
A: You can use Wildcard Match Filter Function to do so.
Then, when you type employee ID in the search box, it will filter it out to dashboard.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,146
|
Quasi-linkage equilibrium (QLE) is a mathematical approximation used in solving population genetics problems. Motoo Kimura introduced the notion to simplify a model of Fisher's fundamental theorem. QLE greatly simplifies population genetic equations whilst making the assumption of weak selection and weak epistasis. Selection under these conditions rapidly changes allele frequencies to a state where they evolve as if in linkage equilibrium. Kimura originally provided the sufficient conditions for QLE in two-locus systems, but recently several researchers have shown how QLE occurs in general multilocus systems. QLE allows theorists to approximate linkage disequilibria by simple expressions, often simple functions of allele or genotype frequencies, thereby providing solutions to highly complex problems involving selection on multiple loci or polygenic traits. QLE also plays an important role in justifying approximations in the derivation of quantitative genetic equations from mendelian principles.
Simple Model
Let , , and represent the frequencies of the four possible genotypes in a haploid two-locus-two-allele model. Kimura's original model showed that
approaches a stable state rapidly if epistatic effects are small relative to recombination. Deviations from will be reduced by the recombination fraction every generation.
References
Evolutionary biology
Population genetics
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 263
|
<?php
/**
* Rss data helper
*
* @category Mage
* @package Mage_Rss
* @author Magento Core Team <core@magentocommerce.com>
*/
class Mage_Rss_Helper_Data extends Mage_Core_Helper_Abstract
{
/**
* Config path to RSS field
*/
const XML_PATH_RSS_ACTIVE = 'rss/config/active';
protected $_rssSession;
protected $_adminSession;
public function __construct(array $params = array())
{
$this->_rssSession = isset($params['rss_session']) ? $params['rss_session'] : Mage::getSingleton('rss/session');
$this->_adminSession = isset($params['admin_session'])
? $params['admin_session'] : Mage::getSingleton('admin/session');
}
/**
* Authenticate customer on frontend
*
*/
public function authFrontend()
{
if (!$this->_rssSession->isCustomerLoggedIn()) {
list($username, $password) = $this->authValidate();
$customer = Mage::getModel('customer/customer')->authenticate($username, $password);
if ($customer && $customer->getId()) {
$this->_rssSession->settCustomer($customer);
} else {
$this->authFailed();
}
}
}
/**
* Authenticate admin and check ACL
*
* @param string $path
*/
public function authAdmin($path)
{
if (!$this->_rssSession->isAdminLoggedIn()) {
list($username, $password) = $this->authValidate();
Mage::getSingleton('adminhtml/url')->setNoSecret(true);
$user = $this->_adminSession->login($username, $password);
} else {
$user = $this->_rssSession->getAdmin();
}
if ($user && $user->getId() && $user->getIsActive() == '1' && $this->_adminSession->isAllowed($path)) {
$this->_rssSession->setAdmin($user);
} else {
$this->authFailed();
}
}
/**
* Validate Authenticate
*
* @param array $headers
* @return array
*/
public function authValidate($headers = null)
{
$userPass = Mage::helper('core/http')->authValidate($headers);
return $userPass;
}
/**
* Send authenticate failed headers
*
*/
public function authFailed()
{
Mage::helper('core/http')->authFailed();
}
/**
* Disable using of flat catalog and/or product model to prevent limiting results to single store. Probably won't
* work inside a controller.
*
* @return null
*/
public function disableFlat()
{
/* @var $flatHelper Mage_Catalog_Helper_Product_Flat */
$flatHelper = Mage::helper('catalog/product_flat');
if ($flatHelper->isAvailable()) {
/* @var $emulationModel Mage_Core_Model_App_Emulation */
$emulationModel = Mage::getModel('core/app_emulation');
// Emulate admin environment to disable using flat model - otherwise we won't get global stats
// for all stores
$emulationModel->startEnvironmentEmulation(0, Mage_Core_Model_App_Area::AREA_ADMINHTML);
}
}
/**
* Check if module was activated in system configurations
*
* @return bool
*/
public function isRssEnabled()
{
return Mage::getStoreConfigFlag(self::XML_PATH_RSS_ACTIVE);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,207
|
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