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Q: CDO - Sending email returns false even though the email is sent I grabbed this example from w3school, however when I add an if statement to check if the email has been sent, it displays the false code even though I receive the email.
Im not sure how asp works, but I'm assuming myMail returns a boolean? Or does it not? How do I check if the email has been sent.
<%
Set myMail=CreateObject("CDO.Message")
myMail.Subject="Sending email with CDO"
myMail.From="mymail@mydomain.com"
myMail.To="examplek@exm.com"
myMail.HTMLBody = "<h1>This is a message.</h1>"
If myMail.Send Then
Response.AddHeader "Content-type", "application/json"
Response.Write "{ request: 'success'}"
Else
Response.AddHeader "Content-type", "application/json"
Response.Write "{ request: 'failed'}"
End If
set myMail=nothing
%>
A: The .Send method just simply sends the message without returning a response.
You can handle an error raised by a failure to send the message something like the code below:
On Error Resume Next
myMail.Send
If Err.Number = 0 then
Response.ContentType="application/json"
Response.Write "{ request: 'success'}"
Else
Response.ContentType="application/json"
Response.Write "{ request: 'failed'}"
End If
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,108 |
title: Java Persistence Architecture Plugin
layout: default
summary: Analyses JPA persistence.xml files and adds any discovered class to the imported packages.
--- | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,618 |
Q: How to convert below XML to Java Object in Java I want to read the values under value tag
i.e. Yes, 750,3500, AL,Real, Approved, G, 140, GT these values i want to map with java object.
<entities>
<entity id="1234" userId="RD" parent="TestID">
<name>AL</name>
<values>
<value id="testA" id="Y">Yes</value>
<value id="testB">750</value>
<value id="testC">3500</value>
<value id="testD">AL</value>
<value id="testE">Real</value>
<value id="testF" ID="A" Changed="true">Approved</value>
<value id="testF">G</value>
<value id="testG">140</value>
<value id="testF">GT</value>
</values>
</entity>
</entities>
A: You can use Jaxb library like below.
Entities.class
@XmlRootElement(name = "entities")
public class Entities {
@XmlElement(name = "entity")
private List<Entity> entityList;
}
Entity.class
@XmlRootElement(name = "entity")
@XmlAccessorType(XmlAccessType.FIELD)
public class Entity {
@XmlAttribute(name = "id")
private String id;
@XmlAttribute(name = "userId")
private String userId;
@XmlAttribute(name = "parent")
private String parent;
@XmlElement(name = "name")
private String name;
@XmlElementWrapper(name = "values")
@XmlElement(name = "value")
private List<Value> valueList;
}
Value.class
@XmlRootElement(name = "value")
public class Value {
@XmlAttribute(name = "id")
private String id;
@XmlAttribute(name = "ID")
private String ID;
@XmlAttribute(name = "Changed")
private String changed;
@XmlValue
private String text;
}
You can see how to use Jaxb library in the reference below.
https://www.baeldung.com/jaxb
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,210 |
package com.zappos.json.data;
import com.zappos.json.annot.JsonKey;
/**
*
* @author Hussachai Puripunpinyo
*
*/
public class JsonKeyBean {
public static class NickName {
@JsonKey("short_name")
private String name = "He";
public String getName() {
return name;
}
public void setName(String name) {
this.name = name;
}
}
@JsonKey("first_name")
private String firstName = "Jason";
@JsonKey("last_name")
private String lastName = "Voorhees";
@JsonKey("nick_name")
private NickName nickname;
public String getFirstName() {
return firstName;
}
public void setFirstName(String firstName) {
this.firstName = firstName;
}
public String getLastName() {
return lastName;
}
public void setLastName(String lastName) {
this.lastName = lastName;
}
public NickName getNickname() {
return nickname;
}
public void setNickname(NickName nickname) {
this.nickname = nickname;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,554 |
Массімо Гоббі (, 31 жовтня 1980, Мілан) — італійський футболіст, що грав на позиції захисника, зокрема, за «Фіорентину», «Парму», а також національну збірну Італії.
Клубна кар'єра
Народився 31 жовтня 1980 року в місті Мілан. Вихованець футбольної школи місцевого «Мілана».
У дорослому футболі дебютував 1998 року виступами за команду «Про Сесто», в якій протягом сезону взяв участь у 6 матчах четвертого італійського дивізіону.
1999 року став гравцем друголігового «Тревізо», до основної команди якого не пробився. Натомість протягом 2001–2003 років грав на умовах оренди за «Джульяно» і «АльбіноЛеффе», команди відповідно четвертої і третьої ліги італійської першості.
Повернувшись до «Тревізо», протягом сезону 2003/04 вже був стабільним гравцем основного складу, після чого отримав запрошення до вищолігового «Кальярі».
Приєднавшись до «Кальярі» влітку 2004 року, наступні 15 сезонів виступав на рівні Серії A, де також захищав кольори «Фіорентини», «Парми» і «К'єво».
Завершив ігрову кар'єру вистпами за «Парму» в сезоні 2018/19.
Виступи за збірну
У серпни 2006 року провів свою єдину офіційну гру у складі національної збірної Італії.
Статистика виступів
Статистика клубних виступів
Статистика виступів за збірну
Примітки
Посилання
Італійські футболісти
Гравці збірної Італії з футболу
Футболісти «Про Сесто»
Футболісти «Тревізо»
Футболісти «АльбіноЛеффе»
Футболісти «Кальярі»
Футболісти «Фіорентини»
Футболісти «Парми»
Футболісти «К'єво»
Уродженці Мілана | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,053 |
Mayfield es una localidad del condado de Sanpete, estado de Utah, Estados Unidos. Según el censo de 2000 la población era de 420 habitantes.
Geografía
Mayfield se encuentra en las coordenadas .
Según la oficina del censo de Estados Unidos, la localidad tiene una superficie total de 2,2 km². No tiene superficie cubierta de agua.
Localidades de Utah | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,108 |
{"url":"https:\/\/diophantus.org\/arxiv\/0704.0388","text":"# diophantus\n\nHello, this is beta version of diophantus. If you want to report about a mistake, please, write to hello@diophantus.org\n\n#### Sterile neutrinos at the CNGS\n\n03 Apr 2007 hep-ph arxiv.org\/abs\/0704.0388\nAbstract. We study the potential of the CNGS beam in constraining the parameter space of a model with one sterile neutrino separated from three active ones by an $\\mathcal{O}(\\eVq)$ mass-squared difference, $\\Dmq_\\Sbl$. We perform our analysis using the OPERA detector as a reference (our analysis can be upgraded including a detailed simulation of the ICARUS detector). We point out that the channel with the largest potential to constrain the sterile neutrino parameter space at the CNGS beam is $\\nu_\\mu \\to \\nu_\\tau$. The reason for that is twofold: first, the active-sterile mixing angle that governs this oscillation is the less constrained by present experiments; second, this is the signal for which both OPERA and ICARUS have been designed, and thus benefits from an extremely low background. In our analysis we also took into account $\\nu_\\mu \\to \\nu_e$ oscillations. We find that the CNGS potential to look for sterile neutrinos is limited with nominal intensity of the beam, but it is significantly enhanced with a factor 2 to 10 increase in the neutrino flux. Data from both channels allow us, in this case, to constrain further the four-neutrino model parameter space. Our results hold for any value of $\\Dmq_\\Sbl \\gtrsim 0.1 \\eVq$, \\textit{i.e.} when oscillations driven by this mass-squared difference are averaged. We have also checked that the bound on $\\theta_{13}$ that can be put at the CNGS is not affected by the possible existence of sterile neutrinos.\n\n# Reviews\n\nThere are no reviews yet.","date":"2021-04-13 11:06:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7660803198814392, \"perplexity\": 873.532942302974}, \"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-2021-17\/segments\/1618038072180.33\/warc\/CC-MAIN-20210413092418-20210413122418-00581.warc.gz\"}"} | null | null |
Q: Button and Text input in JS this is in my PHP file:
<input style="text-align: center; max-width: 500px; margin-top: 0px;" type="text" placeholder="Paste your links here separated by a space" onfocus="this.placeholder = ''" onblur="this.placeholder = 'Paste your links here separated by a space'" name='links' id='links' />
<!--lilbanner-->
<BR>
<button class="button alt" id="submit" type="submit">
<?php printf($obj->lang['sbdown']); ?>
</button>
I want the button to be disabled when the text input is empty, and when the user types text and deletes it, it will be empty again, means button will be disabled again. I have the following code which works exactly as I want to from my other website, but for some reason it doesnt work (empty text input = button..)
<script type="text/javascript">
$(document).ready(function() {
$('.submit').prop('disabled', true);
$('#link').change(function() {
$('.submit').prop('disabled', this.value == "" ? true : false);
})
});
</script>
can you help me out? thanks in advance!
A: There you go. You want 'keyup' and 'paste' rather than 'change':
$(document).ready(function() {
$('#submit').prop('disabled', true);
$('#links').bind('keyup paste', function() {
$('#submit').prop('disabled', this.value == "" ? true : false);
})
});
A: Your selectors was wrong, look the fix:
<script type="text/javascript">
$(document).ready(function() {
$('#submit').prop('disabled', true);
$('#links').change(function() {
$('#submit').prop('disabled', this.value == "" ? true : false);
})
});
</script>
Plunker: https://plnkr.co/edit/ETJiPg6BKDNkqbGHXCfk?p=preview
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,045 |
Габарит Фрейсине () — стандарт, регулирующий размеры шлюзов и речных судов. Указ о введении стандарта был подписан министром общественных работ Франции Шарлем Фрейсине 5 августа 1879 года и поэтому носит его имя.
Согласно указу, камеры шлюзов должны быть не менее длиной, шириной и 2,2 м глубиной, что обеспечивало судоходность 300—350 тонных барж. Как следствие, суда и баржи, например пениши, не должны превышать в длину , в ширину и с осадкой не более 1,8 м. Мосты и прочие сооружения вдоль каналов должны обеспечивать просвет не менее .
Габарит Фрейсине принят за тип I в современной классификации европейских внутренних водных путей. На 2001 год 5800 км водных путей Франции (или 23 % судоходных путей) соответствовало этому стандарту.
Примечания
Незавершённые статьи о речном транспорте
Речной транспорт
Габариты судов | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,592 |
{"url":"https:\/\/bizmeetstech.net\/k2shf\/article.php?a0b939=pascal%27s-triangle-example","text":"For example, x + 2, 2x + 3y, p - q. This can also be found using the binomial theorem: We will know, for example, that. Pascal's Triangle is a number triangle which, although very easy to construct, has many interesting patterns and useful properties. 0 0 1 0 0 0 0. {_0C_0} \\$5px] 1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1 \\newline See all questions in Pascal's Triangle and Binomial Expansion Impact of this question The positive sign between the terms means that everything our expansion is positive. Wiki User Answered . 1 5 10 10 5 1. For example, the fourth row in the triangle shows numbers 1 3 3 1, and that means the expansion of a cubic binomial, which has four terms. So this is the Pascal triangle. Pascal\u2019s triangle is a pattern of triangle which is based on nCr.below is the pictorial representation of a pascal\u2019s triangle. Combinations. {_2C_0} \\quad {_2C_1} \\quad {_2C_2} \\\\[5px] For example- Print pascal\u2019s triangle in C++. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Both $$n$$ and $$k$$ (within $$_nC_k$$) depend on the value of the summation index (I'll use $$\\varphi$$). In this tutorial, we will write a java program to print Pascal Triangle.. Java Example to print Pascal\u2019s Triangle. = (x)6 \u2013 6(x)5(2y2) + 15(x)4(2y2)2 \u2013 20(x)3(2y2)3 + 15(x)2(2y2)4 \u2013 6(x)(2y2)5+ (2y2)6, = x6 \u2013 12x5y2 + 60x4y4 \u2013 160x3y6 + 240x2y8 \u2013 192xy10 + 64y12. If you're familiar with the intricacies of Pascal's Triangle, see how I did it by going to part 2. note: the Pascal number is coming from row 3 of Pascal\u2019s Triangle. Pascal's Triangle can be used to determine how many different combinations of heads and tails you can get depending on how many times you toss the coin. Look at the 4th line. I'm trying to make program that will calculate Pascal's triangle and I was looking up some examples and I found this one. And look at that! The entries in each row are numbered from Example: Input: N = 5 Output: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Method 1: Using nCr formula i.e. So, for example, consider the first five rows of Pascal\u2019s Triangle below, and the path shown between the top number 1 (labelled START) and the left-most 3. Example 6: Using Pascal\u2019s Triangle to Find Binomial Expansions. \\binom{2}{0} \\quad \\binom{2}{1} \\quad \\binom{2}{2} \\newline 1 1. \\[ This algebra 2 video tutorial explains how to use the binomial theorem to foil and expand binomial expressions using pascal's triangle and combinations. As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 1 face, 3 edges, and 3 vertices (the meaning of the final 1 will be explained shortly). See Answer. In this case, the green lines are initially at an angle of $$\\frac{\\pi}{9}$$ radians, and gradually become less steep as $$z$$ increases. If there were 4 children then t would come from row 4 etc\u2026 By making this table you can see the ordered ratios next to the corresponding row for Pascal\u2019s Triangle for every possible combination.The only thing left is to find the part of the table you will need to solve this particular problem( 2 boys and 1 girl): Below is an interesting solution. = a4 \u2013 12a3b + 6a2(9b2) \u2013 4a(27b3) + 81b4. Pascal Triangle and the Binomial Theorem - Concept - Examples with step by step explanation. Popular Problems. (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 + 1y 3 = x 3 + 3x 2 y + 3xy 2 + y 3. \\[ (a+b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{n-k} b^k$. 1 2 1. Get code examples like \"pascals triangle java\" instantly right from your google search results with the Grepper Chrome Extension. EDIT: full working example with register calling convention: file: so_32b_pascal_triangle.asm. Pascal's Triangle can show you how many ways heads and tails can combine. Take a look at Pascal's triangle. The mighty Triangle has spoken. Notes. Pascal's triangle is one of the classic example taught to engineering students. {_1C_0} \\quad {_1C_1} \\$5px] {_4C_0} \\quad {_4C_1} \\quad {_4C_2} \\quad {_4C_3} \\quad {_4C_4} \\\\[5px] In pascal\u2019s triangle, each number is the sum of the two numbers \u2026 n!\/(n-r)!r! Pascal strikes again, letting us know that the coefficients for this expansion are 1, 4, 6, 4, and 1. Generated pascal\u2019s triangle will be: 1.$. $$\\binom{3}{1} = 3\\$4px]$$ We can write the first 5 equations. From the fourth row, we know our coefficients will be 1, 4, 6, 4, and 1. 1 3 3 1. He has noticed that each row of Pascal\u2019s triangle can be used to determine the coefficients of the binomial expansion of ( + ) , as shown in the figure. Examples, videos, worksheets, games, and activities to help Algebra II students learn about the Binomial Theorem and the Pascal's Triangle. \\binom{4}{0} \\quad \\binom{4}{1} \\quad \\binom{4}{2} \\quad \\binom{4}{3} \\quad \\binom{4}{4} \\newline Notice that the sum of the exponents always adds up to the total exponent from the original binomial. In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. So Pascal's triangle-- so we'll start with a one at the top. For example, the fifth row of Pascal\u2019s triangle can be used to determine the coefficients of the expansion of ( + ) . Pascal's triangle can be written as an infintely expanding triangle, with each term being generated as the sum of the two numbers adjacently above it. Given this, we can ascertain that the coefficient $$3$$ choose $$0$$, or $$\\binom{3}{0}$$ = $$1$$. 03:31. As you can see, the $$3$$rd row (starting from $$0$$) includes $$\\binom{3}{0}\\ \\binom{3}{1}\\ \\binom{3}{2}\\ \\binom{3}{3}$$, the numbers we obtained from the binommial expansion earlier. i have a method of proving the fermat's last theorem via the pascal triangle. 1 1. (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 2 The rows of Pascal's triangle are enumerated starting with row r = 1 at the top. n!\/(n-r)!r! Consider again Pascal's Triangle in which each number is obtained as the sum of the two neighboring numbers in the preceding row. The Pascal Integer data type ranges from -32768 to 32767. A while back, I was reintroduced to Pascal's Triangle by my pre-calculus teacher. 17 pascals triangle essay examples from professional writing service EliteEssayWriters.com. ( x + y) 3. Here, is the binomial coefficient . 2008-12-12 00:03:56. It is pretty easy to understand why Pascal's Triangle is applicable to combinations because of the Binomial Theorem. Answer . $$\\binom{n}{k}$$ means $$n$$ choose $$k$$, which has a relation to statistics. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 {\\displaystyle n=0} at the top. At first, Pascal\u2019s Triangle may look like any trivial numerical pattern, but only when we examine its properties, we can find amazing results and applications. It'd be a shame to leave that 3 all on its lonesome. There are various methods to print a pascal\u2019s triangle. What Is Pascal's Triangle? We're not the boss of you. Pascal\u2019s triangle is an array of binomial coefficients. Using Pascal's Triangle Heads and Tails. Get more argumentative, persuasive pascals triangle essay samples and other research papers after sing up Pascal\u2019s triangle. The number of terms being summed up depends on the $$z$$th term. This triangle was among many o\u2026 The top row is numbered as n=0, and in each row are numbered from the left beginning with k = 0. We know that Pascal\u2019s triangle is a triangle where each number is the sum of the two numbers directly above it. Approach #1: nCr formula ie- n!\/(n-r)!r! Asked by Wiki User. \\[z_5 = {_4C_0} + {_3C_1} + {_2C_2} = 5$. The 1 represents the combination of getting exactly 5 heads. Here are some examples of how Pascal's Triangle can be used to solve combination problems: Example 1: The whole triangle can. 2. Sample Problem. Problem : Create a pascal's triangle using javascript. Using the Fibonacci sequence as our main example, we discuss a general method of solving linear recurrences with constant coefficients. For any binomial a + b and any natural number n,(a + b)n = c0anb0 + c1an-1b1 + c2an-2b2 + .... + cn-1a1bn-1 + cna0bn,where the numbers c0, c1, c2,...., cn-1, cn are from the (n + 1)-st row of Pascal\u2019s triangle.Example 1 Expand: (u - v)5.Solution We have (a + b)n, where a = u, b = -v, and n = 5. The first element in any row of Pascal\u2019s triangle \u2026 Q1: Michael has been exploring the relationship between Pascal\u2019s triangle and the binomial expansion. 1. From the above equation, we obtain a cubic equation. Pascal's Triangle can also be used to solve counting problems where order doesn't matter, which are combinations. For example, both $$10$$s in the triangle below are the sum of $$6$$ and $$4$$. $$\\binom{3}{3} = 9\\$4px]$$. A \u2026 Fibonacci\u2019s rabbit problem 9:36. \\[ (x + y)3 = x3 + 3x2y + 3xy2 + y2 The rows of Pascal's triangle are enumerated starting with row r = 1 at the top. How do I use Pascal's triangle to expand the binomial #(a-b)^6#? $$\\binom{3}{2} = 3\\\\[4px]$$ Pascal\u2019s triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y) n.It is named for the 17th-century French mathematician Blaise Pascal, but it is far older.Chinese mathematician Jia Xian devised a triangular representation for the coefficients in the 11th century. You've been inactive for a while, logging you out in a few seconds... Pascal's Triangle and The Binomial Theorem, Use Polynomial Identities to Solve Problems, Using Roots to Construct Rough Graphs of Polynomials, Perfect Square Trinomials and the Difference Between Two Squares. Expand ( x + y) 3. Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. In this program, user is asked to enter the number of rows and based on the input, the pascal\u2019s triangle is printed with the entered number of rows. The coefficients will correspond with line of the triangle. A binomial raised to the 6th power is right around the edge of what's easy to work with using Pascal's Triangle. Example 1. The numbers range from the combination(4,0)[n=4 and r=0] to combination(4,4). Of course, it's not just one row that can be represented by a series of $$n$$ choose $$k$$ symbols. For example, both $$10$$s in the triangle below are the sum of $$6$$and $$4$$. = 1(2x)5 + 5(2x)4(y) + 10(2x)3(y)2 + 10(2x)2(y)3 + 5(2x)(y)4 + 1(y)5, = 32x5 + 80x4y + 80x3y2 + 40x2y3 + 10xy4 + y5. I'll be using this notation from now on. A binomial expression is the sum, or difference, of two terms. Using Pascal\u2019s Triangle you can now fill in all of the probabilities. For a step-by-step walk through of how to do a binomial expansion with Pascal\u2019s Triangle, check out my tutorial \u2b07\ufe0f . The positive sign between the terms means that everything our expansion is positive. Since we are tossing the coin 5 times, look at row number 5 in Pascal's triangle as shown in the image to the right. Like I said, I'm going to be using $$_nC_k$$ symbols to express relationships to Pascal's triangle, so here's the triangle expressed with different symbols. Sample Question Videos 03:30. There are other types which are wider in range, but for now the integer type is enough to hold up our values. All values outside the triangle are considered zero (0). Pascals Triangle \u2014 from the Latin Triangulum Arithmeticum PASCALIANUM ... For position [2], let\u2019s use the above example to demonstrate things. The characteristic equation 8:43. 02:59. \\binom{5}{0} \\quad \\binom{5}{1} \\quad \\binom{5}{2} \\quad \\binom{5}{3} \\quad \\binom{5}{4} \\quad \\binom{5}{5} \\newline #3 Kristofer, July 26, 2012 at 2:31 a.m. Nice illustration! $$\\binom{3}{0} = 1\\\\[4px]$$ This binomial theorem relationship is typically discussed when bringing up Pascal's triangle in pre-calculus classes. Or don't. The program code for printing Pascal\u2019s Triangle is a very famous problems in C language. After using nCr formula, the pictorial representation becomes: This row shows the number of combinations 5 tosses can make. 1 4 6 4 1. Example: Input: N = 5 Output: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 . Pascal's triangle. You will be able to easily see how Pascal\u2019s Triangle relates to predicting the combinations. = 1 x 3 + 3 x 2 y + 3 xy 2 + 1 y 3. To understand this example, you should have the knowledge of the following C++ programming topics: 1 5 10 10 5 1. For example, x+1, 3x+2y, a\u2212 b $$\\binom{3}{0}\\ \\binom{3}{1}\\ \\binom{3}{2}\\ \\binom{3}{3}$$. This is possible as like the Fibonacci sequence, Pascal's triangle adds the two previous (numbers above) to get the next number, the formula if Fn = Fn-1 + Fn-2. \\[ Pascal\u2019s triangle is a pattern of the triangle which is based on nCr, below is the pictorial representation of Pascal\u2019s triangle. = x 3 + 3 x 2 y + 3 xy 2 + y 3. What exactly is this relatiponship? With all this help from Pascal and his good buddy the Binomial Theorem, we're ready to tackle a few problems. Add a Comment. Pascal's Triangle for given n=6: Using equation, pascalTriangleArray[i][j] = BinomialCoefficient(i, j); if j<=i, pascalTriangleArray[i][j] = 0; if j>i. And one way to think about it is, it's a triangle where if you start it up here, at each level you're really counting the different ways that you can get to the different nodes. Doing so reveals an approximation of the famous fractal known as Sierpinski's Triangle. My instructor stated that Pascal's triangle strongly relates to the coefficients of an expanded binomial. So one-- and so I'm going to set up a triangle. The way the entries are constructed in the table give rise to Pascal's Formula: Theorem 6.6.1 Pascal's Formula top Let n and r be positive integers and suppose r \u00a3 n. Then. Input: 6. Pascal\u2019s triangle is a pattern of the triangle which is based on nCr, below is the pictorial representation of Pascal\u2019s triangle.. Examples, videos, worksheets, games, and activities to help Algebra II students learn about the Binomial Theorem and the Pascal's Triangle. 4 5 6. Pascal's Triangle Pascal's triangle is a geometric arrangement of the binomial coefficients in the shape of a triangle. The triangle also shows you how many Combinations of objects are possible. \\binom{1}{0} \\quad \\binom{1}{1} \\newline We hope this article was as interesting as Pascal\u2019s Triangle. Fully expand the expression (2 + 3 ) . Pascal\u2019s triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y) n.It is named for the 17th-century French mathematician Blaise Pascal, but it is far older.Chinese mathematician Jia Xian devised a triangular representation for the coefficients in the 11th century. From Pascal's Triangle, we can see that our coefficients will be 1, 3, 3, and 1. Linear recurrence relations: definition 7:53. You need to find the 6th number (remember the first number in each row is considered the 0th number) of the 10th row in Pascal's triangle. Pascal Triangle in Java | Pascal triangle is a triangular array of binomial coefficients. This is why there is a relationship. In this post, I have presented 2 different source codes in C program for Pascal\u2019s triangle, one utilizing function and the other without using function. Vending machine problem 10:07. Ex #1: You toss a coin 3 times. Example\u2026 C3 Examples: a) For small values of n, it is easier to use Pascal\u2019s triangle, but for large values of n it is easier to use combinations to determine the coefficients in the expansion of (a + b) n. b) If you have a large version of Pascal\u2019s triangle available, then that will immediately give a correct coefficient. If we look closely at the Pascal triangle and represent it in a combination of numbers, it will look like this. The coefficients are given by the eleventh row of Pascal\u2019s triangle, which is the row we label = 1 0.$. Similarly, 3 + 1 = 4 in orange, and 4 + 6 = 10 in blue. $$6$$and $$4$$are directly above each $$10$$. The overall relationship is known as the binomial theorem, which is expressed below. 1 \\quad 3 \\quad 3 \\quad 1\\newline Both of these program codes generate Pascal\u2019s Triangle as per the number of row entered by the user. (x + 3) 2 = (x + 3) (x + 3) (x + 3) 2 = x 2 + 3x + 3x + 9. In the figure above, 3 examples of how the values in Pascal's triangle are related is shown. 8 people chose this as the best definition of pascal-s-triangle: A triangle of numbers in... See the dictionary meaning, pronunciation, and sentence examples. \\]. You should just remove that last row as I think it's a little bit confusing since it makes it less clear that it actually is the Sierpinski triangle we have here. Pascal's Triangle Pascal's triangle is a geometric arrangement of the binomial coefficients in the shape of a triangle. For example, x+1, 3x+2y, a\u2212 b are all binomial expressions. Example Two. Top Answer. In this example, we are going to use the code snippet that we used in our first example. A program that demonstrates the creation of the Pascal\u2019s triangle is given as follows. Expand using Pascal's Triangle (a+b)^6. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. (x + 3) 2 = x 2 + 6x + 9. Pascal\u2019s triangle and various related ideas as the topic. Pascal's Triangle can be displayed as such: The triangle can be used to calculate the coefficients of the expansion of by taking the exponent and adding . These conditions completely spec-ify it. Is it possible to succinctly write the $$z$$th term ($$Fib(z)$$, or $$F(z)$$) of the Fibonacci as a summation of $$_nC_k$$ Pascal's triangle terms? The positive sign between the terms means that everything our expansion is positive. We may already be familiar with the need to expand brackets when squaring such quantities. Although other mathematicians in Persia and China had independently discovered the triangle in the eleventh century, most of the properties and applications of the triangle were discovered by Pascal. The signs for each term are going to alternate, because of the negative sign. You can go higher, as much as you want to, but it starts to become a chore around this point. 1 \\quad 1 \\newline Domino tilings 8:26. A Pascal\u2019s triangle contains numbers in a triangular form where the edges of the triangle are the number 1 and a number inside the triangle is the sum of the 2 numbers directly above it. Be sure to alternate the signs of each term. If we want to raise a binomial expression to a power higher than 2 (for example if we want to \ufb01nd (x+1)7) it is very cumbersome to do this by repeatedly multiplying x+1 by itself. Precalculus. The more rows of Pascal's Triangle that are used, the more iterations of the fractal are shown. From Pascal's Triangle, we can see that our coefficients will be 1, 3, 3, and 1. See if you can figure it out for yourself before continuing! One of the famous one is its use with binomial equations. Examples of Pascals triangle? Pascal\u2019s triangle and the binomial theorem mc-TY-pascal-2009-1.1 A binomial expression is the sum, or di\ufb00erence, of two terms. It has many interpretations. This is a great challenge for Algebra 2 \/ Pre-Calculus students! Be sure to put all of 3b in the parentheses. 8 people chose this as the best definition of pascal-s-triangle: A triangle of numbers in... See the dictionary meaning, pronunciation, and sentence examples. Pascal's Identity states that for any positive integers and . But I don't really understand how the pascal method works. Lesson Worksheet Q1: Michael has been exploring the relationship between Pascal\u2019s triangle and the binomial expansion. 1 2 1. It follows a pattern. From top to bottom, in yellow, the two values are 1 and 1, which sums to 2, the value below. \\binom{3}{0} \\quad \\binom{3}{1} \\quad \\binom{3}{2} \\quad \\binom{3}{3} \\newline Expand (x \u2013 y) 4. This C program for the pascal triangle in c allows the user to enter the number of rows he\/she want to print as a Pascal triangle. Example: You have 16 pool balls. Method 1: Using nCr formula i.e. So values which are not within the specified range cannot be stored by an integer type. The sequence $$1\\ 3\\ 3\\ 9$$ is on the $$3$$rd row of Pascal's triangle (starting from the $$0$$th row). Example 1. 1 4 6 4 1. Example rowIndex = 3 [1,3,3,1] rowIndex = 0 [1] As we know that each value in pascal\u2019s triangle is a binomial coefficient (nCr) where n is the row and r is the column index of that value. The numbers in \u2026 Note that I'm using $$z$$th term rather than $$n$$th term because $$n$$ is used when representing $$_nC_k$$. \\binom{0}{0} \\newline Or don't. The numbers on the fourth diagonal are tetrahedral numbers. PASCAL'S TRIANGLE AND THE BINOMIAL THEOREM A binomial expression is the sum, or difference, of two terms. The green lines represent the division between each term in the Fibonacci sequence and the red terms represent each $$z_{th}$$ term, the sum of all black numbers sandwiched within the green borders. Notice that the sum of the exponents always adds up to the total exponent from the original binomial. He has noticed that each row of Pascal\u2019s triangle can be used to determine the coefficients of the binomial expansion of ( + ) , as shown in the figure. Output: 1. 07_12_44.jpg This path involves starting at the top 1 labelled START and first going down and to the left (code with a 0), then down to the left again (code with another 0), and finally down to the right (code with a 1). For convenience we take 1 as the definition of Pascal\u2019s triangle. You might want to be familiar with this to understand the fibonacci sequence-pascal's triangle relationship. Expand (x + y) 3. 1 \\quad 4 \\quad 6 \\quad 4 \\quad 1 \\newline If you have 5 unique objects and you need to select 2, using the triangle you can find the numbers of unique ways to select them. Precalculus Examples. The numbers in \u2026 One amazing property of Pascal's Triangle becomes apparent if you colour in all of the odd numbers. PASCAL'S TRIANGLE AND THE BINOMIAL THEOREM. 1 3 3 1. Feel free to comment below for any queries \u2026 {_5C_0} \\quad {_5C_1} \\quad {_5C_2} \\quad {_5C_3} \\quad {_5C_4} \\quad {_5C_5} \\$5px] Pascals Triangle Although this is a pattern that has been studied throughout ancient history in places such as India, Persia and China, it gets its name from the French mathematician Blaise Pascal . As you can see, it's the coefficient of the $$k$$th term in the polynomial expansion $$(a+b)^n$$ For example, $$n=3$$ yields the following: \\[ (a+b)^3 = \\sum_{k=0}^{3} \\binom{3}{k} a^{3-k} b^{k}$, $a^3 + 3ab^2 + 3a^2b + 9b^3 = \\binom{3}{0}a^3 + \\binom{3}{1}a^2b + \\binom{3}{2}b^2a + \\binom{3}{3}b^3$. Example: Input : N = 5 Output: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1. {_3C_0} \\quad {_3C_1} \\quad {_3C_2} \\quad {_3C_3} \\\\[5px] do you want to have a look? Alternatively, Pascal's triangle can also be represented in a similar fashion, using $$_nC_k$$ symbols. 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Other types which are not within the specified range can not be stored an! Number triangle which, although very easy to work with using Pascal s... Are shown integer type is enough to hold up our values 1 0 explains to. Combination of getting exactly 5 heads sure to alternate, because of the exponents always adds up to the power! Is numbered as n=0, and 1 examples and I was looking up some examples and I was up. Reveals an approximation of the Pascal number is the pictorial representation of a triangle 1 3. The total exponent from the original binomial two numbers directly above it, so uses same! We look closely at the top row is numbered as n=0, and 1 row shows the of... Algebra 2 \/ pre-calculus students a-b ) ^6 always adds up to the figure above, 3, and.. Can figure it out for yourself before continuing any queries \u2026 the Pascal is! Lines intersecting various rows of the Fibonacci sequence program codes generate Pascal \u2019 s triangle of... Any queries \u2026 the Pascal triangle the above equation, we can see values... 6 = 10 in blue blaise Pascal was born at Clermont-Ferrand, the! To become a chore around this point 3 examples of how the Pascal integer data type from! Is its use with binomial equations ) symbols tutorial explains how to do a binomial is!: Create a Pascal 's triangle is an array of the exponents always up! Tutorial, we 're ready to tackle a few problems coefficients for this expansion are,! Algebra 2 video tutorial explains how to use the binomial theorem, we a. Figure below for any positive integers and triangle and I was looking up some examples and I was looking some... Top to bottom, in the shape of a triangle triangle using javascript note: the Pascal method works determine. From now on 'll be using this notation from now on of each term Chrome! Formula ie- n! \/ ( n-r )! r about Pascal 's triangle, we write. 4, 6, 4, and 1 to do a binomial expansion with Pascal \u2019 s in. We know that the sum of the famous one is its use with binomial equations really how. Worksheet q1: Michael has been exploring the relationship between Pascal \u2019 s triangle and the binomial theorem which. Heads and tails can combine sure to alternate the signs of each term are going to up. To Pascal 's triangle pascal's triangle example, we discuss a general method of solving linear recurrences with coefficients... The 1 represents the combination of getting exactly 5 heads we obtain cubic. Where each number is coming from row 3 of Pascal 's triangle is triangular. Java '' instantly right from your google search results with the need to expand the expression ( 2 6x! Various methods to print Pascal \u2019 s triangle, which are combinations via... Left beginning with k = 0 which sums to 2, the two numbers above it n. Can make by my pre-calculus teacher used in our first example engineering students sum, difference. For example- print pascal's triangle example \u2019 s triangle \\displaystyle n=0 } at the Pascal integer data type from. That for any queries \u2026 the Pascal number is the sum, or difference, of two.! Interesting patterns and useful properties to understand why Pascal 's triangle, check out my tutorial \u2b07\ufe0f right! Algebra 2 video tutorial explains how to use the code snippet that used... Wrote the Treatise on the \\ ( _nC_r\\ ) terms using some (... Fractal are shown n=0 } at the top row is numbered as n=0, and 1, which based..., we 're ready to tackle a few problems expanded binomial and 4 + 6 = in. The topic means that everything our expansion is positive where order does n't matter, which is expressed below a. Sing up Sample Question Videos 03:30 combination ( 4,4 ) squaring such quantities x + 3 ) explains to. Use the binomial theorem relationship is typically discussed when bringing up Pascal 's triangle is a great challenge algebra... 4A ( 27b3 ) + 81b4 end result can look very different from what Pascal initially tells.. Tails can combine is an array of binomial coefficients pascals triangle essay examples from writing. Be sure to alternate, because of the binomial coefficients expansion are,! Terms look like this have some definite evidence that this works where order does n't matter, which based. Of numbers, it will look like this numbers above it it starts to a... Are considered zero ( 0 ) term are going to use the snippet... Terms being summed up depends on the Arithmetical triangle which today is known as definition! Out my tutorial \u2b07\ufe0f Identity states that for any positive integers and triangle ( a+b ) ^6?... Today is known as the binomial theorem - Concept - examples with step step! ) terms using some formula ( starting from 1 ) combinations because of the binomial -. 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In all of 3b in the parentheses 4, 6, 4, 6, 4,,..., it will look like inside the binomial theorem to foil and expand expressions! The Pascal number is found by adding two numbers directly above it 4a ( 27b3 ) + 81b4 that... Tutorial \u2b07\ufe0f using the recursive function to Find binomial Expansions by my pre-calculus teacher in.! Specified range can not be stored by an pascal's triangle example type is enough to hold up our.! Entered by the user Input: n = 0 { \\displaystyle n=0 } at the top triangle in.... 3 1 1 2 1 1 4 6 4 1, 1623 type. Values are 1 and 1 the negative sign code for printing Pascal s... To combination ( 4,4 ) by an integer type is enough to hold our. From the left beginning with k = 0 binomial Expansions 9b2 ) \u2013 4a ( ). ) and \\ ( 4\\ ) are directly above it my pre-calculus teacher this notation from on. To put all of 3b in the Auvergne region of France on June 19 1623! Starts to become a chore around this point you toss a coin 3 times theorem, which is on. Are used, the end result can look very different from what Pascal initially tells us useful.. 27B3 ) + 81b4 do I use Pascal 's triangle and the binomial expansion generate Pascal \u2019 s triangle a! From row 3 of Pascal 's triangle are conventionally enumerated starting with row =.","date":"2021-03-07 07:50:40","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\": 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\": 0.7762048244476318, \"perplexity\": 674.4699758749252}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178376206.84\/warc\/CC-MAIN-20210307074942-20210307104942-00081.warc.gz\"}"} | null | null |
from oslo_log import log as logging
from sqlalchemy import MetaData, Table, Index
from nova.i18n import _LI
LOG = logging.getLogger(__name__)
def upgrade(migrate_engine):
"""Change instances (project_id) index to cover (project_id, deleted)."""
meta = MetaData(bind=migrate_engine)
# Indexes can't be changed, we need to create the new one and delete
# the old one
instances = Table('instances', meta, autoload=True)
for index in instances.indexes:
if [c.name for c in index.columns] == ['project_id', 'deleted']:
LOG.info(_LI('Skipped adding instances_project_id_deleted_idx '
'because an equivalent index already exists.'))
break
else:
index = Index('instances_project_id_deleted_idx',
instances.c.project_id, instances.c.deleted)
index.create()
for index in instances.indexes:
if [c.name for c in index.columns] == ['project_id']:
index.drop()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,568 |
Makua may refer to:
Makua (person), an alaafin of the Oyo Empire
Makua people, an ethnic group in Mozambique and Tanzania
Makhuwa language, a Bantu language spoken in Mozambique
Makua languages, a branch of Bantu languages
Makua Rothman (born 1984), American world champion surfer
See also
Makuv'a language, a language of East Timor
Macuá | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,337 |
/**
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package com.amazonaws.eclipse.core.ansi;
import org.eclipse.swt.custom.StyledText;
import org.eclipse.ui.console.IConsole;
import org.eclipse.ui.console.IConsolePageParticipant;
import org.eclipse.ui.part.IPageBookViewPage;
public class AnsiConsolePageParticipant implements IConsolePageParticipant {
@Override
public Object getAdapter(Class adapter) {
return null;
}
@Override
public void activated() {}
@Override
public void deactivated() {}
@Override
public void dispose() {}
@Override
public void init(IPageBookViewPage page, IConsole console) {
if (page.getControl() instanceof StyledText) {
StyledText viewer = (StyledText) page.getControl();
viewer.addLineStyleListener(new AnsiConsoleStyleListener());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,720 |
By placing the customer first, Labor showing how shared services could work
All of the Labor Department's human resources offices, which at one time numbered 13, are now consolidated into one central, enterprisewide office.
The same is true for procurement and personnel security services.
These three back-office areas are the first of several to come under a new enterprisewide consolidation initiative.
Maylin Jue, the Labor Department's enterprisewide shared services program manager, said this agencywide effort isn't just about consolidation, but making back office or administrative support services better, faster and cheaper.
Maylin Jue, is the Labor Department's enterprisewide shared services program manager.
"We have stood up a new centralized office of human resources. We transitioned all HR staff from across the department into one new office. This allows us to have more team building, training and customer outreach, and it allows us to optimize business processes," Jue said in an interview with Federal News Network. "We have seen improvement in the consistency of the services and making sure everyone is following the same policies across the department. We've gotten a lot of compliments from a lot of the agency about improvements in staffing, which is a huge win for the department to hire more effectively and efficiently."
While the HR consolidation is furthest along, Labor's end goal is to reduce costs and improve efficiencies and consistency across all administrative functions.
"Once transitions are complete, employees across the department will experience improved service delivery from their HR, IT, procurement and personnel security service providers. Service will be more consistent across the department, as it will all fall under one service provider that utilizes standard policies and procedures. Service will be compliant with federal regulations and will strive to meet industry best practices in both the public and private sectors," Jue said. "Creating centralized offices for administrative services will also allow more opportunities for innovation within the department. By improving IT functionality, providing more user friendly HR services, improving procurement efficiency and streamlining personnel security processing, DOL employees will be able to focus more on their mission rather than being burdened by administrative tasks. Lastly, the employees of the service providers will have opportunities to work with a broader portfolio of clients on a variety of challenging projects, receive consistent professional development and training and seek improved advancement within their professional field."
Labor started this back-office consolidation initiative in 2017 with a focus on human resources and soon expanded it to the other areas. Procurement, IT and personnel security now are all well on their way to meeting Labor's goals of agility to meet new requirements from customers and cost savings.
Jue said the first part of the effort was about collecting feedback from users through focus groups that included employees up and down the chain. She said the idea was to understand any and all concerns and business processes, and begin to establish baseline metrics.
Governance boards and SLAs
"Before shared services, a lot of agencies were not necessarily measuring their service levels within their HR or procurement offices. So as part of this, we worked on service level agreements for each area," she said "Time to hire is a good example. We are measuring the components that the HR office is responsible for because there are shared responsibilities when it comes to time to hire."
For IT, metrics included network up time, response rates for help desks tickets and other support services. Under procurement, the metrics include processing of task orders, new procurements and modifications to current contracts.
Jue said the governance boards establish the service level agreements (SLAs) for each administrative function based on customer expectations. She said each administrative function also will depend on client engagement managers who serve as the intermediary between the provider organization and the client offices.
Read more: Shared Services
"These liaisons will discuss business needs directly with customers and facilitate service delivery that meets the standards set in the SLAs," Jue said. "Following a six-month baseline period to calibrate the SLA performance measurements, the SLAs will be reported quarterly to client agencies and linked to senior leadership as part of the annual review process."
Jue said a governance board oversees each of these administrative areas, and Labor is creating an overarching enterprisewide governance board to consider broader strategies for shared services.
The governance boards are a key piece to the "culture change" puzzle. Implementing shared services within an agency many times is dependent on getting both senior executives and mid-tier managers on board with the concepts and changes.
Customer satisfaction survey coming
Agencies like Labor and Commerce have found more success by consolidating functions internally than governmentwide efforts for human resources, financial management and the like.
"Each shared services organization will also have a governing body consisting of functional leadership and rotating client agency representatives to facilitate regular discussion of overall service delivery and departmental needs. These governance boards will typically meet on a bimonthly basis and provide the shared services organizations with real-time feedback," Jue said. "Additionally, client agencies will have the opportunity to provide input into the semi-annual performance reviews of senior leaders of each HR, IT, procurement, and personnel security shared services organization. Adding customer input into the performance appraisals incentivizes the leadership to provide high quality service."
Some of the reasons why Labor is finding success with this enterprisewide shared services approach are these governance boards, the SLAs and the broad input from customers. Jue said her office also will conduct a customer satisfaction survey in fiscal 2021.
She said most of these factors were not part of previous shared services efforts.
"You have to engage with all stakeholders, understand their concerns, their needs and their uniqueness," Jue said. "You also have to look at how you can institute accountability. We did that through performance tracking and the governance board. The agencies need to have input after things are consolidated. You have to think about this as a customer service. At the end of the day, we are delivering services to other parts of Labor and we can't do it well if we don't understand what they need."
When it comes to shared services, Labor isn't eating the apple in one bite
Reporter's Notebook Read more
Labor embraced enterprisewide shared services, now seeing real savings
Commentary Read more
Labor's IT modernization story started with commodity, ends with mission
Ask the CIO Read more
Here is how Commerce is making shared services work
Acquisition Acquisition Policy All News Contracting customer experience IT Modernization Labor Department Maylin Jue Shared Services Technology | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,099 |
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"redpajama_set_name": "RedPajamaC4"
} | 3,202 |
package org.sparkcommerce.openadmin.server.security.domain;
import java.util.ArrayList;
import java.util.List;
/**
* Class to hold the admin menus and sections for which the passed in user has permissions to view.
* @author bpolster
*/
public class AdminMenu {
private List<AdminModule> adminModules = new ArrayList<AdminModule>();
public List<AdminModule> getAdminModules() {
return adminModules;
}
public void setAdminModule(List<AdminModule> adminModules) {
this.adminModules = adminModules;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,515 |
Cnaphalocrocis didialis is een vlinder (nachtvlinder) uit de familie van de grasmotten (Crambidae). De wetenschappelijke naam van deze soort is voor het eerst geldig gepubliceerd in 1958 door Pierre Viette.
De soort komt voor op Madagaskar.
didialis
Dier uit het Afrotropisch gebied | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,138 |
Beaumont-en-Auge é uma comuna francesa na região administrativa da Normandia, no departamento Calvados. Estende-se por uma área de 8 km².
Laplace é natural de Beaumont-en-Auge.
Comunas de Calvados | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,595 |
\section{S1. Non-interacting Hamiltonian for a graphene-insulator heterostructure}
The Hamiltonian for a graphene-insulator heterostructure can be always divided into three parts: graphene part $H_G$, the insulating substrate part $H_S$ and the coupling between them $H_{G\rm{-}S}$. The graphene part can be suitably described by a tight-binding model since we focus on the low-energy physics. As we have explained in the main text, with slight carrier doping band edge, the insulator substrate is supposed to form a long-wavelength charge order on the interface near graphene sheet thanks to Coulomb interactions between electrons occupying the band edge of the insulating substrate (transferred from graphene layer). The insulator substrate part is then modeled by a 2D Hamiltonian for electrons hopping on a 2D superlattice which forms an Wigner-crystal-like or long-wavelength charge ordered insulator state at some proper filling, whose geometry is determined by the long-wavelength order at the interface. Explicitly, the graphene part $H_G$ and the insulator substrate part $H_S$ can be generally written as
\begin{align}
\hat{H}_G &= \sum_{\bm{k}, \sigma, \alpha, \alpha'} \gamma_{\alpha, \alpha'} (\bm{k}) \hat{c}^\dagger_{\sigma \alpha}(\bm{k}) \hat{c}_{\sigma \alpha'}(\bm{k})\\
\hat{H}_S &= \sum_{\widetilde{\bm{k}}, \sigma} \eta (\widetilde{\bm{k}}) \hat{d}^\dagger_{\sigma}(\widetilde{\bm{k}}) \hat{d}_{\sigma}(\widetilde{\bm{k}})
\label{eq:HG_HS}
\end{align}
where $\hat{c}_{\bm{k},\alpha,\sigma}$ ($\hat{c}^{\dagger}_{\bm{k},\alpha,\sigma}$) and $\hat{d}_{\widetilde{\bm{k}},\sigma}$ ($\hat{d}^{\dagger}_{\widetilde{\bm{k}},\sigma}$) are fermionic annihilation (creation) operators for electrons in graphene and the insulator substrate, respectively. In the lower index of these operators, $\alpha$ is the sublattice index for the bipartite lattice of graphene and $\sigma$ is the spin degree of freedom of electrons. To emphasize the fact that graphene and the insulator substrate have different lattices and thus different Brillouin zone, we denote $\bm{k}$ and $\widetilde{\bm{k}}$ as the wavevectors in the Brillouin zone of graphene and that of the long-wavelength superlattice in the substrate, respectively. In our calculations, the lattice for $H_S$ is set to rectangular or triangular, which does not qualitatively change our results.
Since electrons have negligible probability to hop between graphene and the insulator substrate due to rather large distance $d$ between two sheets in the $z$-direction ($d\sim7\,$\AA\ from DFT calculations in CrOCl-graphene heterostructure), we suppose that electrons from two sheets are coupled only via long-ranged Coulomb interactions. Unlike $H_G$ and $H_S$, such long-ranged Coulomb interactions are more easily written in real space. In terms of field operators $\hat{\psi}(\bm{r})$, the inter-sheet coupling reads
\begin{equation}
\hat{H}_{G\rm{-}S} = \int d^2 \bm{r} d^2 \bm{r'} \sum_{\sigma,\sigma '} \hat{\psi}^{\dagger}_{c, \sigma}(\bm{r}) \hat{\psi}^{\dagger}_{d, \sigma'}(\bm{r'}) V (| \bm{r}-\bm{r'} + d \bm{\hat{z}}|) \hat{\psi}_{d, \sigma'}(\bm{r'}) \hat{\psi}_{c, \sigma}(\bm{r})
\end{equation}
where $V(| \bm{r}-\bm{r'}+d \bm{\hat{z}}|)$ is the 3D long-ranged Coulomb potential $e^2/4 \pi \epsilon_0 \epsilon_d r$ and electrons in graphene and the insulating substrate are described by the field operators with lower index $c$ and $d$, respectively. Here $\epsilon_0$ is the vacuum permittivity and $\epsilon_d$ is the dimensionless relative dielectric constant of the insulating substrate. In the spirit of tight-binding formalism, we write the field operators in terms of Wannier functions
\begin{align}
\hat{\psi}^{\dagger}_{c, \sigma}(\bm{r}) &= \sum_{i,\alpha} \phi_\alpha ^* (\bm{r}-\bm{a}_i-\bm{\tau}_\alpha) \chi^{\dagger}_\sigma \hat{c}^{\dagger}_{i,\sigma \alpha} \\
\hat{\psi}^{\dagger}_{d, \sigma}(\bm{r}) &= \sum_{i,\alpha} \widetilde{\phi} ^* (\bm{r}-\bm{R}_i) \chi^{\dagger}_\sigma \hat{d}^{\dagger}_{i,\sigma}
\end{align}
where $\phi_\alpha$ and $\widetilde{\phi}$ are Wannier functions localized on the graphene and the insulator substrate Bravais lattice sites, which are described by $\bm{a}_{i}$ and $\bm{R}_{i}$, respectively. Here $\alpha$ refers to the sublattice index in graphene and $\bm{\tau}_{\alpha}$ is the vector denoting the position of the $\alpha$th sublattice inside the unit-cell. The spin degrees of freedom is included by the index $\sigma$ and also explicitly by spinor $\chi_\sigma$. The Hamiltonian $H_{G\rm{-}S}$ in the Wannier basis reads
\begin{equation}
\hat{H}_{G\rm{-}S} = \sum_{\substack{\sigma, \sigma' \\ \alpha, \alpha'}} \sum_{\substack{i,i' \\ j,j'}} U^{\sigma \sigma'}_{i \alpha j , i' \alpha' j'} \hat{c}^\dagger_{i,\sigma \alpha} \hat{d}^\dagger_{j,\sigma'} \hat{d}_{j',\sigma'} \hat{c}_{i',\sigma \alpha'}
\end{equation}
with
\begin{equation}
U^{\sigma \sigma'}_{i \alpha j, i' \alpha' j'} = \int d^2 \bm{r} d^2 \bm{r'} \phi_\alpha^* (\bm{r}-\bm{a}_i-\bm{\tau}_\alpha) \widetilde{\phi} ^* (\bm{r}-\bm{R}_j) V (| \bm{r}-\bm{r'} + d \bm{\hat{z}}|) \widetilde{\phi} (\bm{r'}-\bm{R}_{j'}) \phi_{\alpha'} (\bm{r}-\bm{a}_{i'}-\bm{\tau}_{\alpha'}) \chi^\dagger_\sigma \chi^\dagger_{\sigma'} \chi_{\sigma'} \chi_\sigma.
\end{equation}
If Wannier functions are so localized such that
\begin{align*}
\phi_\alpha ^* (\bm{r}-\bm{a}_i-\bm{\tau}_\alpha) \phi_{\alpha'} (\bm{r}-\bm{a}_{i'}-\bm{\tau}_{\alpha'}) &\approx 0 \hspace{2cm} \text{if } (i,\alpha) \ne (i',\alpha') \\
\widetilde{\phi} ^* (\bm{r}-\bm{R}_j) \widetilde{\phi} (\bm{r}-\bm{R}_{j'}) &\approx 0 \hspace{2cm} \text{if } j \ne j' \\
|\phi_\alpha (\bm{r}-\bm{a}_i-\bm{\tau}_\alpha)|^2 &\approx \delta^{(2)}(\bm{r}-\bm{a}_i-\bm{\tau}_\alpha) \\
|\widetilde{\phi} (\bm{r}-\bm{R}_j) |^2 &\approx \delta^{(2)}(\bm{r}-\bm{R}_j)
\end{align*}
with $\delta^{(2)}(\bm{r})$ is the 2D Dirac $\delta$-function distribution, we can simplify the previous expression to
\begin{equation}
U^{\sigma \sigma'}_{i \alpha j, i' \alpha' j'} = U_{i \alpha j} \delta_{i,i'} \delta_{\alpha,\alpha'} \delta_{j,j'}
\end{equation}
with $\delta_{\mu, \nu}$ is the Kronecker delta and
\begin{equation}
U_{i \alpha j} = V (|\bm{a}_i+\bm{\tau}_\alpha - \bm{R}_j + d \bm{\hat{z}}|).
\end{equation}
Then, we write $H_{G-S}$ in reciprocal space using the following Fourier transformation
\begin{align}
\hat{c}_{i,\sigma \alpha} &= \frac{1}{\sqrt{N_c}} \sum_{\bm{k}} e^{i \bm{k} \cdot \bm{a}_{i}} \hat{c}_{\sigma \alpha}(\bm{k})\\
\hat{d}_{i,\sigma} &= \frac{1}{\sqrt{N_d}} \sum_{\widetilde{\bm{k}}} e^{i \widetilde{\bm{k}} \cdot \bm{R}_{i}} \hat{d}_{\sigma}(\widetilde{\bm{k}})
\end{align}
where $N_c$ and $N_d$ are the number of lattice sites for electron in graphene and the insulator substrate, respectively. The Hamiltonian $H_{G\rm{-}S}$ in the basis of $\hat{c}_{\sigma \alpha}(\bm{k})$ and $\hat{d}_{\sigma}(\widetilde{\bm{k}})$ reads
\begin{equation}
\hat{H}_{G-S} = \frac{1}{N_c N_d} \sum_{\substack{\sigma,\sigma' \\ i,\alpha,j}} \sum_{\substack{\bm{k},\bm{k} ' \\ \widetilde{\bm{k}}, \widetilde{\bm{k}}'}} U_{i \alpha j} \ e^{i (\bm{k} ' -\bm{k}) \cdot (\bm{a}_{i}-\bm{R}_{j})} \ e^{i (\bm{k} ' - \bm{k} + \widetilde{\bm{k}} ' - \widetilde{\bm{k}}) \cdot \bm{R}_{j}} \ \hat{c}^\dagger_{\sigma \alpha}(\bm{k}) \hat{d}^\dagger_{\sigma'}(\widetilde{\bm{k}}) \hat{d}_{\sigma'} (\widetilde{\bm{k}}') \hat{c}_{\sigma \alpha}(\bm{k}').
\label{eq:hGS_k}
\end{equation}
Now we first define $\widetilde{\mathbf{R}}=\mathbf{a}_i-\mathbf{R}_j$, and let $\bm{k} '-\bm{k}=\bm{q}=\widetilde{\bm{q}} +\mathbf{G}$, where $\mathbf{G}$ is a reciprocal vector of the long-wavelength ordered superlattice and $\widetilde{\bm{q}}$ is the wavevector within the superlattice Brillouin zone. Then we take use of the identity $\sum_j e^{i(\widetilde{\bm{k}} '-\widetilde{\bm{k}} +\bm{q})\cdot\mathbf{R}_j}=\sum_j e^{i(\widetilde{\bm{k}} '-\widetilde{\bm{k}}+\widetilde{\bm{q}}+\mathbf{G})\cdot\mathbf{R}_j}=N_d\delta_{\widetilde{\bm{k}} '-\widetilde{\bm{k}},\widetilde{\bm{q}}}$, Eq.~(\ref{eq:hGS_k}) can be simplified as
\begin{equation}
\hat{H}_{G-S} = \sum_{\substack{\sigma,\sigma' \\ \alpha}} \sum_{\substack{\bm{k},\widetilde{\bm{k}} \\ \widetilde{\bm{q}}, \widetilde{\bm{G}}}} \widetilde{V}(\widetilde{\bm{q}} + \bm{G}) \ \hat{c}^\dagger_{\sigma \alpha}(\bm{k}) \hat{d}^\dagger_{\sigma'}(\widetilde{\bm{k}}) \hat{d}_{\sigma'} (\widetilde{\bm{k}} + \widetilde{\bm{q}}) \hat{c}_{\sigma \alpha}(\bm{k} - \widetilde{\bm{q}} - \bm{G})
\end{equation}
The coupling $\widetilde{V}(\widetilde{\bm{q}}+\mathbf{G})$ reads
\begin{align}
\widetilde{V}(\widetilde{\bm{q}}+\bm{G}) &= \frac{1}{N_c} \sum_{i} V (|\bm{a}_{i}+\bm{\tau}_{\alpha} - \bm{R}_{j} + d \bm{\hat{z}}|) e^{-i (\widetilde{\bm{q}}+\bm{G}) \cdot (\bm{a}_{i} - \bm{R}_{j})} \nonumber \\
&= \frac{1}{N_c} \sum_{\widetilde{\bm{R}}} V (|\widetilde{\bm{R}}+\bm{\tau}_{\alpha} + d \bm{\hat{z}}|) e^{-i (\widetilde{\bm{q}}+\bm{G}) \cdot \widetilde{\bm{R}}} \nonumber \\
&= \frac{1}{N_d} \int \frac{d^2 r}{\Omega_d} V (|\bm{r}+\bm{\tau}_{\alpha} + d \bm{\hat{z}}|) e^{-i (\widetilde{\bm{q}}+\bm{G}) \cdot \bm{r}} \nonumber \\
&= \frac{e^2}{2 \epsilon_0 \epsilon_d N_d \Omega_d} \frac{e^{-|\widetilde{\bm{q}}+\bm{G}|d}}{|\widetilde{\bm{q}}+\bm{G}|}
\end{align}
where $\Omega_d$ is the area of the unit-cell of the surface superlattice of the substrate.
In the third line of the above derivation, we smear the sum over $\widetilde{\bm{R}}=\bm{a}_i-\bm{R}_j$ by replacing it with an integral over the surface $S = N_d \Omega_d = N_c \Omega_c$ with $\Omega_c$ the area of graphene's unit-cell since we are interested in the physics in the length scale of the superlattice $\{ \bm{R}_{j} \}$, which is supposed to much larger than that of graphene. Finally, the last line is the 2D partial Fourier transformation of the 3D Coulomb potential.
Since we focus on the low-energy physics around the Dirac cones of graphene, we can attribute valley index $\mu$ to electrons in graphene and neglect intervalley coupling thanks to the exponential decay of $\widetilde{V}(\bm{q})$ so that
\begin{equation}
\hat{H}_{G-S} = \sum_{\substack{\sigma,\sigma' \\ \alpha}} \sum_{\substack{\bm{k},\widetilde{\bm{k}} \\ \widetilde{\bm{q}}, \bm{G}}} \widetilde{V}(\widetilde{\bm{q}} + \bm{G}) \sum_{\mu} \hat{c}^\dagger_{\sigma \mu \alpha}(\bm{k}) \hat{d}^\dagger_{\sigma'}(\widetilde{\bm{k}}) \hat{d}_{\sigma'} (\widetilde{\bm{k}} + \widetilde{\bm{q}}) \hat{c}_{\sigma \mu \alpha}(\bm{k} - \widetilde{\bm{q}} - \bm{G}) .
\end{equation}
In the meantime, the Hamiltonian for graphene only $H_G$ [see Eq.~\eqref{eq:HG_HS}] can be divided into two valley sectors
\begin{equation}
\hat{H}_G = \sum_{\bm{k}, \sigma, \alpha, \alpha', \mu} \left( \hbar v_F \bm{k} \cdot \bm{\sigma}^\mu \right)_{\alpha, \alpha'} \hat{c}^\dagger_{\sigma \alpha}(\bm{k}) \hat{c}_{\sigma \alpha'}(\bm{k})
\label{eq:H_Gonly}
\end{equation}
where $\bm{\sigma}^\mu = (\mu \sigma_x, \sigma_y)$ with $\sigma_{x,y}$ are the Pauli matrices and the valley index $\mu =\pm 1$.
In the Hartree approximation by pairing $c$ and $d$ separately, we have
\begin{equation}
\hat{H}_{G-S} = \sum_{\substack{\sigma, \alpha, \mu}} \sum_{\substack{\bm{k}, \bm{G}}} \widetilde{V}(\bm{G}) \sum_{\widetilde{\bm{k}}, \sigma '} \langle \hat{d}^\dagger_{\sigma'}(\widetilde{\bm{k}}) \hat{d}_{\sigma'} (\widetilde{\bm{k}}) \rangle \ \hat{c}^\dagger_{\sigma \alpha \mu}(\bm{k}) \hat{c}_{\sigma \alpha \mu}(\bm{k} - \bm{G}).
\end{equation}
Since the long-wavelength charge order state is insulating presumingly with two spin degenerate electrons occupying each supercell, we have
\begin{equation}
\sum_{\widetilde{\bm{k}}, \sigma '} \langle \hat{d}^\dagger_{\sigma'}(\widetilde{\bm{k}}) \hat{d}_{\sigma'} (\widetilde{\bm{k}}) \rangle = 2 N_d.
\end{equation}
Writing $\bm{k} = \widetilde{\bm{k}} + \bm{G}$ with $\bm{G}$ in the superlattice reciprocal lattice, the final form of the coupling between graphene and insulating substrate used in our calculations reads
\begin{equation}
\hat{H}_{G-S} = \sum_{\substack{\sigma, \alpha, \mu}} \sum_{\substack{\bm{G}, \bm{Q} \\ \in \{ \bm{G_i} \} } } \widetilde{U}_d (\bm{Q}) \ \hat{c}^\dagger_{\sigma \mu \alpha, \bm{G}+\bm{Q}}(\widetilde{\bm{k}}) \hat{c}_{\sigma \mu \alpha, \bm{G}}(\widetilde{\bm{k}}).
\label{eq:hamGS_final}
\end{equation}
where
\begin{equation}
\widetilde{U}_d (\bm{Q}) = \frac{e^2}{\epsilon_0 \epsilon_d \Omega_d} \frac{e^{-|\bm{Q}|d}}{|\bm{Q}|}
\label{eq:Udfourier}
\end{equation}
In the meantime, we integrate out the Hamiltonian for insulating substrate $H_S$ [see Eq.~\eqref{eq:HG_HS}] so that it becomes a constant charge density, which is omitted in our calculations. To wrap up, we get the effective non-interacting Hamiltonian in continuum in the valley $\mu$
\begin{equation}
H^\mu_0(\bm{r}) = \hbar v_F \bm{k} \cdot \bm{\sigma}^{\mu} + U_d(\bm{r})
\label{eq:ham_0}
\end{equation}
where the Fourier component of $U_d(\mathbf{r})$ is precisely $\widetilde{U}_d (\bm{G})$ [see Eq.~\eqref{eq:Udfourier}] with $\bm{G}$ in the reciprocal lattice of the underlying insulating substrate's surface superlattice.
In our numerical implementations, the lattice of insulating substrate is set to be rectangular or triangular, from which we obtain qualitatively the same correlated states in the graphene layer. The range of $\{ \bm{G_i} \}$ is limited to $|n_x|,|n_y| \leq 4$ with $\bm{G} = n_x \bm{g_x} + n_y \bm{g_y}$. $\bm{g_{x,y}}$ are the two reciprocal lattice vectors for the rectangular lattice of insulating substrate. The sum over $\bm{Q}$ in Eq.~\eqref{eq:hamGS_final} stops at the limit $|n_x|+|n_y|\leq 2$.
\section{S2. Renormalization group derivations}
The derivation shown in this section is inspired from Ref.~\onlinecite{vafek_prl2020}. The e-e Coulomb interaction operator in our derivations is written as
\begin{equation}
\hat{V}_{\text{int}} = \frac{1}{2} \int d^2 \bm{r} d^2 \bm{r} ' V_c(\bm{r} - \bm{r} ') \hat{\rho} (\bm{r}) \hat{\rho} (\bm{r} ')
\end{equation}
where $V_c(\bm{r}) = e^2/4 \pi \epsilon_0 \epsilon_d r$ and $\hat{\rho}(\bm{r})$ is the density operator of electrons at $\bm{r}$. The Hamiltonian Eq.~\eqref{eq:ham_0} is defined at some high energy cut-off $\pm E_c$. We focus in the valley $\mu=+1$ by the virtue of which the derivation for the valley $\mu=-1$ is immediate and the results are identical. Remember that the parameters $v_F$ and $\widetilde{U}_d(G)$ should be thought of as being fixed by a measurement at $E_c$ without e-e interactions. This also amounts to $\hat{\rho}(\bm{r})= \hat{\psi}^\dagger (\bm{r}) \hat{\psi} (\bm{r})$ with the non-interacting field operator $\hat{\psi} (\bm{r})$
\begin{equation}
\hat{\psi} (\bm{r}) = \sum_{\substack{\sigma, n,
\bm{k};\\ |\epsilon_{n,\bm{k}}| \leq E_c }} \phi_{\sigma n \bm{k}} (\bm{r}) \hat{c}_{\sigma n}(\bm{k})
\end{equation}
where $\phi_{\sigma n \bm{k}} (\bm{r})$ is the wavefunction of an eigenstate of the non-interacting Hamiltonian $H_0$ [see Eq.~\eqref{eq:ham_0}] with energy $\epsilon_{n,\bm{k}}$ and its associated annihilation operator is $\hat{c}_{\sigma n}(\bm{k})$.
\subsection{Electron-electron interaction in a lower energy window}
Now we change the cut-off $E_c$ to a smaller one $E_c '$ and see how these parameters are modified by $\hat{V}_{\text{int}}$. $\hat{V}_{\text{int}}$ can be treated perturbatively when $E_c '$ is much larger than any other energy scale in the system. To do so, we split the field operator $\hat{\psi} (\bm{r}) = \hat{\psi}^{<} (\bm{r}) + \hat{\psi}^{>} (\bm{r})$ where
\begin{align}
\hat{\psi}^{<} (\bm{r}) &= \sum_{\substack{\sigma, n,
\bm{k};\\ |\epsilon_{n,\bm{k}}| \leq E_c ' }} \phi_{\sigma n \bm{k}} (\bm{r}) \hat{c}_{\sigma n}(\bm{k}) \\
\hat{\psi}^{>} (\bm{r}) &= \sum_{\substack{\sigma, n,
\bm{k};\\ E_c ' < |\epsilon_{n,\bm{k}}| \leq E_c }} \phi_{\sigma n \bm{k}} (\bm{r}) \hat{c}_{\sigma n}(\bm{k}). \\
\end{align}
Then, we integrate out the fast modes $\hat{\psi}^{>} (\bm{r})$ in the expansion of $\hat{\rho}(\bm{r}) \hat{\rho}(\bm{r}')$. Note that $\hat{\psi}^{>} (\bm{r})$ and $\hat{\psi}^{> \dagger} (\bm{r})$ must appear equal times in each terms of the expansion otherwise it would vanish by taking the non-interacting mean value $\langle \dots \rangle_0$. Explicitly, these terms are retained up to a constant:
\begin{align*}
\hat{\rho}(\bm{r}) \hat{\rho}(\bm{r}') &= \hat{\rho}^{<} (\bm{r}) \hat{\rho}^{<} (\bm{r}') \\
& \quad + \bar{\rho}^{>} (\bm{r}) \hat{\psi}^{< \dagger} (\bm{r} ') \hat{\psi}^{<} (\bm{r} ') + \bar{\rho}^{>} (\bm{r} ') \hat{\psi}^{< \dagger} (\bm{r}) \hat{\psi}^{<} (\bm{r}) \\
& \quad + \underbrace{\hat{\psi}^{< \dagger} (\bm{r}) \ \langle \hat{\psi}^{>} (\bm{r}) \hat{\psi}^{> \dagger} (\bm{r} ') \rangle_0 \ \hat{\psi}^{<} (\bm{r} ') + \hat{\psi}^{<} (\bm{r}) \ \langle \hat{\psi}^{> \dagger} (\bm{r}) \hat{\psi}^{>} (\bm{r} ') \rangle_0 \ \hat{\psi}^{< \dagger} (\bm{r} ')}_{(*)}
\end{align*}
with
\begin{align}
\hat{\rho}^{<} (\bm{r}) &= \hat{\psi}^{< \dagger} (\bm{r}) \hat{\psi}^{<} (\bm{r})\\
\bar{\rho}^{>} (\bm{r}) &= \sum_{\substack{\sigma, n,
\bm{k};\\ E_c ' < |\epsilon_{n,\bm{k}}| \leq E_c}} \phi_{\sigma n \bm{k}}^* (\bm{r}) \phi_{\sigma n \bm{k}} (\bm{r}).
\end{align}
The first term gives the Coulomb e-e interaction between electrons of the slow modes $\hat{\psi}^{<} (\bm{r})$ below the new cut-off $E_c '$. The second and third term could be omitted if the system has particle-hole (p-h) symmetry as in twisted bilayer graphene \cite{vafek_prl2020}. In our system described by Eq.~\eqref{eq:ham_0}, the first nearest-neighbor coupling in $\widetilde{U}_d (\bm{G})$ preserves p-h symmetry. The p-h symmetry is broken if further-neighbor coupling is included, which is exponentially smaller [see Eq.~\eqref{eq:Udfourier}]. So, it is legitimate in our RG derivation to neglect such weak p-h asymmetry in order to omit the second and the third term in the expansion.
Then, we evaluate the rest of the terms in the expansion, which represents precisely the correction to $H_0$ from the fast modes $\hat{\psi}^{>} (\bm{r})$ via Coulomb e-e interactions. Let us write
\begin{align*}
(*) &= \hat{\psi}^{< \dagger} (\bm{r}) \left( \sum_{\substack{\sigma, n,
\bm{k};\\ E_c ' < \epsilon_{n,\bm{k}} \leq E_c}} \phi_{\sigma n \bm{k}} (\bm{r}) \phi_{\sigma n \bm{k}}^* (\bm{r} ') \right) \hat{\psi}^{<} (\bm{r} ') + \hat{\psi}^{<} (\bm{r}) \left( \sum_{\substack{\sigma, n,
\bm{k};\\ -E_c ' > \epsilon_{n,\bm{k}} \geq -E_c}} \phi_{\sigma n \bm{k}}^* (\bm{r}) \phi_{\sigma n \bm{k}} (\bm{r} ') \right) \hat{\psi}^{< \dagger} (\bm{r} ') \\
&= \hat{\psi}^{< \dagger} (\bm{r}) \left( \sum_{\substack{\sigma, n,
\bm{k};\\ E_c ' < \epsilon_{n,\bm{k}} \leq E_c}} \phi_{\sigma n \bm{k}} (\bm{r}) \phi_{\sigma n \bm{k}}^* (\bm{r} ') \right) \hat{\psi}^{<} (\bm{r} ') + \hat{\psi}^{< \dagger} (\bm{r} ') \left( \sum_{\substack{\sigma, n,
\bm{k};\\ -E_c ' > \epsilon_{n,\bm{k}} \geq -E_c}} -\phi_{\sigma n \bm{k}}^* (\bm{r}) \phi_{\sigma n \bm{k}} (\bm{r} ') \right) \hat{\psi}^{<} (\bm{r})
\end{align*}
where the minus sign in the second line comes from the exchange the two fermionic operators and the constant arising from the exchange is omitted. Then, the e-e interaction $\hat{V}_{\text{int}}$ in the lower energy window delimited by $E_c '$ is
\begin{equation}
\hat{V}_{\text{int}} = \frac{1}{2} \int d^2 \bm{r} d^2 \bm{r} ' V_c(\bm{r} - \bm{r} ') \hat{\rho}^{<} (\bm{r}) \hat{\rho}^{<} (\bm{r} ') + \frac{1}{2} \int d^2 \bm{r} d^2 \bm{r} ' V_c(\bm{r} - \bm{r} ') \hat{\psi}^{< \dagger} (\bm{r}) \mathcal{F}(\bm{r}, \bm{r} ') \hat{\psi}^{<} (\bm{r} ')
\label{eq:vint_RG}
\end{equation}
with
\begin{equation}
\mathcal{F}(\bm{r}, \bm{r} ') = \sum_{\substack{\sigma, n,
\bm{k};\\ E_c ' < |\epsilon_{n,\bm{k}}| \leq E_c}} \text{sign} (\epsilon_{n,\bm{k}}) \phi_{\sigma n \bm{k}} (\bm{r}) \phi_{\sigma n \bm{k}}^* (\bm{r} ').
\label{eq:Frrp}
\end{equation}
\subsection{Evaluation of the correction to the non-interacting Hamiltonian from the fast modes}
In the following, we set $\hbar=1$ for the simplicity in mathematical expressions. Note that $\mathcal{F}(\bm{r}, \bm{r} ')$ has the structure of the residue of the Green's function $\hat{G}(z) = (z - \hat{H}_0)^{-1}$ taking only the valley $\mu=+1$ part in $\hat{H}_0=\hat{H}_G + \hat{H}_{G-S}$ [see Eqs.~\eqref{eq:H_Gonly} and \eqref{eq:hamGS_final}], namely
\begin{equation}
\mathcal{F}(\bm{r}, \bm{r} ') = \oint_\mathcal{C} \frac{dz}{2 \pi i} \bra{\bm{r}} \hat{G}(z) \ket{\bm{r}'}
\end{equation}
where the contour $\mathcal{C}$ encloses the $z$-plane real line segment $[-E_c, -E_c']$ in the clockwise, and segment $[E_c ', E_c ]$ in the counterclockwise, sense.
As long as $E_c '$ dominates over all other energy scales such as $\widetilde{U}_d (\bm{G_0})$ and $v_F G_0$ with $\bm{G_0}$ denoting the primitive reciprocal vector of the underlying superlattice, the dominant contribution to the contour integral can be evaluated perturbatively using $\hat{G}(z) \approx \hat{G}_0(z) + \hat{G}_0(z) \hat{H}_{G-S} \hat{G}_0(z) + \mathcal{O}\left( \widetilde{U}_d^2 (\bm{G_0})/E_c'^2, v_F^2 G_0^2/E_c'^2 \right)$ with $\hat{G}_0 (z) = (z - \hat{H}_G)^{-1}$.
It is easier to calculate the Green's function in the plane wave basis $\ket{\bm{k}}$ \begin{equation}
\mathcal{F}(\bm{r}, \bm{r} ') = \int \frac{d^2 k d^2 k'}{(2 \pi)^4} e^{i (\bm{k} \cdot \bm{r} - \bm{k} ' \cdot \bm{r} ' )} \oint_\mathcal{C} \frac{dz}{2 \pi i} \bra{\bm{k}} \hat{G}(z) \ket{\bm{k}'}
\end{equation}
with
\begin{align}
\ket{\bm{r}} &= \int \frac{d^2 k}{(2 \pi)^2} e^{-i \bm{k} \cdot \bm{r}} \ket{\bm{k}} \\
\bracket{\bm{r}}{\bm{r} '} &= \delta^{(2)} (\bm{r} - \bm{r}') \\
\bracket{\bm{k}}{\bm{k} '} &= (2 \pi)^2 \delta^{(2)} (\bm{k} - \bm{k}')
\end{align}
where $\delta^{(2)}(\bm{x})$ is the 2D Dirac distribution. In the plane wave basis, the evaluation of Green's functions is straightforward
\begin{align}
\bra{\bm{k}} \hat{G}_0(z) \ket{\bm{k}'} &= (2 \pi)^2 \delta^{(2)} (\bm{k} - \bm{k}') \frac{1}{2} \sum_{\lambda=\pm} \frac{1-\lambda \frac{\bm{k}}{k} \cdot \bm{\sigma}}{z - \lambda v_F k} \\
\bra{\bm{k}} \hat{G}_0(z) \hat{H}_{G-S} \hat{G}_0(z) \ket{\bm{k}'} &= (2 \pi)^2 \delta^{(2)} (\bm{k} - \bm{k}' + \bm{G}) \frac{1}{4} \sum_{\bm{G}} \widetilde{U}_d (\bm{G}) \sum_{\lambda,\lambda' = \pm} \frac{\left(1-\lambda \frac{\bm{k}}{k} \cdot \bm{\sigma} \right) \left(1-\lambda' \frac{\bm{k}+\bm{G}}{|\bm{k} + \bm{G}|} \cdot \bm{\sigma} \right)}{ \left( z - \lambda v_F k \right) \left( z - \lambda ' v_F |\bm{k}+\bm{G}| \right)}.
\end{align}
Then, the contour integral can be easily done:
\begin{align}
\oint_\mathcal{C} \frac{dz}{2 \pi i} \bra{\bm{r}} \hat{G}_0(z) \ket{\bm{r}'} &= \int_{ E_c ' < v_F k \leq E_c} \frac{d^2 k}{(2 \pi)^2} e^{i \bm{k} \cdot (\bm{r} - \bm{r} ')} \frac{\bm{k}}{k} \cdot \bm{\sigma} \\
\oint_\mathcal{C} \frac{dz}{2 \pi i} \bra{\bm{r}} \hat{G}_0(z) \hat{H}_{G-S} \hat{G}_0(z) \ket{\bm{r}'} &= \int_{ E_c ' < v_F k \leq E_c} \frac{d^2 k}{(2 \pi)^2} e^{i \bm{k} \cdot (\bm{r} - \bm{r} ') - i \bm{G} \cdot \bm{r} '} \frac{1}{4} \sum_{\bm{G}} \widetilde{U}_d (\bm{G}) \mathcal{I}(\bm{k},\bm{G}) \\
\mathcal{I}(\bm{k},\bm{G}) &= \frac{2}{v_F k + v_F |\bm{k} + \bm{G}|} \left( 1 - \frac{\bm{k} \cdot (\bm{k} + \bm{G})}{k|\bm{k} + \bm{G}|} + \frac{i \sigma_z (\bm{k} \times \bm{G}) \cdot \bm{\hat{z}}}{k|\bm{k} + \bm{G}|} \right).
\end{align}
\subsection{Renormalization group flow equations}
Now we only have to insert the previous results into the second term in Eq.~\eqref{eq:vint_RG} to derive the RG equations for $v_F$ and $\widetilde{U}_d(\bm{G})$. Let us compute first the integral for $\bra{\bm{r}} \hat{G}_0(z) \ket{\bm{r}'}$. After writing the 2D Coulomb potential in Fourier space $\widetilde{V}_{\text{2D}}(\bm{q}) = e^2 / 2 \epsilon_0 \epsilon_r q$, we have
\begin{align*}
&\quad \frac{1}{2} \int d^2 \bm{r} d^2 \bm{r} ' V_c(\bm{r} - \bm{r} ') \oint_\mathcal{C} \hat{\psi}^{< \dagger} (\bm{r}) \frac{dz}{2 \pi i} \bra{\bm{r}} \hat{G}_0(z) \ket{\bm{r}'} \hat{\psi}^{<} (\bm{r} ') \\
&= \int \frac{d^2 q}{(2 \pi)^2} \hat{\widetilde{\psi}}^{< \dagger} (\bm{q}) \underbrace{\left( \int_{ E_c ' < v_F k \leq E_c} \frac{d^2 k}{(2 \pi)^2} \frac{e^2}{4 \epsilon_0 \epsilon_r |\bm{q}-\bm{k}|} \frac{\bm{k}}{k} \cdot \bm{\sigma} \right)}_{\text{(A)}} \hat{\widetilde{\psi}}^{<} (\bm{q})
\end{align*}
with $\hat{\widetilde{\psi}}^{<} (\bm{q})$ is the Fourier transform of $\hat{\psi}^{<} (\bm{r})$. Since $v_F q \ll E_c'$, we can Taylor expand (A) in terms of $q/k$. The leading order reads
\begin{equation}
\text{(A)} = \frac{e^2}{16 \pi \epsilon_0 \epsilon_r} \log \left( \frac{E_c}{E_c'} \right) \,\bm{q} \cdot \bm{\sigma}.
\end{equation}
Therefore, the RG equation reads
\begin{equation}
\frac{d v_F}{ d \log E_c} = - \frac{e^2}{16 \pi \epsilon_0 \epsilon_r}.
\end{equation}
Actually, we find the famous result of the Fermi velocity renormalization in graphene due to the e-e interactions.
In the same way, we calculate the integral for $\bra{\bm{r}} \hat{G}_0(z) \hat{H}_{G-S} \hat{G}_0(z) \ket{\bm{r}'}$:
\begin{align*}
&\quad \frac{1}{2} \int d^2 \bm{r} d^2 \bm{r} ' V_c(\bm{r} - \bm{r} ') \oint_\mathcal{C} \hat{\psi}^{< \dagger} (\bm{r}) \frac{dz}{2 \pi i} \bra{\bm{r}} \hat{G}_0(z) \ket{\bm{r}'} \hat{\psi}^{<} (\bm{r} ') \\
&= \int \frac{d^2 q}{(2 \pi)^2} \hat{\widetilde{\psi}}^{< \dagger} (\bm{q}-\bm{G}) \underbrace{\left( \int_{ E_c ' < v_F k \leq E_c} \frac{d^2 k}{(2 \pi)^2} \widetilde{V}_{\text{2D}}(\bm{q}-\bm{k}-\bm{G}) \frac{1}{8} \sum_{\bm{G}} \widetilde{U}_d (\bm{G}) \mathcal{I}(\bm{k},\bm{G}) \right)}_{\text{(B)}} \hat{\widetilde{\psi}}^{<} (\bm{q}).
\end{align*}
Since $v_F q, v_F G \ll E_c'$, we can Taylor expand (B) in terms of $q/k$ and $G/k$ (considered as if they have the same order of magnitude). The leading order reads
\begin{equation}
\text{(B)} = \frac{e^2}{16 \epsilon_0 \epsilon_r} G^2 \left( \frac{1}{E_c'} - \frac{1}{E_c }\right) + \mathcal{O} \left(\frac{v_F^3 q^3}{E_c'^3},\frac{v_F^3 G^3}{E_c'^3}\right),
\end{equation}
which can be neglected under the first-order RG procedure, namely
\begin{equation}
\frac{d \widetilde{U}_d(\bm{G})}{d \log E_c} = 0.
\end{equation}
In summary, we have shown that the Fermi velocity in graphene $v_F$ is renormalized by the e-e Coulomb interaction in the standard way while the superlattice potential $U_d(\bm{r})$ keep its value unchanged. In our numerical study of e-e interactions, we use the renormalized Fermi velocity $v_F^*$ in the Hartree-Fock calculations, where we have to take a cut-off $n_{\text{cut}}$ to the number of bands, to include the contributions from the higher energy bands outside the cut-off. Technically, we use
\begin{equation}
v_F^* = v_F \left(1 + \frac{e^2}{16 \pi \epsilon_0 \epsilon_r v_F} \log \left( \frac{L_s}{n_{\text{cut}} a_0} \right) \right)
\end{equation}
where $L_s$ and $a_0$ are the lattice constant of the superlattice of $U_d(\bm{r})$ and the carbon-carbon bond length in graphene, respectively. Here, the ratio $L_s/n_{\text{cut}} a_0$ plays the role of $E_c/E_c'$.
\section{S3. Hartree-Fock approximations to electron-electron interactions}
The derivation shown in this section is inspired from Ref.~\onlinecite{zhang_prl2022}. We consider the Coulomb interactions in graphene
\begin{equation}
\hat{V}_\text{int}=\frac{1}{2}\int d^2 r d^2 r' \sum _{\sigma, \sigma '} \hat{\psi}_\sigma ^{\dagger}(\bm{r})\hat{\psi}_{\sigma '}^{\dagger}(\bm{r} ') V_\text{int} (|\bm{r} -\bm{r} '|) \hat{\psi}_{\sigma '}(\bm{r} ') \hat{\psi}_{\sigma}(\bm{r})
\label{eq:coulomb}
\end{equation}
where $\hat{\psi}_{\sigma}(\bm{r})$ is real-space electron annihilation operator at $\bm{r}$ with spin $\sigma$. This interaction can be written as
\begin{equation}
\hat{V}_\text{int}=\frac{1}{2}\sum _{i i' j j'}\sum _{\alpha \alpha '\beta \beta '}\sum _{\sigma \sigma '} \hat{c}^{\dagger}_{i, \sigma \alpha}\hat{c}^{\dagger}_{i', \sigma ' \alpha '} V^{\alpha \beta \sigma , \alpha ' \beta ' \sigma '} _{ij,i'j'}\hat{c}_{j', \sigma ' \beta '} \hat{c}_{j, \sigma \beta}\;,
\end{equation}
where
\begin{align}
V^{\alpha \beta \sigma , \alpha ' \beta ' \sigma '} _{ij,i'j'}=\int d^2 r d^2 r' & V_\text{int} (|\bm{r} -\bm{r} '|) \,\phi ^*_\alpha (\mathbf{r}-\mathbf{R}_i-\tau _\alpha)\,\phi _\beta (\mathbf{r}-\mathbf{R}_j-\tau _\beta) \phi^*_{\alpha '}(\bm{r}-\mathbf{R}_i'-\bm{\tau}_{\alpha '})\phi _{\beta '}(\bm{r}-\mathbf{R}_j'-\bm{\tau} _{\beta '}) \nonumber \\
&\times \chi ^{\dagger}_\sigma \chi ^{\dagger}_{\sigma '}\chi _{\sigma '}\chi _{\sigma} .
\end{align}
Here $i$, $\alpha$, and $\sigma$ refer to Bravis lattice vectors, layer/sublattice index, and spin index. $\phi$ is Wannier function and $\chi$ is the two-component spinor wave function. We further assume that the "density-density" like interaction is dominant in the system, i.e., $V^{\alpha \beta \sigma , \alpha ' \beta ' \sigma '} _{ij,i'j'}\approx V^{\alpha \alpha \sigma , \alpha ' \alpha ' \sigma '} _{ii,i'i'}\equiv V_{i \sigma \alpha ,i' \sigma ' \alpha '}$, then the Coulomb interaction is simplified to
\begin{align}
\hat{V}_\text{int}=&\frac{1}{2}\sum _{i i'}\sum _{\alpha \alpha '}\sum _{\sigma \sigma '}\hat{c}^{\dagger}_{i, \sigma \alpha}\hat{c}^{\dagger}_{i', \sigma' \alpha} V_{i\sigma \alpha, i' \sigma ' \alpha '}\hat{c}_{i', \sigma ' \alpha '}\hat{c}_{i, \sigma \alpha} \nonumber \\
=&\frac{1}{2}\sum _{i\alpha \neq i'\alpha '}\sum _{\sigma \sigma '}\hat{c}^{\dagger}_{i, \sigma \alpha} \hat{c}^{\dagger}_{i', \sigma ' \alpha '} V_{i\alpha,i'\alpha '}\hat{c}_{i', \sigma ' \alpha '}\hat{c}_{i, \sigma \alpha} \nonumber \\
&+\sum _{i\alpha} U_0 \hat{c}^{\dagger}_{i,\uparrow \alpha} \hat{c}^{\dagger}_{i,\downarrow\alpha}\hat{c}_{i,\downarrow\alpha}\hat{c}_{i,\uparrow \alpha}
\end{align}
Here we can see that the Coulomb interaction can be divided into intersite Coulomb interaction and on-site Coulomb interaction. Given that the electron density is low ($10^{11}$ cm$^{-2}$), i.e., a few electrons per supercell, the chance that two electrons meet at the same atomic site is very low. The Coulomb correlations between two electron are mostly contributed by the inter-site Coulomb interactions. Therefore, the on-site Hubbard interaction has been neglected in our calculations.
In order to model the screening effects to the e-e Coulomb interactions from the dielectric environment, we introduce the double gate screening form of $V_\text{int}$, whose Fourier transform is expressed as
\begin{equation}
V_{\rm{int}}(\mathbf{q})=\frac{e^2 \tanh(q d_s)}{2 \Omega_0 \epsilon_r \epsilon_0 q}\;,
\end{equation}
where $\Omega _0$ is the area of the TMO superlattice's primitive cell, $\epsilon_r$ is a background dielectric constant and the thickness between two gates is $d_s=400$ \AA.
Since we are interested in the low-energy bands, the intersite Coulomb interactions can be divided into the intra-valley term and the inter-valley term. The intra-valley term $\hat{V}^{\text{intra}}$ can be expressed as
\begin{equation}
\hat{V}^{\rm{intra}}=\frac{1}{2N_s}\sum_{\alpha\alpha '}\sum_{\mu\mu ',\sigma\sigma '}\sum_{\bm{k} \bm{k} ' \bm{q}} V_{\rm{int}}(\bm{q})\,
\hat{c}^{\dagger}_{\sigma \mu \alpha}(\bm{k}+\bm{q}) \hat{c}^{\dagger}_{\sigma' \mu ' \alpha '}(\bm{k} ' - \bm{q}) \hat{c}_{\sigma ' \mu ' \alpha '}(\bm{k} ')\hat{c}_{\sigma \mu \alpha}(\bm{k})\;,
\label{eq:h-intra}
\end{equation}
with $N_s$ is the total number of the superlattice's sites. The inter-valley term $\hat{V}^{\rm{inter}}$ is expressed as
\begin{equation}
\hat{V}^{\rm{inter}}=\frac{1}{2N_s}\sum_{\alpha\alpha '}\sum_{\mu ,\sigma\sigma '}\sum_{\mathbf{k} \mathbf{k} '\mathbf{q}} V_{\rm{int}}(\vert\mathbf{K}-\mathbf{K}'\vert)\, \hat{c}^{\dagger}_{\sigma \mu \alpha}(\bm{k}+\bm{q}) \hat{c}^{\dagger}_{\sigma' -\mu \alpha '}(\bm{k}' - \bm{q}) \hat{c}_{\sigma' \mu \alpha '}(\bm{k}') \hat{c}_{\sigma -\mu \alpha}(\bm{k})\;.
\label{eq:h-inter}
\end{equation}
$\hat{V}^{\rm{intra}}$ includes the Coulomb scattering processes of two electrons created and annihilated in the same valley, and $\hat{V}^{\rm{inter}}$ includes the processes that two electrons are created in $\mu$ and $-\mu$ and get annihilated in $-\mu$ and $\mu$ valleys. Here the atomic wavevector $\mathbf{k}$ is expanded around the valley $K^{\mu}$ in the big Brillouin zone of graphene, which can be decomposed as $\mathbf{k}=\widetilde{\bm{k}}+\mathbf{G}$, where $\widetilde{\bm{k}}$ is the superlattice wavevector in the superlattice Brillouin zone, and $\mathbf{G}$ denotes a superlattice reciprocal lattice vector.
The electron annihilation operator can be transformed from the original basis to the band basis:
\begin{equation}
\hat{c}_{\sigma\mu\alpha}(\bm{k})=\sum_n C_{\sigma \mu \alpha \mathbf{G},n}(\widetilde{\bm{k}})\,\hat{c}_{\sigma \mu,n}(\widetilde{\bm{k}})\;,
\label{eq:transform}
\end{equation}
where $C_{\sigma \mu \alpha \mathbf{G},n}(\widetilde{\bm{k}})$ is the expansion coefficient in the $n$-th Bloch eigenstate at $\widetilde{\bm{k}}$ of valley $\mu$:
\begin{equation}
\ket{\mu, n; \widetilde{\bm{k}}}=\sum_{\alpha \mathbf{G}}C_{\sigma\mu \alpha \mathbf{G},n}(\widetilde{\bm{k}})\,\ket{ \sigma, \mu, \alpha, \mathbf{G}; \widetilde{\bm{k}} }\;.
\end{equation}
We note that the non-interacting Bloch functions are spin degenerate due to the separate spin rotational symmetry ($SU(2)\otimes SU(2)$ symmetry) of each valley. Using the transformation given in Eq.~(\ref{eq:transform}), the intra- and inter-valley Coulomb interaction can be written in the band basis
\begin{align}
\hat{V}^{\rm{intra}}=&\frac{1}{2N_s}\sum _{\widetilde{\bm{k}} \widetilde{\bm{k}}'\widetilde{\bm{q}}}\sum_{\substack{\mu\mu' \\ \sigma\sigma'}}\sum_{\substack{nm\\ n'm'}} \left(\sum _{\mathbf{Q}}\,V_{\rm{int}} (\mathbf{Q}+\widetilde{\bm{q}})\,\Omega^{\mu \sigma,\mu'\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',\widetilde{\bm{q}},\mathbf{Q})\right) \nonumber \\
&\times \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}}+\widetilde{\bm{q}}) \hat{c}^{\dagger}_{\sigma'\mu',n'})(\widetilde{\bm{k}}'-\widetilde{\bm{q}}) \hat{c}_{\sigma'\mu',m'}(\widetilde{\bm{k}}') \hat{c}_{\sigma\mu,m}(\widetilde{\bm{k}})
\label{eq:Hintra-band}
\end{align}
and
\begin{align}
\hat{V}^{\rm{inter}}=&\frac{1}{2N_s}\sum _{\widetilde{\bm{k}} \widetilde{\bm{k}}'\widetilde{\bm{q}}}\sum_{\substack{\sigma\sigma' \\ \mu}}\sum_{\substack{nm\\ n'm'}}\left(\sum _{\mathbf{Q}}\,V_{\rm{int}}(|\mathbf{K}-\mathbf{K}'|)\,\widetilde{\Omega}^{\mu, \sigma\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',\widetilde{\bm{q}},\mathbf{Q})\right) \nonumber\\
&\times \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}}+\widetilde{\bm{q}}) \hat{c}^{\dagger}_{\sigma'-\mu,n'}(\widetilde{\bm{k}}'-\widetilde{\bm{q}}) \hat{c}_{\sigma'\mu,m'}(\widetilde{\bm{k}}') \hat{c}_{\sigma-\mu,m}(\widetilde{\bm{k}})
\label{eq:Hinter-band}
\end{align}
where the form factors $\Omega ^{\mu \sigma,\mu'\sigma'}_{nm,n'm'}$ and $\widetilde{\Omega}^{\mu, \sigma\sigma'}_{nm,n'm'}$ are written respectively as
\begin{equation}
\Omega ^{\mu \sigma,\mu'\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',\widetilde{\bm{q}},\mathbf{Q})
=\sum _{\alpha\alpha'\mathbf{G}\mathbf{G}'}C^*_{\sigma\mu\alpha\mathbf{G}+\mathbf{Q},n}(\widetilde{\bm{k}}+\widetilde{\bm{q}}) C^*_{\sigma'\mu'\alpha'\mathbf{G}'-\mathbf{Q},n}(\widetilde{\bm{k}}'-\widetilde{\bm{q}})C_{\sigma'\mu'\alpha'\mathbf{G}',m'}(\widetilde{\bm{k}}')C_{\sigma\mu\alpha\mathbf{G},m}(\widetilde{\bm{k}})
\end{equation}
and
\begin{equation}
\widetilde{\Omega}^{\mu, \sigma\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',\widetilde{\bm{q}},\mathbf{Q})
=\sum _{\alpha\alpha'\mathbf{G}\mathbf{G}'}C^*_{\sigma\mu\alpha\mathbf{G}+\mathbf{Q},n}(\widetilde{\bm{k}}+\widetilde{\bm{q}}) C^*_{\sigma'-\mu\alpha'\mathbf{G}'-\mathbf{Q},n}(\widetilde{\bm{k}}'-\widetilde{\bm{q}})C_{\sigma'\mu \alpha'\mathbf{G}',m'}(\widetilde{\bm{k}}')C_{\sigma-\mu\alpha\mathbf{G},m}(\widetilde{\bm{k}}) \;.
\end{equation}
We make Hartree-Fock approximation to Eq.~\eqref{eq:Hintra-band} and Eq.~\eqref{eq:Hinter-band} such that the two-particle Hamiltonian is decomposed into a superposition of the Hartree and Fock single-particle Hamiltonians, where the Hartree term is expressed as
\begin{equation}
\begin{split}
\hat{V}_H^{\rm{intra}}=&\frac{1}{2N_s}\sum _{\widetilde{\bm{k}} \widetilde{\bm{k}}'}\sum _{\substack{\mu\mu'\\ \sigma\sigma'}}\sum_{\substack{nm\\ n'm'}}\left(\sum _{\mathbf{Q}} V_{\rm{int}} (\mathbf{Q}) \, \Omega^{\mu \sigma,\mu'\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',0,\mathbf{Q})\right)\\
&\times \left(\langle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}})\hat{c}_{\sigma\mu,m}(\widetilde{\bm{k}})\rangle \hat{c}^{\dagger}_{\sigma'\mu',n'}(\widetilde{\bm{k}}')\hat{c}_{\sigma'\mu',m'}(\widetilde{\bm{k}}') + \langle \hat{c}^{\dagger}_{\sigma'\mu',n'}(\widetilde{\bm{k}}')\hat{c}_{\sigma'\mu',m'}(\widetilde{\bm{k}}')\rangle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}})\hat{c}_{\sigma\mu,m}(\widetilde{\bm{k}})\right)
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\hat{V}_H^{\rm{inter}}=&\frac{1}{2N_s}\sum _{\widetilde{\bm{k}} \widetilde{\bm{k}}'}\sum _{\substack{\sigma\sigma'\\ \mu}}\sum_{\substack{nm\\ n'm'}}\left(\sum _{\mathbf{Q}} V_{\rm{int}}(|\mathbf{K}-\mathbf{K}'|) \, \widetilde{\Omega}^{\mu, \sigma\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',0,\mathbf{Q})\right)\\
&\times \left(\langle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}})\hat{c}_{\sigma-\mu,m}(\widetilde{\bm{k}})\rangle \hat{c}^{\dagger}_{\sigma'-\mu,n'}(\widetilde{\bm{k}}')\hat{c}_{\sigma'\mu,m'}(\widetilde{\bm{k}}') + \langle \hat{c}^{\dagger}_{\sigma'-\mu,n'}(\widetilde{\bm{k}}')\hat{c}_{\sigma'\mu,m'}(\widetilde{\bm{k}}')\rangle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}})\hat{c}_{\sigma-\mu,m}(\widetilde{\bm{k}})\right) \;.
\end{split}
\end{equation}
The Fock term is expressed as:
\begin{align*}
\hat{V}_F^{\rm{intra}}=&-\frac{1}{2N_s}\sum _{\widetilde{\bm{k}} \widetilde{\bm{k}}'}\sum _{\substack{\mu\mu'\\ \sigma\sigma'}}\sum_{\substack{nm\\ n'm'}}\left(\sum _{\mathbf{Q}} V_{\rm{int}} (\widetilde{\bm{k}}'-\widetilde{\bm{k}}+\mathbf{Q}) \, \Omega^{\mu \sigma,\mu'\sigma'}_{nm,n'm'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',\widetilde{\bm{k}}'-\widetilde{\bm{k}},\mathbf{Q})\right)\\
&\times \left(\langle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}}')\hat{c}_{\sigma'\mu',m'}(\widetilde{\bm{k}}')\rangle \hat{c}^{\dagger}_{\sigma'\mu',n'}(\widetilde{\bm{k}})\hat{c}_{\sigma\mu,m}(\widetilde{\bm{k}}) + \langle \hat{c}^{\dagger}_{\sigma'\mu',n'}(\widetilde{\bm{k}})\hat{c}_{\sigma\mu,m}(\widetilde{\bm{k}})\rangle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}}')\hat{c}_{\sigma'\mu',m'}(\widetilde{\bm{k}}')\right)\;.
\end{align*}
and
\begin{align*}
\hat{V}_F^{\rm{inter}}=&-\frac{1}{2N_s}\sum _{\widetilde{\bm{k}} \widetilde{\bm{k}}'}\sum _{\substack{\sigma \sigma ' \\ \mu}}\sum_{\substack{nm\\ n' m'}}\left(\sum _{\mathbf{Q}} V_{\rm{int}}(|\mathbf{K}-\mathbf{K}'|) \, \widetilde{\Omega}^{\mu, \sigma\sigma'}_{nm,n' m'}(\widetilde{\bm{k}},\widetilde{\bm{k}}',\widetilde{\bm{k}}'-\widetilde{\bm{k}},\mathbf{Q})\right)\\
&\times \left(\langle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}}')\hat{c}_{\sigma' \mu, m'}(\widetilde{\bm{k}}')\rangle \hat{c}^{\dagger}_{\sigma' -\mu,n'}(\widetilde{\bm{k}})\hat{c}_{\sigma -\mu,m}(\widetilde{\bm{k}}) + \langle \hat{c}^{\dagger}_{\sigma' -\mu,n'}(\widetilde{\bm{k}})\hat{c}_{\sigma-\mu,m}(\widetilde{\bm{k}})\rangle \hat{c}^{\dagger}_{\sigma\mu,n}(\widetilde{\bm{k}}')\hat{c}_{\sigma' \mu,m'}(\widetilde{\bm{k}}')\right)\;.
\end{align*}
We note that the typical intravalley interaction energy $\sim 240$ meV for $L_s = 50$~\AA \, and $\epsilon_r = 3$; while the intervalley interaction $\sim 30\,$meV, which is one order of magnitudes smaller than the intravalley interaction, thus we neglect the intervalley term [see Eq.~(\ref{eq:h-inter}] in most of our calculations. We also check \textit{a posteriori} that the intervalley Hartree and Fock energies are at least two orders of magnitude smaller than their intravalley counterpart. However, the intervalley interaction is crucial to lift the degeneracy between many-body ground state, namely their energy difference is within the convergence threshold $10^{-8}$ eV. We show in the section of the Hartree-Fock results that it promotes topologically trivial Hartree-Fock ground states rather than topological Hartree-Fock ground states.
\newpage
\section{S4. Non-interacting energy bands and their topology}
In this section, we show the non-interacting energy spectra and distributions of Berry curvature in the first Brillouin zone for $L_s = 50$, 200, 600 \AA \ with $r=1.2$ and 3. Since the system preserves time-reversal symmetry and the superlattice potential $U_d$ is diagonal in the sublattice subspace, the non-interacting energy spectrum in valley $K'$ is exactly the same as that in valley $K$ so that we only plot the spectrum for valley $K$ here. As shown below, the distribution of Berry curvature of the highest valence and the lowest conduction band in valley $K$ is exactly opposite to that in valley $K'$ as another consequence of time-reversal symmetry of the system.
\begin{figure}[h]
\centering
\begin{subfigure}{\linewidth}
\includegraphics[width=0.8\textwidth]{fig/Ls50_r1.2.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}{\linewidth}
\includegraphics[width=0.8\textwidth]{fig/Ls50_r3.jpg}
\caption{}
\end{subfigure}
\caption{Non-interacting spectrum and distribution of Berry curvature in the first Brillouin zone for $L_s=50$ \AA \ with (a) $r=1.2$ and (b) $r=3$.}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}{\linewidth}
\includegraphics[width=0.8\textwidth]{fig/Ls200_r1.2.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}{\linewidth}
\includegraphics[width=0.8\textwidth]{fig/Ls200_r3.jpg}
\caption{}
\end{subfigure}
\caption{Non-interacting spectrum and distribution of Berry curvature in the first Brillouin zone for $L_s=200$ \AA \ with (a) $r=1.2$ and (b) $r=3$.}
\end{figure}
\begin{figure}[htb]
\centering
\begin{subfigure}{\linewidth}
\includegraphics[width=0.8\textwidth]{fig/Ls600_r1.2.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}{\linewidth}
\includegraphics[width=0.8\textwidth]{fig/Ls600_r3.jpg}
\caption{}
\end{subfigure}
\caption{Non-interacting spectrum and distribution of Berry curvature in the first Brillouin zone for $L_s=600$ \AA \ with (a) $r=1.2$ and (b) $r=3$.}
\end{figure}
We also provide two videos in Supplemental Information, which shows the non-interacting energy spectra and distributions of Berry curvature in the first Brillouin zone for $L_s = 50$, 200, 600 \AA \ with $r=1$-10.
\newpage
\section{S5. Results of Hartree-Fock calculations}
In this section, we gather the results of Hartree-Fock calculations including Hartree-Fock single-particle spectra and distributions of Berry curvature in the first Brillouin zone for $L_s = 50$, 200, 600 \AA.
First, we show the Hartree-Fock single-particle spectrum with a superlattice potential with $r=1.2$ of $L_s = 50$, 200, 600 \AA. Here, we use $n_\text{cut}=5$ and study three types of doping: CNP ($\nu=0$), slight hole doping ($\nu=-0.003$) and slight electron doping ($\nu=+0.003$). As you can see from Table \ref{tab:gap_vf_supp} and the Hartree-Fock single-particle spectra, the results of a slightly electron-doped system is similar to those for a slightly hole-doped one. Note that we include only intravalley Coulomb interactions in these calculations. As shown in the following, the role of intervalley Coulomb interactions is merely to lift the ground state degeneracy and favor the topologically trivial ground state.
\begin{figure}[htb]
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.8\textwidth]{fig/HF_Ls50_r1.2.jpg}
\caption{}
\label{fig:HF_Ls50r1.2}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.8\textwidth]{fig/HF_Ls200_r1.2.jpg}
\caption{}
\label{fig:HF_Ls200r1.2}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.8\textwidth]{fig/HF_Ls600_r1.2.jpg}
\caption{}
\label{fig:HF_Ls600r1.2}
\end{subfigure}
\caption{{Hartree-Fock single-particle spectra for three different dopings with $r=1.2$ for (a) $L_s=50$ \AA, (b) $L_s=200$ \AA\,and (c) $L_s=600$ \AA.}}
\end{figure}
\begin{table}[h]
\caption{Parameters extracted from the Hartree-Fock single-particle spectra: gap opened at the CNP ($\nu=0$) and the ratio between interaction-renormalized Fermi velocity $v_F^*$ and the non-interacting one $v_F$ for different $L_s = 50$, 200, 600 \AA \ with fixed $r=1.2$.}
\label{tab:gap_vf_supp}
\renewcommand{\arraystretch}{1.2}
\centering
\begin{tabular}{c|c|c|c}
\hline
$L_s$(\AA) & 50 & 200 & 600 \\
\hline \hline
Gap at $\nu=0.0$ (meV) & 17 & 1.7 & 0.15 \\
\hline
$v_F^*/v_F$ at $\nu=-0.003$ & 2.1 & 1.8 & 1.7\\
\hline
$v_F^*/v_F$ at $\nu=+0.003$ & 2.1 & 1.8 & 1.7\\
\hline
\end{tabular}
\end{table}
Then, we show below the distributions of Berry curvature in the first Brillouin zone of $r=1.2$ for $L_s = 50$, 200, 600 \AA. Here, $n_\text{cut}=5$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.95\textwidth]{fig/berry_HF_ncut5.jpg}
\caption{Distributions of Berry curvature in the first Brillouin zone of $r=1.2$ for $L_s =$ (a) 50 \AA, (b) 200 \AA, (c) 600 \AA}
\label{fig:berry_HF_ncut5}
\end{figure}
Now we show the effect of intervalley Coulomb interactions by comparing the total energy of topologically trivial ground state with topological one for different $L_s = 50$, 200, 600 \AA \ with fixed $r=1.2$. We calculate the difference (always positive) between them and see how it changes when we include the intervalley Coulomb interactions. Here, we use $n_\text{cut}=3$.
As you can see from Table~\ref{tab:intervalley_lift}, the energy difference between the total energy of the topological ground state and topologically trivial one is enhanced by two orders of magnitude for $L_s=50$ and 200 \AA. However, the energy difference for $L_s=600$ \AA does not benefit anything from intervalley interactions. This suggests that it is plausible to find in practice the topological ground state if one achieves a rather low carrier density ($\sim 10^{10}$ cm$^{-2}$) such that $L_s=600$ \AA.
\begin{table}[h]
\renewcommand{\arraystretch}{1.2}
\centering
\begin{tabular}{c|c|c|c}
\hline
$L_s$(\AA) & 50 & 200 & 600 \\
\hline \hline
$\Delta E$ with only intravalley ($\mu$eV) & 0.024 & 0.005 & 0.05 \\
\hline
$\Delta E$ with intra- and inter-valley ($\mu$eV) & 1.7 & 0.1 & 0.03 \\
\hline
\end{tabular}
\caption{Difference between the total energy of the topological ground state and topologically trivial one, with or without intervalley interactions, for $L_s = 50$, 200, 600 \AA \ with fixed $r=1.2$.}
\label{tab:intervalley_lift}
\end{table}
We also have performed Hartree-Fock calculations on a triangular lattice including three valence and three conduction bands ($n_{\rm{cut}}=3$) for $L_s = 50$, 200, 600 \AA \ using $18 \times 18$ $k$-mesh in the BZ. As shown in Table \ref{tab:gap_vf_tri_supp}, the results on a triangular lattice are qualitatively the same as those on a rectangular lattice. This ensures that our conclusions are lattice-independent.
\begin{table}[!h]
\caption{Parameters extracted from the Hartree-Fock single-particle spectra on a triangular lattice: gap opened at the CNP ($\nu=0$) and the ratio between interaction-renormalized Fermi velocity $v_F^*$ and the non-interacting one $v_F$ for different $L_s = 50$, 200, 600 \AA.}
\label{tab:gap_vf_tri_supp}
\renewcommand{\arraystretch}{1.2}
\centering
\begin{tabular}{c|c|c|c}
\hline
$L_s$(\AA) & 50 & 200 & 600 \\
\hline \hline
Gap at $\nu=0.0$ (meV) & 21 & 1.9 & 0.24 \\
\hline
$v_F^*/v_F$ at $\nu=-0.003$ & 2.2 & 1.7 & 1.7\\
\hline
\end{tabular}
\end{table}
\section{S6 Details of DFT calculations for the substrate materials}
\subsection{Lattice structures, deformation potentials, and band structures of candidate substrate materials}
In this section we present the details for the density function theory (DFT) calculations of the 11 candidate substrate materials presented in Table~\uppercase\expandafter{\romannumeral 2} of main text.
The lattice structures of some of the substrate materials discussed in the main text are presented in Fig.\ref{type1}. The lattice structure of CrI$_3$ is similar to that of YI$_3$ as shown in Fig.~\ref{type1}(d).
The band structures of 10 candidate substrate materials (except for CrOCl) in the bilayer or trilayer structures are presented in Fig.~\ref{type2}, where the green dashed lines mark the energy position of the Dirac point in graphene. We note that the valence band maximum (VBM) of PbO bilayer is energetically close to the Dirac point of graphene; while for the other bilayer or trilayer substrate materials, their conduction band minima (CBM) are close to the Dirac point.
This indicates that charges can easily transferred between graphene and the substrates as controlled by gate voltages. Moreover, we note that the conduction bands and valence bands of these materials are typically flat with large effective masses, which would be very susceptible to $e$-$e$ Coulomb interactions once these substrate materials are slightly charge doped, and may to Wigner-crystal-like state or long-wavelength ordered state as discussed in main text. Another important precondition for the Wigner-crystal state is that the screening effect of substrate materials can not be too strong. For example, the conduction band of ScOBr bilayer has a large effective mass of 2.575$m_0$ ($m_0$ is the bare mass of a free electron), but the dielectric constant $\epsilon_d$ of ScOBr reaches $\sim$13, which makes it difficult to form the Wigner-crystal-like instability in this material under slight charge doping.
We note that all of these proposed substrate materials all have been successfully synthesized in laboratory as listed in Table.~\ref{host2}. Especially, few-layer of ReSe$_2$ as a highly anisotropic material \cite{Jariwala-ReSe2-CM-2016,Yang-ReSe2-NL-2015,Arora-ReSe2-NL-2017}, and few-layer CrI$_3$ system as a 2D magnetic material \cite{Huang-CrI3-nature-2017,Huang-CrI3-NN-2018,Jiang-CrI3-NN-2018,Klein-CrI3-science-2018}, have been extensively studied recently. Moreover, phonon spectra calculations have proved the dynamical stability of these substrate materials in monolayer form \cite{Haastrup-C2DB-2dmater-2018}. Thus the device fabrication of heterostructure consisting of graphene monolayer and one of these candidate substrate materials should be experimentally accessible.
There always exists tension or compression in a heterostructure system. Under some lattice deformation, the variation of conduction band minimum (CBM) or valence band maximum (VBM) is defined as deformation potential. We list the deformation potentials of the candidate substrate materials in Table.~\ref{host2}. We note that the maximum value of the deformation potential is only 5.84\,eV for ScOCl, which means that the energy level of CBM of ScOCl would move down by only 0.063\,eV under 1\% tensile strain. Therefore, even if strain is introduced in the graphene-insulator heterostructure proposed in this work, the band edges (with large effective masses) of those candidate substrate materials are still energetically close to the Dirac point of graphene.
In these candidate materials (except for CrOCl), CrI$_3$ bilayer is the only magnetic system. Previous theoretical studies reveal that the stacking configuration of CrI$_3$ bilayer plays an important role in the magnetic ground state \cite{Sivadas-bilayerCrI3-NL-2018}. Here we use the AB$^{\prime}$-type stacking in the bilayer structure, which is consistent with the stacking configuration in the bulk phase of CrI$_3$. The AB$^{\prime}$-stacked CrI$_3$ bilayer is in an intralayer ferromagnetic and interlayer antiferromagnetic ground state.
\subsection{Electric-field tunable band structures of bilayer CrOCl}
Now we discuss the electronic structure of CrOCl bilayer under vertical electrical fields. Here we consider an intralayer ferromagnetic and interlayer antiferromagnetic state for the bilayer configuration, which turns out to be one of competing low-energy magnetic states, and is the magnetic ground state when the on-site Hubbard $U$ value for the Cr $3d$ orbitals is large.
\footnote{In our DFT+$U$ calculations, the on-site Hubbard $U\!=\!$ 5.48\,eV for the Cr $3d$ orbitals is used in the calculations, and the non-spherical contributions from the gradient corrections are taken into consideration. }
The calculated band gap of CrOCl bilayer with the DFT+$U$ calculation is 3.13\,eV, which is close to that of HSE06 calculation (3.12\,eV)~\cite{Haastrup-C2DB-2dmater-2018}.
The band structure of antiferromagnetic CrOCl bilayer is shown in Fig.~\ref{type3}(a), where the green dashed line marks the energy position of the Dirac point of graphene. Without vertical electric field, the Dirac point is slightly above the CBM of bilayer CrOCl. Applying a vertical electric field of 0.03\,V/nm would push down the CBM as shown in Fig.~\ref{type3}(b). A closer inspection reveals that the top-layer conduction state (red lines) is pushed downwards while the bottom-layer state (blue lines) is pushed upward in energy as shown in Fig.~\ref{type3}(b), such that electron carriers in the graphene layer (if there is any) would be transferred to the top layer of CrOCl substrate, forming a Wigner-crystal-like state at the surface of CrOCl substrate given that the Wigner-Seitz radius of the CBM $\sim 39$ is above the threshold value $\sim 30$ (see Table.~\uppercase\expandafter{\romannumeral 2}\ in main text). Thus, our conjecture is supported by detailed first principles DFT calculations.
In Fig.~\ref{type3}(c) we also present the Fermi surfaces at different Fermi energies above the CBM of bilayer CrOCl. At very low carrier densities with small Fermi energy (CBM is set to zero), the Fermi surface consists of two nearly isotropic circles. For example, at filling factor 1/100 (corresponding to a carrier density $\sim 8\times 10^{12}\,\rm{cm}^{-2}$), the Fermi surface is marked by the red circles. Such isotropic Fermi surface with large effective mass ($\sim 1.308 m_0$) is likely to give rise to Wigner-crystal state as discussed in main text. As the Fermi level further increases, the Fermi surfaces become more and more anisotropic.
\begin{table}[!htbp]
\caption{The experimental works about the ten substrate materials, and the uni-axial deformation potentials of these materials~\cite{Haastrup-C2DB-2dmater-2018}.}
\label{unitcell}
\centering
\begin{tabular}{ccc}
\hline
Materials & References & Deformation potentials \\
\hline
AgScP$_2$S$_6$ & Ref.~\cite{AgScP2S6-Lee1988} & --\\
AgScP$_2$Se$_6$ & Ref.~\cite{AgScP2Se6-2006} & -- \\
IrBr$_3$ & Ref.~\cite{IrBr3-1968} & -3.76\,eV \\
IrI$_3$ & Ref.~\cite{IrI3-1968} & -2.17\,eV \\
YI$_3$ & Ref.~\cite{YI3-1964} & 1.47\,eV \\
YBr$_3$ & Ref.~\cite{YBr3-1980} & 1.43\,eV \\
ReSe$_2$ & Ref.~\cite{ReSe2-1971} & -4.45\,eV \\
ScOCl & Ref.~\cite{ScOCl-1985} & -5.84\,eV \\
PbO & Ref.~\cite{PbO-1989} & -4.60\,eV \\
CrI$_3$ & Ref.~\cite{Huang-CrI3-nature-2017,Huang-CrI3-NN-2018,Jiang-CrI3-NN-2018,Klein-CrI3-science-2018} & -2.20\,eV \\
\hline
\label{host2}
\end{tabular}
\end{table}
\begin{figure*}[!htbp]
\includegraphics[width=0.8\textwidth]{fig/structures.pdf}
\caption{~\label{type1} (a)-(f): top views of the lattice structures of some candidate substrate materials in monolayer form. The primitive cells are remarked with black lines. (g)-(l): the side views of these substrate materials in few-layer form.}
\end{figure*}
\begin{figure*}[!htbp]
\includegraphics[width=1.0\textwidth]{fig/bands.pdf}
\caption{~\label{type2} The calculated energy bands of the candidate substrate materials, where the energy position of the Dirac point of graphene is marked by a dashed green line in the band structure.}
\end{figure*}
\begin{figure*}[!htbp]
\includegraphics[width=1.0\textwidth]{fig/band2.pdf}
\caption{~\label{type3} The calculated energy bands of antiferromagnetic bilayer CrOCl: (a) without electric field, and (b) with an electric field 0.03\,V/nm. In (b), the energy bands from top and bottom layers are marked by red and blue lines, respectively. The energy position of the Dirac point of graphene are remarked with green dashed lines. (c) Fermi surface of bilayer CrOCl at different Fermi levels with respect to the conduction band minimum. The Fermi surface under 1/100 electron filling factor is remarked by red circles.}
\end{figure*}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,953 |
=head1 LICENCE
Copyright 2015 EMBL - European Bioinformatics Institute (EMBL-EBI)
Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
=head1 NAME
add_strain_xrefs.pl
=head1 SYNOPSIS
=cut
use strict;
use warnings;
use DBI;
my (%input_data,%output_data);
## load configuration
eval { require "CFG.pl" } ||
die "Configuration file 'CFG.pl' was not found.\n";
my $dbh = DBI->connect(
$CFG::DSN, $CFG::USER, $CFG::PASSWD,
{
InactiveDestroy => 1, RaiseError => 1, PrintError => 1}
);
# excludes EUCOMM centres and LEX/DEL, Martin Hrabe, Lluis and Yann
my $sql = "set session group_concat_max_len = 10240";
my $sth = $dbh->prepare($sql);
$sth->execute;
$sql = "update strains set mutation_xref = null";
$sth = $dbh->prepare($sql);
$sth->execute;
$sql = "update strains set owner_xref = null";
$sth = $dbh->prepare($sql);
$sth->execute;
$sql = "select group_concat(emma_id) from strains, people where per_id_per=id_per and per_id_per not in (1196,8137,8,7858,8374,8579,8597,9060,8560,7786,7787) and str_access not in ('R','N','C') and str_status not in ('ACCD','EVAL','RJCTD','TNA') group by per_id_per having count(*) > 1 order by emma_id";
$sth = $dbh->prepare($sql);
$sth->execute;
while (my @results = $sth->fetchrow_array){
my @emma_ids = split(/,/,$results[0]);
my $i = 1;
my $max = @emma_ids;
foreach my $emma_id(@emma_ids){
my @xrefs = (@emma_ids[0..$i-2],@emma_ids[$i..$max-1]);
my $owner_xref;
foreach my $xref(@xrefs){
$owner_xref .= "<a href=\"http://www.emmanet.org/mutant_types.php?keyword=$xref\" target=\"_blank\">$xref</a>, ",
}
chop $owner_xref;
chop $owner_xref;
my $sql2 = "UPDATE strains SET owner_xref = '$owner_xref' WHERE emma_id='$emma_id'";
my $sth2 = $dbh->prepare($sql2);
$sth2->execute;
$i++;
}
}
$sql = "select group_concat(emma_id) from strains s, mutations_strains ms, mutations m, alleles a where id_str=ms.str_id_str and id=mut_id and id_allel=alls_id_allel and alls_form != 'NOD' and str_access not in ('R','N','C') and str_status not in ('ACCD','EVAL','RJCTD','TNA') group by alls_id_allel having count(id_str) > 1 order by emma_id";
$sth = $dbh->prepare($sql);
$sth->execute;
while (my @results = $sth->fetchrow_array){
my @emma_ids = split(/,/,$results[0]);
my $i = 1;
my $max = @emma_ids;
foreach my $emma_id(@emma_ids){
my @xrefs = (@emma_ids[0..$i-2],@emma_ids[$i..$max-1]);
my $mutation_xref;
foreach my $xref(@xrefs){
$mutation_xref .= "<a href=\"http://www.emmanet.org/mutant_types.php?keyword=$xref\" target=\"_blank\">$xref</a>, ",
}
chop $mutation_xref;
chop $mutation_xref;
my $sql2 = "UPDATE strains SET mutation_xref = '$mutation_xref' WHERE emma_id='$emma_id'";
my $sth2 = $dbh->prepare($sql2);
#print $sql2."\n";
$sth2->execute;
$i++;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 543 |
4 Weeks to Mother's Day!
Mothers… They give all their lives and expect nothing in return.
For this Mother's Day, give your mom a unique gift: a photo album that will remind her how much you love her!
She will have fun and be a star in a unique photo-shoot and keep the most amazing souvenir of a moment in life!
Gift certificates are available for Mother's Day including a photo-shoot and prints! | {
"redpajama_set_name": "RedPajamaC4"
} | 461 |
Q: Make UI uneditable when we are moving from one page to another I am trying to do userLogin through a webservice. The project is using Bootstrap for css and angular in the backend.
I am having a login Screen with a hidden div that is shoown when user click login button.Its a animated login icon.
Everything is working fine.I just want to make the background that is whole page including the userId, password uneditable as well as signin button unclickable.
A: When you hit the login button you can make the input(s) and button(s) disabled to make sure they remain unchanged.
When you get a result from your REST call you can re-enable them all.
A: You probably already have a variable $scope.loginIn on your controller scope since you are showing an animated login icon while login. If you don't, add it to your login function, something like :
$scope.login = function(...) {
$scope.loginIn = true;
loginService.login(...).finally() {
$scope.loginIn = false;
}
}
then just use it on your input and buttons to disable them :
<div class="login">
<input enable="!loginIn">...</input>
<button enable="!loginIn">...</button>
</div>
(That might be off the context, if you give me some code I could probably be more accurate in my answer)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,482 |
Q: Поиск в массиве. Найти наибольшее в строке и наименьшее в столбце Дан двумерный массив. Найти наибольшее значение в строке и наименьшее значение в столбце. Написал следующую программу. В чём ошибка?
#include <iostream>
using namespace std;
bool min_syun(int [],int,int);
bool max_tox(int[],int,int);
void main ()
{
const int n=3,m=4;
int a[n][m],c[n],i,j,k=0,l=0;
for(i=0;i<n;i++)
for(j=0;j<m;j++)
cin>>a[i][j];
for(i=0;i<n;i++)
for(j=0;j<m;j++)
{
k++;
for(l=0;l<n,k<n;l++,k++)
c[l]=a[k][i];
if(min_syun(c,a[i][j],n) && max_tox(a[i],a[i][j],m))
cout<<a[i][j]<<"-"<<k<<endl;
}
system("pause");
}
bool min_syun(int a[],int x, int m)
{
int min,i;
min=x;
for(i=0;i<m;i++)
if(a[i]<min)
return false;
return true;
}
bool max_tox(int a[], int x, int n)
{
int max,i;
max=x;
for(i=0;i<n;i++)
if(a[i]>max)
return false;
return true;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,888 |
{"url":"https:\/\/web2.0calc.com\/questions\/determine-the-smallest-non-negative-integer-a-that","text":"+0\n\n# Determine the smallest non-negative integer a that satisfies the congruences:\n\n0\n357\n3\n\nDetermine the smallest non-negative integer a\u00a0that satisfies the congruences:\n\na == 2 mod 3,\n\na ==\u00a04 mod 5,\n\na ==\u00a06 mod 7\n\na == 8 mod 9\n\nAug 3, 2018\n\n#1\n0\n\na =314\n\nAug 3, 2018\n#2\n+22884\n0\n\nDetermine the smallest non-negative integer a\u00a0that satisfies the congruences:\n\na == 2 mod 3,\n\na ==\u00a04 mod 5,\n\na ==\u00a06 mod 7\n\na == 8 mod 9\n\n1.\n\n$$\\begin{array}{|rcll|} \\hline & \\mathbf{ a } & \\mathbf{\\equiv}& \\mathbf{8 \\pmod{9}} \\\\\\\\ \\text{or} & a &=& 8+9n \\\\ & a &=& 8 + 3\\cdot 3n \\\\ \\text{or} & a &\\equiv & 8 \\pmod{3} \\quad & | \\quad 8 &\\equiv & 2 \\pmod{3}\\\\\\\\ & \\mathbf{ a } & \\mathbf{\\equiv}& \\mathbf{2 \\pmod{3}} \\\\ \\hline \\end{array}$$\n\n$$\\begin{array}{|lrcll|} \\hline (1) & \\mathbf{ a } & \\mathbf{\\equiv}& \\mathbf{4 \\pmod{5}} \\\\ (2) & \\mathbf{ a } & \\mathbf{\\equiv}& \\mathbf{6 \\pmod{7}} \\\\ (3) & \\mathbf{ a } & \\mathbf{\\equiv}& \\mathbf{8 \\pmod{9}} \\quad & | \\quad \\text{implicit } a \\equiv 2 \\pmod{3} \\\\ \\hline \\end{array}$$\n\nSolve:\n\n$$\\begin{array}{|rcll|} \\hline a &=& 4\\cdot 7 \\cdot 9 \\cdot \\frac{1}{7 \\cdot 9}\\pmod{5} \\\\ &+& 6\\cdot 5 \\cdot 9 \\cdot \\frac{1}{5 \\cdot 9}\\pmod{7} \\\\ &+& 8\\cdot 5 \\cdot 7 \\cdot \\frac{1}{5 \\cdot 7}\\pmod{9} \\\\ &+& 5\\cdot 7 \\cdot 9 n \\qquad n\\in Z \\\\\\\\ a &=& 252 \\cdot \\left( 63^{-1} \\pmod{5} \\right) \\\\ &+& 270 \\cdot \\left( 45^{-1} \\pmod{7} \\right) \\\\ &+& 280 \\cdot \\left( 35^{-1} \\pmod{9} \\right) \\\\ &+& 315 n \\\\\\\\ a &=& 252 \\cdot \\left( 63^{\\phi(5)-1} \\pmod{5} \\right) \\quad &|\\quad \\gcd(63,5) = 1,\\ \\phi(5) = 5-1=4 \\\\ &+& 270 \\cdot \\left( 45^{\\phi(7)-1} \\pmod{7} \\right) \\quad &|\\quad \\gcd(45,7) = 1,\\ \\phi(7) = 7-1=6 \\\\ &+& 280 \\cdot \\left( 35^{\\phi(9)-1} \\pmod{9} \\right) \\quad &|\\quad \\gcd(35,9) = 1,\\ \\phi(9) = 9(1-\\frac13)=6 \\\\ &+& 315 n \\\\\\\\ a &=& 252 \\cdot \\left( 63^{4-1} \\pmod{5} \\right) \\quad &|\\quad \\gcd(63,5) = 1,\\ \\phi(5) = 5-1=4 \\\\ &+& 270 \\cdot \\left( 45^{6-1} \\pmod{7} \\right) \\quad &|\\quad \\gcd(45,7) = 1,\\ \\phi(7) = 7-1=6 \\\\ &+& 280 \\cdot \\left( 35^{6-1} \\pmod{9} \\right) \\quad &|\\quad \\gcd(35,9) = 1,\\ \\phi(9) = 9(1-\\frac13)=6 \\\\ &+& 315 n \\\\\\\\ a &=& 252 \\cdot \\left( 63^{3} \\pmod{5} \\right) \\quad &|\\quad 63^{3} \\pmod{5} = 2 \\pmod{5} \\\\ &+& 270 \\cdot \\left( 45^{5} \\pmod{7} \\right) \\quad &|\\quad 45^{5} \\pmod{7} = 5 \\pmod{7} \\\\ &+& 280 \\cdot \\left( 35^{5} \\pmod{9} \\right) \\quad &|\\quad 35^{5} \\pmod{9} = 8 \\pmod{9} \\\\ &+& 315 n \\\\\\\\ a &=& 252 \\cdot 2 + 270 \\cdot 5 + 280 \\cdot 8 + 315 n \\\\ a &=& 4096 + 315 n \\quad &|\\quad 4096 \\equiv 314 \\pmod{315} \\\\ \\mathbf{a} & \\mathbf{=}&\\mathbf{ 314 + 315 n \\qquad n\\in Z }\\\\ \\hline \\end{array}$$\n\nThe smallest non-negative integer a is 314\n\nAug 3, 2018\n#3\n0\n\nA * 9 + 8 =B * 7 + 6=C * 5 + 4=D * 3 + 2, solve for A, B, C, D\n\nA=34, B =44, C=62, D=104\n\n9*34 + 8 =314 - The smallest positive integer\n\nThe LCM{3, 5, 7, 9} =315\n\n315n + 314, where n =0, 1, 2, 3........etc.\n\nAug 3, 2018\nedited by Guest \u00a0Aug 3, 2018\nedited by Guest \u00a0Aug 3, 2018\nedited by Guest \u00a0Aug 3, 2018\nedited by Guest \u00a0Aug 3, 2018","date":"2019-08-22 21:13:14","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\": 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\": 0.9674370288848877, \"perplexity\": 10105.547198253766}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027317359.75\/warc\/CC-MAIN-20190822194105-20190822220105-00299.warc.gz\"}"} | null | null |
Service
=======
phumbor.url.builder_factory
---------------------------
This service is an instance of \Thumbor\Url\BuilderFactory. It allows its user to access an url builder by calling the url method.
``` php
$thumborUrl = $this
->container
->get('phumbor.url.builder_factory')
->url('test/logo.png')
->resize(50, 50);
```
phumbor.url.transformer
-----------------------
This service wrapped the BuilderFactory to build a thumbor url where we applied a transformation from the config.yml file.
``` yml
jb_phumbor:
transformations:
width_50:
resize: { width: 50, height: 0 }
```
``` php
$thumborUrl = $this
->container
->get('phumbor.url.transformer')
->transform('test/logo.png', 'width_50');
```
Is the same thing as
``` php
$thumborUrl = $this
->container
->get('phumbor.url.transformer')
->url('test/logo.png')
->resize(50, 0);
```
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,466 |
Hi, I am Andrea and I integrate different Energy Healing techniques, EFT (Emotional Freedom Technique), NLP and Reiki to help you heal your toxic emotions and get back into the flow.
I also offer meditation and mindfulness classes.
I hold a safe space of unconditional acceptance and guide clients through a process of becoming aware of their unhelpful beliefs and emotions, facilitating the release of emotions, trauma and patterns which do not serve any more. This is deeply transformative work which allows the client to find a new perspective, to re-frame and gently shift from Ego consciousness to Essence (or True Self), from Fear to Love – a state where healing, peace and forgiveness become possible. The process resolves issues at the deepest level, re-establishing harmony on all levels of our being.
Over the years I have developed a particular interest in the latest cutting edge body-focussed approaches to mental health and their applications for issues such as developmental trauma, depression, anxiety, feelings of never being good enough and much more and continue to train with leading experts in the field.
The bit about me: The search for my true self led through many professional and personal permutations – From advertising executive to film rights trader to holistic practitioner, from party girl to earth mother, from 80s hedonism to environmental activism and a deep commitment to personal and global healing. As I move into eldership in my own life, I draw on this rich tapestry of experience to meet my clients wherever they are in their lives and feel ready and prepared to help others face and overcome their fear and pain, identify their deepest longing and realise their goals.
Integrated Energy Healing – This integrative practice was originally devised by Barbara Brennan ("Hands of Light – A Guide To Healing Through The Human Energy Field" and "Light Emerging") and combines hands-on-healing with a modern psychotherapeutic approach, drawing elements from person centred counselling, transpersonal psychology and core energetics.
Where appropriate, I synergistically add other methods of Energy Psychology in the form of EFT, NLP and Ericksonian Trances to this mix and guide my clients through a process of inner inquiry to get to the core of their issue. Through the combination of deep inquiry and hands-on work, clients are guided to bringing their blockages to light and releasing them. This enables healing on all levels, including physical and emotional well-being, relationships, mental clarity and even spiritual realisation. The tool kit used in Integrated Energy Healing is impressive: techniques include chakra and aura cleansing, balancing and re-structuring, sound and colour healing, working with metaphors, inner child work, relationship cord work, astral clearing, entity release, past life regression, psychic surgery, belief clearing and intention alignment.
EFT – Emotional Freedom Techniques or Tapping is a modern complementary therapy with its roots in Chinese medicine and acupuncture (without using needles).
EFT involves tapping on acupuncture points while focussing on the emotional or physical issue you want to heal. By doing this, you can change the imbalances in your energy system which causes our negative thoughts and feelings. EFT has been proven to be successful in treating a huge range of emotional and health issues and it often works where other treatments have failed. It is a powerful tool for releasing and relieving negative emotions and pain, gently and easily. It is non-invasive and highly effective at healing deep emotional wounds and trauma.
I often combine my Energy Healing/ Reiki sessions with EFT with some incredible results. I also invite my clients to learn the basic EFT routine so they can have this incredible tool "at their fingertips" whenever needed – it's truly empowering! My expertise lies in working with clients in the areas of trauma (often the small "t" trauma from childhood which we don't even recognise as such), childhood abuse, self-harm, low self-esteem, fear of change, procrastination, depression, stress, anxiety and spiritual crisis.
Reiki – The Reiki system was developed by Mikao Usui in Japan in the early 1900s as a path for spiritual development and for natural healing.
Receiving a Reiki healing is a deeply relaxing and pleasant experience. The client remains fully clothed and is lying on a therapy couch. Reiki treatments are very gentle and non-intrusive, they are equally effective hands-on as well as hands-off, with the practitioner holding their hands a few inches away from the body or even as distant healings across the globe.
Today Reiki continues to be taught by Reiki Masters who have trained in the tradition passed down from Master to student. I use Reiki mainly in combination with the other modalities I practise. However, I believe Reiki is a fantastic way to open one's system to energy work and as a Reiki Master, I offer tailor-made attunement workshops upon request.
Please call, text or email me. Further information can be found on my website.
Appointments can be made for my clinic in Muswell Hill, North London or at the Buddha on a Bicycle in Covent Garden. I also work with clients worldwide via Skype.
Please refer to my website or contact me directly for rates. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,000 |
require 'active_support/core_ext'
module YaMap
class ControlPosition
TOP_LEFT = 'YMaps.ControlPosition.TOP_LEFT'
TOP_RIGHT = 'YMaps.ControlPosition.TOP_RIGHT'
BOTTOM_LEFT = 'YMaps.ControlPosition.BOTTOM_LEFT'
BOTTOM_RIGHT = 'YMaps.ControlPosition.BOTTOM_RIGHT'
# ControlPosition.new ControlPosition::TOP_LEFT, :x => 10, :y => 10
def initialize pos, offset = {}
@ancor = pos
@off = offset.reverse_merge :x => 5, :y => 5
end
def to_s mode = :new
case mode
when :new then "new YMaps.ControlPosition(#{@ancor}, new YMaps.Point(#{@off[:x]},#{@off[:y]}))"
end
end
end
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,767 |
\section{Introduction}
The expansion of an H~II region into a surrounding, inhomogeneous
molecular cloud leads to the formation of complex, elongated
``elephant trunk'' structures. Examples of this are the
trunks in the Eagle \citep{hester:96} and the Rosette
\citep{carlqvist:03} nebulae. Examples of similar (but larger)
structures in external galaxies have been found as well,
\citep{carlqvist:10}.
A considerable amount of theoretical work has focused on the
photoevaporation of a single, dense clump, starting with the
paper of \citeauthor{oort:55} (\citeyear{oort:55}). This problem has been addressed
both analytically \citep{bertoldi:89,bertoldi:90}
and numerically \citep{lefloch:94,dale1:07,dale2:07,dale:11,ercolano:12}. Several simulations have
been done including detailed treatments of the radiative transfer
and ionization \citep{mellema:98,raga:09}, and the study
of clumps with low amplitude density inhomogeneities \citep{gonzalez:05},
with self-gravity \citep{esquivel:07} and with magnetic fields
\citep{henney:09}. These models are applicable to photoionized
regions in which individual neutral clumps are clearly visible
(the evident example of this being the Helix Nebula, see, e.~g.,
\citeauthor{odell:05} \citeyear{odell:05}).
However, the observations of elephant trunks suggest the presence
of more complex density structures in the region with neutral gas.
Attempts to address this have included 3D simulations of the propagation
of ionization fronts into ``multi-clump'' structures
\citep{lim:03,raga:09,mackey:10,mackey:11}
and into more complex density distributions \citep{mellema:06,esquivel:07,maclow:07,grithschneder:09,ercolano:11,arthur:11}.
These simulations
took into account the following physical processes:
\begin{itemize}
\item direct gas dynamics+ionizing radiation transfer
\citep{lim:03,mackey:10},
\item self-gravity \citep{esquivel:07,maclow:07,grithschneder:10,ercolano:11},
\item magnetic fields \citep{mackey:11,arthur:11},
\item the diffuse ionizing radiation field \citep{raga:09,ercolano:11}.
\end{itemize}
An important effect that has not been explored in detail
until very recently is the presence of a photodissociating
FUV radiation field (together with the photoionizing EUV radiation).
\citeauthor{arthur:11} (\citeyear{arthur:11}) presented numerical simulations
including the FUV radiation field and obtained that the FUV field can
have a clear dynamical importance in the formation of dense clumps at the edge of an
expanding H~II region. Their simulations include the presence of
a magnetic field, but do not include the self-gravity of the gas.
In the present paper, we discuss 3D simulations which include the
transfer of the EUV and FUV radiation in a self-gravitating medium
with an inhomogeneous initial density distribution. Our simulations
differ from those of \citeauthor{arthur:11} (\citeyear{arthur:11}) in that they do
not include a magnetic field, but do consider the self-gravity of
the gas (see section 2).
We then use the results of our simulations to calculate the number
of clumps (section 3) and mass
distributions of the dense clumps (section 4). Finally,
the results are summarized in section 5.
\section{The numerical simulations}
We have carried out twenty four 3D simulations with the code described
by \citeauthor{lora:09} (\citeyear{lora:09}). This code integrates the gasdynamic equations
in a uniform 3D, Cartesian grid, together with the radiative transfer
of radiation at the Lyman limit, and a hydrogen ionization rate equation,
including the self-gravity of the gas.
The radiative transfer and hydrogen ionization are solved as
follows. We place an ionizing photon source (producing $S_*$
ionizing photons per unit time) far away, outside the computational
grid, and then impose an ionizing photon flux $F_0=S_*/(4\pi R_0^2)$
on the boundary of the computational grid, where $R_0$ is the distance
from the grid boundary to the photon source (assumed to lie along the
$x$-axis). The ionizing photon flux is then marched into the
computational domain as:
\begin{equation}
F_{i+1,j,k}=F_{i,j,k}\,e^{-\Delta \tau_{Ly}}\,,
\label{fp}
\end{equation}
where $F_{i,j,k}$ is the ionizing photon flux at the left boundary
(along the $x$-axis) of the $(i,j,k)$ computational cell and
\begin{equation}
\Delta \tau_{Ly}=n_{HI}\left(\sigma_{H,\nu_0}+\sigma_d\right)\Delta x\,,
\label{dtau}
\end{equation}
where $n_{HI}$ is the neutral H density of cell $(i,j,k)$, $\Delta x$
is the size of the cell (along the $x$-axis),
$\sigma_{H,\nu_0}=6.30\times 10^{-18}$~cm$^2$ is the Lyman limit
photoionization cross section of HI and $\sigma_d=1.1$~cm$^2$
is the FUV/EUV dust absorption cross section. This value of $\sigma_d$
is derived assuming that the EUV/FUV dust absorption
is $A={1.2\times 10^{-21}\rm~cm^{-2}}\,N_{HI}$ (the derivation
of this relation is discussed by \citeauthor{vasconcelos:11} \citeyear{vasconcelos:11}).
We should note that in calculating the Lyman limit optical depth through equation
(\ref{dtau}) we are assuming that the region of photoionized H has no dust. If it
did, one would have to replace $n_{HI}$ by $n_H$ (the total H number density) in
equation (\ref{dtau}). However, because of the low column densities of the ionized
H regions within our computational domain, it makes no difference whether or not
dust is present in these regions.
With the ionizing photon flux $F$, we calculate the H photoionization rate
\begin{equation}
\phi_H=\left(1-e^{-\Delta \tau_{Ly}}\right)F\,.
\label{phih}
\end{equation}
This photoionization rate is then included in a HI continuity equation:
\begin{equation}
\frac{\partial n_{HI}}{\partial t}+ \dot\nabla (n_{HI}\underline{u})=
n_en_{HII}\alpha_H(T)-n_{HI}\phi_H\,,
\label{nhi}
\end{equation}
where $n_{HI}$, $n_{HII}$ and $n_e$ are the neutral H, ionized H and electron densities
(respectivey) and $\alpha_H(T)$ is the recombination coefficient of H. This equation
is integrated together with the standard 3D gasdynamic equations. The diffuse
radiation is included only by considering the ``case B'' recombination (to all levels
with energy quantum number $N>1$).
To equations (\ref{fp}-\ref{nhi}), which were included in the code
of \citeauthor{lora:09} (\citeyear{lora:09}), we have now
added the transfer of FUV radiation, and an ionization rate equation
for CI. This has been done in the following way.
For the FUV photons, we solve the radiative transfer problem
at $\lambda=1100$~\AA\ (the ionization edge of the ground state of CI),
considering the absorption due to CI photoionization and to dust
extinction. We therefore calculate the optical depth for the FUV
photons in each computational cell as:
\begin{equation}
\Delta \tau_{FUV}=\left(n_{CI}\sigma_{C,\nu_0}+n_{HI}\sigma_d\right)\Delta x\,,
\label{tfuv}
\end{equation}
where $n_{CI}$ is the CI number density and $\sigma_{C,\nu_0}=1.22\times 10^{-17}$~cm$^2$
is the photoionization cross section at the CI ionization edge. We have
assumed that the dust extinction cross section (per H atom) has the same
values at the CI and HI ionization edges (i.e. and $\lambda\sim 1100$ and 900~\AA,
respectively). With this value for the FUV optical depth of the computational
cells, we then use equations equivalent to (\ref{fp}) and (\ref{phih}) to calculate
the CI photoionization rate. We then integrate a continuity equation for CI (with the
appropriate source terms, such as equation \ref{nhi} but for CI) together with
the gasdynamic and HI continuity equations.
Instead of integrating an energy equation (with appropriate heating and cooling
terms), we compute the temperature of the gas as:
\begin{equation}
T=(T_1-T_2)x_{HII}+T_2x_{CII}+T_3(1-x_{CII})\,,
\label{t}
\end{equation}
where $x_{HII}$ and $x_{CII}$ are the H and C ionization
fractions (respectively), and $T_1=10^4$~K, $T_2=10^3$~K
and $T_3=10$~K are the typical temperatures of photoionized,
photodissociated and molecular regions respectively.
Therefore, we do not calculate the photodissociation of H$_2$, and
assume following \citeauthor{richling:00} (\citeyear{richling:00}),
that it approximately follows the ionization of CI.
From the paper of \citeauthor{diaz:98} (\citeyear{diaz:98}), we now take the EUV
($\lambda>912$~\AA) photon rate $S_I$ and the FUV
(912~\AA$<\lambda<$1100~\AA) photon
rate $S_D$ for a set of three main sequence O stars (of effective
temperatures $T=50000$, 45000 and 40000~K). For the EUV photons,
we solve the radiative transfer problem (parallel to the $x$-axis
of the computational grid) using the Lyman-limit H absorption
coefficient $\sigma_{\nu_0}(H)=6.30\times 10^{-18}$~cm$^2$, and
using the EUV flux to calculate the HI photoionization rate
(this is completely equivalent to the models of \citeauthor{lora:09} \citeyear{lora:09}).
For our simulations, we consider a computational domain
of $(3.0,\,1.5,\,1.5)\times 10^{18}$~cm (along
the $x$-, $y$- and $z$-axes, respectively) resolved with
$256\times 128\times 128$ grid points. An outflow boundary
condition is applied on the $x$-axis boundaries, and reflection
conditions in all of the other boundaries.
This domain is initially filled with an inhomogeneous density
structure with a power law power-spectrum index of $-11/3$
(i.e., $P(k)\propto k^{-11/3}$, where $k$ is the wave number,
see \citeauthor{esquivel:03} \citeyear{esquivel:03}). This results in a density distribution
with a dispersion of $\approx 2$ times the mean density. We
have chosen four different realizations of the density
distribution, which we use to compute models
identified with the letters M, O, D and E (each letter corresponding
to one of the chosen initial density distributions).
The medium is initially at rest.
The computational domain is divided
into an initially ionized region (with ionized H and C)
for $x<x_0=4\times 10^{17}$~cm and a neutral region
(with neutral H and C) for $x>x_0$. The average
density in the neutral medium is 100 times the average
density in the ionized medium, and the transition between
the two follows a tanh profile with a width
of $\sim 10$ pixels. The resulting neutral structure has
a total mass of 228~M$_\odot$. This initial setup is identical to
the one used by \citeauthor{lora:09} (\citeyear{lora:09}).
Other authors have used more complex initial conditions for this
kind of simulation, in particular, including an initial velocity field.
This was done, e.g. by \citeauthor{arthur:11} (\citeyear{arthur:11}), who
took as initial conditions the output from a 3D, turbulent cloud simulation.
Other examples of initial conditions can be found in \citeauthor{dale:11} (\citeyear{dale:11}),
\citeauthor{dale:12} (\citeyear{dale:12}) and \citeauthor{ercolano:12} (\citeyear{ercolano:12}).
However, as the flow motions induced by the photoionization and photoevaporation
have velocities which are much larger than the ones of the initial turbulent
motions, including these initial motions is unlikely to produce large
effects on the results.
We assume that we have a stellar source situated $3\times 10^{18}$~cm
from the edge of the computational domain in the $-x$ direction. For
this photon source we consider three possibilities, an O3 ($T_{eff}=50000$~K),
an O5.5 ($T_{eff}=45000$~K) and an O7.5 ($T_{eff}=40000$~K) main sequence
star. The EUV and FUV fluxes computed by \citeauthor{diaz:98} (\citeyear{diaz:98})
for such stars are given in Table~1.
We then compute simulations with the EUV and FUV fluxes given
in Table 1 (models M1-3, O1-3, D1-3 and E1-3, with letter M-E
corresponding to the four different initial density distributions, see
above), and simulations with the
same EUV fluxes but with zero FUV flux (models
M1B-3B, O1B-3B, D1B-3B and E1B-3B). As can be seen from Table 1, the
models with the same number (second character of the model identification)
have the same impinging EUV field.
The set of models with zero FUV fields (models M1B-E3B)
has a transition from $T_3=10$~K in the region
with neutral H to $T_1=10^4$~K in the photoionized region (see
equation \ref{t}), with no intervening photodissociated region,
and are therefore equivalent to the models of \citeauthor{lora:09} (\citeyear{lora:09}).
\section{Results}
In Figures 1 and 2, we show the mid-plane density stratifications
and the positions of the H and C ionization fronts
obtained for a $t=2.5$~kyr (Figure 1) and $t=100$~kyr (Figure 2)
integration time
for models M1, O1, D1 and E1. In Figures 3 and 4, we show the
same density stratifications, obtained for a $t=2.5$~kyr (Figure 3)
and $t=100$~kyr (Figure 4) integration time
for the models with zero FUV fluxes: M1B, O1B, D1B and E1B.
In the models with non-zero FUV fluxes (see Figures 1 and 2)
we see that the C and H ionization fronts are separated by a
photodissociated region (with CII and HI) with a width of $\sim
10^{18}$~cm in the $t=2.5$ and $100$~kyr frames. The neutral region
(to the right of the C ionization front) develops progressively
denser regions which collapse under the combined effects
of the compression ahead of the C ionization front and the
self-gravity of the gas. As expected, the higher temperature,
photodissociated region does not develop such dense structures.
For the models with zero FUV fluxes (models M1B-E1B see Figures 3 and 4)
there is of course no photodissociated
region.
Qualitatively similar time-evolutions are obtained for all
of the computed models. Of course, the details of the fragmentation
of the neutral gas into dense clumps differ in all simulations.
These differences are quantified in the following subsection.
\begin{table}
\caption{Model Parameters}
\label{table:1}
\centering
\begin{tabular}{lcccc}
\hline\hline\noalign{\smallskip}
Models & $T_{eff}$ & $\log_{10} S_I$ & $\log_{10} S_D$ & $S_D/S_I$\\
& [$10^3$~K] & \multispan2{\hfil [photons~s$^{-1}$]\hfil} & \\
\hline\noalign{\smallskip}
M1, O1, D1, E1$^{\mathrm{a,b}}$ & 50 & 49.89 & 49.54 & 0.45 \\
M2, O2, D2, E2$^{\mathrm{a,b}}$ & 45 & 49.35 & 49.16 & 0.65 \\
M3, O3, D3, E3$^{\mathrm{a,b}}$ & 40 & 48.78 & 48.76 & 0.96 \\
\hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] models with letters M through E have identical
FUV and EUV fluxes, and correspond to different initial
density distributions (see the text),
\item[$^{\mathrm{b}}$] models M1B-E1B, M2B-E2B and M3B-E3B have the same values
of $S_I$ as models M1-E1, M2-E2 and M3-E3 (respectively), but have $S_D=0$.
\end{list}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.55]{fig1.eps}
\caption{The $t=2.5$~kyr, $xy$-mid-plane density stratifications
of models M1, O1, D1 and E1 (with non-zero FUV fields, see Table 1).
The density stratifications are shown with the logarithmic gray scale
given (in g~cm$^{-3}$) by the top right bar. In the four frames, we show
the contour corresponding to an H ionization fraction of $50\%$ (black
line), which indicates the position of the HI/II ionization front. The
blue lines show the width of the HI/II region. The contour corresponding to
a C ionization fraction (white line) of $50\%$, which indicates the position
of the CI/II ionization front, is also shown. The pink lines show the width
of the CI/II region. The $x$ and $y$-axes are labeled in cm.}
\label{fig1}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.55]{fig2.eps}
\caption{Same as Figure 1, but for $t=100$~kyr (see Table 1).}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.55]{fig3.eps}
\caption{The $t=2.5$~kyr, $xy$-mid-plane density stratifications
of models M1B, O1B, D1B and E1B (with zero FUV fields,
see Table 1). The density stratifications are shown
with the logarithmic gray scale given (in g~cm$^{-3}$) by the top right
bar. In the four frames, we show the contour corresponding to an H
ionization fraction of $50\%$ (black line), which indicates the position
of the HI/II ionization front, and also the
contour corresponding to a C ionization fraction (white line) of $50\%$, which
indicates the position of the CI/II ionization front. The
blue lines show the width of the HI/II region. The $x$ and $y$-axes
are labeled in cm.}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.55]{fig4.eps}
\caption{Same as Figure 3, but for $t=100$~kyr (see Table 1).}
\end{figure}
\subsection{Number of clumps as a function of time}
We take the density stratifications resulting from
our 24 simulations (see Table 1), and compute the number
of clumps present at different integration times. To
calculate the number of clumps, we define a cutoff
density $\rho_c$, and count all spatially contiguous
structures with densities $\rho\geq \rho_c$.
Following \citeauthor{lora:09} (\citeyear{lora:09}),
we choose three different cutoff densities
$\rho_c=10^{-20}$, $10^{-19}$ and $3\times 10^{-18}$~g~cm$^{-3}$
(corresponding to number densities of atomic nuclei
$\sim$4600, 46000 and $1.4\times 10^6$~cm$^{-3}$).
For the models without a FUV field (models M1B-E3B),
we count clumps in the neutral H region. For the models with
non-zero FUV fields (models M1-E3, see Table 1), we
count clumps that satisfy one of the two following conditions~:
\begin{enumerate}
\item that their material has neutral H,
\item that they have neutral C.
\end{enumerate}
Notably, these two criteria result in identical clump numbers
for the $\rho_c=10^{-19}$ and $3\times 10^{-18}$~g~cm$^{-3}$
cutoff densities (see Figures 5, 6 and 7). For $\rho_c=10^{-20}$~g~cm$^{-3}$, the
second criterion results in somewhat larger clump numbers.
In the computation of the clump numbers, we calculate
averages over each of the sets of four models with identical
parameters (see Table 1) but with different initial
density structures.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.5]{fig5.eps}
\caption{The top panel shows the number of neutral
clumps obtained from models M1, O1, D1 and E1
(with non-zero FUV fluxes, see Table 1) as a function of time,
for the three chosen density cutoffs. The bottom
panel shows the number of neutral clumps for the same density cutoffs but for
models M1B, O1B, D1B and E1B (with zero FUV, see Table 1).
In the top panel, the two solid lines correspond
to a $\rho_c=10^{-20}$~g~cm$^{-3}$ cutoff density, with clumps with neutral
H (thin line) and with neutral C (thick line).
}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.5]{fig6.eps}
\caption{Same as Figure 5, but for models M2, O2, D2 and E2 (top frame)
and M2B, O2B, D2B and E2B (bottom frame).}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.5]{fig7.eps}
\caption{Same as Figure 5, but for models M3, O3, D3 and E3 (top frame)
and M3B, O3B, D3B and E3B (bottom frame).}
\end{figure}
The results obtained from this clump counting exercise are
given in Figures 5 (models M1-E1 and M1B-E1B), 6 (models M2-E2
and M2B-E2B) and 7 (models M3-E3 and M3B-E3B). These Figures show the number of
clumps (averaged over 20 kyr time-intervals) as a function
of the integration time, obtained with the three chosen cutoff densities.
From Figures 5-7, we see that for the $\rho_c=10^{-20}$~g~cm$^{-3}$ cutoff
density, the number of clumps first decreases rapidly,
and then stabilizes (for $t>20$~kyr) at a value of $\approx 100$
for most of the models. Actually, if for the models with
nonzero FUV field we count clumps with neutral C,
the number of clumps stabilizes at a value of $\sim 200$ (see
the top panels of Figures 5-7).
For the $\rho_c=10^{-19}$~g~cm$^{-3}$ cutoff
density, the number of clumps starts at a value of $\sim 3\times10^3$,
and in all models decreases to $\sim 200$ within the first
$\sim 50$~kyr of the time-evolution. For larger times, in
the models with zero FUV flux (M1B-E1B, M2B-E2B and M3B-E3B, bottom panels
of Figures 5, 6 and 7, respectively), the clump number decreases
and then stabilizes (for $t>30$~kyr) at a value of $\approx 50$.
For $\rho_c=3\times 10^{-18}$~g~cm$^{-3}$, the models
with zero FUV flux take $\sim 50\to 70$~kyr to develop the first clump,
and by $t=130$~kyr they have developed $\sim 40$ clumps. The models with
non-zero FUV flux (top panels of Figures 5-7) develop the first
clump much earlier, after only $\sim 10$~kyr, and have about
$70$ clumps at $t=130$~kyr.
Even though our models with zero FUV flux (M1B, M2B and M3B)
cover a factor of $\sim 13$
in EUV photon rates (see Table 1), except for relatively small effects
(e.g., the earlier appearance of high $\rho_c$ clumps in model M1B-E1B,
see the bottom panel of Figure 5) they show qualitatively similar
trends of number of clumps as a function of time. Our models with
non-zero FUV flux (M1-E1, M2-E2 and M3-E3, see Table 1) also cover a factor
of $\sim 6$ of FUV photon production rates, and these three
models also show
qualitatively similar time evolution of the number of clumps
(top panels of Figures 5-7).
From this, we conclude that for the range of early to late O-type stars
chosen, the fragmentation of the neutral structure into
clumps presents a similar behavior regardless of the spectral subclass of the star.
We also see that including the effect of the FUV photons does produce
important differences (see the above list of 3 items). The effect
of introducing the FUV flux is
to increase the number of $\rho_c = 3\times 10^{-18}$~g~cm$^{-3}$ clumps
(from $\sim 40$ to $\sim 70$) at the final, $t=240$~kyr integration
time of our simulations.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.4]{fig8.eps}
\caption{Neutral clump mass distribution for two cutoff densities
($\rho_c=10^{-19}$ and $3\times10^{-18}$~g~cm$^{-3}$) for the integration
time $t=150$~kyr for models M1, O1, D1 and E1 (left panels)
and M1B, O1B, D1B and E1B (right panels). }
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.4]{fig9.eps}
\caption{Same as Figure 8, but for models M2, O2, D2 and E2 (left
panels) and M2B, O2B, D2B and E2B (right panels).}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.4]{fig10.eps}
\caption{Same as Figure 8, but for models M3, O3, D3 and E3 (left
panels) and M3B, O3B, D3B and E3B (right panels).}
\end{figure}
\subsection{The clump mass distributions}
We now focus on a $t=150$~kyr evolutionary time, in which
a sizable population of clumps has developed in all models
(see Figures 5-7). For the stratifications resulting from
all models at this time, we compute the mass distributions
of the clumps obtained with the $\rho_c=10^{-19}$ and
$3\times 10^{-18}$~g~cm$^{-3}$ cutoff densities. We do
not compute the distributions for $\rho_c=10^{-20}$~g~cm$^{-3}$
because at $t=150$~kyr they only have one clump which includes
basically all of the neutral region of the flow.
The resulting clump number vs. mass distributions are shown
in Figures 8 (models M1-E1 and M1B-E1B, see Table 1), 9 (models M2-E2 and M2B-E2B)
and 10 (models M3-E3 and M3B-E3B). If we look at the $\rho_c=10^{-19}$~g~cm$^{-3}$
clump mass distributions (top panels of Figures 8, 9 and 10), we observe that the
distributions of all models have two clumps in the
high mass, $10^2\to 10^3$~M$_\odot$ bin, and for models with non zero FUV flux 2-4 clumps
in the low mass, $10^{-3}\to 10^{-2}$~M$_\odot$ bin.
The distributions of the models with zero FUV flux
(M1B-E1B, M2B-E2B and M3B-E3B) have 0-1 clump in the $10^{-2}\to 10^{-1}$~M$_\odot$
range, while the distributions of the non-zero FUV flux
models (M1-E1, M2-E2 and M3-E3) have 4-6 clumps in this mass
range (except for model M1, with one clump in this mass range,
see Figure 8).
Therefore, the main effect of including a non-zero FUV field
is mostly to enhance the $\rho_c=10^{-19}$~g~cm$^{-3}$ number of clumps in
the $10^{-3}\to 10^{0}$~M$_\odot$ mass range.
If we look at the $\rho_c=3\times 10^{-18}$~g~cm$^{-3}$
clump mass distributions (bottom panels of Figures 8, 9 and 10), we see that
the three models with zero FUV flux (M1B-E1B, M2B-E2B and M3B-E3B) show
very similar clump mass distributions, with
one clump in the $10^{-1}\to 10^2$~M$_\odot$ mass range for
model M1B-E1B, and no clumps in this mass range for models
M2B-E2B and M3B-E3B.
In the $10^{-2}\to 10^{-1}$~M$_\odot$
range, all of the models with non-zero FUV flux have 7-8 clumps,
generally having 3 more
clumps in this range than the zero FUV models (4-5 clumps).
Therefore, the presence of an FUV flux allows the formation
of more low mass, $\rho_c=3\times 10^{-18}$~g~cm$^{-3}$ clumps,
than in the zero FUV flux models.
\section{Summary}
In a previous paper \citep{lora:09}, we have studied
the formation of dense clumps in the interaction of a
photoionizing radiation field with an inhomogeneous medium
(with an initial power law spectrum of density fluctuations).
The applicability of these models to real astrophysical flows
(associated with expanding HII regions) was questionable because
of the absence (in the models) of a photodissociation region
preceding the HI/II ionization front.
In this work, we present a set of numerical simulations
which explore the effect of a FUV radiative field, which
produces a photodissociation region outside the HII region.
To this effect, we compute 3D simulations which include
the photoionization of H and C, assuming that the CI/II ionization
front approximately coincides with the outer edge of the
photodissociation region (as initially suggested by
\citeauthor{richling:00} \citeyear{richling:00}). \citeauthor{arthur:11} (\citeyear{arthur:11}) calculate
numerical simulations in which a different approximation
is used for determining the outer edge of the photodissociation
region, but which leads to similar results.
From our simulations, we obtain the masses and the number of
clumps (defined as contiguous regions of density higher than
a given cutoff density $\rho_c$, see section 3). We calculate
models with photoionizing/photodissociating stars at $\approx 1$~pc
from the computed region, and explore a range from
early to late-type O stars (using the EUV and FUV photon
rates presented by \citeauthor{diaz:98} \citeyear{diaz:98}). We also
compute models setting the FUV flux to zero, in order
to isolate the effects of having a non-zero FUV flux.
We find that including the photodissociation
region produced by a non-zero FUV flux has the following
main effects:
\begin{itemize}
\item clumps with low cutoff densities ($\rho_c=10^{-19}$~g~cm$^{-3}$)
are slightly depleted (see sections 3 and 4),
\item denser clumps (with $\rho_c=3\times 10^{-18}$~g~cm$^{-3}$)
develop earlier than in the models with zero FUV (see Figures
5-7),
\item a larger number of dense clumps
(with $\rho>\rho_c=3\times 10^{-18}$~g~cm$^{-3}$)
is produced, and these clumps have a broader mass distribution than
in the zero FUV models (see section 4).
\end{itemize}
From this, we conclude that the presence of an outer
photodissociation region has an important effect on the
formation of dense structures due to the expansion of
an HII region in an initially inhomogeneous medium. In particular,
including a FUV field leads to the earlier formation of
a larger number of dense clumps. This in principle, might lead
to the formation of more young stars. However, our simulations
do not have the resolution nor include the physical processes
necessary for determining whether or not the clumps actually
collapse to form one or more stars.
\acknowledgments
VL gratefully acknowledges support from the Alexander von
Humboldt Foundation fellowship and H.B-L.
We acknowledge support from the CONACyT grant 61547, 101356 and
101975.
AHC would like to thank the CNPq for partial financial
support (307036/2009-0).
The authors would like to also thank the anonymous referee, for very useful comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,580 |
Die Hand in der Falle (Originaltitel: La mano en la trampa) ist ein argentinisch-spanischer Spielfilm in Schwarzweiß des Regisseurs Leopoldo Torre Nilsson aus dem Jahr 1961 mit Francisco Rabal, Elsa Daniel und Leonardo Favio in den Hauptrollen. Das Drehbuch von Beatriz Guido und Ricardo Muñoz Suay basiert auf einem Roman von Ricardo Muñoz Suay. Erstmals gezeigt wurde der Streifen im Mai 1961 bei den Internationalen Filmfestspielen von Cannes. In der Bundesrepublik Deutschland kam er am 15. April 1966 in die Kinos.
Handlung
Aus dem Internat in das Haus ihrer Mutter und ihrer Tante zurückgekehrt, möchte Laura Lavigne das Geheimnis ergründen, das seit Langem das Obergeschoss des herrschaftlichen Wohnsitzes der Familie umgibt. Angeblich soll dort ein geistesschwaches nichteheliches Kind des alten Lavigne verborgen gehalten werden. In Wirklichkeit ist dieses Kind längst tot. Dagegen entdeckt Laura dort ihre Tante Ines, die vor zwanzig Jahren von ihrem angesehenen Verlobten Cristobal Archaval sitzen gelassen worden war und sich aus Angst vor dem Gespött der argentinischen Bourgeoisie in ihrem Zimmer lebendig begraben hat. Bei den Nachforschungen über die genauen Gründe für das Schicksal ihrer Tante bedient Laura sich halb aus Neugier, halb aus Hingezogenheit der Hilfe eben jenes Cristobal Archaval. Sie führt ihn sogar zu Ines, die darüber so erregt ist, dass ihr Herz versagt. Laura bleibt bei dem Manne, der ihr in der Hauptstadt eine Wohnung einrichtet. In ihrem Bewusstsein wird ihr Zimmer plötzlich zu dem von Ines. Sie ist gefangen wie ihre Tante.
Kritik
Auszeichnungen
FIPRESCI-Preis 1961
Regiepreis in Santa Margherita Ligure
Prädikat «Besonders Wertvoll», Filmbewertungsstelle Wiesbaden
Weblinks
Einzelnachweise
Filmtitel 1961
Argentinischer Film
Spanischer Film
Schwarzweißfilm
Literaturverfilmung
Filmdrama | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,704 |
Working at Dosign? Our internal positions at a glance!
Currently, our internal positions are suitable for Dutch-speakers only. Please consult our Dutch section for details.
If your vacancy is not listed, please send us an open application. We will then contact you. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,981 |
What grows almost anywhere and is a vegetable, but tastes like a fruit? If you guessed rhubarb you are right! Rhubarb is this healthy little plant that gives us gorgeous red stems that taste tart and sweet like a granny smith apple. You can bake with it, make jam with it, and even cook it in stews, but my favorite way to eat rhubarb is in a Berry & Barb or Strawberry Rhubarb Pie.
My grandmother loved her rhubarb. It was a family treasure for her and many stories were told of how her great grandparents planted rhubarb on the prairies of Nebraska as their only source of fruit. Grandma tended those plants like they were something of great value and from her harvest she made the best jams, pies and puddings. Rhubarb grows from the beginning of spring through fall and you can find rhubarb plants in most of our local garden shops ready to go. If you harvest it as soon as the stems are about a foot high, you can get several harvests in one year. Trust me, if I can grow it in Woodland Hills you can grow it anywhere!
The easiest way to preserve rhubarb is to freeze it. It will last up to a year in a freezer and will stay good in your refrigerator for 3 to 4 weeks wrapped in a paper towel. I chop mine up and freeze them for whenever I am craving a Berry and Barb Pie or Grandma Abbie's Rhubarb Pudding. If you don't have a garden you can pick up frozen rhubarb at your local grocery store in the freezer section and it will be just as good!
2. Roll out bottom crust into a 9" pie plate.
3. Wash and chop rhubarb into small 1/4" pieces. Place into medium size bowl.
4. Wash berry of choice. We love our strawberries sliced in half, but you can use blackberries, cherries or any other berry you like. Add to bowl.
5. Add dry ingredients and lemon juice and toss together.
6. Pour mixture into pie plate.
7. Roll out top pie pastry and decorate in strips or with cookie press shapes. Pinch edges around and brush with a egg white wash and sprinkle with coarse sugar.
8. Bake at 350 degrees for 60 to 90 minutes until crust is golden brown.
9. Let cool and enjoy with fresh cream.
Sift the flour, salt, and sugar together in a food processor. Add the butter pieces pulsing to break up the chunks into the dry ingredients. The consistency should be of pea sized crumbles. Be careful not to over blend.
Combine the water and vinegar together and slowly sprinkle it into the mixture while pulsing your processor until it is incorporated together. Divide the dough into two balls and shape the dough into round discs. Place between wax paper sheets and place in the fridge for one hour. Once it has rested a bit in the cold it is ready to roll out and use!
I hope you enjoy this pie as much as we do. For me it reminds me of home and tastes like a whole lot of love! I will be sharing Grandma Abbie's Rhubarb Pudding recipe on Sunday. It is so easy to make and a treat you'll love.
It's still a little chilly and gloomy outside, and spring is trying it's hardest to burst through. The other day I needed some color and decided to paint a few more of the Pretty Poppy Collection and get them on the wall.
I just love this little set on mini barn quilts all hung together. It is like a garden burst out of my wall! Each mini barn quilt is hand painted and sealed for use indoor or out. I hung mine in my mudroom, but I jeep imagining them together in a little girls room or a laundry room, breakfast nook, sunroom, etc. Of course they would look equally charming over a potting bench on a porch or shed.
As always custom colors are available upon request. Once they are hung together it makes a 38" x 38" wall "quilt". The Pretty Poppy Wall Collection is available in The Shop today and if ordered before midnight tonight...it's Free Shipping Friday!
I hope you have a fabulous holiday weekend and that you are feeling the little burst of spring wherever you are!!!
Twice a year we have Free Shipping Friday... which means that all products in Our Shop have free shipping!!! It begins at midnight MST on April 13th and runs through midnight MST on April 14th.
We have added a truck load of new barn quilts, patterns, and accessories to the shop for this sale! There are many option for colors and sizes...it is ahhmazing. So don't miss this chance to save on shipping. Have a wonderful Easter...all our love!
Next week April 21st and 22nd we will be in Oklahoma teaching our barn quilt class at the fabulous Oklahoma Quiltworks in Oklahoma City! There are three class times offered over the two days we are there...so if you are in the Oklahoma area and have wanted to take one of our classes you can sign up here.
We will be covering the history of barn quilts and barn stars, how to build, paint, finish and use these pieces of American folk art. There are still a few spots available so contact them, sign up and come by to say hi to us!
We hope to see y'all next week!
It is April and spring is here and I promised a new mini-barn quilt for the season and I think these little Pretty Poppy Barn Quilts will last well into fall with no water, weeding, or fuss!
They are a darling 12" x 12" square, hand painted and sealed for indoor or outdoor use. They also come with a convenient hanger in the back so they are ready to go.
The only trouble with these little cuties is one isn't enough. They look so cute together on my fence. They are $30.00 a piece or $95.00 for the collection of four!
I just may add a few other colors to the collection and make a nine square wall quilt with them. What do you think? I think it would be darling in a little girls room, a laundry room, hallway and of course on a barn/shed. We can't forget they are called barn quilts.
I hope you love these Pretty Poppy Barn Quilts as much as I do. They are available in The Shop today and ready to ship this week! Happy spring!!! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,493 |
Modern life can be stressful. All at once we're expected to be parents, spouses, children, professionals – the list goes on. Often, the pressure of trying to fulfil all of these roles simultaneously can leave us feeling that we have no strength or time left for ourselves. Now and then, we all need the occasional break from our daily-responsibilities for the sake of our sanity – but how can we make this happen?
Make yourself a priority – Realise that you are worthy of being added to your own list of priorities. It's easy to think catching some time for yourself isn't important, but ultimately, this time will give you the opportunity to recharge your batteries so that you can carry out all of your other roles and tasks to a higher level and with a clear head.
Schedule your 'me' time – Treat your personal time like you would do a dentist appointment or an important meeting. Add it to your diary and make a pact with yourself not to shuffle it around to suit others.
Take deep breaths – Whether you are sitting at your desk or in the car, use your 5 minutes to take slow deep breaths. Focus on the sensation of breathing in and out and follow your breath. If your mind wanders, try to bring it back.
Listen to your favourite song – Plug into your iPod and just enjoy listening to a few of your favourite tunes. Don't do anything else during this time, just sit back, close your eyes, relax and enjoy the music.
Have a cuppa – Usually when we have a cuppa, we drink it whilst working, watching TV or reading etc. Today, just try sitting with a cup of tea or coffee and peacefully sip away with no distractions.
Call a friend – Usually we ring our friends with a purpose – to arrange a lunch date or to share some news. Today, call and talk to a friend with no agenda.
If you are finding it difficult to achieve harmony in your life and you find you have very little time to yourself, a life coach could help. For further information, visit our work/life balance page. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,884 |
White Oak Florists. White Oak TX flower shops.
Dr. Seuss' "The Lorax" began pouring in.According to the post, the phenomena aren't even really flowers, but are called "wool sower gall," and they grow when wool sower wasps lay eggs in a white oak, the rangers posted.When the eggs hatch in springtime, chemicals are released that stimulate the plant to grow the fluffy ball, which gives the tiny wasp grubs food and protection. The photos show the puffy pink plant growing out of dead leaves and branches, perched on the end of a thin twig.The post describing Ranger Steve's find has received more than 1,100 likes and has been shared almost 2,000 times since Tuesday.Because of all the interest in the gall, the park decided to host a hike Friday so locals can see the natural phenomenon and learn more about them, ., wool sower galls allow the wasps to have a parasitic relationship to the oak tree, but the galls don't overtake or damage the tree. Wool sower galls have also been spotted in North Carolina, quite a distance from Texas. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,142 |
Q: array indexing with MIPS I have a bit of trouble with array indexing in MIPS. Lets say I have the following C code:
void main() {
.
.
int[2] a; # or any other length
.
.
a[1] = 7; # or any other number
.
.
}
Lets say I know that 'a' offset from the frame pointer is for example 12, so that:
lw t0, -12($fp)
Gives me the base address of 'a'.
Now lets say the array access index value (in this case 1) is stored in $t1. But I don't what it is. How can I store 7 in a[1]? I am looking for something like:
mul $t1, $t1, -4 # since each integer takes 4 bytes
addi $t1, $t1, -12 # t1 = exact offset from $fp to a[1]
li $t2, 7 # t2 = 7
sw $t2, $t1($fp)
The problem is the last operation is illegal (despite Integer array indexing with MIPS assembly). How can I do this? Thank you
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 970 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-12x-2-6x-0","text":"# How do you solve 12x^2 - 6x = 0?\n\nMar 13, 2018\n\nSee a solution process below:\n\n#### Explanation:\n\nFirst, factor an $6 x$ from each term on the left side of the equation:\n\n$\\left(6 x \\cdot 2 x\\right) - \\left(6 x \\cdot 1\\right) = 0$\n\n$6 x \\left(2 x - 1\\right) = 0$\n\nNow, solve each term on the left for $0$:\n\nSolution 1:\n\n$6 x = 0$\n\n$\\frac{6 x}{\\textcolor{red}{6}} = \\frac{0}{\\textcolor{red}{6}}$\n\n$\\frac{\\textcolor{red}{\\cancel{\\textcolor{b l a c k}{6}}} x}{\\cancel{\\textcolor{red}{6}}} = 0$\n\n$x = 0$\n\nSolution 2:\n\n$2 x - 1 = 0$\n\n$2 x - 1 + \\textcolor{red}{1} = 0 + \\textcolor{red}{1}$\n\n$2 x - 0 = 1$\n\n$2 x = 1$\n\n$\\frac{2 x}{\\textcolor{red}{2}} = \\frac{1}{\\textcolor{red}{2}}$\n\n$\\frac{\\textcolor{red}{\\cancel{\\textcolor{b l a c k}{2}}} x}{\\cancel{\\textcolor{red}{2}}} = \\frac{1}{2}$\n\n$x = \\frac{1}{2}$\n\nThe Solution Set Is: $x = \\left\\{0 , \\frac{1}{2}\\right\\}$","date":"2019-11-21 05:50:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 16, \"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\": 0.8943830132484436, \"perplexity\": 1690.6270149664895}, \"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-2019-47\/segments\/1573496670731.88\/warc\/CC-MAIN-20191121050543-20191121074543-00228.warc.gz\"}"} | null | null |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace SecureAuth.Sdk.Models
{
public class ApiVersion
{
private ApiVersion(string value) { Value = value; }
public string Value { get; private set; }
public static ApiVersion V1 { get { return new ApiVersion("v1"); } }
public static ApiVersion V2 { get { return new ApiVersion("v2"); } }
}
}
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Home » News » When Disaster Strikes Locally, Urban Networks Spread the Damage Globally
Disasters that occur in one place can trigger costs in cities across the world due to the interconnectedness of the global urban trade network. In fact, these secondary impacts can be three times greater than the local impacts, a Yale study finds.
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When cyclones and other natural disasters strike a city or town, the social and economic impacts locally can be devastating. But these events also have ripple effects that can be felt in distant cities and regions — even globally — due to the interconnectedness of the world's urban trade networks.
In fact, a new study by researchers at the Yale School of Forestry & Environmental Studies finds that local economic impacts — such as damage to factories and production facilities — can trigger secondary impacts across the city's production and trade network. For the largest storms, they report, these impacts can account for as much as three-fourths of the total damage.
"Cities are strongly connected by flows of people, of energy, and ideas — but also by the flows of trade and materials."
— Chris Shughrue, lead author
According to their findings, published in the journal Nature Sustainability, the extent of these secondary costs depends more on the structure of the production and supply networks for a particular city than on its geographic location. Regional cities that are dependent on their urban network for industrial supplies — and that have access to relatively few suppliers— are most vulnerable to these secondary impacts. Larger, global cities such as New York and Beijing, meanwhile, are more insulated from risks.
"Cities are strongly connected by flows of people, of energy, and ideas — but also by the flows of trade and materials," said Chris Shughrue '18 Ph.D., lead author of the study which is based on his dissertation work at Yale. He is now a data scientist at StreetCred Labs in New York. "These connections have implications for vulnerability, particularly as we anticipate cyclones and other natural hazards to become more intense and frequent as a result of climate change over the coming decades."
The paper was co-authored by Karen Seto, a professor of geography and urbanization science at F&ES, and B.T. Werner, a professor from the Scripps Institution of Oceanography.
"This study is especially important in the context of climate impacts on urban areas," Seto said. "Whereas we tend to consider a city's vulnerability to climate change as limited to local events, this study shows that we need to rethink this conceptualization. It shows that disasters have a domino effect through urban networks."
Using a simulation coupled with a global urban trade network model — which maps the interdependencies of cities worldwide — the researchers show how simulated disasters in one location can trigger a catastrophic domino effect.
The global spread of damage was particularly acute when cyclones occurred in cities of North America and East Asia, largely because of their outsize role in global trade networks — as purchasers and suppliers, respectively — and because these regions are particularly susceptible to cyclone events.
"To be resilient to climate change is not only about building dikes and sea walls, but understanding a city's supply chains and how they are linked to other cities that may be vulnerable."
— Karen Seto, Frederick C. Hixon Professor of Geography and Urbanization Science
Often, adverse impacts are primarily caused by a spike in material prices, followed by production losses to purchasers. These production losses eventually can cause industrial shortages, which can then induce additional cycles of price spikes and shortages throughout the production chain.
Similar outcomes have been borne out following real world disasters. For instance, when catastrophic flooding occurred in Queensland, Australia, the impact on coking coal production prompted a 25-percent spike in the global costs. And the economic impacts of Hurricane Katrina extended far beyond New Orleans for several years after the historic storm.
While the example of cyclones can act as a proxy for other isolated disasters — such as the 2011 tsunami in Japan which caused global economic disruptions, particularly in the auto sector — the researchers say the findings are particularly relevant in terms of climate-related natural events.
"To be resilient to climate change is not only about building dikes and sea walls, but understanding a city's supply chains and how they are linked to other cities that may be vulnerable," Seto said.
– Kevin Dennehy kevin.dennehy@yale.edu 203 436-4842
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©2020 Yale School Forestry & Environmental Studies
195 Prospect Street, New Haven, CT 06511 | {
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{"url":"https:\/\/stats.libretexts.org\/Courses\/Lumen_Learning\/Book%3A_Concepts_in_Statistics_(Lumen)\/06%3A_Probability_and_Probability_Distributions\/6.10%3A_Another_Look_at_Probability_(2_of_2)","text":"# 6.10: Another Look at Probability (2 of 2)\n\n\nSource: GeoGebra, license: CC BY SA\n\nNote that by clicking the GeoGebra link above you can launch a new window with this simulation in it if you would like to position it closer to the questions you\u2019ll be answering below to avoid scrolling so much.\n\n1. Make sure Coins = 1 and P(heads) = 0.2.\n2. Click the Auto button and watch the count of heads and tails change.\n3. Click the Pause (II) button once Total Flips is over 100 or so.\n4. Record the total number of Heads (1\u2019s) and the total number of flips.\n5. Calculate P(H) (Number of heads \/ Total Flips) when Total Flips is about 100.\n6. Click the Auto button again to continue the flips.\n7. Click the Pause (II) once Total Flips is over 1,000 or so.\n8. Record the total number of Heads (1\u2019s) and the total number of flips.\n9. Calculate P(H) (Number of heads \/ Total Flips) when Total Flips is about 1,000.\n\nLet\u2019s summarize what we have learned from these activities:\n\n\u2022 The empirical probability will approach the theoretical probability after a large number of repetitions. In some situations, such as in flipping an unfair coin, we cannot calculate the theoretical probability. In these cases, we have to depend on data.\n\u2022 There is less variability in a large number of repetitions. This means that in the long run, we will see a pattern, so we are more confident about estimating the probability of an event using empirical probability with a large number of repetitions.\n\n## What Do We Mean When We Say an Event Is Random or Due to Chance?\n\nIn the discussion of the role of probability in the Big Picture of Statistics, we said that probability is the machinery that allows us to draw conclusions about a population on the basis of a random sample. To understand why we can trust random selection in an observational study and random assignment in an experiment, we need to look more closely at what we mean by random or chance behavior.\n\nWhen we say that an event is random or due to chance, we mean that the event is unpredictable in the short run but has a regular and predictable behavior in the long run. This is obviously true for the coin-tossing activity. We cannot predict whether an individual toss will be heads, but in the long run, the outcomes have a predictable pattern. The relative frequency of heads is very close to 0.5 for a fair coin.\n\nWe can make probability statements only about random events.\n\n## What Is the Connection between the Coin-Flipping Activities and the Discussion of Probability in the Previous Module?\n\nLet\u2019s look at two probability questions that we might answer using the familiar data set from Relationships in Categorical Data with Intro to Probability. Recall that 6,198 of the 12,000 students at a West Coast community college are female. Previously, we calculated P(female) = 6,198 \/ 12,000 = 0.5165. What is the random event in this case? Let\u2019s be very specific about the question this calculation is meant to answer.\n\nWhat is the probability that a student at the West Coast community college is a female?\n\n\u2022 In this case, the relative frequency 6,198 \/ 12,000 is the actual proportion of females at the college. This is like the fair coin situation. Because we know the gender distribution at the college, we can think of 0.5165 as the theoretical probability that a randomly selected student at this particular college is a female. Tossing the fair coin in the simulation is like randomly selecting a student from the spreadsheet of data. We do not know if a randomly selected student will be female. But if we repeat this process many, many times, in the long run, the relative frequency of females will have a predictable pattern. The relative frequency will be very close to the proportion of females in the data set.\n\nWhat is the probability that a community college student in the United States is female?\n\n\u2022 In this case, we are using the data from the 12,000 West Coast community college students to represent students at all community colleges in the United States. The relative frequency is an estimate for the chance that a randomly selected U.S. student is female. This is like tossing the unfair coin 12,000 times and using the relative frequency of heads as an estimate of P(head). We do not know P(female) for all community colleges, just as we did not know the P(heads) with an unfair coin. But if the sample is random, we can use the relative frequency of females in the sample as an estimate of P(female) in all community colleges.\n\nThe main points are these:\n\n\u2022 We can make probability statements only about random events.\n\u2022 Probability of an event A is the relative frequency with which that event occurs in a long series of repetitions.\n\n6.10: Another Look at Probability (2 of 2) is shared under a not declared license and was authored, remixed, and\/or curated by LibreTexts.","date":"2022-12-03 13:22:45","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\": 0.6754891872406006, \"perplexity\": 407.73700840706493}, \"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-2022-49\/segments\/1669446710931.81\/warc\/CC-MAIN-20221203111902-20221203141902-00349.warc.gz\"}"} | null | null |
(1065) Amundsenia es un asteroide perteneciente al grupo de los asteroides que cruzan la órbita de Marte descubierto el 4 de agosto de 1926 por Serguéi Ivánovich Beliavski desde el observatorio de Simeiz en Crimea.
Está nombrado en honor de Roald Amundsen (1872-1928), explorador noruego del siglo XX.
Véase también
Lista de asteroides del (1001) al (1100)
Referencias
Enlaces externos
Asteroides que cruzan la órbita de Marte
Objetos astronómicos descubiertos por Serguéi Beliavski
Objetos astronómicos descubiertos desde el Observatorio de Simeiz
Objetos astronómicos descubiertos en 1926
Wikiproyecto:Asteroides/Artículos de asteroides | {
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Q: Calculate the remainder of $11^{2020} / 7$ I tried to get a pattern by doing
$11^1 / 7 = 1 , r=4$
$11^2/7 = 17, r=2$
$11^3 / 7 = 190, r=1$
but the numbers keep getting larger and larger and I think this is not the way to go about this problem. Can someone please explain the correct way on how to deal with these problems?
A: $$11^3\equiv 1 \pmod{7}$$
$$11^{2020}\equiv 11^{3\cdot 673}\cdot 11 \equiv (1)^{673}\cdot {11}\equiv 4 \pmod 7$$
| {
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Our Homes Are Health Delivery Platforms – The New Home Health/Care at CES 2021
By Jane Sarasohn-Kahn on 18 January 2021 in Aging, Aging and Technology, Baby health, Big data and health, Bio/life sciences, Bioethics, Boomers, Broadband, Business and health, Connected health, Consumer electronics, Consumer experience, Consumer-directed health, Coronavirus, COVID-19, Data analytics and health, Demographics and health, Dental care, Design and health, Diagnostics, Digital health, Digital therapeutics, Electronic medical records, Exercise, Fitness, Food and health, GDPR, Grocery stores, Guns and health, Health and Beauty, Health at home, Health care marketing, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health ecosystem, Health engagement, Health equity, Health marketing, Health media, Health Plans, Health policy, Health politics, Health privacy, Health regulation, Healthcare access, Healthcare DIY, Heart disease, Heart health, HIPAA, Home care, Hospitals, Housing and health, Internet and Health, Internet of things, Medical device, Medical innovation, Nutrition, Obesity, Oral care, Patient engagement, Patient experience, Pharmacy, Physicians, Popular culture and health, Prevention and wellness, Primary care, Privacy and security, Public health, Remote health monitoring, Retail health, Robots and health, Safety net and health, SDoH, Self-care, Sensors and health, Sleep, Smartwatches, Social determinants of health, Social responsibility, Sustainability, Telehealth, Telemedicine, Transparency, Trust, Virtual health, Wearable tech, Wearables, Wellbeing
The coronavirus pandemic disrupted and re-shaped the annual CES across so many respects — the meeting of thousands making up the global consumer tech community "met" virtually, both keynote and education sessions were pre-recorded, and the lovely serendipity of learning and meeting new concepts and contacts wasn't so straightforward. But for those of us working with and innovating solutions for health and health care, #CES2021 was baked with health goodness, in and beyond "digital health" categories. In my consumer-facing health care work, I've adopted the mantra that our homes are our health hubs. Reflecting on my many conversations during CES
Trust Plummets Around the World: The 2021 Edelman Trust Barometer in #CES2021 and Microsoft Context
By Jane Sarasohn-Kahn on 14 January 2021 in Big data and health, Bio/life sciences, Bioethics, Business and health, Cognitive computing and health, Computers and health, Connected health, Consumer electronics, Coronavirus, COVID-19, Data analytics and health, Design and health, Digital health, Global Health, Health and safety, Health care industry, Health care marketing, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health education, Health engagement, Health equity, Health literacy, Health marketing, Health media, Health policy, Health politics, Health privacy, HIPAA, Internet and Health, Internet of things, media and health, Medical technology, Patient engagement, Popular culture and health, Privacy and security, Public health, Risk management, Sensors and health, Social media and health, Social networks and health, Social responsibility, Social security, Stress, Telehealth, Transparency, Trust, Virtual health, Wellbeing
Citizens around the world unite around the concept that Trust is Dead. This is no truer than in the U.S., where trust in every type of organization and expert has plummeted in the wake of the COVID-19 pandemic, political and social strife, and an economic downturn. Welcome to the sobering 2021 Edelman Trust Barometer, released this week as the world's technology innovators and analysts are convening at CES 2021, and the annual JP Morgan Healthcare meetup virtually convened. As the World Economic Forum succinctly put the situation, "2020 was the year of two equally destructive viruses: the pandemic and the
The Digital Consumer, Increasingly Connected to Health Devices; Parks Associates Kicking Off #CES2021
By Jane Sarasohn-Kahn on 11 January 2021 in Broadband, Computers and health, Connected health, Consumer electronics, Consumer experience, Coronavirus, COVID-19, Cybersecurity, Design and health, Digital health, Financial health, Financial wellness, Health apps, Health at home, Health care marketing, Health citizenship, Health Consumers, Health ecosystem, Health engagement, Health equity, Health literacy, Health media, Health policy, Health privacy, Healthcare DIY, Heart disease, High deductibles, HIPAA, Home care, Hospitals, Internet and Health, Internet Demographics, Internet of things, Medical technology, Mobile health, Money and health, Patient engagement, Patient experience, Popular culture and health, Privacy and security, Public health, Race and health, Remote health monitoring, Retirement and health, SDoH, Self-care, Sensors and health, Smart homes, Smartphones, Smartwatches, Social determinants of health, Social networks and health, Social responsibility, Telehealth, Telemedicine, Transparency, User experience UX, Virtual health, Voice technology, Wearable tech, Wearables
In 2020, the COVID-19 pandemic drove U.S. consumers to increase spending on electronics, notably laptops, smartphones, and desktop computers. But the coronavirus era also saw broadband households spending more on connecting health devices, with 42% of U.S. consumers owning digital health tech compared with 33% in 2015, according to research discussed in Supporting Today's Connected Consumer from Parks Associates. developed for Sutherland, the digital transformation company. Consumer electronics purchase growth was, "likely driven by new social distancing guidelines brought on by COVID-19, which requires many individuals to work and attend school from home. Among the 26% of US broadband households
"The virus is the boss" — U.S. lives and livelihoods at the beginning of 2021
By Jane Sarasohn-Kahn on 8 January 2021 in Children's health, Coronavirus, COVID-19, Death, Demographics and health, Financial health, Financial wellness, Health and wealth, Health at home, Health citizenship, Health Consumers, Health Economics, Health ecosystem, Health equity, Health policy, Love and health, Moms and health, Money and health, Nurses, Public health, Race and health, SDoH, Self-care, Seniors and health, Social determinants of health, Vaccines, Wellbeing, Women and health
"The virus is the boss," Austan Goolsbee, former Chair of the Council of Economic Advisers under President Obama, told Stephanie Ruhle this morning on MSNBC. Goolsbee and Jason Furman, former Chair of Obama's Council of Economic Advisers, tag-teamed the U.S. economic outlook following today's news that the U.S. economy lost 140,000 jobs — the greatest job loss since April 2020 in the second month of the pandemic. The 2020-21 economic recession is the first time in U.S. history that a downturn had nothing to do with the economy per se. As Uwe Reinhardt, health economist guru, is whispering in my
The 2021 Shkreli Awards: Lown Institute Counts Down the Top 10 Healthcare Industry Abuses in the Coronavirus Pandemic
By Jane Sarasohn-Kahn on 6 January 2021 in Bio/life sciences, Bioethics, Business and health, Coronavirus, Corporate responsibility, COVID-19, Global Health, Health care industry, Health citizenship, Health Consumers, Health costs, Health Economics, Health ecosystem, Health equity, Health policy, Health politics, Health Quality, HIV/AIDS, Medicines, Money and health, Mortality, Pharmaceutical, Popular culture and health, Prescription drugs, Public health, Social responsibility, Transparency, Trust
The first year of the coronavirus pandemic in America was a kind of stress test on the U.S. health care system, revealing weak links and opportunities for bad behavior. "These are not just about individual instances or bad apples," Dr. Vikas Saini, President of The Lown Institute, explained, referring to them as "cautionary tales" of the current state of U.S. health care. Dr. Saini and his colleague Shannon Brownlee released the annual Lown Institute 2020 Shkreli Awards this week, highlighting their ten most egregious examples of the worst events in U.S. health care that happened in the past year —
Nurses, doctors, pharmacists join with teachers in Gallup's 2021 honest and ethics poll
By Jane Sarasohn-Kahn on 30 December 2020 in Business and health, Coronavirus, Corporate responsibility, COVID-19, Doctors, Education and health, Employers, Health at home, Health benefits, Health care industry, Health care marketing, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health education, Health equity, Health media, Health policy, Health social networks, Hospitals, Infectious disease, Love and health, Medical innovation, Money and health, Nurses, Patient engagement, Patient experience, Transparency
Each year, Americans rank nurses as the most honest and ethical professionals along, generally followed by doctors and pharmacists. In the middle of the coronavirus pandemic in the U.S., grade school teachers join the three medical professions in the annual Gallup Poll on the top-ranked professions for honest and ethical behavior in America as we enter 2021 with many U.S. hospitals' intensive care units at full capacity….and schools largely emptied of students. The three health care professions scored their highest marks ever achieved in this Gallup Poll, which has been assessing honesty and ethics in America since 1999. Nurses are
The Comforts of Home Drive Demand for Healthcare There
By Jane Sarasohn-Kahn on 1 December 2020 in Aging and Technology, Broadband, Connected health, Consumer electronics, Consumer experience, Coronavirus, COVID-19, Demographics and health, Design and health, Digital health, Doctors, Employers, Grocery stores, Health apps, Health at home, Health citizenship, Health Consumers, Health ecosystem, Health engagement, Health Plans, Health policy, Health privacy, Health regulation, Healthcare access, Home care, Hospitals, Housing and health, Internet and Health, Internet of things, medical home, Mental health, Nutrition, Prevention and wellness, Primary care, Public health, Rehabilitation, Remote health monitoring, Retail health, Robots and health, SDoH, Self-care, Seniors and health, Sensors and health, Smart homes, Smartphones, Smartwatches, Social determinants of health, Social isolation, Telehealth, Telemedicine, Value based health, Virtual health, Voice technology, Wearable tech
Two in three U.S. consumers skipped or delayed getting in-person medical care in 2020. One in 2 people had a telehealth visit int he last year. Most would use virtual care again. The coronavirus pandemic has mind-shifted how patients envision a health care visit. Today, most consumers prefer the idea of getting health care at home compared with going to a doctor's office. Most Americans also like the idea of recovering at home instead of at a medical facility after a major medical event, according to the report, Health-at-Home 2020: The New Standard of Care Delivery from CareCentrix. COVID-19 has
Vaccine Hesitancy Is Greatest Among Those at Highest Risk of Dying from COVID-19: Black People
By Jane Sarasohn-Kahn on 24 November 2020 in Behavior change, Bioethics, Business and health, Children's health, Consumer experience, Coronavirus, COVID-19, Demographics and health, Design and health, Education and health, FDA, Global Health, Health and safety, Health at home, Health care industry, Health care marketing, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health education, Health engagement, Health equity, Health literacy, Health marketing, Health media, Health policy, Health politics, Kids' health, Moms and health, Nurses, Patient safety, Popular culture and health, Prevention, Prevention and wellness, Public health, Race and health, Safety net and health, Schools and health, Self-care, Social determinants of health, Transparency, Trust, Vaccines
While 85% of people are open to receiving a COVID-19 vaccine, over one-half of them would want to wait some time to observe if after-effects developed in people who took the jab, according to a new study from Acxiom, the data analytics-marketing company. Not all people are as enthused about getting a coronavirus vaccine at all, Acxiom discovered: in fact, those hardest hit by the virus — Black people — would be the least-likely to want to get a COVID-19 vaccine, discussed in in Vaccine Hesitancy in the U.S., a survey the company conducted among 10,000 people in the U.S.
The COVID Healthcare Consumer – 5 Trends Via The Medecision Liberation Blog
By Jane Sarasohn-Kahn on 23 November 2020 in Anxiety, Behavioral health, Broadband, Connected health, Coronavirus, COVID-19, Depression, Financial health, Financial wellness, GDPR, Health at home, Health care industry, Health citizenship, Health Consumers, Health costs, Health disparities, Health equity, Health insurance, Health policy, Health politics, Health privacy, Healthcare access, Internet and Health, Jobs and health, Loneliness, Love and health, Mental health, Money and health, Mortality, Prevention, Prevention and wellness, Public health, Race and health, Retail health, SDoH, Self-care, Seniors and health, Social determinants of health, Social isolation, Stress, Telehealth, Telemedicine, Trust, Universal health care, Virtual health
The first six months into the coronavirus pandemic shocked the collective system of U.S. consumers for living, learning, laboring, and loving. I absorbed all kinds of data about consumers in the wake of COVID-19 between March and mid-August 2020, culminating in my book, Health Citizenship: How a virus opened hearts and minds, published in September on Kindle and in print in October. In this little primer, I covered the five trends I woven based on all that data-immersion, following up the question I asked at the end of my previous book, HealthConsuming: when and how would Americans claim their health
Rebuilding Resilience, Trust, and Health – Deloitte's Latest on Health Care and Sustainability
By Jane Sarasohn-Kahn on 11 November 2020 in Amazon, Bio/life sciences, Business and health, Coronavirus, Corporate responsibility, Corporate wellness, COVID-19, Cybersecurity, Design and health, Digital health, Financial wellness, Global Health, Health and safety, Health at home, Health care industry, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health equity, Health policy, Health social networks, Hospital finance, Hospitals, Mental health, Money and health, Public health, Race and health, SDoH, Self-care, Social determinants of health, Social networks and health, Social responsibility, Supply chain, Telehealth, Telemedicine, Transparency, Vaccines, Virtual health
The COVID-19 pandemic has accelerated health care providers' and plans' investment in digital technologies while reducing capital spending on new physical assets, we learn in Building resilience during the COVID-19 pandemic and beyond from the Deloitte Center for Health Solutions. What must be built (or truly re-built), health care leaders believe, is first and foremost trust, followed by financial viability to ensure long-term resilience and sustainability — for the workforce, the organization, the community, and leaders themselves. For this report, Deloitte interviewed 60 health care chief financial officers to gauge their perspectives during the pandemic looking at the future of
Will We See A Field of Dreams for the COVID-19 Vaccine in the U.S.?
By Jane Sarasohn-Kahn on 10 November 2020 in Business and health, Consumer experience, Coronavirus, COVID-19, Doctors, Health and safety, Health care industry, Health care marketing, Health citizenship, Health Consumers, Health ecosystem, Health education, Health engagement, Health literacy, Health marketing, Health media, Health policy, Health politics, Health social networks, Infectious disease, Moms and health, Patient engagement, Patient safety, Pediatrics, Pharmaceutical, Physicians, Popular culture and health, Prevention, Prevention and wellness, Public health, Race and health, Retail health, Self-care, Social media and health, Social networks and health, Trust, Vaccines
"If you build it, he will come," the voice of James Earl Jones echoes in our minds when we recall the plotline of the film, Field of Dreams. A quick summary if you don't know the movie: the "it" was a baseball field to be built in a rural cornfield. The "he" was a baseball player, ultimately joined by a dream-team of ball players who would convene on that dreamy field to play an amazing game. Today, the day after Pfizer announced a 90% benefit for its coronavirus vaccine, bolstering Wall Street returns on 9th November 2020, two new consumer
Voting for Health in 2020
By Jane Sarasohn-Kahn on 2 November 2020 in ACA, Anxiety, Behavioral economics, Big data and health, Coronavirus, COVID-19, Data analytics and health, Design and health, Financial health, Financial wellness, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health engagement, Health law, Health policy, Health politics, Health privacy, Health reform, Health social networks, Love and health, Medicines, Mental health, Moms and health, Nurses, Personal health finance, Pharmaceutical, Physicians, Prescription drugs, Prevention, Public health, Race and health, Retail health, Safety net and health, Shopping and health, Social determinants of health, Social networks and health, Women and health
In the 2018 mid-term elections, U.S. voters were driven to polls with health care on their minds. The key issues for health care voters were costs (for care and prescription drugs) and access (read: protecting pre-existing conditions and expanding Medicaid). Issue #2 for 2018 voters was the economy. In 2020, as voting commences in-person tomorrow on 3rd November, U.S. voters have lives and livelihoods on their minds. It's the pandemic – our physical lives looming largest in the polls – coupled with our fiscal and financial lives. Health is translating across all definitions for U.S. voters in November 2020: for
Masks Work. A Picture From Kansas That Tells A Story in Two Words.
By Jane Sarasohn-Kahn on 28 October 2020 in Behavior change, Coronavirus, COVID-19, Design and health, Health and safety, Health citizenship, Health education, Health engagement, Health law, Health policy, Health politics, Health social networks, Infectious disease, Kids' health, Love and health, Popular culture and health, Prevention, Public health, Self-care, Social responsibility, Wearables
It is said that a picture tells a thousand words. This picture tells an even quicker story that can save lives: "Masks work." The backstory: Kansas Governor Laura Kelly issued a mask mandate on July 2, 2020. The rationale: That was two days before Independence Day, the holiday weekend when she and state public health officials anticipated health citizens would abandon their personal efforts to physically distance and cover faces to avoid contracting or spreading the coronavirus. This was the message directly communicated to U.S. residents by the White House Coronavirus Task Force that week before Independence Day. The backlash:
Health Care Providers Are More Politically Engaged in the 2020 Elections
By Jane Sarasohn-Kahn on 27 October 2020 in Consumer experience, Demographics and health, Doctors, Emergency care, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health engagement, Health equity, Health insurance, Health policy, Health politics, Nurses, Physicians, SDoH, Social determinants of health
Until 2016, physicians' voting rates in U.S. elections had not changed since the late 1970s. Then in 2018, two years into President Donald Trump's four-year term, the mid-term elections drove U.S. voters to the election polls…including health care providers. Based on the volume and intensity of medical professional societies' editorials on the 2020 Presidential Election, we may be in an inflection-curve moment for greater clinician engagement in politics as doctors and nurses take claim of their health citizenship. A good current example of this is an essay published in the AMA's website asserting, "Why it's okay for doctors to ask
Stress in America, Like COVID-19, Impacts All Americans
By Jane Sarasohn-Kahn on 21 October 2020 in Anxiety, Behavioral health, Boomers, Coronavirus, COVID-19, Deaths of despair, Demographics and health, Depression, Financial health, Financial wellness, Health and wealth, Health citizenship, Health costs, Health ecosystem, Health equity, Health policy, Health politics, Health reform, Health social networks, Infectious disease, Kids' health, Loneliness, Love and health, Mental health, Money and health, Opioids, Stress
With thirteen days to go until the U.S. #2020Elections day, 3rd November, three in four Americans say the future of America is a significant source of stress, according to the latest Stress in America 2020 study from the American Psychological Association. Furthermore, seven in 10 U.S. adults believe that "now" is the lowest point in the nation's history that they can remember. "We are facing a national mental health crisis that could yield serious health and social consequences for years to come," APA introduces their latest read into stressed-out America. Two in three people in the U.S. say that the
Women's Health Policy Advice for the Next Occupant of the White House: Deal With Mental Health, the Pandemic, and Health Care Costs
By Jane Sarasohn-Kahn on 20 October 2020 in Anxiety, Baby health, Behavioral health, Caregivers, Children's health, Coronavirus, COVID-19, Depression, Family, Fertility, Financial health, Financial wellness, Health and wealth, Health at home, Health Consumers, Health costs, Health disparities, Health finance, Health insurance, Health policy, Kids' health, Loneliness, medical home, Mental health, Moms and health, Money and health, Popular culture and health, Prescription drugs, Prevention and wellness, Public health, Race and health, SDoH, Self-care, Social determinants of health, Stress, Telehealth, Universal health care, Wellbeing, Women and health
2020 marked the centennial anniversary of the 19th Amendment to the U.S. Constitution, giving women the right to vote. In this auspicious year for women's voting rights, as COVID-19 emerged in the U.S. in February, women's labor force participation rate was 58%. Ironic timing indeed: the coronavirus pandemic has been especially harmful to working women's lives, the Brookings Institution asserted last week in their report in 19A: The Brookings Gender Equality Series. A new study from Tia, the women's health services platform, looks deeply into COVID-19's negative impacts on working-age women and how they would advise the next occupant of
Black Health Should Matter More in America: The Undefeated Survey on Race and Health
By Jane Sarasohn-Kahn on 15 October 2020 in Anxiety, Behavioral health, Coronavirus, Corporate responsibility, COVID-19, Demographics and health, Depression, Financial health, Financial wellness, Food and health, Health care industry, Health citizenship, Health disparities, Health ecosystem, Health equity, Health policy, Health reform, Mental health, Money and health, Public health, Race and health, SDoH, Social determinants of health
In 2020, most Black people, men and women alike, feel it is a bad time to be Black in America. More than twice as many Black men believed that in 2020 compared with 2006. More than four times as many Black women believed that it's a bad time to be Black in America in 2020 versus 2011, we learn in The Undefeated Survey on Race and Health from Kaiser Family Foundation (KFF). KFF collaborated with The Undefeated, ESPN's project that focuses on sports, race, and culture. The Undefeated program was started in May 2016, and has become a thought leader
Two in Three Americans Cite the U.S. Presidential Election is a Significant Source of Stress, Even More than the 2016 Race
By Jane Sarasohn-Kahn on 13 October 2020 in ACA, Anxiety, Chronic disease, Coronavirus, COVID-19, Demographics and health, Health citizenship, Health Consumers, Health law, Health policy, Health politics, Health reform, Health social networks, Stress
If you perceive you are more stressed in the COVID-19 pandemic than you were in 2019, you are one in many millions feeling so. If you felt stressed during the 2016 Presidential election season, you were also one in a million. Now, as the 2020 Election converges with the coronavirus crisis, even more Americans are feeling significant stress in a double-whammy impact. Our friends at the American Psychological Association have assessed Stress in America for many years of studies. The latest, published 7 October, finds that the 2020 Presidential election is a source of significant stress for more Americans than
In the Past Ten Years, Workers' Health Insurance Premiums Have Grown Much Faster Than Wages
By Jane Sarasohn-Kahn on 9 October 2020 in ACA, Affordable Care Act, Anxiety, Behavioral health, Business and health, Coronavirus, COVID-19, Employee benefits, Employers, Financial health, Financial toxicity, Financial wellness, Health and wealth, Health benefits, Health Consumers, Health costs, Health insurance, Health Plans, Health policy, High deductibles, High-Deductible Health Plans, Mental health, Money and health, Uninsured, Workplace benefits, Workplace wellness
For a worker in the U.S. who benefits from health insurance at the workplace, the annual family premium will average $21,342 this year, according to the 2020 Employer Health Benefits Survey from the Kaiser Family Foundation. The first chart illustrates the growth of the premium shares split by employer and employee contributions. Over ten years, the premium dollars grew from $13,770 in 2010 to $21K in 2020. The worker's contribution share was 29% in 2010, and 26% in 2020. Single coverage reached $7,470 in 2020 and was $5,049 in 2010. Roughly the same proportion of companies offered health benefits to
Financial Health Is On Americans' Minds Just Weeks Before the 2020 Elections
By Jane Sarasohn-Kahn on 8 October 2020 in ACA, Anxiety, Banks and health, Coronavirus, COVID-19, Demographics and health, Depression, Financial toxicity, Financial wellness, Health benefits, Health citizenship, Health Consumers, Health costs, Health Economics, Health insurance, Health policy, Health politics, Mental health
Financial health is part of peoples' overall health. As Americans approach November 3, 2020, the day of the real-time U.S. Presidential and down-ballot elections, personal home economics are front-of-mind. Twenty-seven days before the 2020 elections, 7 in 10 Americans say their financial health will influence their votes this year, according to the doxoINSIGHTS survey which shows personal financial health as a key voter consideration in the Presidential election. Doxo, a consumer payments company, conducted a survey among 1,568 U.S. bill-paying households in late September 2020. The study has a 2% margin of error. U.S. voters facing this year's election are
Our Home Is Our Health Hub: CTA and CHI Align to Address Digital and Health Equity
By Jane Sarasohn-Kahn on 1 October 2020 in Broadband, Business and health, Connected health, Consumer electronics, Coronavirus, Corporate responsibility, COVID-19, Data analytics and health, Deaths of despair, Digital health, Digital therapeutics, Health and wealth, Health apps, Health at home, Health care industry, Health citizenship, Health Consumers, Health disparities, Health equity, Health policy, Housing and health, Medical innovation, Money and health, Public health, Race and health, Smartphones, Social determinants of health, Social responsibility, Telehealth, Virtual health, Wearable tech
In the pandemic, I've been weaving together data to better understand how people as consumers are being re-shaped in daily life across their Maslow Hierarchies of Needs. One of those basic needs has been digital connectivity. People of color have faced many disparities in the wake of the pandemic: the virus itself, exacting greater rates of mortality and morbidity being the most obvious, dramatic inequity. Another has been digital inequity. Black people have had a more difficult time paying for phone and Internet connections during the COVID-19 crisis, we learned in a Morning Consult poll fielded in June 2020. In
The Emergence of Health Citizens – Part of Liberating Health, on the Medecision Blog
By Jane Sarasohn-Kahn on 30 September 2020 in ACA, Coronavirus, COVID-19, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health equity, Health policy, Health politics, Love and health
I have always appreciated Medecision's mission as expressed in the company's tagline: "Liberating," as in "liberate healthcare." I began collaborating with Medecision as a client several years ago, a couple of years after I heard Todd Park, then Chief Technology Officer in the White House, joyfully assert the phrase, "Data Liberación!" at an early Health 2.0 Conference. Medecision published my blog about health citizenship this week on the company's Liberate.Health site. The emergence of health citizens in the U.S. — people re-claiming their health, health care, and control of data — is part of liberating health in America. Please check
Redefining PPE As Primary Care, Public Health, and Health Equity – The Community PPE Index
By Jane Sarasohn-Kahn on 29 September 2020 in Coronavirus, COVID-19, Data analytics and health, Demographics and health, Design and health, Doctors, End of life care, Financial wellness, Food and health, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health equity, Health policy, Health social networks, Healthcare access, Housing and health, Infectious disease, Love and health, Mental health, Money and health, Nutrition, Patient engagement, Physicians, Pre-existing conditions, Prevention, Prevention and wellness, Primary care, Public health, Race and health, Safety net and health, SDoH, Social determinants of health, Social networks and health, Social responsibility, Wellbeing
In May 2020, the Oxford English Dictionary (OED) re-visited the acronym, "PPE." As OED evolves the definition of PPE, the wordsmiths could borrow from OSHA's website, noting that PPE, "is equipment worn to minimize exposure to hazards that cause serious workplace injuries and illnesses. These injuries and illnesses may result from contact with chemical, radiological, physical, electrical, mechanical, or other workplace hazards. Personal protective equipment may include items such as gloves, safety glasses and shoes, earplugs or muffs, hard hats, respirators, or coveralls, vests and full body suits." Perhaps Definition 3 in the OED could be updated by a blog
Health Citizenship in the America. If Not Now, When?
By Jane Sarasohn-Kahn on 18 September 2020 in Anxiety, Behavioral health, Broadband, Connected health, Coronavirus, Corporate responsibility, COVID-19, Demographics and health, Depression, Digital health, Financial health, Financial wellness, Food and health, Grocery stores, Health at home, Health care industry, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health ecosystem, Health equity, Health policy, Health politics, Health reform, Health social networks, Internet and Health, Jobs and health, Love and health, medical home, Mental health, Money and health, Popular culture and health, Primary care, Public health, Race and health, Retail health, SDoH, Self-care, Shopping and health, Smart homes, Social determinants of health, Social isolation, Social networks and health, Social responsibility, Stress, Telehealth, Trust, Virtual health, Wellbeing, Women and health, Workplace benefits
On February 4th, 2020, in a hospital in northern California, the first known inpatient diagnosed with COVID-19 died. On March 11th, the World Health Organization called the growing prevalence of the coronavirus a "pandemic." On May 25th, George Floyd, a 46-year-old Black man, died at the hands of police in Minneapolis. This summer, the Dixie Chicks dropped the "Dixie" from their name, and NASCAR cancelled the confederate flag from their tracks. Today, nearly 200,000 Americans have died due to the novel coronavirus. My new book, Health Citizenship: How a virus opened hearts and minds, launched this week. In it, I
Only in America: The Loss of Health Insurance as a Toxic Financial Side Effect of the COVID-19 Pandemic
By Jane Sarasohn-Kahn on 17 September 2020 in ACA, Affordable Care Act, Business and health, Coronavirus, Corporate wellness, COVID-19, Demographics and health, Employee benefits, Employers, Financial health, Financial toxicity, Financial wellness, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health equity, Health finance, Health insurance, Health insurance marketplaces, Health Plans, Health policy, Health politics, Health reform, Health social networks, Money and health, Mortality, Prevention, Prevention and wellness, Primary care, Public health, Race and health, SDoH, Social determinants of health, Social responsibility, Social security, Trust, Universal health care, Wellbeing, Workplace benefits
In terms of income, U.S. households entered 2020 in the best financial shape they'd been in years, based on new Census data released earlier this week. However, the U.S. Census Bureau found that the level of health insurance enrollment fell by 1 million people in 2019, with about 30 million Americans not covered by health insurance. In fact, the number of uninsured Americans rose by 2 million people in 2018, and by 1.9 million people in 2017. The coronavirus pandemic has only exacerbated the erosion of the health insured population. What havoc a pandemic can do to minds, bodies, souls, and wallets. By September 2020,
50 Days Before the U.S. Elections, Voters Say Health Care Costs and Access Top Their Health Concerns — More than COVID-19
By Jane Sarasohn-Kahn on 14 September 2020 in ACA, Coronavirus, COVID-19, Financial health, Health benefits, Health citizenship, Health Consumers, Health costs, Health Economics, Health Plans, Health policy, Health politics, Money and health, Prevention, Public health, Social responsibility, Social security, Uninsured, Workplace benefits
The coronavirus pandemic has revealed deep cracks and inequities in U.S. health care in terms of exposure to COVID-19 and subsequent outcomes, with access to medical care and mortality rates negatively impacting people of color to a greater extent than White Americans. The pandemic has also led to economic decline that, seven weeks before the 2020 elections in America, is top-of-mind for health citizens with the virus-crisis itself receding to second place, according to the Kaiser Family Foundation September 2020 Health Tracking Poll. KFF polled 1,199 U.S. adults 18 years of age and older between August 28 and September 3,
Behavioral Health Side-Effects in the COVID Era
By Jane Sarasohn-Kahn on 9 September 2020 in Anxiety, Behavioral health, Coronavirus, COVID-19, Deaths of despair, Demographics and health, Depression, Digital therapeutics, Health at home, Health citizenship, Health Consumers, Health costs, Health disparities, Health ecosystem, Health equity, Health policy, Health politics, Health social networks, Loneliness, Mental health, Money and health, Popular culture and health, Population health, Pre-existing conditions, Prevention and wellness, Public health, Race and health, Remote health monitoring, Risk management, Safety net and health, Self-care, Sleep, Social determinants of health, Social isolation, Social networks and health, Stress, Suicide, Telehealth, Value based health, Virtual health, Wellbeing, Women and health
"This surge of people experiencing acute behavioral health problems…has the potential to further impact the healthcare system for years to come," a report from McKinsey expects looking at the hidden costs of COVID-19's impact on U.S. health care. The coronavirus pandemic has taken a toll on Americans' mental health, with anxiety and depression growing as a side-effect to worries about the virus itself, the long Great Lockdown in much of the country, and the economic recession that has particularly impacted women and people of color. I covered depression impacts due to COVID-19 here in Health Populi yesterday, and wanted to
The Burden of Depression in the Pandemic – Greater Among People With Fewer Resources
By Jane Sarasohn-Kahn on 8 September 2020 in Anxiety, Caregivers, Coronavirus, COVID-19, Demographics and health, Depression, Employee benefits, Financial health, Financial toxicity, Financial wellness, Health and wealth, Health at home, Health benefits, Health Consumers, Health costs, Health disparities, Health Economics, Health ecosystem, Health insurance, Health policy, Health social networks, Jobs and health, Loneliness, Love and health, Mental health, Moms and health, Money and health, Prevention, Prevention and wellness, Public health, SDoH, Self-care, Sex and health, Social determinants of health, Social isolation, Social networks and health, Social security, Wellbeing, Wellness, Women and health, Workplace benefits
In the U.S., symptoms of depression were three-times greater in April 2020 in the COVID-19 pandemic than in 2017-2018. And rates for depression were even higher among women versus men, along with people earning lower incomes, losing jobs, and having fewer "social resources" — that is, at greater risk of isolation and loneliness. America's health system should be prepared to deal with a "probable increase" in mental illness after the pandemic, researchers recommend in Prevalence of Depression Symptoms in US Adults Before and During the COVID-19 Pandemic in JAMA Network Open. A multidisciplinary team knowledgeable in medicine, epidemiology, public health,
The Next-Normal Health Care Consumer After the Pandemic
By Jane Sarasohn-Kahn on 3 September 2020 in Coronavirus, COVID-19, FDA, Health citizenship, Health Consumers, Health policy, Health politics, Public health, Vaccines
McKinsey invites us to Meet the next-normal consumer in a recently-published research report describing changing consumer behavior responding to the COVID-19 lockdown and aftermath. The report gives us insights into the next-normal health consumer, which I'll discuss in today's post. Note the massive digital shift every person living in a country touched by the coronavirus has experienced, illustrated in the first graphic from the report. Tele-work, tele-education, ecommerce, and streaming entertainment all grew so fast within a matter of a few weeks. And telemedicine, McKinsey points out, was adopted at a rate of ten times growth over 15 days, the
Data Well-Being: A Pillar of Health Citizenship for US Consumers
By Jane Sarasohn-Kahn on 28 August 2020 in AI, Artificial intelligence, Big data and health, Bioethics, Cognitive computing and health, Computers and health, Connected health, Coronavirus, COVID-19, Data analytics and health, Demographics and health, Design and health, Digital health, GDPR, Health care information technology, Health citizenship, Health Consumers, Health IT, Health policy, Health politics, Health privacy, HIPAA, Privacy and security, Public health, Trust, Wellbeing
In the COVID-19 era, most U.S. consumers believe they have an obligation to share personal health information to stop the spread of the coronavirus. However, only 44% would be willing to share their personal data with a national database, a MITRE study learned. Only one-third of Americans would be willing to share their temperature, 29% their location, and one-fourth information about their chronic conditions. The Harris Poll conducted the study among 2,065 U.S. adults 18 and over in mid-June 2020 to gauge peoples' perspectives on health data and privacy. Three-quarters of people in the U.S. believe that data privacy "is a
My ABCovid-19 Journal – Day 4 of 5, Letters "P" through "T"
By Jane Sarasohn-Kahn on 13 August 2020 in Anxiety, Consumer experience, Coronavirus, COVID-19, Demographics and health, Depression, Design and health, Global Health, Grocery stores, Health at home, Health citizenship, Health Consumers, Health ecosystem, Health engagement, Health policy, Health politics, Housing and health, Loneliness, Mental health, Money and health, Patient experience, Popular culture and health, Prevention, Prevention and wellness, Public health, Retail health, Schools and health, Shopping and health, Social determinants of health, Social isolation, Stress, Wellbeing, Wellness
While I'm on holiday this week, restoring and re-setting, I've been sharing pages from my ABCovid-19 Journal with readers of Health Populi. I created this journal during the early phase of the pandemic in the U.S., as a form of art therapy, creative outlet, and learning. Today is Day 4 of sharing: we consider the letters "P" through "T," and what I saw in the early coronavirus era. P is for pandemic This "P" was self-evidence in our collective early COVID-19 lexicon. The "P" word was uttered by the Secretary General of the World Health Organization on March 11, confirming
My ABCovid-19 Journal – Day 3 of 5, Letters "K" through "O"
By Jane Sarasohn-Kahn on 12 August 2020 in Anxiety, Art and health, Coronavirus, COVID-19, Death, Demographics and health, Doctors, Global Health, Health Consumers, Health ecosystem, Health literacy, Health policy, Health politics, Hospitals, Infectious disease, Mental health, Mortality, Nurses, Patient safety, Public health, Self-care, Social isolation
Welcome back to my ABCovid-19 Journal, which I created/curated in the early weeks of the coronavirus pandemic. This week, I'm sharing all the letters of the alphabet with you which reminded me keywords and themes emerging as we were learning about this dastardly public health threat beginning early in 2020. In today's Health Populi blog I bring you letters "K" through "O," continuing through the rest of the alphabet tomorrow and Friday while I'm on a lake-side holiday that's good for mind, body, and spirit. K is for Kirkland, Washington state In the U.S., one of the earliest hotspots for
The She-Cession – a Financially Toxic Side-Effect of the Coronavirus Pandemic
By Jane Sarasohn-Kahn on 4 August 2020 in Business and health, Caregivers, Coronavirus, COVID-19, Demographics and health, Family, Health at home, Health benefits, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health policy, Home care, Jobs and health, Kids' health, Moms and health, Money and health, Public health, SDoH, Social determinants of health, Women and health, Workplace benefits, Workplace wellness
Along with the life-threatening impact of the coronavirus on physical health, and the accompanying mental health distress activated by self-distancing comes a third unintended consequence with the pandemic: a hard hit on women's personal economies. The recession of the pandemic is considered by many economists as a "She-Cession," a downturn in the economy that's negatively impacting women more acutely than men. This is markedly different than the Great Recession of 2008, the last major financial crisis: that financial decline was coined a "ManCession," taking a more significant toll out of more typically men's jobs like construction and manufacturing where fewer
Return-To-School Is Stressing Out U.S. Parents Across Income, Race and Political Party
By Jane Sarasohn-Kahn on 24 July 2020 in Children's health, Coronavirus, COVID-19, Demographics and health, Education and health, Financial health, Financial wellness, Food and health, Health at home, Health citizenship, Health Consumers, Health disparities, Health education, Health equity, Health policy, Health politics, Healthcare access, Housing and health, Kids' health, Moms and health, Personal health finance, Public health, Safety net and health, Schools and health, Social determinants of health, Social responsibility, Women and health
The worse of the coronavirus pandemic is yet to come, most Americans felt in July 2020. That foreboding feeling is shaping U.S. parents' concerns about their children returning to school, with the calendar just weeks away from educators opening their classrooms to students, from kindergarten to the oldest cohort entering senior year of high school. The Kaiser Family Foundation's July 2020 Health Tracking Poll focuses on the COVID-19 pandemic, return-to-school, and the governments' response. KFF polled 1,313 U.S. adults 18 and older between July 14 to 19, 2020. The first line chart illustrates Americans' growing concerns about the coronavirus, shifting
More Americans Pivot to Distancing and Mask-Wearing in the Hot Summer of 2020
By Jane Sarasohn-Kahn on 22 July 2020 in Coronavirus, COVID-19, Demographics and health, Health care marketing, Health citizenship, Health Consumers, Health disparities, Health education, Health engagement, Health literacy, Health marketing, Health policy, Health politics, Health social networks, Infectious disease, Popular culture and health, Prevention, Public health, Self-care, Social determinants of health, Social responsibility, Wearable tech
With growing coronavirus case hotspots in southern and western states, more Americans perceive the pandemic is worsening this summer, shown by a Gallup poll published 20 July 2020. Gallup titles the analysis, Americans' social distancing steady as pandemic worsens. The first table organizes Gallup's data by demographics, illustrating a significant gap between how women perceive the exacerbating pandemic compared with men. In early June, roughly one-third of both men and women saw COVID-19 was getting "worse"; five weeks later, in the second week of July, men and women's perceptions were 12 points apart with more women concerned about the situation
Most Virtual Care Consumers, Satisfied With Visits in the COVID Era, Expect It for Future Care
By Jane Sarasohn-Kahn on 10 July 2020 in Behavioral health, Chronic care, Chronic disease, Connected health, Consumer experience, Coronavirus, COVID-19, Digital health, Health citizenship, Health Consumers, Health insurance, Health policy, Medicare, Mental health, Patient experience, Primary care, Telehealth, Telemedicine, Virtual health
Within days of the coronavirus pandemic emerging in the U.S., health care providers set up virtual care arrangements to convene with patients. Three months into the COVID-19 crisis, how have patients felt about these telehealth visits? In Patient Perspectives on Virtual Care, Kyruus answers this question based on an online survey of 1,000 patients 18 years of age and older, conducted in May 2020. Each of these health consumers had at least one virtual care visit between February and May 2020. The key findings were that: Engaging in a virtual visit was a new-new thing for 72% of people Patients'
Beyond Care and Outcomes, Hospitals Must Deliver on Civics, Inclusivity, Equity, and Value – Lown Institute's Best Hospitals
By Jane Sarasohn-Kahn on 8 July 2020 in Consumer experience, Coronavirus, COVID-19, Demographics and health, Design and health, Health disparities, Health ecosystem, Health equity, Health policy, Hospital finance, Hospitals, Social determinants of health
The core business of hospitals is patient care, often baked with teaching and research. But wait — there's more, asserts the Lown Institute in their approach to ranking America's Best Hospitals in 2020. The Institute's methodology for assessing what's "best" addresses ten pillars. Several of these are the "stick-to-the-knitting" components of the Webster Dictionary definition of hospital work: patient outcomes, clinical outcomes, avoiding overuse, patient safety, and a recent focus, patient satisfaction. Community benefit has been part of a hospital's life, especially in the not-for-profit world where hospitals must demonstrate goodwill generated for and provided in the neighborhoods in which
Stressed Out By COVID and Civil Unrest – the APA's Stress in America Survey, Part 2
By Jane Sarasohn-Kahn on 30 June 2020 in Anxiety, Consumer experience, Coronavirus, COVID-19, Depression, Health citizenship, Health Consumers, Health equity, Health policy, Health politics, Mental health, Stress
"Now" is the lowest point in history that most Americans can remember: 7 in 10 people in the U.S. feel this way, up from 56% in 2018 and 2019. Furthermore, 4 in 5 people in the U.S. say the future of America is a significant source of stress, as discussed in Stress in the Time of COVID-19, Volume Two, a report covering a poll of U.S. adults sponsored by the American Psychological Association. APA's Stress in America research has been one of my annual go-to's for better understanding U.S. residents through the lens of health consumers and, especially this year
Health Insurance and Demand for Masking, Testing and Contact Tracing – New Data from The Commonwealth Fund
By Jane Sarasohn-Kahn on 25 June 2020 in Coronavirus, COVID-19, Health insurance, Health Plans, Health policy, Health politics, Medicaid, Social determinants of health, Uninsured
The coronavirus pandemic occasioned the Great Lockdown for people to shelter-at-home, tele-work if possible, and shut down large parts of the U.S. economy considered "non-essential." As health insurance for working-age people is tied to employment, COVID-19 led to disproportionate loss of health plan coverage especially among people earning lower incomes, as well as non-white workers, explained in the Commonwealth Fund Health Care Poll: COVID-19, May-June 2020. The Commonwealth Fund commissioned interviews with 2,271 U.S. adults 18 and over between 13 May and 2nd June 2020 for this study. The survey has two lenses: first, on health insurance coverage among working
Juneteenth 2020: Inequality and Injustice in Health Care in America
By Jane Sarasohn-Kahn on 19 June 2020 in Deaths of despair, Demographics and health, Education and health, Financial wellness, Health and safety, Health and wealth, Health citizenship, Health Consumers, Health disparities, Health Economics, Health equity, Health policy, Health politics, Health reform, Housing and health, Jobs and health, Love and health, Money and health, Mortality, Nutrition, Popular culture and health, Public health, Safety net and health, SDoH, Social determinants of health, Social networks and health, Trust, Wellbeing
"Of all the forms of inequality, injustice in health is the most shocking and the most inhuman because it often results in physical death," Martin Luther King, Jr., asserted at the second meeting of the Medical Committee for Human Rights in Chicago on March 25, 1966. This quote has been shortened over the five+ decades since Dr. King told this truth, to the short-hand, "Of all the forms of inequality, injustice in health care is the most shocking and inhumane." Professor Charlene Galarneau recently enlightened me on Dr. King's original statement in her seminal essay, "Getting King's Words Right." Among
Americans Across Political Party Worry About Prescription Drug Prices – Especially to Deal with COVID-19
By Jane Sarasohn-Kahn on 18 June 2020 in Bio/life sciences, Financial toxicity, Health Consumers, Health costs, Health policy, Health politics, media and health, Medicaid, Medicare, Medicines, Pharmaceutical, Pharmacy, Prescription drugs, Retail health, Specialty drugs, Transparency, Women and health
Nine in ten Americans is concerned about the price of prescription drugs in the wake of the coronavirus pandemic, Gallup and West Health found in their survey on the cost of healthcare, published today. A majority of people across political party share this concern: overall, 88% of U.S. adults are concerned about rising drug prices in response to COVID19, split across party ID with: 94% of Democrats, 86% of Independents, and 84% of Republicans. By demographics, more women than men are concerned about rising costs for the three health care spending categories the survey studied: drug prices, insurance premiums, and
Economic Anxieties Rise, Medical and Vacation Plans Delayed: the COVID-19 Consumer in June 2020
By Jane Sarasohn-Kahn on 16 June 2020 in Broadband, Connected health, Consumer-directed health, Coronavirus, COVID-19, Digital health, Health at home, Health care industry, Health Consumers, Health costs, Health ecosystem, Health policy, Health politics, Healthcare access, Internet Demographics, Money and health, Retail health, Self-care, Social determinants of health, Telehealth, Telemedicine
Some 6 in 10 people in the U.S. have been financially impacted by COVID-19. Those most negatively affected by the pandemic tend to be younger, Gen Z age group and African-American, 63% of whom felt financial pressure directly due from the virus and the national economic lockdown. By late May 2020, 34% of black Americans had lost their jobs compared with 21% in late April, compared with 18% of white consumers, reported in The COVID-19 Pandemic's Financial Impact on U.S. Consumers, survey research from TransUnion. This post describes data from TransUnion's Wave 9 report, which polled 2,086 U.S. adults 18
Americans' Concerns About the US Healthcare System Loom Larger Than Worries About Their Own Care
By Jane Sarasohn-Kahn on 9 June 2020 in Bio/life sciences, Business and health, Coronavirus, COVID-19, Education and health, Health benefits, Health care industry, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health insurance, Health Plans, Health policy, Health politics, media and health, Medical innovation, Medicines, Money and health, Personal health finance, Pharmaceutical, Pre-existing conditions, Prescription drugs, Public health, SDoH, Shared decision making, Social determinants of health, Universal health care, Wellbeing
The coronavirus pandemic has further opened the kimono of the U.S. healthcare system to Americans: four months into the COVID-19 outbreak, most consumers (62%) of people in the U.S. are more concerned about other people not having access to high quality health care versus themselves. This is a 16 point increase in concern in May 2020 compared with the response to the same question asked in February in a poll conducted by the University of Chicago Harris School of Public Policy and The Associated Press-NORC Center for Public Affairs Research (the AP-NORC Center). The AP-NORC Poll found more of this
Addressing Health Equity Must Include Digital Equity Beyond Access To Medical Services and Insurance
By Jane Sarasohn-Kahn on 5 June 2020 in Broadband, Computers and health, Connected health, Consumer experience, Coronavirus, COVID-19, Demographics and health, Design and health, Digital health, Health apps, Health at home, Health care information technology, Health citizenship, Health Consumers, Health disparities, Health engagement, Health equity, Health IT, Health policy, Medical technology, Mobile health, Patient engagement, Patient experience, Public health, Remote health monitoring, Safety net and health, SDoH, Self-care, Smartphones, Social determinants of health, Telehealth, Telemedicine, Trust, Virtual health
The 21st Century Cures Act emphasizes patients' control of personal health information. ONC rules issues in March 2020 called for more patient-facing health tools and apps to bolster health consumer engagement and empowerment. But the emergence of the coronavirus in the U.S. revealed many weakness in the American health care system, one of which has been health inequities faced by millions of people — especially black Americans, who have sustained higher rates morbidity and mortality for COVID-19. There have also been digital health divides found in the COVID-19 pandemic, discussed in a timely essay in JAMA, Digital Health Equity as
Stress in America – COVID-19 Takes Toll on Finances, Education, Basic Needs and Parenting
By Jane Sarasohn-Kahn on 21 May 2020 in Anxiety, Behavioral health, Coronavirus, COVID-19, Employers, Financial health, Financial wellness, Health disparities, Health Economics, Health policy, Health politics, Housing and health, Mental health, Money and health, Prevention and wellness, Schools and health, SDoH, Social determinants of health, Social isolation, Stress, Vaccines, Wellbeing, Wellness
"The COVID-19 pandemic has altered every aspect of American life, from health and work to education and exercise," the new Stress in America 2020 study from the American Psychological Association begins. The APA summarizes the impact of these mass changes on the nation: "The negative mental health effects of the coronavirus may be as serious as the physical health implications," with COVID-19 stressors hitting all health citizens in the U.S. in different ways. Beyond the risk of contracting the virus, the Great Lockdown of the U.S. economy has stressed the U.S. worker and the national economy, with 7 in 10
Health Care In the COVID-19 Era – PwC Finds Self-Rationing of Care and Meds Especially for Chronic Care
By Jane Sarasohn-Kahn on 20 May 2020 in Chronic care, Chronic disease, Coronavirus, COVID-19, Diabetes, Digital health, Health at home, Health benefits, Health care industry, Health Consumers, Health engagement, Health finance, Health insurance, Health Plans, Health policy, Heart disease, Home care, Hospital finance, Hospitals, medical home, Medicare, Medication adherence, Medicines, Mental health, Pharmaceutical, Pharmacy, Prescription drugs, Prevention and wellness, Primary care, Self-care, Telehealth, Telemedicine, Virtual health
Patients in the U.S. are self-rationing care in the era of COVID-19 by cutting spending on health care visits and prescription drugs. The coronavirus pandemic's impact on health consumers' spending varies depending on whether the household is generally a healthy family unit, healthy "enthusiasts," dealing with a simple or more complex chronic conditions, or managing mental health issues. PwC explored how COVID-19 is influencing consumers' health care behaviors in survey research conducted in early April by the Health Research Institute. The findings were published in a May 2020 report, detailing study findings among 2,533 U.S. adults polled in early April
How COVID-19 Is Driving More Deaths of Despair
By Jane Sarasohn-Kahn on 11 May 2020 in Behavioral health, Broadband, Connected health, Coronavirus, COVID-19, Deaths of despair, Financial toxicity, Financial wellness, Health and wealth, Health disparities, Health policy, Jobs and health, Loneliness, Mental health, Money and health, Mortality, Primary care, Risk management, SDoH, Social determinants of health, Social networks and health, Suicide, Wellbeing
In the current state of the COVID-19 pandemic, we all feel like we are living in desperate times. If you are a person at-risk of dying a Death of Despair, you're even more at-risk of doing so in the wake of the Coronavirus in America. Demonstrating this sad fact of U.S. life, the Well Being Trust and Robert Graham Center published Projected Deaths of Despair from COVID-19. The analysis quantifies the impact of isolation and loneliness combined with the dramatic economic downturn and mass unemployment with the worsening of mental illness and income inequity on the epidemic of Deaths of
Health, Wealth & COVID-19 – My Conversation with Jeanne Pinder & Carium, in Charts
By Jane Sarasohn-Kahn on 23 April 2020 in Anxiety, Banks and health, Business and health, Consumer-directed health, Coronavirus, COVID-19, Employee benefits, Financial health, Financial toxicity, Financial wellness, Health and wealth, Health benefits, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health equity, Health insurance, Health policy, Health politics, High deductibles, Hospital finance, Money and health
The coronavirus pandemic is dramatically impacting and re-shaping our health and wealth, simultaneously. Today, I'll be brainstorming this convergence in a "collaborative health conversation" hosted by Carium's Health IRL series. Here's a link to the event. Jeanne founded ClearHealthCosts nearly ten years ago, having worked as a journalist with the New York Times and other media. She began to build a network of other journalists, each a node in a network to crowdsource readers'-patients' medical bills in local markets. Jeanne started in the NYC metro and expanded, one node at a time and through many sources of funding from not-for-profits/foundations,
Americans' Sense of Well-Being Falls to Great Recession Levels, Gallup Finds
By Jane Sarasohn-Kahn on 14 April 2020 in Coronavirus, COVID-19, Demographics and health, Depression, Financial health, Financial wellness, Global Health, Health Consumers, Health costs, Health Economics, Health equity, Health finance, Health policy, Health politics, Mental health, Population health, Prevention and wellness, Public health, Stress, Wellbeing, Wellness
It's déjà vu all over again for Americans' well-being: we haven't felt this low since the advent of the Great Recession that hit our well-well-being hard in December 2008. As COVID-19 diagnoses reached 200,000 in the U.S. in April 2020, Gallup gauged that barely 1 in 2 people felt they were thriving. In the past 12 years, the percent of Americans feeling they were thriving hit a peak in 2018, as the life evaluations line graph illustrates. Gallup polled over 20,000 U.S. adults in late March into early April 2020 to explore Americans' self-evaluations of their well-being. FYI, Gallup asks consumers
The Coronavirus Impact on American Life, Part 2 – Our Mental Health
By Jane Sarasohn-Kahn on 7 April 2020 in Addiction, Anxiety, Behavioral health, Coronavirus, COVID-19, Deaths of despair, Demographics and health, Depression, Digital therapeutics, Exercise, Financial health, Financial wellness, Guns and health, Happiness, Health benefits, Health citizenship, Health Consumers, Health costs, Health disparities, Health ecosystem, Health education, Health equity, Health law, Health literacy, Health policy, Health politics, Health social networks, Heart disease, Heart health, Infectious disease, Integrative medicine, Kids' health, Loneliness, Love and health, medical home, Meditation, Mental health, Mindfulness, Moms and health, Money and health, Mortality, Opioids, Pain, Patient engagement, Pets and health, Prevention, Public health, Retail health, Schools and health, Self-care, Sleep, Social determinants of health, Social isolation, Social networks and health, Suicide, Telehealth, Trust, Violence and health, Voice technology, Wearable tech, Wearables, Wellbeing, Women and health
As the coronavirus pandemic's curve of infected Americans ratchets up in the U.S., people are seeking comfort from listening to Dolly Parton's bedtime stories, crushing on Dr. Anthony Fauci's science-wrapped-with-empathy, and streaming the Tiger King on Netflix. These and other self-care tactics are taking hold in the U.S. as most people are "social distancing" or sheltering in place, based on numbers from the early April 2020 Kaiser Family Foundation health tracking poll on the impact of the coronavirus on American life. While the collective practice of #StayHome to #FlattenTheCurve is the best-practice advice from the science leaders at CDC, the NIAID
The Coronavirus Impact on American Life, Part 1 – Life Disrupted, and Money Concerns
By Jane Sarasohn-Kahn on 6 April 2020 in Behavior change, Coronavirus, COVID-19, Demographics and health, Employers, Financial health, Financial wellness, Health and wealth, Health Consumers, Health Economics, Health ecosystem, Health policy, Health politics, Infectious disease, Mental health, Money and health, Personal health finance, Public health, Wellbeing
Nearly 3 in 4 Americans see their lives disrupted by the coronavirus pandemic, according to the early April Kaiser Family Foundation Health Tracking Poll. This feeling holds true across most demographic factors: among both parents and people without children; men and women alike; white folks as well as people of color (although fewer people identifying as Hispanic, still a majority). There are partisan differences, however, in terms of who perceives a life-disruption due to COVID-19: 76% of Democrats believe this, 72% of Independents, and 70% of Republicans. Interestingly, only 30% of Republicans felt this way in March 2020, more than
Honor Your Doctor – It's National Doctors Day Today (and EveryDay)
By Jane Sarasohn-Kahn on 30 March 2020 in Coronavirus, COVID-19, Depression, Doctors, Health care industry, Health Economics, Health ecosystem, Health policy, Hospitals, Medical education, Medical school, Physicians
Today, March 30, is National Doctors Day. We honor doctors annually on this day. But every day, we must honor physicians for bolstering the health and wellness of our fellow Americans, our beloved families and friends, and our selves. The Coronavirus Pandemic reminds us of the precious and scarce resource that is our national supply of physicians in America — numbering about 750,000 active clinicians in the U.S. according to the U.S. Bureau of Labor Statistics. Even before the COVID-19 crisis, physicians in America had been feeling increasingly burned out and depressed. The 2020 WebMD survey on the state of
West Virginia Was the Last State to ID a COVID-19 Positive Patient; The States' Residents Are At Highest Risk for Severe Reaction to C19
By Jane Sarasohn-Kahn on 27 March 2020 in Aging, Broadband, Cancer, Coronavirus, COVID-19, Death, Demographics and health, Diabetes, Health policy, Health politics, Heart disease, Heart health, Prevention, Public health, Social determinants of health, Wellbeing
Gallup has estimated 11 Million in U.S. at Severe Risk If Infected With COVID-19 in research published today. And the health citizens of West Virginia would be at greatest risk for having a severe reaction to the coronavirus. A "severe reaction" here means being critically ill or dying. The forecast doesn't focus on the whole number of people in the US. who would be at-risk of contracting the coronavirus; the 11 million is the total number of Americans who have a "very high chance of becoming critically ill or dying" if 100% of the country were infected with C19. This
Wistful Thinking: The National Health Spending Forecast In a Land Without COVID-19
By Jane Sarasohn-Kahn on 26 March 2020 in Coronavirus, COVID-19, Health care industry, Health costs, Health Economics, Health ecosystem, Health insurance, Health policy, Health politics, High deductibles, Hospitals, Infectious disease, Loneliness, Medicines, Money and health, Public health, Universal health care
U.S. health care spending will grow to 20% of the national economy by 2028, forecasted in projections pre-published in the April 2020 issue of Health Affairs, National Health Expenditure (NHE) Projections. 2019-28: Expected Rebound in Prices Drives Rising Spending Growth. NHE will grow 5.4% in the decade, the model expects. But…what a difference a pandemic could make on this forecast. This year, NHE will be $3.8 trillion, growing to $6.2 trillion in 2028. Hospital care spending, the largest single component in national health spending, is estimated at $1.3 trillion in 2020. These projections are based on "current law," the team
In the US COVID-19 Pandemic, A Tension Between the Fiscal and the Physical
By Jane Sarasohn-Kahn on 24 March 2020 in Bioethics, Coronavirus, COVID-19, Financial health, Financial toxicity, Health citizenship, Health Consumers, Health costs, Health Economics, Health equity, Health policy, Health politics, High deductibles, Money and health, Mortality, Public health
"Act fast and do whatever it takes," insists the second half of the title of a new eBook with contributions from forty leading economists from around the world. The first half of the title is, Mitigating the COVID Economic Crisis. The book is discussed in a World Economic Forum essay discussing the economists' consensus to "act fast." As the U.S. curve adds new American patients testing positive for the coronavirus, the book and essay illustrate the tension between health consumer versus the health citizen in the U.S. For clinical context, as I write this post on 24th March 2020, today's U.S.
In A Nation "At War" with the C19 Virus, Partisan Healthcare Differences Persist
By Jane Sarasohn-Kahn on 23 March 2020 in Anxiety, Coronavirus, COVID-19, Education and health, FDA, Global Health, Health benefits, Health care industry, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health engagement, Health policy, Health politics, Infectious disease, Mental health, Mortality, Prevention, Public health
More Democrats would want to get tested for the coronavirus (C19) than would Republicans. And, more women than men believe that a vaccine to address the COVID-19 pandemic believe that treatment would be offered at no-or-low-cost under a Democratic president versus President Trump. These are two key insights gleaned from a look into U.S. adults' perspectives on the C19 virus in the second week of March 2020. What Are Americans' Views on the Coronavirus Pandemic? asks and answers an NBC News/Commonwealth Fund Health Care Poll published on 20th March 2020. NBC News and the Commonwealth Fund polled 1,006 people 18
The COVID19 Consumer: #AloneTogether and More Health Aware
By Jane Sarasohn-Kahn on 20 March 2020 in Anxiety, Connected health, Coronavirus, COVID-19, Financial health, Food and health, Grocery stores, Health Consumers, Health ecosystem, Health literacy, Health policy, Health politics, Health privacy, Jobs and health, media and health, Mental health, Money and health, Popular culture and health, Prevention and wellness, Public health, Risk management, Self-care, Social networks and health, Trust, Wellbeing, Wellness
The number of diagnoses of people testing positive with the coronavirus topped 14,000 today in the U.S., Johns Hopkins COVID-19 interactive map told us this morning. As tests have begun to come on stream from California on the west coast to New York state on the east, the U.S. COVID-19 positives will continue to ratchet up for weeks to come, based on the latest perspectives shared by the most-trusted expert in America, Dr. Anthony Fauci. This report from the U.S. Department of Health and Human Services on the nation's response to the coronavirus pandemic, published March 13, 2020, forecasts a
Estimates of COVID-19 Medical Costs in the US: $20K for inpatient stay, $1300 OOP costs
By Jane Sarasohn-Kahn on 18 March 2020 in Banks and health, Broadband, CDHPs, Coronavirus, COVID-19, Employee benefits, Employers, Financial health, Financial wellness, Health benefits, Health citizenship, Health Consumers, Health costs, Health finance, Health insurance, Health law, Health Plans, Health policy, Health politics, Health reform, Hospital finance, Infectious disease, Money and health, Personal health finance, Public health, Seniors and health, Social determinants of health, Telehealth, Telemedicine, Universal health care, Workplace benefits
In the midst of growing inpatient admissions and test results for COVID-19, Congress is working as I write this post to finalize a round of legislation to help Americans with the costs-of-living and (hopefully) health care in a national, mandated, clarifying way. Right now in the real world, real patients are already being treated for COVID-19 in American hospitals. Patients are facing health care costs that may result in multi-thousand dollar bills at discharge (or death) that will decimate households' financial health, particularly among people who don't have health insurance coverage, covered by skinny or under-benefited plans, and/or lack banked
Lockdown Economics for U.S. Health Consumers
By Jane Sarasohn-Kahn on 17 March 2020 in Anxiety, Behavioral health, Connected health, Coronavirus, COVID-19, Depression, Grocery stores, Health at home, Health benefits, Health care industry, Health citizenship, Health Consumers, Health costs, Health Economics, Health insurance, Health Plans, Health policy, Health politics, Health social networks, Hospital finance, Hospitals, Infectious disease, Loneliness, Mental health, Personal health finance, Prevention, Public health, Self-care, Shopping and health, Social determinants of health, Social isolation, Social networks and health, Universal health care
The hashtag #StayHome was ushered onto Twitter by 15 U.S. national healthcare leaders in a USA Today editorial yesterday. The op-ed co-authors included Dr. Eric Topol, Dr. Leana Wen, Dr. Zeke Emanuel, Dr. Jordan Shlain, Dr. Vivek Murthy, Andy Slavitt, and other key healthcare opinion leaders. Some states and regions have already mandated that people stay home; at midnight last night, counties in the Bay Area in California instituted this, and there are tightening rules in my area of greater Philadelphia. UBS economist Paul Donovan talked about "Lockdown Economics" in his audio commentary today. Paul's observations resonated with me as
Telehealth and COVID-19 in the U.S.: A Conversation with Ann Mond Johnson, ATA CEO
By Jane Sarasohn-Kahn on 13 March 2020 in Business and health, Chronic disease, Computers and health, Connected health, Consumer electronics, Coronavirus, COVID-19, Data analytics and health, Digital health, Digital therapeutics, Global Health, Health apps, Health care industry, Health Consumers, Health costs, Health ecosystem, Health insurance, Health Plans, Health policy, Health politics, Healthcare access, Hospitals, Medical technology, Mental health, Mobile health, Patient experience, Primary care, Remote health monitoring, Retail health, Rural health, Self-care, Smartphones, Telehealth, Telemedicine, Virtual health, Voice technology
Will the coronavirus inspire greater adoption of telehealth in the U.S.? Let's travel to Shanghai, China where, "the covid-19 epidemic has brought millions of new patients online. They are likely to stay there," asserts "The smartphone will see you now," an article in the March 7th 2020 issue of The Economist. The article returns to the advent of the SARS epidemic in China in 2003, which ushered in a series of events: people stayed home, and Chinese social media and e-commerce proliferated. The coronavirus spawned another kind of gift to China and the nation's health citizens: telemedicine, the essay explains. A
Waking Up a Health Consumer in the COVID-19 Era
By Jane Sarasohn-Kahn on 12 March 2020 in Business and health, Consumer-directed health, Coronavirus, COVID-19, Digital health, Financial health, Financial wellness, Global Health, Health at home, Health benefits, Health care industry, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health engagement, Health policy, Health politics, Health social networks, High deductibles, Home care, Hospitals, Infectious disease, Money and health, Patient engagement, Popular culture and health, Prevention and wellness, Retail health, Self-care, Shopping and health, Social networks and health, Supply chain, Telehealth, Telemedicine, Virtual health, Wellbeing
With President Trump's somber speech from the Oval Office last night, we wake up on 12th March 2020 to a ban on most travel from Europe to the U.S., recommendations for hygiene, and call to come together in America. His remarks focused largely on an immigration and travel policy versus science, triaging, testing and treatment of the virus itself. Here is a link to the President's full remarks from the White House website, presented at about 9 pm on 11 March 2020. Over the past week, I've culled several studies and resources to divine a profile of the U.S. consumer
The Book on Deaths of Despair – Deaton & Case On Education, Pain, Work and the Future of Capitalism
By Jane Sarasohn-Kahn on 10 March 2020 in Addiction, Anxiety, Deaths of despair, Demographics and health, Depression, Education and health, Financial health, Financial wellness, Food and health, Happiness, Health citizenship, Health Consumers, Health disparities, Health Economics, Health equity, Health insurance, Health policy, Health politics, Healthcare access, Loneliness, Mental health, Money and health, Mortality, Pain, Popular culture and health, Prevention, Public health, Rural health, Safety net and health, Self-care, Social determinants of health, Social isolation, Social networks and health, Suicide, Wellbeing
Anne Case and Angus Deaton were working in a cabin in Montana the summer of 2014. Upon analyzing mortality data from the U.S. Centers for Disease Control, they noticed that death rates were rising among middle-aged white people. "We must have hit a wrong key," they note in the introduction of their book, Deaths of Despair and the Future of Capitalism. This reversal of life span in America ran counter to a decades-long trend of lower mortality in the U.S., a 20th century accomplishment, Case and Deaton recount. In the 300 pages that follow, the researchers deeply dive into and
"How's Life?" for American Women? The New OECD Report Reveals Financial Gaps on International Women's Day 2020
By Jane Sarasohn-Kahn on 9 March 2020 in Affordable Care Act, Aging, Business and health, Demographics and health, Financial health, Financial wellness, Health and wealth, Health Consumers, Health costs, Health disparities, Health equity, Health finance, Health policy, Money and health, Women and health
March 8 is International Women's Day. In the U.S., there remain significant disparities between men and women, in particular related to financial well-being. The first chart comes from the new OECD "How's Life?" report published today (March 9th) measuring well-being around the country members of the OECD. This chart focuses on women versus men in the United States based on over a dozen key indicators. Top-line, many fewer women feel safe in America, and earnings in dollars and hours worked fall short of men's incomes. This translates into lower socioeconomic status for women, which diminishes overall health and well-being for
Most Americans Concerned About Coronavirus Impact on Economy & Families, and Not a "Hoax"
By Jane Sarasohn-Kahn on 5 March 2020 in ACA, Affordable Care Act, Anxiety, Health benefits, Health citizenship, Health Consumers, Health literacy, Health policy, Health politics, Infectious disease, media and health, Popular culture and health, Prevention and wellness, Public health, Self-care, Social media and health, Telehealth, Trust, Vaccines
Seven in 10 Americans are concerned about the coronavirus outbreak's impact on the economy, and 6 in 10 people worried about someone they love getting sick from COVID-19. But most Americans also get the politicized nature of the coronavirus and say they're less likely to vote for President Trump in November based on his handling of the public health threat, according to a just-released survey from Protect Our Care fielded by Public Policy Polling. Some of the data points which demonstrate that Americans are taking the emerging coronavirus pandemic quite seriously are that: 53% disagree that President Trump and his
How Coronavirus Is Re-Shaping Consumer Behavior, From the Amusement Park to the Voting Booth
By Jane Sarasohn-Kahn on 4 March 2020 in Amazon, Behavior change, Broadband, Connected health, Consumer-directed health, Digital health, Grocery stores, Health at home, Health Consumers, Health costs, Health policy, Health politics, Home care, Internet and Health, Mobile health, Money and health, Patient engagement, Self-care, Shopping and health, Social determinants of health, Telehealth, Telemedicine
The coronavirus has shaken U.S. consumer confidence, both in terms of financial markets and personal health risks. COVID-19 is re-shaping peoples' behavior and daily choices, from using public transit to choosing where to shop, based on Morning Consult's National Tracking Poll #200276 conducted February 28-March 1, 2020. Morning Consult surveyed 2,200 U.S. adults, finding that 3 in 4 Americans were concerned about the coronavirus outbreak. The first chart from the survey shows various consumer activities by peoples' likelihood of choosing to do them. Clearly, our daily life-flows outside of our homes have been impacted by our perceived risks of the coronavirus:
Job #1 for Next President: Reduce Health Care Costs – Commonwealth Fund & NBC News Poll
By Jane Sarasohn-Kahn on 2 March 2020 in Aging, Demographics and health, Financial health, Financial toxicity, Financial wellness, Health and wealth, Health Consumers, Health costs, Health Economics, Health finance, Health insurance, Health policy, Health politics, Money and health, Retirement and health, Seniors and health
Four in five U.S. adults say lowering the cost of health care in America should be high priority for the next American president, according to a poll from The Commonwealth Fund and NBC News. Health care costs continue to be a top issue on American voters' minds in this 2020 Presidential election year, this survey confirms. The first chart illustrates that lowering health care costs is a priority that crosses political parties. This is true for all flavors of health care costs, including health insurance deductibles and premiums, out-of-pocket costs for prescription drugs, and the cost of long-term care. While
The High Cost-of-Thriving and the Evolving Social Contract for Health Care
By Jane Sarasohn-Kahn on 28 February 2020 in Aging, Business and health, Deaths of despair, Demographics and health, Employers, Financial health, Financial wellness, Global Health, Health and wealth, Health citizenship, Health Consumers, Health costs, Health Economics, Health ecosystem, Health equity, Health finance, Health insurance, Health policy, Health politics, Health reform, Money and health, Personal health finance, Public health, Retirement and health, Seniors and health, Social determinants of health, Suicide, Wellbeing
Millions of Americans have to work 53 weeks to cover a year's worth of household expenses. Most Americans haven't saved much for their retirement. Furthermore, the bullish macroeconomic outlook for the U.S. in early 2020 hasn't translated into individual American's optimism for their own family budgets. (Sidebar and caveat: yesterday was the fourth day in a row of the U.S. financial markets losing as much as 10% of market cap, so the global economic outlook is being revised downward by the likes of Goldman Sachs, Vanguard, and Morningstar, among other financial market prognosticators. MarketWatch called this week the worst market
Americans' Top 2 Priorities for President Trump and Congress Are To Lower Health Care and Rx Costs
By Jane Sarasohn-Kahn on 20 February 2020 in ACA, Affordable Care Act, Consumer experience, Financial health, Financial toxicity, Financial wellness, Health citizenship, Health Consumers, Health costs, Health Economics, Health education, Health finance, Health literacy, Health policy, Health politics, Health reform, High deductibles, media and health, Money and health, Personal health finance, Pharmaceutical, Prescription drugs
Health care pocketbook issues rank first and second place for Americans in these months leading up to the 2020 Presidential election, according to research from POLITICO and the Harvard Chan School of Public Health published on 19th February 2020. This poll underscores that whether Democrat or Republican, these are the top two domestic priorities among Americans above all other issues polled including immigration, trade agreements, infrastructure and regulations. The point that Robert Blendon, Harvard's long-time health care pollster, notes is that, "Even among Democrats, the top issues…(are) not the big system reform debates…They're worried about their own lives, their own
The Ill Health of Rural Hospitals in Four Charts
By Jane Sarasohn-Kahn on 18 February 2020 in ACA, Business and health, Digital health, Financial health, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health policy, Health politics, Hospital finance, Hospitals, Medicaid, Primary care, Retail health, Rural health, Social determinants of health, Telehealth
There are 1,844 rural hospitals operating in the U.S. That number is down by 19 in the 2019 calendar year, the worst year of rural hospital closings seen in the past decade. That hockey-stick growth of closures is shown in the first chart, where 34 rural hospitals shut down in the past 2 years. Rural U.S. hospitals are in poor fiscal health. "The accelerated rate at which rural hospitals are closing continues to unsettle the rural healthcare community and demands a more nuanced investigation into rural hospital performance," threatening the stability of the rural health safety net, according to the
The Federal Reserve Chairman Speaks Out on Health Care Costs: "Spending But Getting Nothing"
By Jane Sarasohn-Kahn on 17 February 2020 in Anxiety, Banks and health, Financial health, Financial toxicity, Financial wellness, Health and wealth, Health Consumers, Health costs, Health Economics, Health finance, Health policy, Health politics, Health reform, Money and health, Personal health finance
On February 12, 2020, the Chairman of the Federal Reserve Bank of the U.S. submitted the Semiannual Monetary Policy Report to Congress and testified to the Senate Banking Committee. Chairman Jerome Powell detailed the current state of the economy, discussing the state of the macroeconomy, GDP growth, unemployment, inflation, and projections for 2022 and beyond. The top line data points are shown in the first chart. After his prepared remarks, Chairman Powell responded to questions from members of the Senate Banking Committee. Senator Ben Sasse (R-Neb.) asked him about health care costs' impact on the national U.S. economy. The Chairman
Health Care Costs Concern Americans Approaching Retirement – Especially Women and Sicker People
By Jane Sarasohn-Kahn on 10 February 2020 in Aging, Baby Boomers and Health, Boomers, Demographics and health, Financial health, Financial wellness, Health and wealth, Health citizenship, Health costs, Health disparities, Health Economics, Health finance, Health insurance, Health policy, Health politics, Medication adherence, Medicines, Money and health, Personal health finance, Pharmacy, Seniors and health, Women and health
Even with the prospect of enrolling in Medicare sooner in a year or two or three, Americans approaching retirement are growing concerned about health care costs, according to a study in JAMA Network Open. The paper, Health Insurance Affordability Concerns and health Care Avoidance Among US Adults Approaching Retirement, explored the perspectives of 1,028 US adults between 50 and 64 years of age between November 2018 and March 2019. The patient survey asked one question addressing two aspects of "health care confidence:" "Please rate your confidence with the following:" Being able to afford the cost of your health insurance nad
USA Today Finds Hidden Common Ground Among Americans For Health Care
By Jane Sarasohn-Kahn on 7 February 2020 in ACA, Affordable Care Act, Health citizenship, Health Consumers, Health costs, Health Economics, Health policy, Health politics, Health privacy, Nurses, Social determinants of health
"We need to demand our health citizenship. That means our nation must approach medical treatment and data privacy as civil rights that protect everyone." This is the start of my column, Americans, let's claim our health care rights, published by USA Today today. USA Today is publishing a 10-part series called "Hidden Common Ground" addressing key issues where Americans can come together. Thus far, the series has covered climate change, and this month health care. Going forward, we'll see analyses on jobs, gun rights and violence, and immigration through May's publishing schedule. In their study conducted in December 2020, USA Today
Come Together – A Health Policy Prescription from the Bipartisan Policy Center
By Jane Sarasohn-Kahn on 6 February 2020 in ACA, Affordable Care Act, Business and health, Education and health, Employee benefits, Employers, Health benefits, Health citizenship, Health Consumers, Health costs, Health insurance, Health insurance exchanges, Health insurance marketplaces, Health law, Health policy, Health politics, Health privacy, Health reform, Medicaid, Medicare, Medicines, Nutrition, Pharmaceutical, Pre-existing conditions, Prescription drugs, Primary care, Privacy and security, Self-care, Single-payer health care, Social determinants of health, Social responsibility, Workplace benefits
Among all Americans, the most popular approach for improving the health care in the U.S. isn't repealing or replacing the Affordable Care Act or moving to a Medicare-for-All government-provided plan. It would be to improve the current health care system, according to the Bipartisan Policy Center's research reported in a Bipartisan Rx for America's Health Care. The BPC is a truly bipartisan organization, co-founded by Former Democratic Senate Majority Leaders Tom Daschle and George Mitchell, and Former Republican Senate Majority Leaders Howard Baker and Bob Dole. While this political week in America has revealed deep chasms between the Dems and
What's Causing Fewer Primary Care Visits in the US?
By Jane Sarasohn-Kahn on 5 February 2020 in ACA, Affordable Care Act, Design and health, Employee benefits, Employers, Health benefits, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health Plans, Health policy, Health politics, Health reform, Physicians, Primary care, Retail health, Self-care, Social determinants of health, Telehealth
Americans who have commercial health insurance (say, through an employer or union) are rarely thought to face barriers to receiving health care — in particular, primary care, that front line provider and on-ramp to the health care system. But in a new study published in the Annals of Internal Medicine, commercially-insured adults were found to have visited primary care providers (PCPs) less often, and 1 in 2 had no PCP visits in one year. In Declining Use of Primary Care Among Commercially Insured Adults in the United States, 2008-2016, the researchers analyzed data from a national sample of adult health
The State of the Union for Prescription Drug Prices
By Jane Sarasohn-Kahn on 4 February 2020 in Bio/life sciences, Business and health, Cancer, Children's health, Diabetes, Financial health, Financial toxicity, Financial wellness, Health benefits, Health Consumers, Health costs, Health insurance, Health law, Health policy, Health politics, Pharmaceutical, Pharmacy, Prescription drugs
Tonight, President Trump will present his fourth annual State of the Union address. This morning we don't have a transcript of the speech ahead of the event, but one topic remains high on U.S. voters' priorities, across political party – prescription drug prices. Few issues unite U.S. voters in 2020 quite like supporting Medicare's ability to negotiate drug prices with pharmaceutical companies, shown by the October 2019 Kaiser Family Foundation Health Tracking Poll. Whether Democrat, Independent, or Republican, most people living in America favor government intervention in regulating the cost of medicines in some way. In this poll, the top
A Uniting Issue in the United States is Lowering Prescription Drug Costs
By Jane Sarasohn-Kahn on 31 January 2020 in ACA, Health citizenship, Health costs, Health Economics, Health insurance, Health policy, Health politics, Health reform, Health regulation, Medicare, Money and health, Pharmaceutical, Pharmacy, Prescription drugs, Trust
Health care continues to be the top-ranked voting issue in the U.S. looking to the November 2020 Presidential and Congressional elections. The Kaiser Family Foundation conducts the monthly poll which gauges U.S. adults' perspectives on health care, and this month's January 2020 Kaiser Health Tracking Poll explores Americans' views on broad healthcare reform plans and specific medical policy issues. Overall, Americans point to prescription drug costs and the preservation of the Affordable Care Act's protections for people with pre-existing conditions, the first chart tells us. Third and fourth on voters' minds are protecting patients from surprise medical bills and better
Most Americans Regardless of Income Say It's Unfair for Wealthier People to Get Better Health Care
By Jane Sarasohn-Kahn on 30 January 2020 in Behavioral economics, Cancer, Deaths of despair, Demographics and health, Financial health, Financial wellness, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health equity, Health insurance, Health policy, Health politics, Medicines, Money and health, Pharmaceutical, Pharmacy, Prescription drugs
In America, earning lower or middle incomes is a risk factor for having trouble accessing health care and/or paying for it. But most Americans, rich or not, believe that it's unfair for wealthier people to get better health care, according to a January 2020 poll from NPR, the Robert Wood Johnson Foundation and Harvard Chan School of Public Health, Life Experiences and Income Equality in the United States. The survey was conducted in July and August 2019 among 1,885 U.S. adults 18 or older. Throughout the study, note the four annual household income categories gauged in the research: Top 1%
The Pace of Tech-Adoption Grows Among Older Americans, AARP Finds – But Privacy Concerns May Limit Adoption
By Jane Sarasohn-Kahn on 28 January 2020 in Aging and Technology, GDPR, Health apps, Health care information technology, Health citizenship, Health Consumers, Health law, Health policy, Health privacy, Health social networks, HIPAA, Internet of things, Medication adherence, Patient experience, Privacy and security, Remote health monitoring, Retirement and health, Self-care, Seniors and health, Sensors and health, Shopping and health, Smart homes, Smartphones, Social media and health, Social networks and health, Telehealth, Trust, User experience UX, Voice technology, Wearable tech, Wearables
One in two people over 50 bought a piece of digital technology in the past year. Three in four people over fifty in America now have a smartphone. One-half of 50+ Americans use a tablet, and 17% own wearable tech. The same percentage of people over 50 own a voice assistant, a market penetration rate which more than doubled between 2017 and 2019, AARP noted in the 2020 Tech and the 50+ Survey published in December 2019. For this research, AARP worked with Ipsos to survey (online) 2,607 people ages 50 and over in June and July 2019. Across all
Income Inequality is Fostering Mis-Trust, the Edelman 2020 Trust Barometer Observes
By Jane Sarasohn-Kahn on 23 January 2020 in AI, Business and health, Corporate responsibility, Employers, Financial health, Financial wellness, Guns and health, Health and wealth, Health care industry, Health citizenship, Health costs, Health ecosystem, Health policy, Hospitals, Jobs and health, media and health, Money and health, Primary care, Public health, Social responsibility, Sustainability, Transparency, Trust
Economic development has historically built trust among nations' citizens. But in developed, wealthier parts of the world, like the U.S., "a record number of countries are experiencing an all-time high 'mass-class' trust divide," according to the 2020 Edelman Trust Barometer. For 20 years, Edelman has released its annual Trust Barometer every year at the World Economic Forum in Davos, recognizing the importance of trust in the global economy and society. Last year, it was the employer who was the most-trusted touch-point in citizens' lives the world over, I discussed in Health Populi one year ago. This year, even our employers can't
Consumers Seek Benefits From Food, a Personal Social Determinant of Health
By Jane Sarasohn-Kahn on 22 January 2020 in Behavior change, Children's health, Chronic disease, Consumer-directed health, Deaths of despair, Demographics and health, Diabetes, Food and health, Grocery stores, Health and wealth, Health at home, Health benefits, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health ecosystem, Health education, Health policy, Heart health, Moms and health, Money and health, Noncommunicable disease, Nutrition, Obesity, Prevention, Prevention and wellness, Retail health, Self-care, Shopping and health, Social determinants of health, Suicide, Weight loss, Wellbeing, Wellness
As consumers in the U.S. wrestle with accessing and paying for medical benefits, there's another sort of health benefit people increasingly understand, embrace, and consume: food-as-medicine. More people are taking on the role of health consumers as they spend more out-of-pocket on medical care and insurance, and seeking food to bolster their health is part of this behavior change. One in four Americans seek health benefits from food, those who don't still seek the opportunity to use food for weight loss goals, heart health and energy boosting, according to the 2019 Food & Health Survey from the International Food Information
The 2020 Social Determinants of Health: Connectivity, Art, Air and Love
By Jane Sarasohn-Kahn on 30 December 2019 in Anxiety, Art and health, Banks and health, Behavioral health, Broadband, Business and health, Chronic disease, Connected health, Consumer electronics, Corporate responsibility, Death, Demographics and health, Depression, Design and health, Education and health, Environment and heatlh, Financial health, Financial toxicity, Financial wellness, Food and health, Guns and health, Happiness, Health and safety, Health and wealth, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health engagement, Health equity, Health insurance, Health Plans, Health policy, Health politics, Health social networks, Hospitals, Internet and Health, Loneliness, Love and health, Mental health, Mobile health, Money and health, Mortality, Music and health, Nutrition, Pain, Popular culture and health, Primary care, Remote health monitoring, Retail health, Self-care, Social determinants of health, Social isolation, Social responsibility, Stress, Suicide, Sustainability, Telehealth, Violence and health, Wellbeing, Wellness
Across the U.S., the health/care ecosystem warmly embraced social determinants of health as a concept in 2019. A few of the mainstreaming-of-SDoH signposts in 2019 were: Cigna studying and focusing in on loneliness as a health and wellness risk factor Humana's Bold Goal initiative targeting Medicare Advantage enrollees CVS building out an SDOH platform, collaborating with Unite US for the effort UPMC launching a social impact program focusing on SDoH, among other projects investing in social factors that bolster public health. As I pointed out in my 2020 Health Populi trendcast, the private sector is taking on more public health
In 2020, PwC Expects Consumers to Grow DIY Healthcare Muscles As Medical Prices Increase
By Jane Sarasohn-Kahn on 11 December 2019 in Amazon, Connected health, Consumer experience, Consumer-directed health, Data analytics and health, Design and health, Digital health, Employers, Financial health, Financial toxicity, GDPR, Health benefits, Health care industry, Health care information technology, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health engagement, Health marketing, Health Plans, Health policy, Health politics, Health privacy, Health reform, Health regulation, Hospitals, Internet of things, Jobs and health, Medical innovation, Money and health, Patient experience, Pharmaceutical, Pharmacy, Prescription drugs, Primary care, Retail health, Robots, Robots and health, Self-care, Social responsibility, User experience UX, Value based health, Venture capital and health, Workplace benefits, Workplace wellness
The new year will see a "looming tsunami" of high prices in healthcare, regulation trumping health reform, more business deals reshaping the health/care industry landscape, and patients growing do-it-yourself care muscles, according to Top health industry issues of 2020: Will digital start to show an ROI from the PwC Health Research Institute. I've looked forward to reviewing this annual report for the past few years, and always learn something new from PwC's team of researchers who reach out to experts spanning the industry. In this 14th year of the publication, PwC polled executives from payers, providers, and pharma/life science organizations. Internally,
Food As Medicine: Grocery Stores Expand as Health Destinations While the Federal Government Cuts Food Stamps
By Jane Sarasohn-Kahn on 9 December 2019 in Diabetes, Food and health, Grocery stores, Health at home, Health Consumers, Health ecosystem, Health engagement, Health marketing, Health policy, Health politics, HealthDIY, Heart health, Medication adherence, Money and health, Noncommunicable disease, Nutrition, Patient engagement, Personalized medicine, Pharmacy, Prescription drugs, Prevention and wellness, Primary care, Retail health, Rural health, Self-care, Shopping and health, Social determinants of health, Telehealth, Virtual health, Weight loss
There's something like cognitive dissonance as I prepare my 2020 Health Populi TrendCast of what to expect in the health/care ecosystem in the new year. One of my key pillars for health-making is food-as-medicine, and that opportunity in this moment resonates in this holiday season with Dickens' "Best of Times, Worst of Times" context-setting that kicks off Great Expectations. In the "best of times" part of the food+health equation, we recognize the growing role of grocery stores, food-tech and food manufacturers in the health/care landscape. A current example comes from Kroger, partnering with Ascension's health system in Tennessee, enhancing the organization's
Being Transparent About Healthcare Transparency – My Post on the Medecision Blog
By Jane Sarasohn-Kahn on 4 December 2019 in Consumer-directed health, Health benefits, Health citizenship, Health Consumers, Health costs, Health finance, Health insurance, Health Plans, Health policy, Health politics, Hospitals, Transparency
With new rules emanating from the White House this month focusing on health care price transparency, health care costs are in the spotlight at the Centers for Medicare and Medicaid Services. A hospital transparency mandate will go into effect in January 2021 as a final rule, and a second rule with a focus on health plans and friendly explanations-of-benefits will receive comments in the Federal Register until January 14, 2020. As patients continue to grow muscles as payors and health consumers, transparency is one key to enabling people to "shop" for those health care and medical products and services that
Despite Greater Digital Health Engagement, Americans Have Worse Health and Financial Outcomes Than Other Nations' Health Citizens
By Jane Sarasohn-Kahn on 2 December 2019 in Artificial intelligence, Behavioral economics, Caregivers, Chronic disease, Computers and health, Connected health, Data analytics and health, Digital health, Financial health, Financial wellness, Fitness, Global Health, Health apps, Health at home, Health care industry, Health care information technology, Health citizenship, Health Consumers, Health costs, Health ecosystem, Health engagement, Health policy, Health privacy, Health ratings and report cards, Heart health, High deductibles, Hospitals, Internet and Health, Medical technology, Medication adherence, Medicines, Mobile health, Patient engagement, Pharmaceutical, Pharmacy, Prescription drugs, Prevention, Privacy and security, Remote health monitoring, Robots, Robots and health, Self-care, Telehealth, Virtual health, Voice technology, Wearable tech, Wellness
The idea of health care consumerism isn't just an American discussion, Deloitte points out in its 2019 global survey of healthcare consumers report, A consumer-centered future of health. The driving forces shaping health and health care around the world are re-shaping health care financing and delivery around the world, and especially considering the growing role of patients in self-care — in terms of financing, clinical decision making and care-flows. With that said, Americans tend to be more healthcare-engaged than peer patients in Australia, Canada, Denmark, Germany, the Netherlands, Singapore, and the United Kingdom, Deloitte's poll found. Some of the key behaviors
Hospitals Suffer Decline in Consumer Satisfaction
By Jane Sarasohn-Kahn on 27 November 2019 in Consumer experience, Consumer-directed health, Design and health, Doctors, Emergency care, Financial health, Financial toxicity, Grocery stores, Health apps, Health benefits, Health care industry, Health care marketing, Health Consumers, Health costs, Health engagement, Health finance, Health law, Health Plans, Health policy, High deductibles, Hospitals, Money and health, Patient engagement, Patient experience, Personal health finance, Physicians, Popular culture and health, Primary care, Retail health, Self-care, Shopping and health, Transparency, Value based health
While customer satisfaction with health insurance plans slightly increased between 2018 and 2019, patient satisfaction with hospitals fell in all three settings where care is delivered — inpatient, outpatient, and the emergency room, according to the 2018-2019 ACSI Finance, Insurance and Health Care Report. ACSI polls about 300,000 U.S. consumers each year to gauge satisfaction with over 400 companies in 46 industries. For historic trends, you can check out my coverage of the 2014 version of this study here in Health Populi. The 2019 ACSI report bundles finance/banks, insurance (property/casualty, life and health) and hospitals together in one document. Health
Longevity Stalls Around the World And Wealth, More Concentrated
By Jane Sarasohn-Kahn on 22 November 2019 in Baby health, Cancer, Chronic disease, Death, Demographics and health, Diabetes, Education and health, Financial health, Financial wellness, Food and health, Global Health, Health and wealth, Health benefits, Health citizenship, Health Consumers, Health costs, Health Economics, Health ecosystem, Health education, Health equity, Health policy, Health social networks, Mental health, Money and health, Noncommunicable disease, Nutrition, Obesity, Opioids, Population health, Primary care, Public health, Safety net and health, Social determinants of health, Wellbeing
Two separate and new OECD reports, updating health and the global economic outlook, raise two issues that are inter-related: that gains in longevity are stalling, with chronic illnesses and mental ill health affecting more people; and, as wealth grows more concentrated among the wealthy, the economic outlook around most of the world is also slowing. First, we'll mine the Health at a Glance 2019 annual report covering data on population health, health system performance, and medical spending across OECD countries. The first chart arrays the x-y data points of life expectancy versus health spending for each of the OECD countries
Social Determinants of Health – My Early Childhood Education and Recent Learnings, Shared at the HealthXL Global Gathering
By Jane Sarasohn-Kahn on 20 November 2019 in Children's health, Consumer experience, Corporate responsibility, Death, Demographics and health, Education and health, Environment and heatlh, Financial health, Fitness, Food and health, Grocery stores, Health and wealth, Health citizenship, Health Consumers, Health costs, Health disparities, Health ecosystem, Health equity, Health Plans, Health policy, Health politics, Health social networks, Healthcare access, Hospitals, Kids' health, Loneliness, Love and health, Medicare, Mental health, Money and health, Mortality, Noncommunicable disease, Nutrition, Pain, Pharmaceutical, Physicians, Popular culture and health, Prescription drugs, Prevention and wellness, Public health, Safety net and health, Schools and health, Seniors and health, Social determinants of health, Social isolation, Social networks and health, Social responsibility, Violence and health, Wellbeing
My cousin Arlene got married in Detroit at the classic Book Cadillac Hotel on July 23, 1967, a Sunday afternoon wedding. When Daddy drove us back out to our suburban home about 30 minutes from the fancy hotel, the car radio was tuned to WWJ Newsradio 950, all news all the time. As soon as Daddy switched on the radio, we were shocked by the news of a riot breaking out in the city, fires and looting and gunshots and chaos in the Motor City. Two days later, my father, who did business with Mom-and-Pop retail store owners in the
More Evidence of Self-Rationing as Patients Morph into Healthcare Payors
By Jane Sarasohn-Kahn on 19 November 2019 in ACA, Affordable Care Act, Boomers, Demographics and health, Financial health, Financial wellness, Health Consumers, Health costs, Health Economics, Health insurance, Health policy, Health politics, High deductibles, Hospitals, Medicare, Medication adherence, Medicines, Money and health, Personal health finance, Prescription drugs, Shopping and health, Transparency
Several new studies reveal that more patients are feeling and living out their role as health care payors as medical spending vies with other household line items. This role of patient-as-the-payor crosses consumers' ages and demographics, and is heating up health care as the top political issue for the 2020 elections at both Federal and State levels. In research from HealthPocket, 2 in 5 Americans said they needed to reduce other household expenses to be able to afford their monthly insurance premiums. Four in ten consumers said their monthly health insurance premiums were increasing. One in four people in the
Will Technology Cure Americans' Health Care System Ills? Considering Google and Ascension Health's Data Deal
By Jane Sarasohn-Kahn on 12 November 2019 in Artificial intelligence, Bioethics, Business and health, Caregivers, Cognitive computing and health, Computers and health, Connected health, Consumer experience, Corporate responsibility, Data analytics and health, Design and health, Digital health, Health care information technology, Health citizenship, Health Consumers, Health disparities, Health ecosystem, Health engagement, Health equity, Health IT, Health law, Health policy, Health privacy, Patient engagement, Patient experience, Popular culture and health, Privacy and security, Social determinants of health, Social responsibility, Transparency, Trust
"Google's 'Project Nightingale' Gathers Personal Health Data on Millions of Americans," the Wall Street Journal reported in today's paper and on the WSJ.com website. The story started with the scenario that, "Search giant is amassing health records from Ascension facilities in 21 states; patients not yet informed." Here's Ascension's press release on the collaboration, described in the title as "healthcare transformation." Note: this release was written after the Wall Street Journal published this story. And, according to the WSJ reporting, "Neither patients nor doctors have been notified. At least 150 Google employees already have access to much of the data
A Tale of Two Americas as Told by the 2019 OECD Report on Health
By Jane Sarasohn-Kahn on 7 November 2019 in Chronic disease, Death, Demographics and health, Diabetes, Doctors, Employers, Exercise, Financial health, Financial toxicity, Financial wellness, Food and health, GDPR, Global Health, Health and wealth, Health benefits, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health ecosystem, Health equity, Health policy, Health politics, Health privacy, Health reform, Hospitals, Mental health, Money and health, Obesity, Primary care, Privacy and security, Public health, Social determinants of health
It was the best of times, It was the worst of times, It was the age of wisdom, it was the age of foolishness, It was the epoch of belief, it was the epoch of incredulity, … starts Dickens' Tale of Two Cities. That's what came to my mind when reading the latest global health report from the OECD, Health at a Glance 2019, which compares the United States to other nations' health care outcomes, risk factors, access metrics, and spending. Some trends are consistent across the wealthiest countries of the world, many sobering, such as: Life expectancy rates fell in 19 of the
The Link Between Wellness & Wealth Is Powerful for Everyone – and Especially Women
By Jane Sarasohn-Kahn on 6 November 2019 in Affordable Care Act, Chronic disease, Financial health, Financial wellness, Guns and health, Health and wealth, Health benefits, Health Consumers, Health costs, Health finance, Health insurance, Health insurance marketplaces, Health Plans, Health policy, Health politics, Money and health, Stress
In the U.S., the link between wellness and wealth, money and health, is strong and common across people, young and old. But the impacts of money on health, well-being, and life choices varies across the ages, based on a study from Lively, a company that builds platforms for health savings accounts. The first chart illustrates that health care costs challenge people in many ways: the most obvious health care cost problems prevent people from saving more for retirement or paying down debt. Eight in 10 Americans concur that rising health care costs challenge their ability to save for retirement. Beyond the
Great Expectations for Health Care: Patients Look for Consumer Experience and Trust in Salesforce's Latest Research
By Jane Sarasohn-Kahn on 5 November 2019 in Anxiety, Business and health, Connected health, Consumer experience, Design and health, Digital health, Financial health, Financial toxicity, Financial wellness, Health apps, Health at home, Health benefits, Health care industry, Health Consumers, Health costs, Health ecosystem, Health education, Health engagement, Health IT, Health Plans, Health policy, Health politics, Health reform, Hospitals, Medical technology, Medication adherence, Medicines, Mental health, Money and health, Patient experience, Personal health finance, Pharmaceutical, Self-care, Trust
On the demand side of U.S. health care economics, patients are now payors as health consumers with more financial skin in paying medical bills. As consumers, people have great expectations from the organizations on the supply side of health care — providers (hospitals and doctors), health insurance plans, pharma and medical device companies. But as payors, health consumers face challenges in getting care, so great expectations are met with frustration and eroding trust with the system, according to the latest Connected Healthcare Consumer report from Salesforce published today as the company announced expansion of their health cloud capabilities. This is
Thinking About Health Care One Year From the 2020 Presidential Election
By Jane Sarasohn-Kahn on 4 November 2019 in ACA, Business and health, Corporate responsibility, Corporate wellness, Doctors, Employee benefits, Employers, Financial wellness, Health benefits, Health care industry, Health citizenship, Health Consumers, Health costs, Health disparities, Health Economics, Health ecosystem, Health equity, Health insurance, Health law, Health Plans, Health policy, Health politics, Medicare, Nurses, Public health, Social responsibility, Trust
Today is 4th November 2019, exactly one year to the day that Americans can express their political will and cast their vote for President of the United States. Health care will be a key issue driving people to their local polling places, so it's an opportune moment to take the temperature on U.S. voters' perspectives on healthcare reform. This post looks at three current polls to gauge how Americans are feeling about health care reform 365 days before the 2020 election, and one day before tomorrow's 2019 municipal and state elections. Today's Financial Times features a poll that found two-thirds
Will Consumers Cross the Cost-and-Trust Chasm Between Prescription Drugs and Hospitals?
By Jane Sarasohn-Kahn on 29 October 2019 in Bio/life sciences, Biotech, Consumer experience, Financial toxicity, Health care industry, Health Consumers, Health costs, Health Economics, Health ecosystem, Health policy, Health politics, Hospitals, Medicare, Medication adherence, Medicines, Money and health, Pharmaceutical, Physicians, Prescription drugs, Retail health, Social determinants of health, Specialty drugs, Transparency
People in the U.S. rank prescription drugs, lab tests, emergency room visits, dental and vision care, preventive services, chronic disease management and mental health care as the "most essential" health care services, according to the 2019 Survey of America's Patients conducted by The Physicians Foundation. When asked what factors contribute to rising health care costs in America, most consumers cite the cost of prescription drugs. Taken together, these two data points demonstrate the potent political import of prescription drug prices as the U.S. approaches the 2020 Presidential election. The Physicians Foundation surveyed 2,001 U.S. adults between 27 and 75 years
While Costs Are A Top Concern Among Most U.S. Patients, So Are Challenges of Poverty, Food, and Housing
By Jane Sarasohn-Kahn on 28 October 2019 in Financial health, Financial toxicity, Financial wellness, Food and health, Health care industry, Health care marketing, Health citizenship, Health Consumers, Health costs, Health Economics, Health ecosystem, Health engagement, Health policy, Health politics, Health reform, Money and health, Single-payer health care, Social determinants of health, Social responsibility, Universal health care
Rising health care costs continue to concern most Americans, with one in two people believing they're one sickness away from getting into financial trouble, according to the 2019 Survey of America's Patients conducted for The Physicians Foundation. In addition to paying for "my" medical bills, most people in the U.S. also say that income inequality and inadequate social services significantly contribute to high medical spending for every health citizen in the nation. The Physicians Foundation conducts this study into Americans' views on the U.S. health care system every other year. This year's poll was conducted in September 2019 and included input
Learning from Dr. Eric Topol, Live from Medecision Liberation 2019
By Jane Sarasohn-Kahn on 24 October 2019 in AI, Artificial intelligence, Big data and health, Bio/life sciences, Bioethics, Broadband, Cognitive computing and health, Computers and health, Connected health, Data analytics and health, Design and health, Digital health, Doctors, Health at home, Health citizenship, Health Consumers, Health costs, Health Economics, Health equity, Health policy, Health privacy, Heart health, Hospitals
"Bold thinking is great. Bold doing is better," Dr. Eric Topol introduced his talk yesterday at Medecision's Liberation 2019 conference. I have the opportunity, for which I'm so grateful, of not only attending this meeting but playing a role as a speaker, a sometimes stage "emcee," and a keynote speaker. And as an attendee, I learn so much from other speakers, fellow attendees, and Medecision staff all sharing perspectives during breakouts and networking breaks. In mode of attendee (and self-confessed collegial-groupie of Dr. Topol's), I took in his remarks taking notes as fast as I could thanks to Mom teaching
What the 2019 Nobel Prize Winners in Economics Teach Us About Health
By Jane Sarasohn-Kahn on 14 October 2019 in Baby health, Children's health, Education and health, Financial health, Food and health, Global Health, Health and wealth, Health costs, Health Economics, Health ecosystem, Health education, Health equity, Health policy, Healthcare access, Mobile health, Nutrition, Public health, Schools and health, Social determinants of health, Vaccines
The three winners of the 2019 Nobel Prize for Economics — Banerjee and Duflo (both of MIT) and Kremer (working at Harvard) — were recognized for their work on alleviating global poverty." "Over 700 million people still subsist on extremely low incomes. Every year, five million children still die before their fifth birthday, often from diseases that could be prevented or cured with relatively cheap and simple treatments," The Nobel Prize website notes. To respond to this audaciously huge challenge, Banerjee, Duflo and Kremer asked quite specific, granular questions that have since shaped the field of development economics — now
Wasted: $1 of Every $4 Spent on Health Care In America
By Jane Sarasohn-Kahn on 8 October 2019 in ACA, Affordable Care Act, Business and health, Employers, Health care industry, Health citizenship, Health Consumers, Health costs, Health Economics, Health ecosystem, Health insurance, Health policy, Health politics, Hospitals, Medicare, Medicines, Pharmaceutical, Pharmacy, Prescription drugs, Transparency, Value based health
A study in JAMA published this week analyzed research reports that have measured waste in the U.S. health care system, calculating that 25% of medical spending in America is wasted. If spending is gauged at $3.8 trillion, waste amounts to nearly $1 trillion. If spending is 18% of the American gross domestic product (GDP), then some 4.5% of the U.S. economy is wasted spending by the health care system and its stakeholders. In "Waste in the US Health Care System," a team from Humana and the Univrsity of Pittsburgh recalibrated the previous finding of 30% of wasted spending to the 25%,
Prelude to Health 2.0 2019: Thinking Consumers At the Center of Digital Health Transformation
By Jane Sarasohn-Kahn on 16 September 2019 in AI, Artificial intelligence, Blockchain, Cognitive computing and health, Connected health, Consumer experience, Data analytics and health, Design and health, Digital health, Health at home, Health care information technology, Health Consumers, Health costs, Health ecosystem, Health engagement, Health equity, Health IT, Health literacy, Health policy, Health privacy, Healthcare access, HIPAA, Home care, Hospitals, Internet of things, Medical innovation, Patient engagement, Patient experience, Physicians, Population health, Primary care, Privacy and security, Remote health monitoring, Safety net and health, Self-care, Shared decision making, Smart homes, Social determinants of health, Telehealth, Trust, User experience UX, Value based health, Virtual health, Wearable tech, Wearables, Wellbeing, Workflow
"Digital transformation" is the corporate strategy flavor of the moment across industries, and the health are sector isn't immune from the trend. As this 13th year of the annual Health 2.0 Conference kicks off this week, I'm focused on finding digital health innovations that engage people — consumers, caregivers, patients, health citizens all. This year's conference will convene thought leaders across a range of themes, and as is the Health 2.0 modus operandi, live demo's of new-new things. As Health 2.0 kicks off today in pre-conference sessions, there is useful context described in a new report from the American Hospital
Jane's forecast on health care consumers published in AHA's FutureScan 2021-2026 - 01/23/2021
Every year, the American Hospital Association (AHA) publishes a health care forecast, FutureScan. The AHA asked Jane to share her read on the consumer health care tea leaves for the new 2021-2026 publication, available through the AHA.
How WiFi Connectivity Bolsters Health Citizenship – Jane's essay in USA Today/MediaPlanet - 01/25/2021
Without connectivity to the Internet, people have been hard-pressed to take on their full health citizenship in the pandemic era. Grateful that Media Planet placed my essay on WiFi as a social determinant of health in a USA Today insert on 29th December 2020.
Jane will lead a TweetChat with Microsoft to brainstorm AI in health care in and beyond the pandemic - 01/26/2021
Jane will moderate a chat on Twitter to brainstorm artificial intelligence in health care, in and beyond the COVID-19 pandemic. I'll be joined by Microsoft's Tom Lawry (@TCLawry) on the tweetchat; Tom is Microsoft's National Director for AI, Health & Life Sciences. To follow our discussion, we'll use the hashtag #MicrosoftAI.
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together. However, I think I may have misunderstood the nature of how they work and I used
cin.ignore(INT_MAX, '\n')
before using the first getline function, and it was causing my program to pause. I think that I am supposed to use this only if i use cin before, and when I want to use a getline function in order to prevent the getline function from taking in the empty space?
at the start, and it is causing me errors, I'm not sure when I use this.
I think this part might be the error... but I'm not quite for how the || and the && operators work
else if (donorGender != "Male" || "Female" || "Trans Male" || "Trans Female" || "Queer" || "Different")
is this it the way I do it?
else if (donorGender != "Male" && donorGender != "Female" && donorGender != "Trans Male" && donorGender != "Trans Female" && donorGender != "Queer" && donorGender != "Different")
or is this the way I do it
Please help...
A: Your code says that it will test donorName else donorGender and so on. You need to check all the conditions. and your donorGender checking if statement is not correctly formatted. What if user enters both name and gender invalid!
I think you should not check other conditions if one if false. if you want to tell all the wrong things then approach is different. but in your case below code can help. try it out!
Nested Conditional Statements
if(donorName != "")
{
if(donorGender == "Male"
or donorGender == "Female"
or donorGender == "Trans Male"
or donorGender == "Trans Female"
or donorGender == "Queer"
or donorGender =="Different")
{
if(donorAge >= 0) {
if(donorWeight >= 0) {
if (donorHeight >= 0)
{
cout << "--- You must enter a valid height." << endl;
return (-1);
}
else {
}
}
else {
cout << "--- You must enter a valid weight." << endl;
return (-1);
}
}
else {
cout << "--- You must enter a valid age." << endl;
return (-1);
}
}
else {
cout << "--- You must enter a valid gender." << endl;
return(-1);
}
}
else {
cout << " --- You must enter a valid name." << endl;
return (-1);
}
A: The statement
cin.ignore(INT_MAX, '\n');
is pausing your program at the start until you press Enter key.
From this:
std::istream::ignore
istream& ignore (streamsize n = 1, int delim = EOF);
Extract and discard characters
Extracts characters from the input sequence and discards them, until either n characters have been extracted, or one compares equal to delim.
Generally, cin.ignore(INT_MAX, '\n') used with getline if you have some other input statements using cin before calling getline because when a user inputs something with cin, they hit Enter key and a \n (newline) character gets into the input buffer. Then if your program calling getline, it gets the newline character instead of the string you want. In your program you are is using getline for the first input from the user, so you don't need it. You can safely remove cin.ignore statement from your program.
This statement is wrong:
else if (donorGender != "Male" || "Female" || "Trans Male" || "Trans Female" || "Queer" || "Different")
You need to compare donorGender with all the possible valid values and not only with just one valid value. Even if you compare donorGender != with all valid values this will not work because || operator in the condition will always evaluate to true as a valid value of donorGender will be != rest of all valid values. Change it to:
else if (donorGender != "Male" && donorGender != "Female" && donorGender != "Trans Male" && donorGender != "Trans Female" && donorGender != "Queer" && donorGender != "Different")
With these changes, your program should work as expected.
Also, I would suggest you to add some input validation for all the inputs you are taking from the user.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,213 |
{"url":"https:\/\/deepai.org\/publication\/tail-bounds-for-volume-sampled-linear-regression","text":"# Tail bounds for volume sampled linear regression\n\nThe n \u00d7 d design matrix in a linear regression problem is given, but the response for each point is hidden unless explicitly requested. The goal is to observe only a small number k \u226a n of the responses, and then produce a weight vector whose sum of square loss over all points is at most 1+\u03f5 times the minimum. A standard approach to this problem is to use i.i.d. leverage score sampling, but this approach is known to perform poorly when k is small (e.g., k = d); in such cases, it is dominated by volume sampling, a joint sampling method that explicitly promotes diversity. How these methods compare for larger k was not previously understood. We prove that volume sampling can have poor behavior for large k - indeed worse than leverage score sampling. We also show how to repair volume sampling using a new padding technique. We prove that padded volume sampling has at least as good a tail bound as leverage score sampling: sample size k=O(d d + d\/\u03f5) suffices to guarantee total loss at most 1+\u03f5 times the minimum with high probability. The main technical challenge is proving tail bounds for the sums of dependent random matrices that arise from volume sampling.\n\n## Authors\n\n\u2022 6 publications\n\u2022 24 publications\n\u2022 64 publications\n\u2022 ### Reverse iterative volume sampling for linear regression\n\nWe study the following basic machine learning task: Given a fixed set of...\n06\/06\/2018 \u2219 by Michal Derezinski, et al. \u2219 0\n\n\u2022 ### Unbiased estimators for random design regression\n\nIn linear regression we wish to estimate the optimum linear least square...\n07\/08\/2019 \u2219 by Micha\u0142 Derezi\u0144ski, et al. \u2219 4\n\n\u2022 ### Minimax experimental design: Bridging the gap between statistical and worst-case approaches to least squares regression\n\nIn experimental design, we are given a large collection of vectors, each...\n02\/04\/2019 \u2219 by Michal Derezinski, et al. \u2219 16\n\n\u2022 ### Correcting the bias in least squares regression with volume-rescaled sampling\n\nConsider linear regression where the examples are generated by an unknow...\n10\/04\/2018 \u2219 by Michal Derezinski, et al. \u2219 38\n\n\u2022 ### Generalized Leverage Score Sampling for Neural Networks\n\nLeverage score sampling is a powerful technique that originates from the...\n09\/21\/2020 \u2219 by Jason D. Lee, et al. \u2219 0\n\n\u2022 ### L1 Regression with Lewis Weights Subsampling\n\nWe consider the problem of finding an approximate solution to \u2113_1 regres...\n05\/19\/2021 \u2219 by Aditya Parulekar, et al. \u2219 0\n\n\u2022 ### Stable recovery and the coordinate small-ball behaviour of random vectors\n\nRecovery procedures in various application in Data Science are based on ...\n04\/17\/2019 \u2219 by Shahar Mendelson, et al. \u2219 0\n\n##### This week in AI\n\nGet the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.\n\n## 1 Introduction\n\nConsider a linear regression problem where the input points in are provided, but the associated response for each point is withheld unless explicitly requested. The goal is to sample the responses for just a small subset of inputs, and then produce a weight vector whose total square loss on all points is at most times that of the optimum.111The total loss of the algorithm being at most times loss of the optimum can be rewritten as the regret being at most times the optimum. This scenario is relevant in many applications where data points are cheap to obtain but responses are expensive. Surprisingly, with the aid of having all input points available, such multiplicative loss bounds are achievable without any range dependence on the points or responses common in on-line learning (see, e.g., onlineregr, ).\n\nA natural and intuitive approach to this problem is volume sampling, since it prefers \u201cdiverse\u201d sets of points that will likely result in a weight vector with low total loss, regardless of what the corresponding responses turn out to be\u00a0(unbiased-estimates, ). Volume sampling is closely related to optimal design criteria\u00a0(optimal-design-book, ; dual-volume-sampling, ), which are appropriate under statistical models of the responses; here we study a worst-case setting where the algorithm must use randomization to guard itself against worst-case responses.\n\nVolume sampling and related determinantal point processes are employed in many machine learning and statistical contexts, including linear regression\n\n(dual-volume-sampling, ; unbiased-estimates, ; regularized-volume-sampling, ), clustering and matrix approximation\u00a0(pca-volume-sampling, ; efficient-volume-sampling, ; avron-boutsidis13, ), summarization and information retrieval\u00a0(dpp, ; k-dpp, ; dpp-shopping, ), and fairness\u00a0(celis2016fair, ; celis2018fair, ). The availability of fast algorithms for volume sampling\u00a0(dual-volume-sampling, ; unbiased-estimates, ) has made it an important technique in the algorithmic toolbox alongside i.i.d.\u00a0leverage score sampling\u00a0(drineas2006sampling, ) and spectral sparsification\u00a0(batson2012twice, ; lee2015constructing, ).\n\nIt is therefore surprising that using volume sampling in the context of linear regression, as suggested in previous works\u00a0(unbiased-estimates, ; dual-volume-sampling, ), may lead to suboptimal performance. We construct an example in which, even after sampling up to half of the responses, the loss of the weight vector from volume sampling is a fixed factor larger than the minimum loss. Indeed, this poor behavior arises because for any sample size , the marginal probabilities from volume sampling are a mixture of uniform probabilities and leverage score probabilities, and uniform sampling is well-known to be suboptimal when the leverage scores are highly non-uniform.\n\nA possible recourse is to abandon volume sampling in favor of leverage score sampling\u00a0(drineas2006sampling, ; woodruff2014sketching, ). However, all i.i.d.\u00a0sampling methods, including leverage score sampling, suffer from a coupon collector problem that prevents their effective use at small sample sizes\u00a0(regularized-volume-sampling, ). Moreover, the resulting weight vectors are biased (regarded as estimators for the least squares solution using all responses), which is a nuisance when averaging multiple solutions (e.g., as produced in distributed settings). In contrast, volume sampling offers multiplicative loss bounds even with sample sizes as small as and it is the only known non-trivial method that gives unbiased weight vectors\u00a0(unbiased-estimates, ).\n\nWe develop a new solution, called leveraged volume sampling, that retains the aforementioned benefits of volume sampling while avoiding its flaws. Specifically, we propose a variant of volume sampling based on rescaling the input points to \u201ccorrect\u201d the resulting marginals. On the algorithmic side, this leads to a new determinantal rejection sampling procedure which offers significant computational advantages over existing volume sampling algorithms, while at the same time being strikingly simple to implement. We prove that this new sampling scheme retains the benefits of volume sampling (like unbiasedness) but avoids the bad behavior demonstrated in our lower bound example. Along the way, we prove a new generalization of the Cauchy-Binet formula, which is needed for the rejection sampling denominator. Finally, we develop a new method for proving matrix tail bounds for leveraged volume sampling. Our analysis shows that the unbiased least-squares estimator constructed this way achieves a approximation factor from a sample of size , addressing an open question posed by unbiased-estimates .\n\n#### Experiments.\n\nFigure\u00a01 presents experimental evidence on a benchmark dataset (cpusmall from the libsvm collection libsvm ) that the potential bad behavior of volume sampling proven in our lower bound does occur in practice. Appendix E shows more datasets and a detailed discussion of the experiments. In summary, leveraged volume sampling avoids the bad behavior of standard volume sampling, and performs considerably better than leverage score sampling, especially for small sample sizes .\n\n#### Related work.\n\nDespite the ubiquity of volume sampling in many contexts already mentioned above, it has only recently been analyzed for linear regression. Focusing on small sample sizes, (unbiased-estimates, ) proved multiplicative bounds for the expected loss of size volume sampling. Because the estimators produced by volume sampling are unbiased, averaging a number of such estimators produced an estimator based on a sample of size with expected loss at most times the optimum. It was shown in regularized-volume-sampling\n\nthat if the responses are assumed to be linear functions of the input points plus white noise, then size\n\nvolume sampling suffices for obtaining the same expected bounds. These noise assumptions on the response vector are also central to the task of A-optimal design, where volume sampling is a key technique (optimal-design-book, ; symmetric-polynomials, ; tractable-experimental-design, ; proportional-volume-sampling, ). All of these previous results were concerned with bounds that hold in expectation; it is natural to ask if similar (or better) bounds can also be shown to hold with high probability, without noise assumptions. Concentration bounds for volume sampling and other strong Rayleigh measures were studied in pemantle2014concentration , but these results are not sufficient to obtain the tail bounds for volume sampling.\n\nOther techniques applicable to our linear regression problem include leverage score sampling\u00a0(drineas2006sampling, ) and spectral sparsification\u00a0(batson2012twice, ; lee2015constructing, ). Leverage score sampling is an i.i.d. sampling procedure which achieves tail bounds matching the ones we obtain here for leveraged volume sampling, however it produces biased weight vectors and experimental results (see regularized-volume-sampling and Appendix E) show that it has weaker performance for small sample sizes. A different and more elaborate sampling technique based on spectral sparsification\u00a0(batson2012twice, ; lee2015constructing, ) was recently shown to be effective for linear regression\u00a0(chen2017condition, ), however this method also does not produce unbiased estimates, which is a primary concern of this paper and desirable in many settings. Unbiasedness seems to require delicate control of the sampling probabilities, which we achieve using determinantal rejection sampling.\n\n#### Outline and contributions.\n\nWe set up our task of subsampling for linear regression in the next section and present our lower bound for standard volume sampling. A new variant of rescaled volume sampling is introduced in Section 3. We develop techniques for proving matrix expectation formulas for this variant which show that for any rescaling the weight vector produced for the subproblem is unbiased.\n\nNext, we show that when rescaling with leverage scores, then a new algorithm based on rejection sampling is surprisingly efficient (Section 4): Other than the preprocessing step of computing leverage scores, the runtime does not depend on (a major improvement over existing volume sampling algorithms). Then, in Section 4.1 we prove multiplicative loss bounds for leveraged volume sampling by establishing two important properties which are hard to prove for joint sampling procedures. We conclude in Section 5 with an open problem and with a discussion of how rescaling with approximate leverage scores gives further time improvements for constructing an unbiased estimator.\n\n## 2 Volume sampling for linear regression\n\nIn this section, we describe our linear regression setting, and review the guarantees that standard volume sampling offers in this context. Then, we present a surprising lower bound which shows that under worst-case data, this method can exhibit undesirable behavior.\n\n### 2.1 Setting\n\nSuppose the learner is given input vectors , which are arranged as the rows of an input matrix . Each input vector\n\nhas an associated response variable\n\nfrom the response vector . The goal of the learner is to find a weight vector that minimizes the square loss:\n\n w\u2217\\tiny{def}=argminw\u2208RdL(w),whereL(w)% \\tiny{def}=n\u2211i=1(x\u22a4iw\u2212yi)2=\u2225Xw\u2212y\u22252.\n\nGiven both matrix and vector , the least squares solution can be directly computed as , where is the pseudo-inverse. Throughout the paper we assume w.l.o.g.\u00a0that has (full) rank .222Otherwise just reduce to a subset of independent columns. Also assume has no rows of all zeros (every weight vector has the same loss on such rows, so they can be removed).\n\nIn our setting, the learner is only given the input matrix , while response vector remains hidden. The learner is allowed to select a subset of row indices in for which the corresponding responses are revealed. The learner constructs an estimate of using matrix and the partial vector of observed responses. The learner is evaluated by the loss over all rows of (including the ones with unobserved responses), and the goal is to obtain a multiplicative loss bound, i.e., that for some ,\n\n L(\u02c6w)\u2264(1+\u03f5)L(w\u2217).\n\n### 2.2 Standard volume sampling\n\nGiven and a size , standard volume sampling jointly chooses a set of indices in with probability\n\n Pr(S)=det(X\u22a4SXS)(n\u2212dk\u2212d)det(X\u22a4X),\n\nwhere is the submatrix of the rows from indexed by the set . The learner then obtains the responses , for , and uses the optimum solution for the subproblem as its weight vector. The sampling procedure can be performed using reverse iterative sampling (shown on the right), which, if carefully implemented, takes time (see unbiased-estimates ; regularized-volume-sampling ).\n\nThe key property (unique to volume sampling) is that the subsampled estimator is unbiased, i.e.\n\n E[w\u2217S]=w\u2217,wherew\u2217=argminwL(w).\n\nAs discussed in unbiased-estimates\n\n, this property has important practical implications in distributed settings: Mixtures of unbiased estimators remain unbiased (and can conveniently be used to reduce variance). Also if the rows of\n\nare in general position, then for volume sampling\n\n E[(X\u22a4SXS)\u22121]=n\u2212d+1k\u2212d+1(X\u22a4X)\u22121. (1)\n\nThis is important because in A-optimal design bounding is the main concern. Given these direct connections of volume sampling to linear regression, it is natural to ask whether this distribution achieves a loss bound of times the optimum for small sample sizes .\n\n### 2.3 Lower bound for standard volume sampling\n\nWe show that standard volume sampling cannot guarantee multiplicative loss bounds on some instances, unless over half of the rows are chosen to be in the subsample.\n\n###### Theorem 1\n\nLet be an least squares problem, such that\n\nLet be obtained from size volume sampling for . Then,\n\n lim\u03b3\u21920E[L(w\u2217S)]L(w\u2217)\u22651+n\u2212kn\u2212d, (2)\n\nand there is a such that for any ,\n\n Pr(L(w\u2217S)\u2265(1+12)L(w\u2217))>14. (3)\n\nProof \u00a0In Appendix A we show part (2), and that for the chosen we have (see (8)), where is the -th leverage score of . Here, we show (3). The marginal probability of the -th row under volume sampling (as given by unbiased-estimates-journal ) is\n\n Pr(i\u2208S)=\u03b8\u00a0li+(1\u2212\u03b8)\u00a01=1\u2212\u03b8\u00a0(1\u2212li),\u00a0where\u00a0\u03b8=n\u2212kn\u2212d. (4)\n\nNext, we bound the probability that all of the first input vectors were selected by volume sampling:\n\n Pr([d]\u2286S) (\u2217)\u2264d\u220fi=1Pr(i\u2208S)=d\u220fi=1(1\u2212n\u2212kn\u2212d(1\u2212li))\u2264exp(\u2212n\u2212kn\u2212d\u2211di=1(1\u2212li)\ue150\ue154\ue154\ue153\ue152\ue154\ue154\ue151L(w\u2217)),\n\nwhere follows from negative associativity of volume sampling (see dual-volume-sampling ). If for some we have , then . So for such that and any :\n\n Pr(L(w\u2217S)\u2265(1+12)2\/3\ue150\ue154\ue154\ue153\ue152\ue154\ue154\ue151L(w\u2217))\n\nNote that this lower bound only makes use of the negative associativity of volume sampling and the form of the marginals. However the tail bounds we prove in Section 4.1 rely on more subtle properties of volume sampling. We begin by creating a variant of volume sampling with rescaled marginals.\n\n## 3 Rescaled volume sampling\n\nGiven any size , our goal is to jointly sample row indices with replacement (instead of a subset of of size , we get a sequence ). The second difference to standard volume sampling is that we rescale the -th row (and response) by , where is any discrete distribution over the set of row indices , such that and for all . We now define -rescaled size volume sampling as a joint sampling distribution over , s.t.\n\n q-rescaled size\u00a0k\u00a0volume sampling:Pr(\u03c0)\u223cdet(k\u2211i=11q\u03c0ix\u03c0ix\u22a4\u03c0i)k\u220fi=1q\u03c0i. (5)\n\nUsing the following rescaling matrix we rewrite the determinant as . As in standard volume sampling, the normalization factor in rescaled volume sampling can be given in a closed form through a novel extension of the Cauchy-Binet formula (proof in Appendix B.1).\n\n###### Proposition 2\n\nFor any , and , such that , we have\n\n \u2211\u03c0\u2208[n]kdet(X\u22a4Q\u03c0X)k\u220fi=1q\u03c0i=k(k\u22121)...(k\u2212d+1)det(X\u22a4X).\n\nGiven a matrix , vector and a sequence , we are interested in a least-squares problem , which selects instances indexed by , and rescales each of them by the corresponding . This leads to a natural subsampled least squares estimator\n\n w\u2217\u03c0=argminwk\u2211i=11q\u03c0i(x\u22a4\u03c0iw\u2212y\u03c0i)2=(Q\\sfrac12\u03c0X)+Q\\sfrac12\u03c0y.\n\nThe key property of standard volume sampling is that the subsampled least-squares estimator is unbiased. Surprisingly this property is retained for any -rescaled volume sampling (proof in Section 3.1). As we shall see this will give us great leeway for choosing to optimize our algorithms.\n\n###### Theorem 3\n\nGiven a full rank and a response vector , for any as above, if is sampled according to (5), then\n\n E[w\u2217\u03c0]=w\u2217,wherew\u2217=argminw\u2225Xw\u2212y\u22252.\n\nThe matrix formula (1), discussed in Section 2 for standard volume sampling, has a natural extension to any rescaled volume sampling, turning here into an inequality (proof in Appendix B.2).\n\n###### Theorem 4\n\nGiven a full rank and any as above, if is sampled according to (5), then\n\n E[(X\u22a4Q\u03c0X)\u22121]\u2aaf1k\u2212d+1(X\u22a4X)\u22121.\n\n### 3.1 Proof of Theorem 3\n\nWe show that the least-squares estimator produced from any -rescaled volume sampling is unbiased, illustrating a proof technique which is also useful for showing Theorem 4, as well as Propositions 2 and 5. The key idea is to apply the pseudo-inverse expectation formula for standard volume sampling (see e.g., unbiased-estimates ) first on the subsampled estimator , and then again on the full estimator . In the first step, this formula states:\n\nwhere and denotes a subsequence of indexed by the elements of set . Note that since is of size , we can decompose the determinant:\n\n det(X\u22a4Q\u03c0SX)=det(X\u03c0S)2\u220fi\u2208S1q\u03c0i.\n\nWhenever this determinant is non-zero, is the exact solution of a system of linear equations:\n\n 1\u221aq\u03c0ix\u22a4\u03c0iw=1\u221aq\u03c0iy\u03c0i,fori\u2208S.\n\nThus, the rescaling of each equation by cancels out, and we can simply write . Note that this is not the case for sets larger than whenever the optimum solution incurs positive loss. We now proceed with summing over all . Following Proposition 2, we define the normalization constant as , and obtain:\n\n ZE[w\u2217\u03c0] (1)=(kd)\u2211\u00af\u03c0\u2208[n]ddet(X\u00af\u03c0)2(X\u00af\u03c0)+y\u00af\u03c0\u2211~\u03c0\u2208[n]k\u2212dk\u2212d\u220fi=1q~\u03c0i\n\nNote that in we separate into two parts (subset and its complement, ) and sum over them separately. The binomial coefficient counts the number of ways that can be \u201cplaced into\u201d the sequence . In we observe that whenever has repetitions, determinant is zero, so we can switch to summing over sets. Finally, again uses the standard size volume sampling unbiasedness formula, now for the least-squares problem , and the fact that \u2019s sum to 1.\n\n## 4 Leveraged volume sampling: a natural rescaling\n\nRescaled volume sampling can be viewed as selecting a sequence of rank-1 covariates from the covariance matrix . If are sampled i.i.d. from , i.e. , then matrix is an unbiased estimator of the covariance matrix because . In rescaled volume sampling (5), , and the latter volume ratio introduces a bias to that estimator. However, we show that this bias vanishes when is exactly proportional to the leverage scores (proof in Appendix B.3).\n\n###### Proposition 5\n\nFor any and as before, if is sampled according to (5), then\n\n E[Q\u03c0]=(k\u2212d)I+diag(l1q1,\u2026,lnqn),whereli\\tiny{def}=x\u22a4i(X\u22a4X)\u22121xi.\n\nIn particular, if and only if for all .\n\nThis special rescaling, which we call leveraged volume sampling, has other remarkable properties. Most importantly, it leads to a simple and efficient algorithm we call determinantal rejection sampling: Repeatedly sample indices i.i.d. from , and accept the sample with probability proportional to its volume ratio. Having obtained a sample, we can further reduce its size via reverse iterative sampling. We show next that this procedure not only returns a -rescaled volume sample, but also exploiting the fact that is proportional to the leverage scores, it requires (surprisingly) only a constant number of iterations of rejection sampling with high probability.\n\n###### Theorem 6\n\nGiven the leverage score distribution and the determinant for matrix , determinantal rejection sampling returns sequence distributed according to leveraged volume sampling, and w.p. at least finishes in time .\n\nProof\u00a0 We use a composition property of rescaled volume sampling (proof in Appendix B.4):\n\n###### Lemma 7\n\nConsider the following sampling procedure, for :\n\n \u03c0 s\u223cX (q-rescaled size\u00a0s\u00a0volume sampling), S k\u223c\u00a0\u239b\u239c \u239c \u239c \u239c\u239d1\u221aq\u03c01x\u22a4\u03c01\u20261\u221aq\u03c0sx\u22a4\u03c0s\u239e\u239f \u239f \u239f \u239f\u23a0=(Q\\sfrac12[1..n]X)\u03c0 (standard size\u00a0k\u00a0volume sampling).\n\nThen is distributed according to -rescaled size volume sampling from .\n\nFirst, we show that the rejection sampling probability in line 5 of the algorithm is bounded by :\n\n =det(1sX\u22a4Q\u03c0X(X\u22a4X)\u22121)(\u2217)\u2264(1dtr(1sX\u22a4Q\u03c0X(X\u22a4X)\u22121))d =(1dstr(Q\u03c0X(X\u22a4X)\u22121X\u22a4))d=(1dss\u2211i=1dlix\u22a4i(X\u22a4X)\u22121xi)d=1,\n\nwhere\n\nfollows from the geometric-arithmetic mean inequality for the eigenvalues of the underlying matrix. This shows that sequence\n\nis drawn according to -rescaled volume sampling of size . Now, Lemma 7 implies correctness of the algorithm. Next, we use Proposition 2 to compute the expected value of acceptance probability from line 5 under the i.i.d. sampling of line 4:\n\n \u2211\u03c0\u2208[n]s(s\u220fi=1q\u03c0i)det(1sX\u22a4Q\u03c0X)det(X\u22a4X) =s(s\u22121)\u2026(s\u2212d+1)sd\u2265(1\u2212ds)d\u22651\u2212d2s\u226534,\n\nwhere we also used Bernoulli\u2019s inequality and the fact that (see line 2). Since the expected value of the acceptance probability is at least , an easy application of Markov\u2019s inequality shows that at each trial there is at least a 50% chance of it being above . So, the probability of at least trials occurring is less than . Note that the computational cost of one trial is no more than the cost of SVD decomposition of matrix (for computing the determinant), which is . The cost of reverse iterative sampling (line 7) is also with high probability (as shown by regularized-volume-sampling ). Thus, the overall runtime is , where w.p. at least .\n\n### 4.1 Tail bounds for leveraged volume sampling\n\nAn analysis of leverage score sampling, essentially following (woodruff2014sketching, , Section 2) (which in turn draws from sarlos-sketching, ), highlights two basic sufficient conditions on the (random) subsampling matrix that lead to multiplicative tail bounds for .\n\nIt is convenient to shift to an orthogonalization of the linear regression task by replacing matrix with a matrix . It is easy to check that the columns of have unit length and are orthogonal, i.e., . Now, is the least-squares solution for the orthogonal problem and prediction vector for is the same as the prediction vector for the original problem . The same property holds for the subsampled estimators, i.e., , where . Volume sampling probabilities are also preserved under this transformation, so w.l.o.g. we can work with the orthogonal problem. Now can be rewritten as\n\n (6)\n\nwhere follows via Pythagorean theorem from the fact that lies in the column span of and the residual vector is orthogonal to all columns of , and follows from . By the definition of , we can write as follows:\n\n \u2225v\u2217\u03c0\u2212v\u2217\u2225=\u2225(U\u22a4Q\u03c0U)\u22121U\u22a4Q\u03c0(y\u2212Uv\u2217)\u2225\u2264\u2225(U\u22a4Q\u03c0U)\u22121d\u00d7d\u2225\u2225(U\u22a4Q\u03c0U)\u22121U\u22a4Q\u03c0rd\u00d71\u2225, (7)\n\nwhere\n\ndenotes the matrix 2-norm (i.e., the largest singular value) of\n\n; when is a vector, then is its Euclidean norm. This breaks our task down to showing two key properties:\n\n1. Matrix multiplication:\u2003Upper bounding the Euclidean norm ,\n\n2. Subspace embedding:\u2003Upper bounding the matrix 2-norm .\n\nWe start with a theorem that implies strong guarantees for approximate matrix multiplication with leveraged volume sampling. Unlike with i.i.d. sampling, this result requires controlling the pairwise dependence between indices selected under rescaled volume sampling. Its proof is an interesting application of a classical Hadamard matrix product inequality from hadamard-product-inequality (Proof in Appendix\u00a0C).\n\n###### Theorem 8\n\nLet be a matrix s.t. . If sequence is selected using leveraged volume sampling of size , then for any ,\n\nNext, we turn to the subspace embedding property. The following result is remarkable because standard matrix tail bounds used to prove this property for leverage score sampling are not applicable to volume sampling. In fact, obtaining matrix Chernoff bounds for negatively associated joint distributions like volume sampling is an active area of research, as discussed in\n\nharvey2014pipage . We address this challenge by defining a coupling procedure for volume sampling and uniform sampling without replacement, which leads to a curious reduction argument described in Appendix D.\n\n###### Theorem 9\n\nLet be a matrix s.t. . There is an absolute constant , s.t. if sequence is selected using leveraged volume sampling of size , then\n\nTheorems 8 and 9 imply that the unbiased estimator produced from leveraged volume sampling achieves multiplicative tail bounds with sample size .\n\n###### Corollary 10\n\nLet be a full rank matrix. There is an absolute constant , s.t.\u00a0if sequence is selected using leveraged volume sampling of size , then for estimator\n\n w\u2217\u03c0=argminw\u2225Q\\sfrac12\u03c0(Xw\u2212y)\u22252,\n\nwe have with probability at least .\n\nProof \u00a0Let . Combining Theorem 8 with Markov\u2019s inequality, we have that for large enough , w.h.p., where . Finally following (6) and (7) above, we have that w.h.p.\n\n L(w\u2217\u03c0) \u2264L(w\u2217)+\u2225(U\u22a4Q\u03c0U)\u22121\u22252\u2225U\u22a4Q\u03c0r\u22252\u2264L(w\u2217)+82k2\u03f5k282\u2225r\u22252=(1+\u03f5)L(w\u2217).\n\n## 5 Conclusion\n\nWe developed a new variant of volume sampling which produces the first known unbiased subsampled least-squares estimator with strong multiplicative loss bounds. In the process, we proved a novel extension of the Cauchy-Binet formula, as well as other fundamental combinatorial equalities. Moreover, we proposed an efficient algorithm called determinantal rejection sampling, which is to our knowledge the first joint determinantal sampling procedure that (after an initial preprocessing step for computing leverage scores) produces its samples in time , independent of the data size . When is very large, the preprocessing time can be reduced to by rescaling with sufficiently accurate approximations of the leverage scores. Surprisingly the estimator stays unbiased and the loss bound still holds with only slightly revised constants. For the sake of clarity we presented the algorithm based on rescaling with exact leverage scores in the main body of the paper. However we outline the changes needed when using approximate leverage scores in Appendix F.\n\nIn this paper we focused on tail bounds. However we conjecture that expected bounds of the form also hold for a variant of volume sampling of size .\n\n## References\n\n\u2022 [1] Nir Ailon and Bernard Chazelle. The fast johnson\u2013lindenstrauss transform and approximate nearest neighbors. SIAM Journal on computing, 39(1):302\u2013322, 2009.\n\u2022 [2] Zeyuan Allen-Zhu, Yuanzhi Li, Aarti Singh, and Yining Wang. Near-optimal design of experiments via regret minimization. In Doina Precup and Yee\u00a0Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume\u00a070 of Proceedings of Machine Learning Research, pages 126\u2013135, International Convention Centre, Sydney, Australia, 2017. PMLR.\n\u2022 [3] T\u00a0Ando, Roger A.\u00a0Horn, and Charles R.\u00a0Johnson. The singular values of a hadamard product: A basic inequality. 21:345\u2013365, 12 1987.\n\u2022 [4] Haim Avron and Christos Boutsidis. Faster subset selection for matrices and applications. SIAM Journal on Matrix Analysis and Applications, 34(4):1464\u20131499, 2013.\n\u2022 [5] Joshua Batson, Daniel\u00a0A Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704\u20131721, 2012.\n\u2022 [6] L\u00a0Elisa Celis, Amit Deshpande, Tarun Kathuria, and Nisheeth\u00a0K Vishnoi. How to be fair and diverse? arXiv preprint arXiv:1610.07183, 2016.\n\u2022 [7] L\u00a0Elisa Celis, Vijay Keswani, Damian Straszak, Amit Deshpande, Tarun Kathuria, and Nisheeth\u00a0K Vishnoi. Fair and diverse dpp-based data summarization. arXiv preprint arXiv:1802.04023, 2018.\n\u2022 [8] N.\u00a0Cesa-Bianchi, P.\u00a0M. Long, and M.\u00a0K. Warmuth. Worst-case quadratic loss bounds for on-line prediction of linear functions by gradient descent.\n\nIEEE Transactions on Neural Networks\n\n, 7(3):604\u2013619, 1996.\nEarlier version in 6th COLT, 1993.\n\u2022 [9] Chih-Chung Chang and Chih-Jen Lin.\n\nLIBSVM: A library for support vector machines.\n\nACM Transactions on Intelligent Systems and Technology, 2:27:1\u201327:27, 2011. Software available at http:\/\/www.csie.ntu.edu.tw\/~cjlin\/libsvm.\n\u2022 [10] Xue Chen and Eric Price. Condition number-free query and active learning of linear families. CoRR, abs\/1711.10051, 2017.\n\u2022 [11] Micha\u0142 Derezi\u0144ski and Manfred\u00a0K Warmuth. Unbiased estimates for linear regression via volume sampling. In I.\u00a0Guyon, U.\u00a0V. Luxburg, S.\u00a0Bengio, H.\u00a0Wallach, R.\u00a0Fergus, S.\u00a0Vishwanathan, and R.\u00a0Garnett, editors, Advances in Neural Information Processing Systems 30, pages 3087\u20133096. Curran Associates, Inc., 2017.\n\u2022 [12] Micha\u0142 Derezi\u0144ski and Manfred\u00a0K. Warmuth. Unbiased estimates for linear regression via volume sampling. CoRR, abs\/1705.06908, 2017.\n\u2022 [13] Micha\u0142 Derezi\u0144ski and Manfred\u00a0K. Warmuth.\n\nSubsampling for ridge regression via regularized volume sampling.\n\nIn\n\nProceedings of the 21st International Conference on Artificial Intelligence and Statistics\n\n, 2018.\n\u2022 [14] Amit Deshpande and Luis Rademacher. Efficient volume sampling for row\/column subset selection. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS \u201910, pages 329\u2013338, Washington, DC, USA, 2010. IEEE Computer Society.\n\u2022 [15] Amit Deshpande, Luis Rademacher, Santosh Vempala, and Grant Wang. Matrix approximation and projective clustering via volume sampling. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA \u201906, pages 1117\u20131126, Philadelphia, PA, USA, 2006. Society for Industrial and Applied Mathematics.\n\u2022 [16] Petros Drineas, Malik Magdon-Ismail, Michael\u00a0W. Mahoney, and David\u00a0P. Woodruff. Fast approximation of matrix coherence and statistical leverage. J. Mach. Learn. Res., 13(1):3475\u20133506, December 2012.\n\u2022 [17] Petros Drineas, Michael\u00a0W Mahoney, and S\u00a0Muthukrishnan. Sampling algorithms for regression and applications. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 1127\u20131136. Society for Industrial and Applied Mathematics, 2006.\n\u2022 [18] Valerii\u00a0V. Fedorov, William\u00a0J. Studden, and E.\u00a0M. Klimko, editors. Theory of optimal experiments. Probability and mathematical statistics. Academic Press, New York, 1972.\n\u2022 [19] Mike Gartrell, Ulrich Paquet, and Noam Koenigstein. Bayesian low-rank determinantal point processes. In Proceedings of the 10th ACM Conference on Recommender Systems, RecSys \u201916, pages 349\u2013356, New York, NY, USA, 2016. ACM.\n\u2022 [20] David Gross and Vincent Nesme. Note on sampling without replacing from a finite collection of matrices. arXiv preprint arXiv:1001.2738, 2010.\n\u2022 [21] Nicholas\u00a0JA Harvey and Neil Olver. Pipage rounding, pessimistic estimators and matrix concentration. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 926\u2013945. SIAM, 2014.\n\u2022 [22] Wassily Hoeffding.\n\nProbability inequalities for sums of bounded random variables.\n\nJournal of the American statistical association, 58(301):13\u201330, 1963.\n\u2022 [23] Alex Kulesza and Ben Taskar. k-DPPs: Fixed-Size Determinantal Point Processes. In Proceedings of the 28th International Conference on Machine Learning, pages 1193\u20131200. Omnipress, 2011.\n\u2022 [24] Alex Kulesza and Ben Taskar. Determinantal Point Processes for Machine Learning. Now Publishers Inc., Hanover, MA, USA, 2012.\n\u2022 [25] Yin\u00a0Tat Lee and He\u00a0Sun. Constructing linear-sized spectral sparsification in almost-linear time. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 250\u2013269. IEEE, 2015.\n\u2022 [26] Chengtao Li, Stefanie Jegelka, and Suvrit Sra. Polynomial time algorithms for dual volume sampling. In I.\u00a0Guyon, U.\u00a0V. Luxburg, S.\u00a0Bengio, H.\u00a0Wallach, R.\u00a0Fergus, S.\u00a0Vishwanathan, and R.\u00a0Garnett, editors, Advances in Neural Information Processing Systems 30, pages 5045\u20135054. Curran Associates, Inc., 2017.\n\u2022 [27] Michael\u00a0W. Mahoney. Randomized algorithms for matrices and data. Found. Trends Mach. Learn., 3(2):123\u2013224, February 2011.\n\u2022 [28] Zelda\u00a0E. Mariet and Suvrit Sra. Elementary symmetric polynomials for optimal experimental design. In I.\u00a0Guyon, U.\u00a0V. Luxburg, S.\u00a0Bengio, H.\u00a0Wallach, R.\u00a0Fergus, S.\u00a0Vishwanathan, and R.\u00a0Garnett, editors, Advances in Neural Information Processing Systems 30, pages 2136\u20132145. Curran Associates, Inc., 2017.\n\u2022 [29] Aleksandar Nikolov, Mohit Singh, and Uthaipon\u00a0Tao Tantipongpipat. Proportional volume sampling and approximation algorithms for a-optimal design. CoRR, abs\/1802.08318, 2018.\n\u2022 [30] Robin Pemantle and Yuval Peres. Concentration of lipschitz functionals of determinantal and other strong rayleigh measures. Combinatorics, Probability and Computing, 23(1):140\u2013160, 2014.\n\u2022 [31] Tamas Sarlos. Improved approximation algorithms for large matrices via random projections. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS \u201906, pages 143\u2013152, Washington, DC, USA, 2006. IEEE Computer Society.\n\u2022 [32] Joel\u00a0A. Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, 12(4):389\u2013434, Aug 2012.\n\u2022 [33] David\u00a0P Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends\u00ae in Theoretical Computer Science, 10(1\u20132):1\u2013157, 2014.\n\n## Appendix A Proof of part (2) from Theorem\u00a01\n\nFirst, let us calculate . Observe that\n\n (X\u22a4X)\u22121 =c\ue150\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue153\ue152\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue151(1+n\u2212dd\u03b32)\u22121\u00a0I, andw\u2217 =cX\u22a4y=c1d.\n\nThe loss of any can be decomposed as , where is the total loss incurred on all input vectors or :\n\n Li(w\u2217)=(1\u2212c)2+1c\u22121\ue150\ue154\ue154\ue154\ue153\ue152\ue154\ue154\ue154\ue151n\u2212dd\u03b32c2=1\u2212c,\n\nNote that -th leverage score of is equal , so we obtain that\n\n L(w\u2217)=d(1\u2212c)=d\u2211i=1(1\u2212li). (8)\n\nNext, we compute . Suppose that is produced by size standard volume sampling. Note that if for some we have , then and therefore . Moreover, denoting ,\n\n (X\u22a4SXS)\u22121 \u2ab0(X\u22a4X)\u22121=cI,andX\u22a4SyS=(b1,\u2026,bd)\u22a4,\n\nso if , then and\n\n Li(w\u2217S)\u2265n\u2212dd\u03b32c2=(1c\u22121)c2=cLi(w\u2217).\n\nPutting the cases of and together, we get\n\n Li(w\u2217S) \u2265cLi(w\u2217)+(1\u2212cLi(w\u2217))(1\u2212bi) \u2265cLi(w\u2217)+c2(1\u2212bi).\n\nApplying the marginal probability formula for volume sampling (see (4)), we note that\n\n E[1\u2212bi] =1\u2212Pr(i\u2208S)=n\u2212kn\u2212d(1\u2212c)=n\u2212kn\u2212dLi(w\u2217).\n\nTaking expectation over and summing the components over , we get\n\n E[L(w\u2217S)]\u2265L(w\u2217)(c+c2n\u2212kn\u2212d).\n\nNote that as , we have , thus showing (2).\n\n## Appendix B Properties of rescaled volume sampling\n\nWe give proofs of the properties of rescaled volume sampling which hold for any rescaling distribution . In this section, we will use as the normalization constant for rescaled volume sampling.\n\n### b.1 Proof of Proposition 2\n\nFirst, we apply the Cauchy-Binet formula to the determinant term specified by a fixed sequence :\n\nNext, we compute the sum, using the above identity:\n\n \u2211\u03c0\u2208[n]kdet(X\u22a4Q\u03c0X)k\u220fi=1q\u03c0i =(kd)\u2211\u00af\u03c0\u2208[n]ddet(X\u00af\u03c0)2\u2211~\u03c0\u2208[n]k\u2212dk\u2212d\u220fi=1q~\u03c0i =(kd)\u2211\u00af\u03c0\u2208[n]ddet(X\u00af\u03c0)2\u00a0(n\u2211i=1qi)k\u2212d =(kd)d!\u2211S\u2208([n]d)det(XS)2=k(k\u22121)...(k\u2212d+1)det(X\u22a4X),\n\nwhere the steps closely follow the corresponding derivation for Theorem 3, given in Section 3.1.\n\n### b.2 Proof of Theorem 4\n\nWe will prove that for any vector ,\n\n E[v\u22a4(X\u22a4Q\u03c0X)\u22121v]\u2264v\u22a4(X\u22a4X)\u22121vk\u2212d+1,\n\nwhich immediately implies the corresponding matrix inequality. First, we use Sylvester\u2019s formula, which holds whenever a matrix is full rank:\n\n det(A+vv\u22a4)=det(A)(1+v\u22a4A\u22121v).\n\nNote that whenever the matrix is not full rank, its determinant is (in which case we avoid computing the matrix inverse), so we have for any :\n\n \u2264det(X\u22a4Q\u03c0X+vv\u22a4)\u2212det(X\u22a4Q\u03c0X)\n\nwhere follows from applying the Cauchy-Binet formula to both of the determinants, and cancelling out common terms. Next, we proceed in a standard fashion, summing over all :\n\n Z\u00a0E[v\u22a4(X\u22a4Q\u03c0X)\u22121v] =\u2211\u03c0\u2208[n]kv\u22a4(X\u22a4Q\u03c0X)\u22121vdet(X\u22a4Q\u03c0X)k\u220fi=1q\u03c0i =d!(kd)k\u2212d+1(det(X\u22a4X+vv\u22a4)\u2212det(X\u22a4X))=Zv\u22a4(X\u22a4X)\u22121vk\u2212d+1.\n\n### b.3 Proof of Proposition 5\n\nFirst, we compute the marginal probability of a fixed element of sequence containing a particular index under -rescaled volume sampling:\n\n Z\n\nwhere the first term can be computed by following the derivation in Appendix B.1, obtaining","date":"2021-10-18 23:32:36","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8728430271148682, \"perplexity\": 1160.7032026665813}, \"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-2021-43\/segments\/1634323585215.14\/warc\/CC-MAIN-20211018221501-20211019011501-00442.warc.gz\"}"} | null | null |
In pictures: Sheikha Sana Al Maktoum's visit to Expo 2020
Over the last three months, Expo 2020 Dubai has had over eight million visits from people around the globe.
One of the most recent visitors to the mega-event was Her Highness Sheikha Sana Al Maktoum, who toured the Italy Pavilion earlier this week.
Sharing a series of photos on Instagram from her visit, Her Highness noted how brilliantly the relationship between Italy and the UAE was blended together in the pavilion.
Sheikha Sana also applauded the team behind the pavilion for focussing on sustainability as much of the design included recycled materials.
"It was a pleasure to be welcomed to the Italy Pavillion at Expo 2020," she said on Instagram. "I enjoyed seeing the attention to detail and how they have blended the relationship between the UAE and Italy through the design of the space.
A post shared by H.H. Sheikha Sana Al Maktoum (@hhshsam)
"The innovation and focus on sustainability of this pavilion is commendable, from the citrus peel floors to the ropes made with recycled plastic bottles."
This visit marked Her Highness' first official one of 2022.
At the end of 2021, to mark the UAE's 50th National Day, Sheikha Sana visited the Al Noor Training Centre and spent time with the students of determination.
Discussing the visit, Sheikha Sana described it as an "honour and privilege" to meet the students of the Al Noor Training Centre to celebrate with them and mark 50 years of the UAE.
– For more on luxury lifestyle, news, fashion and beauty follow Emirates Woman on Facebook and Instagram
Images: Sheikha Sana Al Maktoum Instagram | {
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{"url":"https:\/\/socratic.org\/questions\/561a938b11ef6b27250a651a","text":"# Question a651a\n\nOct 11, 2015\n\n${8.3}^{\\circ} \\text{C}$\n\n#### Explanation:\n\nNotice that no mention was made about the number of moles of gas and the volume of the cylinder, which means that you can ssume them to be constant.\n\nIn this case, you know that pressur eand temperature have a direct relationship when the number of moles of gas and the volume of the container are kept constant - this is known as Gay Lussac's Law.\n\nIn other words, if the temperature increases, the pressure increases as well. If the temperature decreases, the pressure decreases as well.\n\nIn your case, the pressure of the sample of gas decreased from $\\text{4.88 atm}$ the first day, to $\\text{4.69 atm}$ the second day.\n\nThis means that the temperature of the sample decreased as well,so you can expect the temperature on th previous day to be a little higher than ${8}^{\\circ} \\text{C}$.\n\nMathematically, Gay Lussac's Law can be written like this\n\n$\\frac{{P}_{1}}{T} _ 1 = {P}_{2} \/ {T}_{2} \\text{ }$, where\n\n${P}_{1}$, ${T}_{1}$ - the pressure and temperature of the gas at an initial state;\n${P}_{2}$, ${T}_{2}$ - the pressure and temperature of the gas at a final state.\n\nPlug in your values and solve for ${T}_{1}$\n\n${T}_{1} = {P}_{1} \/ {P}_{2} \\cdot {T}_{2}$\n\nT_1 = (4.88color(red)(cancel(color(black)(\"atm\"))))\/(4.69color(red)(cancel(color(black)(\"atm\")))) * 8^@\"C\" = 8.324^@\"C\"#\n\nI'll leave the answer rounded to two sig figs, despite the fact that you only gave one sig fig for the temperature of the gas\n\n${T}_{1} = \\textcolor{g r e e n}{{8.3}^{\\circ} \\text{C}}$","date":"2021-06-15 10:24:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 13, \"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\": 0.8544645309448242, \"perplexity\": 356.7811569438732}, \"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\/1623487620971.25\/warc\/CC-MAIN-20210615084235-20210615114235-00637.warc.gz\"}"} | null | null |
The Bill & Melinda Gates Foundation sold all its shares of Apple and Twitter right before the billionaire couple announced their divorce
Ben Gilbert
May 24, 2021, 11:48 AM ·1 min read
Bill and Melinda Gates in 2019. Elaine Thompson/AP
The Bill & Melinda Gates Foundation sold all its shares of Apple and Twitter earlier this year.
In early May, Bill and Melinda Gates announced plans to divorce.
The foundation's trust sold the shares for hundreds of millions of dollars.
Sign up for the 10 Things in Tech daily newsletter.
Weeks before Bill and Melinda Gates announced their divorce in early May, the foundation bearing their names sold off hundreds of millions of dollars' worth of stock in Apple and Twitter.
Securities and Exchange Commission filings from the Bill & Melinda Gates Foundation, first reported by Barron's on Sunday, revealed that the foundation's trust had sold its entire stakes in Apple and Twitter as of the end of March.
Since the divorce announcement, about $4 billion in stock has been transferred from Bill to Melinda. Bill is one of the world's richest people, with a net worth estimated at about $146 billion.
Read more: He took his boss' money and then his head, police say. The disturbing story of the assistant accused of murdering and decapitating a tech CEO in New York.
Bill and Melinda reportedly don't have a prenuptial agreement and are relying on a "separation contract" to divide assets. A prenuptial agreement establishes a contractual agreement in the event of a divorce, but a separation contract is less formal - it's a legal agreement that stipulates each party's rights and obligations, like child support and custody, that doesn't involve a court.
The Gateses' divorce filing said their marriage was "irretrievably broken" and asked that their assets be split according to the separation contract.
Got a tip? Contact Insider senior correspondent Ben Gilbert via email (bgilbert@insider.com), or Twitter DM (@realbengilbert). We can keep sources anonymous. Use a non-work device to reach out. PR pitches by email only, please.
Mount Vernon man indicted for manslaughter in 2021 crash that killed two, injured a third
Lohud | The Journal News
What's inside your licensed pot?
Love Island viewers convinced producers are 'trying to separate' two contestants | {
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The 4th 10 Hours of Messina was a sports car race, held on 25 July 1955 in the street circuit of Messina, Italy.
Final standings
Started: 23
Classified: 6
See also
Messina Grand Prix (auto race that replaced it)
References
External links
La 10 Ore di Messina, la storia
10 Hours of Messina | {
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Q: How to show preceding zero in csv export? I have this block of code that saves info to csv. The problem I'm facing is the preceding 0's are not being shown in excel although they show in notepad.
Is this a programming problem or excel csv conversion problem. Any suggestions are appreciated.
code:
public bool SaveToCSV(DataTable dt, string FileName)
{
try
{
var lines = new List<string>();
string[] columnNames = dt.Columns.Cast<DataColumn>().
Select(column => column.ColumnName).
ToArray();
var header = string.Join(",", columnNames);
lines.Add(header);
var valueLines = dt.AsEnumerable()
.Select(row => string.Join(",", row.ItemArray));
lines.AddRange(valueLines);
File.WriteAllLines(FileName, lines);
return true;
}
catch (Exception ex)
{
MessageBox.Show(ex.Message.ToString());
return false;
}
}
A:
the preceding 0's are not being shown in excel although they show in notepad
You can:
*
*Import (instead of Open) the data in your csv and make sure to set the column(s) that you want treated as text to be imported as text, and not interpreted by Excel as numeric, or;
*change the way you output your csv to include an apostrophe ' prefix to numeric data you want to be treated as text by Excel when it opens the file - e.g. '0123456
Hth.
A: Not the most elegant solution, but if you are interested in adding an All-0's row, consider the following code that simply adds a row of 0s to your CSV:
public bool SaveToCSV(DataTable dt, string FileName)
{
try
{
var lines = new List<string>();
string[] columnNames = dt.Columns.Cast<DataColumn>().
Select(column => column.ColumnName).
ToArray();
var header = string.Join(",", columnNames);
lines.Add(header);
//New Code Starts Here
List<string> zeroList = new List<string>();
for(int i = 0; i < columnNames.Count; i++){
zeroList.Add("0");
}
lines.AddRange(zeroList);
//New Code Ends Here
var valueLines = dt.AsEnumerable()
.Select(row => string.Join(",", row.ItemArray));
lines.AddRange(valueLines);
File.WriteAllLines(FileName, lines);
return true;
}
catch (Exception ex)
{
MessageBox.Show(ex.Message.ToString());
return false;
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,458 |
# Uncivil Agreement
# Uncivil Agreement
# How Politics Became Our Identity
Lilliana Mason
The University of Chicago Press
Chicago and London
The University of Chicago Press, Chicago 60637
The University of Chicago Press, Ltd., London
© 2018 by The University of Chicago
All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations in critical articles and reviews. For more information, contact the University of Chicago Press, 1427 East 60th Street, Chicago, IL 60637.
Published 2018
Printed in the United States of America
27 26 25 24 23 22 21 20 19 18 1 2 3 4 5
ISBN-13: 978-0-226-52440-5 (cloth)
ISBN-13: 978-0-226-52454-2 (paper)
ISBN-13: 978-0-226-52468-9 (e-book)
DOI: 10.7208/chicago/9780226524689.001.0001
Library of Congress Cataloging-in-Publication Data
Names: Mason, Lilliana, author.
Title: Uncivil agreement : how politics became our identity / Lilliana Mason.
Description: Chicago ; London : The University of Chicago Press, 2018. | Includes bibliographical references and index.
Identifiers: LCCN 2017046162 | ISBN 9780226524405 (cloth : alk. paper) | ISBN 9780226524542 (pbk. : alk. paper) | ISBN 9780226524689 (e-book)
Subjects: LCSH: Party affiliation—United States. | Political parties—United States. | Political activists—United States. | United States—Politics and government—21st century.
Classification: LCC JK2271 .M312 2018 | DDC 324.273—dc23
LC record available at <https://lccn.loc.gov/2017046162>
This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).
# Contents
Acknowledgments
ONE / Identity-Based Democracy
TWO / Using Old Words in New Ways
THREE / A Brief History of Social Sorting
FOUR / Partisan Prejudice
FIVE / Socially Sorted Parties
SIX / The Outrage and Elation of Partisan Sorting
SEVEN / Activism for the Wrong Reasons
EIGHT / Can We Fix It?
Appendix
Notes
References
Index
# Acknowledgments
First and foremost I thank the people of the University of Chicago Press for working tirelessly to help this book come into the world. John Tryneski kept this project alive, and Chuck Myers made it real. The anonymous reviewers of the manuscript made it much, much better.
The National Science Foundation funded a portion of my research under grant no. SES-1065054. This made my data collection possible.
None of this work would have happened without the help of my graduate advisors Stanley Feldman and Leonie Huddy, whose continuing advice, support, and faith in me were at times what kept me writing.
My deep and endless thanks go to Rick Lau, who read every chapter of the first draft of this book, providing crucial and insightful guidance and wisdom, and also a desk to write from.
Profound thanks go to many years of attendees of the New York Area Political Psychology meeting, particularly those who commented directly on a distilled version of the theory presented here, including John Bullock, Adam Levine, George Marcus, Tali Mendelberg, Diana Mutz, and Julie Wronski.
To the attendees of the National Capital Area Political Science Association, I thank you for your welcome and your feedback.
For inviting me to their home institutions to present many of these ideas, I am extremely grateful to Nick Valentino, Ted Carmines, Hans Noel, Jon Ladd, Jaime Settle, Carin Robinson, Sean Westwood, Cornell Clayton, and Torben Lütjen.
I would like to thank discussants and reviewers of many versions of the ideas presented here, including those at the University of Göttingen Symposium on the 2016 Elections, the Foley Institute Symposium, the Princeton Center for the Study of Democratic Politics, the University of Michigan Center for Political Studies, the Indiana University Center on American Politics, the Georgetown American Government Seminar, and the University of Maryland American Politics Workshop.
For being reliable and kind colleagues, I thank my mentors Antoine Banks, Mike Hanmer, and Frances Lee, as well as the rest of the extremely supportive Department of Government and Politics at the University of Maryland.
To Bill Bishop, my fairy godfather, you found my research out in the wilds of academic scholarship and helped me spread the word.
For one surprise email early on in this process, I thank Morris Fiorina, who helped me believe that this whole idea was worthwhile.
To my parents: you are the reason I chose academia, and your continuing encouragement and support make every day possible. To my brothers, Joe and Keith: your talented ears helped me write this book in my own voice.
And beyond all others, I thank the people who truly sacrificed their time and my attention. To my family—Dave, Penny, and Mabel: you are the reason I write and the reason I can. This book is dedicated to you. To my daughters in particular: your curiosity, endless enthusiasm, and faith give me hope for the next generation of Americans. It's up to you to fix this. I can't think of anyone more qualified.
# ONE
# Identity-Based Democracy
In the summer of 1954, the social psychologist Muzafer Sherif and his colleagues recruited twenty-two fifth-grade boys from Oklahoma City and sent them to two adjacent campsites in Robbers Cave State Park. The boys were carefully selected to be nearly identical to each other in social, educational, physical, and emotional fitness. They were all white, Protestant, and middle class. None had ever met the others before. They were carefully divided into two equal-sized teams, designed to be similar to each other in every possible way. The two teams came to call themselves the Eagles and the Rattlers, and without knowing it they participated in a three-week-long psychological experiment.
During the first week, the teams were kept separate. The boys on each team grew to know each other and to form, from scratch, a sense of being a group. In the second week, each team learned of the other's existence. Having never laid eyes on the other team, the boys on each side immediately began referring to the others as "outsiders," "intruders," and "those boys at the other end of the camp." They grew impatient for a challenge. The experimenters arranged a tournament between the Eagles and the Rattlers. When they came into contact for the very first time—to play baseball—a member of the Eagles immediately called one of the Rattlers "dirty shirt." By the second day of the tournament, both teams were regularly name-calling and using derogatory terms such as _pigs_ , _bums_ , and _cheaters_ , and they began to show reluctance to spend time with members of the other team. Even boys who were compelled to sit out the competitions hurled insults from the sidelines.
In the next few days, the relations between the teams quickly degraded. The Eagles burned the Rattlers' flag. The Rattlers raided the Eagles' cabin in the middle of the night. The Eagles raided the Rattlers' cabin in the middle of the day. Boys from both sides began to collect rocks to use in combat, fistfights broke out, and the staff decided to "stop the interaction altogether to avoid possible injury" (Sherif et al. 1988, 115). They were sent back to their separate camps. By the end of the second week, twenty-two highly similar boys who had met only two weeks before had formed two nearly warring tribes, with only the gentle nudge of isolation and competition to encourage them.
By the start of the third week, the conflict had affected the boys' abilities to judge objective reality. They were given a task to collect as many beans off the ground as possible. Each boy's collection was viewed by both groups on an overhead projector for five seconds. The campers were asked to quickly estimate the number of beans collected by each child. Every boy estimated more beans for their own teammates than for the children on the opposing team. The experimenters had shown them the same number of beans every time.
The Robbers Cave experiment was one of the first to look at the determinants and effects of group membership and intergroup conflict. It inspired years of increasingly precise and wide-ranging research, looking into exactly how our group memberships shape us, affect our relationships with outsiders, and distort our perceptions of objective reality. The following chapters will discuss many of these results. But the simplicity of the Robbers Cave experiment is itself telling. The boys at Robbers Cave needed nothing but isolation and competition to almost instantaneously consider the other team to be "dirty bums," to hold negative stereotypes about them, to avoid social contact with them, and to overestimate their own group's abilities. In very basic ways, group identification and conflict change the way we think and feel about ourselves and our opponents.
We, as modern Americans, probably like to think of ourselves as more sophisticated and tolerant than a group of fifth-grade boys from 1954. In many ways, of course, we are. But the Rattlers and the Eagles have a lot more in common with today's Democrats and Republicans than we would like to believe. Recently, the presidential campaign and election of Donald Trump laid bare some of the basest motivations in the American electorate, and they provide a compelling demonstration of the theory underlying this book.
The Trump phenomenon is particularly rooted in identity and intergroup competition—something that Trump himself often highlights. In September 2015, then-candidate Trump told a crowd, "We will have so much winning if I get elected that you may get bored with the winning" (Schwartz 2015). Trump's ultimately successful rhetoric, while often criticized for its crudeness and lack of ideological coherence, is consistent in its most important message: we will win. The "we" that is promised to win is a crucial element for understanding the election of Donald Trump and, more broadly, recent politics in the American electorate as a whole.
The election of Trump is the culmination of a process by which the American electorate has become deeply socially divided along partisan lines. As the parties have grown racially, religiously, and socially distant from one another, a new kind of social discord has been growing. The increasing political divide has allowed political, public, electoral, and national norms to be broken with little to no consequence. The norms of racial, religious, and cultural respect have deteriorated. Partisan battles have helped organize Americans' distrust for "the other" in politically powerful ways. In this political environment, a candidate who picks up the banner of "us versus them" and "winning versus losing" is almost guaranteed to tap into a current of resentment and anger across racial, religious, and cultural lines, which have recently divided neatly by party.
Across the electorate, Americans have been dividing with increasing distinction into two partisan teams. Emerging research has shown that members of both parties negatively stereotype members of the opposing party, and the extent of this partisan stereotyping has increased by 50 percent between 1960 and 2010 (Iyengar, Sood, and Lelkes 2012). They view the other party as more extreme than their own, while they view their own party as not at all extreme (Jacobson 2012). In June 2016, a Pew study found that for the first time in more than twenty years, majorities of Democrats and Republicans hold _very_ unfavorable views of their partisan opponents (Pew 2016). American partisans today prefer to live in neighborhoods with members of their own party, expressing less satisfaction with their neighborhood when told that opposing partisans live there (Hui 2013).
Increasing numbers of partisans don't want party leaders to compromise, blaming the other party for all incivility in the government (Wolf, Strachan, and Shea 2012), even though, according to a 2014 Pew poll, 71 percent of Americans believe that a failure of the two parties to work together would harm the nation "a lot" (Pew 2014). Yet, as a 2016 Pew poll reports, "Most partisans say that, when it comes to how Democrats and Republicans should address the most important issues facing the country, their party should get more out of the deal" (Pew 2016).
Democrats and Republicans also view objective economic conditions differently, depending on which party is in power (Enns and McAvoy 2012). In the week before the 2016 election, 16 percent of Republicans and 61 percent of Democrats believed the US economy was getting better. In the week after the election, 49 percent of Republicans and 46 percent of Democrats believed the economy was improving (Gallup 2016).
These attitudes are all strikingly reminiscent of the relations between the Rattlers and the Eagles. Those boys desperately wanted to defeat each other, for no reason other than that they were in different groups. Group victory is a powerful prize, and American partisans have increasingly seen that goal as more important than the practical matters of governing a nation. Democrats and Republicans do not like each other. But unlike the Rattlers and the Eagles, the Democrats and Republicans today make up 85 percent of the American population.
This book looks at the effects of our group identities, particularly our partisan identities and other party-linked identities, on our abilities to fairly judge political opponents, to view politics with a reasoned and unbiased eye, and to evaluate objective reality. I explain how natural and easy it can be for Democrats and Republicans to see the world through partisan eyes and why we are increasingly doing so. Just like the Rattlers and the Eagles, American partisans today are prone to stereotyping, prejudice, and emotional volatility, a phenomenon that I refer to as social polarization. Rather than simply disagreeing over policy outcomes, we are increasingly blind to our commonalities, seeing each other only as two teams fighting for a trophy.
Social polarization is defined by prejudice, anger, and activism on behalf of that prejudice and anger. These phenomena are increasing quickly—more quickly, in fact, than the level of our policy disagreements. We act like we disagree more than we really do. Like the Rattlers and the Eagles, our conflicts are largely over who we think we are rather than over reasoned differences of opinion.
The separation of the country into two teams discourages compromise and encourages an escalation of conflict, with no camp staff to break up the fights. The cooperation and compromise required by democracy grow less attainable as partisan isolation and conflict increase. As political scientist Seth Masket wrote in December 2016, "The Republican Party is demonstrating every day that it hates Democrats more than it loves democracy" (Masket 2016). That is, the election of Donald Trump and the policy and party conflicts his campaign engendered has revealed a preference for party victory over real policy outcomes that has only been building over time.
## The First Step Is to Admit There Is a Problem
In 1950, the American Political Science Association (APSA) assembled a Committee on Political Parties that produced a report arguing for a "responsible two-party system" (American Political Science Association 1950). As they argued, "popular government in a nation of more than 150 million people requires political parties which provide the electorate with a proper range of choice between alternatives of action" (APSA Report 1950, 15). Parties, therefore, simplify politics for people who rightly do not have the time or resources to be political experts. In fact, E. E. Schattschneider argued in 1942 that "political parties created democracy and that modern democracy is unthinkable save in terms of parties" (Schattschneider 1942, 1).
Sean Theriault, in his 2008 book on congressional polarization, described the context of the APSA report this way:
> When the report was released (the 81st Congress, 1950), the average Democrat in the House was less than 3 standard deviations away from the average Republican. In the Senate, the distance was less than 2.25 standard deviations. Little changed in the ensuing 25 years. . . . As a result of both polarization between the parties and homogenization within the parties, by the 108th Congress (2003–4), the average party members were separated by more than 5 standard deviations in the House and almost 5 standard deviations in the Senate. . . . Now, political scientists, in claiming that party polarization has drastic consequences, are offering reforms to weaken the party leadership inside Congress. . . . Although polarized parties may be ugly for the legislative process, they were the prescription for a responsible electorate. No longer are constituents forced to make the complicated vote choice between a liberal Republican and a conservative Democrat. Additionally, voters need not wonder whom to credit or blame for the way that Congress operates. (Theriault 2008, 226)
Political parties are indeed important elements of democracy. Parties simplify the voting decision. The vast majority of American citizens are not, and cannot be expected to be, political experts. They do not read legislation; many do not even know which party is currently in the majority. But most voters have a sense of party loyalty. They know, either through a lifetime of learning, from parental socialization, from news media, or through some combination thereof, that one party is better suited to them. This acts as a heuristic, a cognitive shortcut that allows voters to make choices that are informed by some helpful truth. According to Schattschneider (1942), this is a crucial element of representative democracy.
Even better, when people feel linked to a party, they tend to more often participate in politics, just like sports fans attend a game and cheer. Partisanship, then, is one important link between individuals and political action. It encourages citizens to participate and feel involved in their own democracy.
So why write a book about the problems generated by partisan identity? It should be clarified at the start that this book is not opposed to all partisanship, all parties, party systems, or even partisan discord. There has been, and can be, a responsible two-party system in American politics. Instead, this book explains how the responsible part of a two-party system can be called into question when the electorate itself begins to lose perspective on the differences between opponents and enemies. If the mass electorate can be driven to insulate themselves from their partisan opponents, closing themselves off from cordial interaction, then parties become a tool of division rather than organization. Parties can help citizens construct and maintain a functioning government. But if citizens use parties as a social dividing line, those same parties can keep citizens from agreeing to the compromise and cooperation that necessarily define democracy.
Partisanship grows irresponsible when it sends partisans into action for the wrong reasons. Activism is almost always a good thing, particularly when we have so often worried about an apathetic electorate. But if the electorate is moved to action by a desire for victory that exceeds their desire for the greater good, the action is no longer, as regards the general electorate, responsible.
In the chapters that follow, I demonstrate how partisan, ideological, religious, and racial identities have, in recent decades, moved into strong alignment, or have become "sorted." This means that each party has grown increasingly socially homogeneous. It is not a new finding. Matthew Levendusky (2009) wrote a thorough review of how partisan and ideological identities, in particular, have grown increasingly sorted. Alan Abramowitz (2011) wrote a full summary of the polarization of various demographic groups in the American electorate. Both authors note the increasing divide in the electorate but generally come to the conclusion that, on balance, this sorting or demographic polarization could be read as a source for good, as it has simplified our electoral choices and increased political engagement.
I take a more cautious, even cautionary, view of the effects of the social, demographic, and ideological sorting that has occurred during recent decades. In line with Bill Bishop's (2009) book _The Big Sort_ , I argue that this new alignment has degraded the cross-cutting social ties that once allowed for partisan compromise. This has generated an electorate that is more biased against and angry at opponents, and more willing to act on that bias and anger.
There is a very wide line between a political rally and an angry mob. At some point, however, there must be an assessment of how closely a responsible party can or should approach that line. When parties grow more socially homogeneous, their members are quicker to anger and tend toward intolerance. I argue here that, despite clearer partisan boundaries and a more active public, the polarizing effects of social sorting have done more harm than good to American democracy.
Robert Kagan, a prominent neoconservative, wrote in spring 2016, "Here is the other threat to liberty that Alexis de Tocqueville and the ancient philosophers warned about: that the people in a democracy, excited, angry and unconstrained, might run roughshod over even the institutions created to preserve their freedoms" (Kagan 2016).
As American partisans find themselves in increasingly socially isolated parties, it is worth examining what kind of effects this social isolation may have on their political behavior and sense of civic responsibility.
## Cross-Pressures
For decades, political scientists have understood that the effects of partisanship are mitigated by what are called "cross-cutting cleavages." These are attitudes or identities that are not commonly found in the partisan's party. If a person is a member of one party and also a member of a social group that is generally associated with the opposing party, the effect of partisanship on bias and action can be dampened. However, if a person is a member of one party and also a member of another social group that is mostly made up of fellow partisans, the biasing and polarizing effect of partisanship can grow stronger.
Since the earliest studies of political behavior, scholars have found that those with "cross-pressures" on their partisanship would be less likely to participate in politics. In 1944, Paul Lazarsfeld and his colleagues and, in 1960, Angus Campbell and his colleagues suggested that partisans who identify with groups associated with the opposing party would be less likely to vote (Lazarsfeld, Berelson, and Gaudet 1944; Campbell et al. 1960). Lipset ([1960] 1963) went so far as to call these cross-pressured voters "politically impotent," suggesting that "the more pressures brought to bear on individuals or groups which operate in opposing directions, the more likely are prospective voters to withdraw from the situation by 'losing interest' and not making a choice" (211). Further research found that these voters would be less strongly partisan (Powell 1976), and that these "cross-cutting cleavages" would mitigate social conflict (Nordlinger 1972).
Berelson, Lazarsfeld, and McPhee (1954), in their seminal book _Voting_ , wrote, "For those who change political preferences most readily are those who are least interested, who are subject to conflicting social pressures, who have inconsistent beliefs and erratic voting histories. Without them—if the decision were left only to the deeply concerned, well-integrated, consistently principled ideal citizens—the political system might easily prove too rigid to adapt to changing domestic and international conditions" (316).
While the traditional view of cross-pressured voters is that they are generally uninvolved and uninterested, some of the foundational literature of political behavior suggests that those with cross-cutting social identities are an important segment of the American electorate. Democracy needs these voters. Berelson and colleagues found them to be an important source of flexibility in American policy responses to changing conditions. Not only are cross-pressured voters a source of popular responsiveness, they are also a buffer against social polarization.
Cross-cutting religious, racial, and partisan identities tend to allow partisans to engage socially with their fellow citizens and partisan opponents. On the other end of the social-sorting spectrum, those with highly aligned religious, racial, and partisan identities are less prepared to engage with their partisan opponents.
But we don't have to go back to 1954 to find positive references to cross-pressured partisans. More recently, Lavine, Johnston, and Steenbergen (2012) described another group of responsive voters, looking directly at what happens when a partisan holds some negative opinions about their own party. They call this "partisan ambivalence." In line with prior research, they find that these ambivalent partisans are in fact more likely to defect from the party in voting and, further, that they tend to think more carefully about their political decisions, rather than taking partisan identity as a simple cue. These voters are far more like what is normatively desirable in a voter—they are open to new information. Unfortunately, they are also less likely to participate.
The ambivalent, however, are not the voters I focus on in the current study. Here, rather than looking at a clash between partisans and their evaluations of their own party, I look at the relationship between partisan identities and other social identities that are to greater or lesser degrees associated with the party.
The reason I focus on the clash of identities, rather than the clash between party and attitudes, is that social identities have a special power to affect behavior. First, scholars Betsy Sinclair (2012) and Samara Klar (2014) have found that social environments can dramatically affect partisanship and political behavior. Partisans are responsive to the identities and ideas of the people around them.
Second, and more central to the theme of the book, the identities themselves have psychological effects of their own. Green, Palmquist, and Schickler (2002) make a strong argument for the social elements of partisan identity but explicitly reject the psychological theory of social identity. I believe that this rejection misses out on a wealth of information provided by the social identity literature. I therefore follow in the footsteps of Steven Greene (1999, 2002, 2004), who has repeatedly made the case for using the psychological definition of a social identity to better understand partisanship and political behavior. This is, in fact, the key to truly taking advantage of the cross-cutting-cleavage literature from decades ago. The power of cross-pressures (or the lack thereof) is far easier to see when social-psychological theory is employed to explain it.
This explanation must begin with a look, first, at the psychological effects of holding a single social identity.
## The Origins of Group Conflict
That was the first time we got together and decided we were a group, and not just a bunch of pissed-off guys.
—Mick Mulvaney, Director of the Office of Management and Budget, founding member of the Freedom Caucus (quoted in Lizza 2015)
Humans are hardwired to cling to social groups. There are a few good reasons for us to do so. First, without a sense of social cohesion, we would have had a hard time creating societies and civilizations. Second, and even more basic, humans have a need to categorize (Tajfel et al. 1971). It is how we understand the world. This includes categorizing people. Third, our social categories don't simply help us understand our social environment, they also help us understand ourselves and our place in the world. Once we are part of a group, we know how to identify ourselves in relation to the other people in our society, and we derive an emotional connection and a sense of well-being from being group members. These are powerful psychological motivations to form groups.
However, simple social cohesion creates boundaries between those in our group and those outside it. Marilynn Brewer has argued that as human beings we have two competing social needs: one for inclusion and one for differentiation. That is, we want to fit in, but we don't want to disappear within the group. If we create clear boundaries between our group and outsiders, we can satisfy our needs for both inclusion and exclusion (Brewer 1991). This means that humans are motivated not only to form groups but to form exclusive groups. The exclusivity of group identities isn't necessarily based in animosity. As the psychologist Gordon Allport described in 1954, people automatically tend to spend time with people like themselves. Much of the reasoning for this is simple convenience. He explains, "it requires less effort to deal with people who have similar presuppositions" (Allport [1954] 1979, 17). However, once this separation occurs, we are psychologically inclined to evaluate our various groups with an unrealistic view of their relative merits. This is true of nearly any social group that can exist. One famous experiment makes this abundantly clear.
### Minimal Group Paradigm
In the late 1960s, a social psychologist named Henri Tajfel wanted to know more about the origins of conflict between groups. He grew interested in the work of Muzafer Sherif, who, based on his research at Robbers Cave and other experiments, had formed a theory that discrimination between groups naturally arises out of a simple conflict of interest between them. Tajfel and his colleagues wanted to know whether the conflict of interest was necessary for creating discrimination between groups, or whether intergroup discrimination grew out of something even simpler. They ran a number of experiments in order to find a baseline intergroup relationship in which there were two distinct groups with so little conflict between them that they did not engage in discrimination or bias. The design and outcome of these experiments became known as the minimal group paradigm.
The original baseline condition required that subjects in the experiments remain isolated in a laboratory, unaware of who was in their ingroup or in their outgroup, unable to even see or hear any of the other subjects. The groups were designed to be meaningless and value-free—no group was objectively superior to the other. In one experiment, subjects were shown a number of dots on a screen, and asked to estimate the number of dots. Some were then told they were overestimators, some that they were underestimators. In a second experiment, the subjects were shown a number of abstract paintings and asked to choose their favorites. Some were told that they preferred the paintings of Klee, others that they preferred the paintings of Kandinsky. These group labels were, in fact, randomly assigned.
After being informed of their group label, the subjects were then asked to allocate money to other subjects (not to themselves), each identified only by a subject identification number and a group label. They allocated money by writing numbers on a sheet of paper. In one experiment, they were explicitly invited to choose between two scenarios: (1) everyone receives the maximum amount of money; or (2) the subject's own group receives less than the maximum, but the outgroup receives even less than that. They still had never seen another subject's face. They did not stand to gain any benefit themselves.
Tajfel did not expect to find intergroup discrimination in these experiments. He was looking for a design that generated no discrimination and hoping to slowly add conditions until discrimination was achieved (Turner 1996). He expected that with no conflict, no value differences, no contact, and no personal utility gained from group cohesiveness, the group names would not matter in determining the amount of money allocated at the end of the experiment. He expected the common good of the whole to be more attractive than turning the teams against each other. He was incorrect in this expectation.
Even in the most basic definition of a group, Tajfel and his colleagues found evidence of ingroup bias: a preference for or privileging of the ingroup over the outgroup. In every conceivable iteration of this experiment, people privileged the group to which they had been randomly assigned. Ingroup bias emerged even when Billig and Tajfel in 1973 explicitly told respondents that they had been randomly assigned to two groups, because it was "easier this way." The ingroup bias still appeared, simply because the experimenters distinguished two groups. These respondents were not fighting for tangible self-interest, the money they allocated went to other people, not themselves. They simply felt psychologically motivated to privilege members of their own imaginary and ephemeral group—a group of people they had never met and would never meet, and whose existence they had only learned of minutes earlier. People react powerfully when they worry about a group losing status, even when the group is "minimal."
The ingroup bias that results from even minimal group membership is very deeply rooted in human psychological function and is perhaps impossible to escape. Adults, children, and even monkeys have automatic negative associations with outgroup individuals (Greene 2013). Simply being part of a group causes ingroup favoritism, with or without objective competition between the groups over real resources. Even when there is nothing to fight over, group members want to win.
Tajfel points out that one of the most important lessons of the minimal group experiments is that when the subjects are given a choice between providing the maximum benefit to all of the subjects, including those in their own group, or gaining less benefits for their group but seeing their team win, " _it is the winning that seems more important to them_ " (Tajfel et al. 1971, 172). This is a crucial discovery for understanding American partisan politics today. The privileging of victory over the greater good is a natural outcome of even the most meaningless group label.
These natural, even primal human tendencies toward group isolation and group comparison open the door to group conflict. The human inclination is to prefer and privilege members of the ingroup. The primary result of group membership is simply to hold positive feelings for the ingroup, and no positive feelings toward outsiders. Even this difference can cause discrimination, but it is not distinctly hostile. Under circumstances of perceived threat or competition, however, the preference for the ingroup can lead to outright hostility toward the outgroup, particularly when the competition is a zero-sum game (Brewer 2001a). The Rattlers and Eagles were involved in a zero-sum competition, as are Democrats and Republicans every election. Only one team can win, and the other team loses. This threat of loss will prove to be an essential ingredient in modern polarization.
## Physical Evidence of Group Attachment
It is important to be clear that group identities are not simply factual memberships. Emerging research is finding repeated instances of physical effects of group membership on human bodies and brains. Avenanti, Sirigu, and Aglioti (2010) showed respondents video of hands being pricked by pins. People tended to unconsciously twitch their own hand when watching these videos, except when the hand belonged to a member of a racial outgroup.
Scheepers and Derks (2016) explained that it is possible to observe changes in brain activity within 200 milliseconds after a face is shown to a person, and that these changes depend on the social category of the face. Furthermore, they found that people who identify with a group use the same parts of their brain to process group-related and self-related information, but a different part of the brain to process outgroup-related information.
People learn differently depending on whether an ingroup member or an outgroup member is observing them. Hobson and Inzlicht (2016) found that when learning a new task, a person will learn more slowly if he or she is being observed by an outgroup member.
You can find evidence of group membership in saliva. Sampasivam et al. (2016) found that when people's group identity is threatened, they secrete higher levels of cortisol in their saliva, indicating stress.
Even our emotions are neurally connected to our groups. People's brains respond similarly when people are sad and when they are observing a sad ingroup member, but when they are observing a sad outgroup member, their brains respond by activating areas of positive emotion. As Scheepers and Derks (2016) explain, "favoring the ingroup is not a conscious choice. Instead, people automatically and preferentially process information related to their ingroup over the outgroup" (8).
This is an important point for all of the analyses that follow. Group-based reactions to events and information are not entirely voluntary. A person cannot simply turn off his or her preference for the ingroup. It should not be considered an insult to point out the inherent ingroup bias shared by all humans. Ingroup bias is deeply rooted in the physical body as well as the thoughtful mind, and no person is immune.
## Invented Conflicts
Social identities can alter the way people see the world. Zero-sum conflict between groups is easily exacerbated and can be based in both real and invented conflicts. During the Robbers Cave experiment, the boys from both teams began accusing each other of sabotage that had never occurred. The Rattlers accused the Eagles of throwing trash on their beach (they had forgotten that they themselves had left the trash behind the day before). The Eagles erroneously accused the Rattlers of throwing ice and stones into their swimming hole after one of them considered the water to be colder than the day before, and another stubbed his toe.
Allport ([1954] 1979) explains that group members "easily exaggerate the degree of difference between groups, and readily misunderstand the grounds for it. And, perhaps most important of all, the separateness may lead to genuine conflicts of interest as well as to many imaginary conflicts" (19). Allport's words were meant to describe the conflicts between racial, religious, or class-based groups. The previous passage, however, is almost eerily prescient in its descriptions of the current conflict between Democrats and Republicans in American politics. Partisan conflict today is characterized by an exaggerated and poorly understood difference between the parties, based in both genuine and imaginary conflicts of interest.
Political psychologists Milton Lodge and Charles Taber in 2013 wrote a comprehensive review of the effects of motivated reasoning on voters. Motivated reasoning is the process by which individuals rationalize their choices in a way that is consistent with what they prefer to believe, rather than with what is actually true. Lodge and Taber (2013) write that "political behavior and attitudes are very much a function of the unconscious mechanisms that govern memory accessibility" (1). Motivated reasoning is not exactly "inventing" conflicts, but it is the brain's way of making preexisting attitudes easier to believe. This occurs not by choice, but at a subconscious level in the brain, where the things a person wants to believe are easier to locate than the things that contradict a person's worldview. In this way, imaginary and exaggerated conflicts are very difficult to remedy. The human brain prefers not to revise erroneous beliefs about opponents. Eric Groenendyk (2013) suggests that these often-elaborate justifications in defense of the party can occasionally be broken down by reminding partisans of civic values and a desire for accuracy. The tendency toward motivated reasoning, however, remains prominent.
American politics has always been characterized by real differences between the two parties and by true conflicts of interest. As the APSA committee on responsible two-party government explained, the parties should be distinguishable and unique. They should represent real differences in governing philosophy, so that citizens can choose between them. A partisan's natural inclination, once he or she has chosen sides, is to engage strongly in claiming victory for his or her own side. In fact, politics, along with religion, has long been one of the most famous dinner-party topics to avoid if you want the discussion to remain polite. None of this is the major problem with American political identities today.
The trouble arises when party competitions grow increasingly impassioned due to the inclusion of additional, nonpartisan social identities in every partisan conflict. The American political parties are growing socially polarized. Religion and race, as well as class, geography, and culture, are dividing the parties in such a way that the effect of party identity is magnified. The competition is no longer between only Democrats and Republicans. A single vote can now indicate a person's partisan preference _as well as_ his or her religion, race, ethnicity, gender, neighborhood, and favorite grocery store. This is no longer a single social identity. Partisanship can now be thought of as a mega-identity, with all the psychological and behavioral magnifications that implies.
American citizens currently believe that they are in a partisan competition against a socially homogeneous group of outsiders, sometimes to an exaggerated degree (Ahler and Sood 2016). At a dinner party today, talking about politics is increasingly also talking about religion and race. They are wrapped together in a new way. Social sorting is not simply a score on a scale, it is a general trend of partisan homogenization. Ironically, politics and religion may be increasingly acceptable topics at a dinner party today, because most of our dinner parties include mainly socially and politically similar people. When we limit our exposure to outgroup individuals, the differences we perceive between parties grow increasingly exaggerated, and imaginary conflicts of interest rival genuine ones.
## Why Does This Matter?
In this binary tribal world, where everything is at stake, everything is in play, there is no room for quibbles about character, or truth, or principles. If everything—the Supreme Court, the fate of Western civilization, the survival of the planet—depends on tribal victory, then neither individuals nor ideas can be determinative.
—Charles Sykes, "Charlie Sykes on Where the Right Went Wrong"
Unlike the Rattlers and the Eagles, the Democrats and Republicans aren't fighting over a simple trophy. Their job, as the only two governing parties, is to enact real policies that benefit or harm real people. When winning becomes as important as or more important than the content of those policies, real people feel the consequences.
As American social identities grow increasingly party linked, parties become more influential in American political decision-making, behavior, and emotion. Two separate factors drive these changes. The first is the effect of partisanship on policy opinion itself. Policy opinion is defined here as the collection of attitudes that an individual holds about how the government should (or should not) address particular public problems. It could be argued that partisanship encourages more consistency in political attitudes and that this helps democracy. However, in the extreme this consistency can also be a signal that American voters are no longer thinking independently, that they are less open to alternative ideas. In the latter case, the policy opinions of Americans become a reflexive response to party cues, and deliberation or reasoned disagreement grows increasingly difficult.
The second effect is the main concern of this book, and that is the power of social identities to affect party evaluations, levels of anger, and political activism, _independently of a person's policy opinions_. When megaparties form, social polarization increases in the American electorate. Both social and issue-based polarization have recently been shown to decrease public desire for compromise (Wolf, Strachan, and Shea 2012), decrease the impact of substantive information on policy opinions (Druckman, Peterson, and Slothuus 2013), increase income inequality (Bonica et al. 2013), discourage economic investment and output, increase unemployment, and inhibit public understanding of objective economic information (Enns and McAvoy 2012), among other things. Polarization is generally not considered to be a helpful political development.
The increase in social and issue-based polarization has been blamed on elected officials, the primary system, gerrymandering, the partisan media, and a host of other influences. This book takes account of these generally structural and outward-looking explanations for social polarization but adds to the discussion the possibility that one source of our polarized politics is a psychological motivation that most Americans share. Social polarization is an increasingly intense conflict between our two partisan groups. It is based in the same impulses that drive racial and religious prejudice. And just as in the case of racial or religious prejudice, there are institutional, outward-looking explanations, as well as individual psychological explanations. These inner sources of social polarization are less visible, but they are Americans' responsibility to observe and understand.
As citizens, we may not be able to change the primary rules or tone down the partisan media, but we can begin to understand how much of our political behavior is driven by forces that are not rational or fair-minded. This book lays out the evidence for the current state of social polarization, in which our political identities are running circles around our policy preferences in driving our political thoughts, emotions, and actions. I explain how this came to be, illustrate the extent of the problem, and offer some suggestions on how to bring American politics back to a state of civil competition, rather than a state of victory-centric conflict.
# TWO
# Using Old Words in New Ways
The goal of this book is to examine the effect of social sorting on social polarization. In the social-scientific study of politics the term _polarization_ traditionally describes an expansion of the distance between the issue positions of Democrats and Republicans. The process of polarization is defined by Democrats acquiring more extremely liberal issue positions and Republicans acquiring more extremely conservative issue positions. In the same vein, _sorting_ is usually defined as an increasing alignment between party and ideology, where _ideology_ indicates a set of issue positions or values. The process of sorting is traditionally understood as Democrats holding more consistently liberal issue positions and Republicans holding more consistently conservative issue positions.
In this book, one major goal is to make the point that each of these terms— _polarization_ , _sorting_ , and _ideology_ —include within them both a social meaning and an issue-based meaning. The social definition focuses on people's feelings of social attachment to a group of others, not their policy attitudes. The issue-based definition is limited to individual policy attitudes, excluding group attachments. The fact that these two elements can be separated from each other at all is the basis on which this entire argument rests. In the following pages, I examine literature that supports this division, but for now it is simply important to understand that social attachments and policy preferences, while related, are not the same concept, and can have different downstream effects on political behavior.
Following this principle, I discuss two types of polarization, one that is social (or affective) and one that is issue based. _Social polarization_ refers to an increasing social distance between Democrats and Republicans. This is made up of three phenomena: increased partisan bias, increased emotional reactivity, and increased activism. _Issue-based polarization_ is closer to the traditional understanding of the term _polarization_ , and indicates an increasing distance between the average issue positions of Democrats and Republicans.
Similarly, I discuss two types of ideology, one that is identity based, and one that is issue based. This is described more fully below, but for now it should be made clear that _identity-based_ (or _symbolic_ ) _ideology_ is the sense of belonging to the groups called liberal and conservative, regardless of policy attitudes. _Issue-based_ (or _operational_ ) _ideology_ is a set of policy attitudes and the extent to which they tend to be on the liberal or conservative end of the spectrum.
Finally, I discuss two types of sorting, one that is social, and one that is issue based. _Social sorting_ involves an increasing social homogeneity within each party, such that religious, racial, and ideological divides tend to line up along partisan lines. _Issue-based sorting_ is closer to the traditional understanding of sorting, meaning that Democrats hold liberal issue positions and Republicans hold conservative issue positions.
The difference between _issue-based sorting_ and _issue-based polarization_ is simply that issue-based sorting occurs when partisans hold policy preferences that are increasingly consistent with their party's positions. There are fewer cross-cutting policy attitudes. Issue-based polarization involves the policy preferences of Democrats and Republicans growing increasingly bimodal and moving toward extremely liberal or conservative policy choices. For more than a decade, this distinction has been the subject of debate between, among others, Morris Fiorina (Fiorina, Abrams, and Pope 2005) and Alan Abramowitz (2011), who disagree over whether issue-based sorting or issue-based polarization is occurring in the American electorate at large. I choose here not to engage in this debate but to suggest a new, social-identity-based approach to these familiar ideas.
The common theme here is that in each case—polarization, sorting, and ideology—I am separating the identity-based social elements from the issue-based policy elements. This separation is borne out by the data, and it allows me to explain how American partisans can grow increasingly socially distant from one another even if their policy disagreements are not profound.
## A Different Identity Politics
A traditional view of identity politics takes individual social identities such as race or religion and examines how each identity is capable of driving political behavior in relation to that specific group. As Conover (1984, 761) explains,
> Relatively few Americans think "ideologically" in the sense that they order their political beliefs according to certain basic ideological principles. Thus, as Kinder (1982) has pointed out, the key question is no longer "do people think ideologically?" but rather simply, "how _do_ people think about politics?" In addressing this question one approach is to return to "basics," to go back to those ideas that originally fueled research on political behavior. _And, one of the more appealing of those is the notion that people's ties to various groups help to structure their political thinking_. (emphasis added)
Using individuals' ties to their distinct social groups has in fact been a productive way to help political scientists understand how Americans organize their political views. Klandermans (2014), for example, explains that "collective identity becomes politically relevant when people who share a specific identity take part in political action on behalf of that collective" (2). In other words, social identities translate into political ones when the group expresses political demands. At times, even as a replacement for ideological sophistication, a strong racial or religious identity can motivate individuals to take political action on behalf of racial or religious issues, respectively. Yet these are inherently limited entries into politics because they force us to consider each identity in isolation from the other identities that compose a person's worldview.
This book represents a revised approach to identity politics in the sense that these single social identities not only have effects on politics in isolation but they have significantly different effects when understood in relation to each other. Imagine how much more intense the Robbers Cave conflict would have been had the Rattlers all been Catholic, northern, and white, while the Eagles were Protestant, southern, and black.
A single group identity can have powerful effects, but multiple identities all playing for the same team can lead to a very deep social and even cultural divide. Those with cross-cutting partisan, religious, and racial identities are more likely than socially sorted citizens to welcome the opposing team into their lives and to consider them as fellow citizens. Identity politics is a far more powerful concept if we consider how a collection of identities is working in concert, rather than isolating each one and examining them in turn. The relationship between identities is also identity politics, and it may be a more powerful way to understand political involvement. Not only do our identities work together in powerful ways but a threat to one identity makes it easier to dislike multiple additional outgroups.
The American electorate has sorted itself into two increasingly homogeneous parties, with a variety of social, economic, geographic, and ideological cleavages falling in line with the partisan divide. This creates two megaparties, with each party representing not only policy positions but also an increasing list of other social cleavages. Parties, then, draw convenient battle lines between an array of social groups. Isolation and competition, the two sources of intergroup conflict in the Robbers Cave experiment, increase between the two parties. Policy preferences, over time, take a back seat to the team loyalty that is bound to grow out of these increasingly homogeneous and isolated partisan collections. Remember that the Rattlers and the Eagles were competing only for a trophy. Neither pursued any agenda but victory.
## Identity and Policy
Theoretically, a democracy should represent the will of its citizens. In the most common understanding of American democracy, citizens' opinions regarding policies are related to their own self-interest or values. These citizens evaluate which party is closest to their own position. They support that party through voting or activism and, ideally, change parties when the other party moves closer to their position. V. O. Key, in 1966, described an electorate "moved by concern about central and relevant questions of public policy, of governmental performance, and of executive personality" (7). At the very least, according to E. E. Schattschneider ([1960] 1975), democracy is defined as a "competitive political system in which competing leaders and organizations define the alternatives of public policy in such a way that the public can participate in the decision-making process" (141). Informed public choice over policies is an ideal element of a well-functioning democracy. But it isn't exactly how American democracy works. In fact, this view of American democracy is what Christopher Achen and Larry Bartels (2016a) call the "folk theory" of democracy.
Self-interest and political values do enter into policy opinions, but they are not the only ingredients. Quietly, behind the scenes of reasoned, analytical thought, some subtle but powerful forces are at work. Primal psychological influences such as motivated reasoning and social identity are capable of shifting and sometimes entirely determining the policies that citizens support. As Achen and Bartels (2016b) explain, "Decades of social-scientific evidence show that voting behavior is primarily a product of inherited partisan loyalties, social identities and symbolic attachments. Over time, engaged citizens may construct policy preferences and ideologies that rationalize their choices, but those issues are seldom fundamental."
Partisans edit their list of reasons for holding particular attitudes in order to defend the position that is faithful to the party. More often than not, citizens do not choose which party to support based on policy opinion; they alter their policy opinion according to which party they support. Usually they do not notice that this is happening, and most, in fact, feel outraged when the possibility is mentioned. All citizens want to believe that their political values are solid and well reasoned. More often, though, policy attitudes grow out of group-based defense. Partisanship muddies the folk pathway from interests to outcomes, sometimes sending a person in a wrong direction or further down a path than self-interest and values alone would dictate.
In this identity-based democracy, arguments over policy are partly about important issues of the day and partly about which side is winning. Political action is driven not only by policy concerns but also, powerfully, by the need to feel victorious. Parties and their policies are evaluated unfairly, with a biased and distorted eye. Citizens are easily angered, somewhat by policy defeats but more intensely by party defeats. In American politics, individual interests and partisanship have always vied for influence in determining political behavior. Recently, however, political identities have been increasingly dominant.
## Identity and Ideology
Importantly, the distinction between social identity and policy preference does not mean that ideology cannot have a social component. There is a difference between a Democrat who holds increasingly liberal policy positions and a Democrat who increasingly self-identifies as a liberal. In the first case, the Democrat's policy positions are changing. He or she is changing his or her mind about whether affirmative action is fair, welfare is helpful, or health care should be supported by government. This is a policy-driven change.
In the second case, when a Democrat increasingly identifies with a group called liberals, the individual's group identity shifts. The individual's policy opinion might also shift, although not necessarily. This seems counterintuitive, but it is a concept key to the entire scheme of the book and must therefore be made very explicit. Feeling more strongly connected to a group called conservatives does not automatically mean that a person holds more conservative policy positions. In a study of Americans that I conducted in 2011 (described more fully in chapter 3), conservative identification was correlated with conservative policy positions at only r = 0.24, suggesting that there is a weak positive relationship between the two. In other words, those with intense conservative identities do tend to have more intensely conservative policy positions, but one value cannot be precisely predicted from the other. Similarly, liberal identification was correlated with liberal policy positions at r = 0.25.
These are significant correlations, but they leave a great deal of wiggle room between identifying with an ideological group and holding ideologically extreme or consistent policy positions. Ideology is not simply a system of values and preferences that constrain policy positions. It is also an identity that, like party identity, can guide political behavior without relying on policy preferences.
Other scholars have found this to be the case. Christopher Ellis and James Stimson in their 2012 book discuss the difference between what they call symbolic ideology and operational ideology. The first is an identification with liberals or conservatives, and the second is a list of policy positions and values. Americans, it seems, are generally liberal in our policy positions and values, but generally conservative in what we like to call ourselves. Going back to the classic work of Philip Converse in 1964, American policy attitudes are relatively unrelated to each other, in the sense that there is little understanding of what ideological consistency, or constraint, requires. It is important to understand that American ideological _identities_ are not synonymous with policy preferences. Identities function as the boys at Robbers Cave demonstrated, by affecting prejudice, emotion, and collective action. Policy positions, on the other hand, are caught up somewhere between individual values and interests, the policies that our group leaders demand, and whatever misunderstandings emerge between the two.
In the following pages, when I discuss partisan, religious, or racial identities, the meaning of each will be clear. These are all simple identifications with a group. However, the meaning of ideological identities can easily be confused. Ideology has, in fact, been an embattled concept essentially since its first definition. However, as Frances Lee said in 2009, "The difficulty in devising operational definitions of ideology has not prevented the concept from becoming central to political science" (50). The key point of this book depends upon a clear divide between ideological identity and the collection of policy positions and values that is often referred to as ideology. When I refer to ideology, it will be in reference to ideological identity, or the intensity with which individuals identify with the groups that are labeled liberals or conservatives. This is distinct from the set of individual policy preferences that each person holds, which will generally be referred to as policy attitudes or issue positions. The fact that the two can be separated is a large part of why American partisans have become more biased, intolerant, angry, and politically active than their policy disagreements can explain.
As American partisan and ideological identities have grown more sorted, partisans have grown more intolerant of their political opponents. This new prejudice and distrust does not come simply from more extreme and intense policy disagreements. It comes out of the simple power of two or more social identities lining up together. The Catholic Rattlers and Protestant Eagles would fight harder for the trophy, even though the trophy remained the same. This is what has happened to American politics. The Rattlers and the Eagles have grown racially, religiously, geographically, and ideologically divided. They now have more to fight for, not in terms of the trophy itself, but in terms of their own commitment to the group and the stakes of losing. These stakes include the potential sense of humiliation in seeing your group be the loser. As multiple groups line up behind one party or the other, they all win or lose together. The humiliation of loss is amplified. Victory, then, becomes more important than policy outcomes. Even when both sides hold the same policy positions, the priority is often to make sure the "dirty shirts" don't win.
## Three Elements of Social Polarization
Social identities generate distinct psychological and behavioral outcomes. Three of these make up social polarization.
First, in line with Tajfel and Turner's social identity theory, when two groups are in a zero-sum competition, they treat each other with bias and even prejudice. The first element of social polarization is therefore partisan prejudice.
Second, those who identify with a social group are more likely to take action to defend it. When a group's status is threatened, a strongly identified group member will fight to maintain the status of the group. This group member's individual sense of esteem is tied to the group's status, and therefore any reduction in that status would be painful to experience. In other words, when a party might lose, a strongly identified partisan will take action to defend the group. This political action is the second element of social polarization.
Third, one outgrowth of social identity theory is intergroup emotions theory, which specifies that group members can and do feel emotions on behalf of the group (Mackie, Devos, and Smith 2000). In particular, the most strongly identified group members will feel heightened anger in the face of a threat to the group and greater enthusiasm when the group is victorious. This emotional reactivity is the third element of social polarization. Through the remainder of this book, I explain how a well-sorted set of social and partisan identities are uniquely capable of motivating these three types of polarization.
# THREE
# A Brief History of Social Sorting
The chief oppositions which occur in society are between individuals, sexes, ages, races, nationalities, sections, classes, political parties and religious sects. Several such may be in full swing at the same time, but the more numerous they are the less menacing is any one. _Every species of conflict interferes with every other species in society at the same time, save only when their lines of cleavage coincide; in which case they reinforce one another_. . . . A society, therefore, which is riven by a dozen oppositions along lines running in every direction, may actually be in less danger of being torn with violence or falling to pieces than one split along just one line.
—Edward Alsworth Ross, _The Principles of Sociology_
For the past hundred years, sociologists such as Edward Alsworth Ross (1920) have understood that a society is sewn together by the complexity of its social divisions. Though the nature of humans is to categorize themselves into self-contained groups, and the nature of those groups is to produce division and conflict, societies are stabilized by the great number and diversity of group divisions. The more divisions there are, and the less organized those groups are around any one division, the more peaceful and cooperative a society can be. Each group conflict is tamped down by a separate group allegiance. A wealthy Republican can find common ground with a working-class Democrat by joining together at a Protestant church. Once the chaotic mess of group loyalties begins to organize itself around a single line of cleavage, however, society is in danger of "falling to pieces," according to Ross's 1920 account.
Political scientists have agreed with this view of the roots of a stable society, using the concept to explain the unique stability of the American political system. Seymour Lipset argued in 1960 that "the available evidence suggests that the chances for stable democracy are enhanced to the extent that groups and individuals have a number of cross-cutting, politically relevant affiliations" (77). In his 1981 analysis of the American political system, Robert Dahl described the "Normal System" of democracy as one of "crosscutting cleavages and low polarization" (284). He praised this system's democratic cooperation and compromise as being assured by two conditions. First, party loyalties are not related to social, economic, or geographical differences. Second, party loyalties are not consistently related to policy opinion differences.
Unfortunately, the conditions for political and social stability that characterized the last century are increasingly unmet in American politics today. The sorting of American social groups into two partisan camps has intensified in recent decades, leading to a distinct decrease in the number of cross-cutting cleavages. Social cleavages today have become significantly linked to our two political parties, with each party taking consistent sides in racial, religious, ideological, and cultural divides.
Decades ago, social divisions between Americans over party, ideology, religion, class, race, and geography did not align neatly, so that particular social groups were friends in some circumstances and opponents in others. In the South, strong geographical attachments to the Democratic Party were unaligned with ideological differences, so that southern, Democratic, and conservative identities were tied together for many Americans. These alignments pitted the generally more liberal Democratic identity against a strong conservative identity, allowing southern Democrats to feel some compassion and understanding for both Republicans and liberals. Similarly, those holding liberal, northern, and Republican identities were motivated to cooperate occasionally with Democrats and conservatives. After all, they often needed to work together with them, sometimes within their own parties, in order to accomplish their goals.
In the 1950s, when the APSA commission called for two responsible parties, the Democrats and Republicans in government were sending mixed messages to the electorate. To be clear, this was not a time of social peace. McCarthyism, the red scare, and fights over civil rights were only a few of the decade's major conflicts. But due to the parties' many cross-cutting cleavages, the voters received ambiguous cues, making it more difficult to make clear electoral choices. The upside of this, however, was that since voters did not receive clear cues about their partisan ingroups and outgroups, they did not treat their fellow citizens as enemies simply because of their party affiliation. Plenty of racial and anticommunist animosity existed, but these did not perfectly match partisan lines. Voters, therefore, could engage in social prejudice and vitriol, but this was decoupled from their political choices.
The 1950s, while certainly not a golden era of social justice, were a time of less social polarization between American partisans. Cross-cutting cleavages, or conflicting party-linked affiliations, aligned partisans with people not entirely like themselves. Conservative Democrats were on the same team as liberal Democrats. Though their ideological identities, and also policy platforms, were in distinct opposition, they were all Democrats, and their party won or lost together. They had superordinate goals. They also had a number of psychological motivations to be charitable to one another, which are elaborated in chapter 4.
The social cleavages between the parties during this time were messily arranged. Not only did party and ideology conflict in the 1950s, but party did not align with class in the South—upper-class southerners were most likely to be Democrats. In the nonsouthern states, the opposite was true—upper-class northerners were more likely to be Republican (Nadeau and Stanley 1993). Since then, the relationship between class and party in the South has come to match and even surpass the class-party alignment in the rest of the country. By 2000, class-party alignment had increased in all of the American states, but the largest increase has been in the South, leading to a highly class-consistent set of parties (Nadeau et al. 2004).
Today, Democrats and Republicans have a lot more information about who their social and partisan enemies are, and have little reason to find common ground. They have become increasingly homogeneous parties, with Democrats now firmly aligned with identities such as liberal, secular, urban, low-income, Hispanic, and black. Republicans are now solidly conservative, middle class or wealthy, rural, churchgoing, and white. These identities are increasingly aligned so that fewer identities affiliated with either party are also associated with the other side. White, religious, and conservative people have little incentive to reach across to the nonwhite, secular, and liberal people in the other party. What superordinate goals do they have? In which places do they mix with opposing partisans? Few of today's salient social groups help either party to reach across the aisle.
## Ideological Sorting
The first thing that pops into most political scientists' minds when they hear the word "sorting" is specifically issue-based partisan sorting. Democrats have become more liberal and Republicans have grown more conservative in their policy preferences. Confusingly, this type of sorting is typically described with a measure that asks survey respondents to place themselves on a seven-point scale, ranging from extremely conservative to extremely liberal. Pamela Conover and Stanley Feldman found in 1981 that this measure largely assesses ideological identification, but it nonetheless remains the most common measure of ideology, whether understood as an identity or a set of issue positions. It is therefore useful to examine trends in this variable before attempting to break ideology down into its identification and issue-based elements. Figure 3.1 illustrates the self-reported ideology of Democrats and Republicans in the American electorate over time. Ideological sorting among legislators has increased substantially more than in the electorate as a whole, but the sorting I am most interested in is that of the general population. These are the people who ultimately have the power to change American politics. Whether they wish to do so is partially determined by the sorting I describe here.
In figure 3.1, Democrats and Republicans in the American National Election Studies (ANES) have been growing increasingly ideologically distinct since at least 1972, when ideology was first included in the study. As noted above, ideology is measured as a seven-point scale allowing respondents to identify themselves as extremely conservative, conservative, slightly conservative, moderate, slightly liberal, liberal, or extremely liberal (recoded to range from 0 to 1). The score of 0.5 on the ideology scale marks the purely moderate partisans. Since 1972, Democrats have grown about 7 percentage points more liberal on the total ideology scale, while Republicans have grown about 10 percentage points more conservative. Republicans have also been more distinctly ideological than Democrats during the entire span of time, with scores farther from the moderate midpoint of the scale. This type of figure, though, does not clarify whether partisans are talking about their ideological identities or their policy positions—two separate phenomena. Fortunately, the ANES provides evidence that allows for separating these two types of ideology.
Figure 3.1. Traditional measure of ideology
Note: Data drawn from the American National Election Studies cumulative file 1948–2008 (fully weighted) and the 2012 file (fully weighted). Ideology item available from 1972 and later. Democrats and Republicans include independents who lean toward one party.
Figure 3.2 demonstrates what happens when, instead of placing themselves on a seven-point ideological scale, partisans are asked whether they feel particularly close to liberals and conservatives. They are asked whether liberals and conservatives are "people who are most like you in their ideas and interests and feelings about things." This comes as close as possible in the ANES data to a measure of a social identity. Unfortunately this measure was available only in select years, but can be examined in 1972, 1992, and 2000. When measured this way, Democratic social identification with liberals increased by about 24 percentage points between 1972 and 2000, while Republican social identification with conservatives increased by about 35 percentage points. Here again, Republicans are more strongly socially identified with conservatives than Democrats are identified with liberals, and the Republican increase in ideological identity strength over time is substantially larger than that of Democrats. Among Republicans in 2000, more than 60 percent of them point to conservatives as the people most like them. In 1972 and 1992, this figure was only around 25 percent. Among Democrats, a similar pattern has occurred, though on a smaller scale. By 2000, 35 percent of Democrats point to liberals as the people most like them, compared to 11 percent in 1972. By 2000, Republicans are nearly twice as likely as Democrats to feel socially connected to their ideological group.
Figure 3.2. Identity-based ideology
Note: Data drawn from the American National Election Studies from 1972, 1992, and 2000. Democrats and Republicans include independents who lean toward one party. Vertical axis represents the percentage of Democrats/Republicans who identify with liberals/conservatives, respectively.
Specifying ideology as a social identity makes it clear that identity-based ideological sorting is increasing more robustly than the traditional measure of ideology can indicate. Democrats and especially Republicans are feeling closer to liberals and conservatives, considering them to be groups of people that are most like them "in their ideas and interests and feelings about things." To a much larger extent than in 1972, partisans in 2000 had found their ideological clans.
In comparison, considering ideology as a set of issue positions also shows a growing partisan divide, though a smaller one. Figure 3.3 depicts the difference between the average Democratic and Republican positions on a variety of issues, with all available issues combined into a single scale at the bottom of each graph. Unfortunately, only three of the six issues are available in 1972. The difference between the three-issue scale in 1972 and the six-issue scale in 2012 is an increase of about 15 percentage points in the difference between Democratic and Republican policy positions in the electorate at large. A more fair assessment is to compare the six-item scale in 1982 with the same six-item scale in 2012, indicating a 16 percentage point increase in the difference between the two parties' policy positions. These partisan policy differences are not much different from the partisan-ideological differences as measured with the seven-point ideology measure. Both measures, however, produce significantly smaller effects than the identity-based ideological differences between the parties. The increase in partisan-ideological-identity differences is more than twice as large as the increase in partisan policy differences.
Figure 3.3. Difference between the average policy positions of Democrats and Republicans
Note: Data drawn from the American National Election Studies cumulative file 1948–2008 (fully weighted) and the 2012 ANES file (fully weighted). Democrats and Republicans include independents who lean toward one party. Values represent the difference between the mean Democratic and mean Republican position on each issue. Bars to the left of zero indicate that Democrats and Republicans differ in a party-inconsistent direction.
The concept of ideology as a set of policy preferences or a placement of a person on a seven-point scale obscures the depth of the increasing ideological divide between Democrats and Republicans. The average opinions of the two parties have certainly diverged. A much larger division, however, is growing between them in their sense of themselves as liberals and conservatives. Democrats and Republicans have chosen ideological teams, and their sense of belonging to one side has divided them more powerfully than their policy differences have. This sense of social division is one that radiates out to many other social cleavages between the parties. More than simply disagreeing, Democrats and Republicans are feeling like very different kinds of people.
## Theories of Sorting
The explanations for the increase in ideological sorting (vaguely defined, but generally referring to issue-based polarization) in the American electorate are varied and have been widely examined. A number of theories attempt to explain issue-based sorting, beginning with an issue-based story and ending with a more complete sorting of multiple social identities along partisan lines.
The most popular explanation for issue-based sorting involves the changes in the Democratic Party that occurred as a result of the civil rights movement, culminating in the Civil Rights Act of 1964. James Sundquist, in his study of partisan realignments in America, argues that the transition of the South from Democratic to Republican began "on the day in 1948 that President Truman sent Congress his civil rights proposals, reversing the moderate policy of his predecessor, Franklin Roosevelt" (Sundquist 1983, 353). This caused a deep division within the Democratic Party, which was papered over for a few years but reopened in 1954 after the Supreme Court ruled against segregation in _Brown v. Board of Education_. The northern Democrats strongly supported this decision and demanded that it be carried out, while the southern Democrats engaged in "massive resistance." By the time of the 1960 Democratic convention, the pro–civil rights forces were loudly advocating for reform. President Kennedy remained relatively centrist to placate the intransigent southern contingent of the Democratic Party. After Kennedy's death, however, Lyndon Johnson, the first president from a southern state, chose to align the Democratic Party with the northern contingent on the issue of civil rights.
As a result, the southern states voted heavily for Republican Barry Goldwater in the 1964 election. Stanley, Bianco, and Niemi (1986) point out that, though this was technically a policy difference, the timing of partisan changes by blacks and southern whites in the 1960s "suggest[s] a racially inspired shift in the group basis of the Democratic party" (975). The new alignment of the Democratic Party with the rights of blacks caused vast numbers of conservative white southern Democrats to move gradually to the Republican Party, leaving average Democrats more liberal and average Republicans more conservative. This racial-policy shift in the Democratic Party and subsequent realignment of conservative southern Democrats is generally understood to be one major reason that Democrats and Republicans today are more ideologically pure than they were in 1950. In this explanation, a racial-policy change in the Democratic Party is the root of all of today's partisan polarization. In some sense, this may be true. Without the civil rights policy change, nothing that came later would likely have happened in the same way. But if a policy change started the path to a more socially sorted nation, the effects of that change drew American politics further away from policy and toward an increasingly social partisan divide.
## Social Sorting
During this time of change in the Democratic Party, the Republican Party experienced major changes as well. Not only were southern whites leaving the Democratic Party, white citizens all over the country were growing more Republican. As Carmines and Stimson said in 1982, "The 1964 presidential election not only widened the gulf between the Republican Party and the Democratic Party on a number of policy issues, but it drove a wedge between the parties on issues of race for the first time in this century" (6). By 1988, 49 percent of northern whites identified with the Republican Party, a 17 percentage point increase in the Republican partisanship of northern whites since 1972 (Carmines and Stanley 1992). In places where race was highly salient to white voters (like the heavily black South), they changed their partisanship to match their race (Giles and Hertz 1994).
After the civil rights debates of the 1960s, black voters had clear policy reasons to be loyal to the Democratic Party, and racially biased white voters had clear reasons to prefer the Republican Party. But this policy-based affiliation has since grown into a distinctly social partisan divide. As of 2013, party identity is strongly predicted by racial identity, not racial-policy positions (Mangum 2013). The parties have grown so divided by race that simple racial identity, without policy content, is enough to predict party identity. The policy division that began the process of racial sorting is no longer necessary for Democrats and Republicans to be divided by race. Their partisan identities have become firmly aligned with their racial identities, and decoupled from their racial-policy positions.
As this process of racial sorting was in full swing, another social cleavage lined up along the partisan divide. The formerly nonpolitical religious right found common cause with the Republican leadership. In 1989, after a failed presidential bid by televangelist Pat Robertson, a group that called itself the Christian Coalition emerged. It was made up of conservative Christians, many of whom identified as evangelical or fundamentalist (Schnabel 2013). The Christian Coalition grew steadily in support and influence in the following years. By 1995 they had publicly paired with prominent Republican senators and congressmen, including the Republican Speaker of the House, Newt Gingrich, to announce the Contract with the American Family, which sought the legislation of Christian morality. All of the Contract with the American Family goals were incorporated into the Republican Party platform by 2000. The Republican Party became firmly affiliated with conservative Christianity. A new religious/nonreligious gap between the Republicans and Democrats grew as their activist bases responded to the new religious difference between the parties, leading candidates to take more extreme positions on religious issues and, thus, changing the public perceptions of the two parties (Layman 2001). Once the public was aware of a religious divide between the parties, their electoral composition changed, divided by religion-linked issues and images that both reflected and reinforced religious identities. Gradually, the religious/secular divide was added to the growing list of social cleavages drawing the parties apart.
The changing social alignments in the Democratic and Republican parties are clearly visible in figure 3.4. This figure uses ANES data to tell the story of how the two parties have changed and grown increasingly socially distinct. To do this, I first identify the percentage of each constituent group in each party (e.g., union members made up 31 percent of the Democratic Party in 1952 but only 22 percent of the Republican Party). Each bar represents the difference between the two parties on this measure (in the union example, I subtract 0.22 from 0.31 and end up with −0.09). To the right of the zero line, the Republican Party has a higher percentage of group members than the Democratic Party, and the degree of the difference is represented by the length of the line. To the left of the zero line, the Democratic Party has a higher percentage of group members than the Republican Party. Figure 3.4 presents these partisan social differences every twenty years, beginning in 1952.
Figure 3.4. Social-group sorting
Note: Bars represent the difference between the percentage of group members in the Republican Party and the percentage of group members in the Democratic Party (including partisan leaners). Income percentile is the difference between the mean income percentile of Republicans and Democrats. Data are drawn from the ANES cumulative data file for the 1952, 1972, and 1992 graphs (fully weighted) and from the 2012 ANES (fully weighted) for the 2012 graph. There is also a gender gap between the parties, but this does not follow a clear pattern over time. Though Democrats were slightly more male (by 3 points) in 1952, Republicans have been consistently more male ever since, with a partisan gap of 4 points in 1972, 10 points in 1992, and 7 points in 2012.
### 1952
Though I argue that social sorting has increased in recent decades, it does not mean there were no sociopartisan alignments in the past. In 1952, the two parties were each affiliated with different social groups. Republicans were significantly more Protestant than Democrats, and Democrats were more strongly southern, Catholic, and union affiliated. To a small degree, Republicans were more white and Democrats were more black. The difference between then and now is that although the parties did attract separate social groups, the extent of the partisan division between these groups was relatively small in 1952. With the exception of the partisan affiliations of southerners and Protestants, no group saw more than a 10 percentage point difference in the percentage of its members represented within each party.
### 1972
By 1972, two years before _Roe v. Wade_ and less than a decade after the Civil Rights Act of 1964, partisan-group affiliations were not significantly different from those of 1952, with the exception of racial-group and southern affiliations. In comparison with 1952, Republicans had grown slightly more Protestant and wealthy, and Democrats slightly more Catholic and union affiliated. Racial sorting, however, had significantly increased, with Republicans now about 13 percentage points more white than Democrats, and Democrats about 11 percentage points more black than Republicans (in comparison, these racial differences were only 6 points and 6 points respectively in 1952). The data revealed another marked change in the percentages of each party that were southerners. In 1952, Democrats were 21 percentage points more southern than Republicans. This difference had decreased to 10 percentage points in 1972. In the twenty-year span between 1952 and 1972, the racial differences between the parties doubled while the southern divide was cut in half.
The ANES also added measures of ideology and church attendance in 1972, which are included in the figure from 1972 through 2012. In 1972, 10 percent more Democrats than Republicans called themselves liberal, but 20 percent more Republicans than Democrats called themselves conservative. The difference between the parties on conservatism is the largest difference in 1972. The lopsidedness of the difference between liberalism and conservatism is not surprising, considering that the Democratic Party still contained a large number of conservative southern Democrats, while the Republican Party, by comparison, was far more ideologically coherent.
The religious difference between the parties, however, was still Protestant versus Catholic. In fact, this divide had even grown slightly since 1952. There was, however, no difference whatsoever between the parties in levels of weekly church attenders in 1972. Though the racial divide had started to grow, the religious/nonreligious divide had not yet begun.
### 1992
Twenty years later, by 1992, the religious divide had cracked open, and the beginnings of our current state of sorting had become evident. The difference between the parties on the percentage of weekly churchgoers had increased to an 11 percentage point gap, with Republicans more churchgoing than Democrats. Connected to this new divide, Democrats in 1992 were only 2 percent more Catholic than Republicans. Twenty years earlier the difference had been 13 percentage points. The conservative religious were moving toward the Republican Party. The difference between the parties in the number of Protestants decreased from 17 percentage points in 1972 to 12 percentage points in 1992, as simply identifying as a Protestant was no longer a cohesive partisan indicator. A partisan rift was occurring among Protestants, splitting the traditional, conservative Protestants from the more progressive, liberal, mainline Protestants. As the conservative Christians became affiliated with the Republican Party, many liberal, mainline Protestants felt alienated by the new Republican identity and moved toward the Democrats. Many moved away from religion entirely, though still identified themselves as Protestants. While the Republican Party still remained more Protestant than the Democratic Party in the aggregate, this difference began to diminish as Protestants themselves divided politically.
In many other ways, the parties were becoming increasingly divided by 1992. The difference between the parties in the percentage of liberals had nearly doubled, moving from 10 percentage points to nearly 19 percentage points. In the number of self-identified conservatives, the parties were now nearly 34 points apart.
Geographically, the defection of southern Democrats finally tipped the partisan balance, so that by 1992, Republicans had become 2 percentage points more southern than Democrats. Racial differences were also growing rapidly. Republicans in 1992 were 20 percent more white than Democrats, and Democrats were 17 percent more black than Republicans. In the twenty years between 1972 and 1992, the parties had again almost doubled the racial divide between them. Also in 1992, the income differences between the parties began to grow. In 1952, the two parties were only 5 percentage points apart in average income percentile, by 1992 this gap had doubled in size. The Republicans were increasingly the party of the wealthy.
### 2012
By 2012, the old Protestant/Catholic, southern/nonsouthern, and union/nonunion divisions between the parties had largely disappeared. The differences between the parties in southern residency and Catholic religion had faded entirely. The previous 10 percentage point difference between the parties in union membership declined to barely 4 percentage points. But these old divisions had been firmly replaced by much larger ideological, religious, income, and racial differences. This was not simply a matter of swapping one set of partisan alignments with another. The new partisan alignments divided the parties far more powerfully than the divides of the 1950s had. The partisan difference in the percentage of liberals had again doubled, from 18 percentage points in 1992 to a 37-point difference between Democrats and Republicans in 2012. The difference between the parties in the number of constituents self-identifying as conservative rose from 34 percentage points in 1992 to nearly 50 points in 2012.
The parties differed by 14 percentage points in how many attend religious services each week. Levels of income continued to divide the parties, with the influence of wealth slightly increasing in its power to divide the wealthy Republicans from the less-wealthy Democrats. But by far the most powerful social divide between the parties, rivaling the difference in ideology, was race. By 2012, Republicans were, on average, 30 percentage points more white than Democrats. Democrats were, on average, 21 points more black than Republicans. Racial, religious, and ideological divisions separated the parties, and these divisions ran far deeper than in the previous sixty years.
## Identity-Based Social Sorting
The partisan shifts shown in figure 3.4, however, cannot tell the story of how closely American partisans feel affiliated with their various groups. Many of the social groups represented in figure 3.4 are essentially ascribed groups: race, southern residence, and income percentile are objective facts about a person. For the remaining groups—ideology, church attendance, religion, and union affiliation—a respondent has some leeway in deciding whether or not to identify him or herself as part of that group, but, in a response to a yes-or-no item, many people will simply provide the generally true response—that they are technically group members. They were raised Catholic. They are union members. They attend church once a week. These types of identities are also often considered objective group identities, ones to which members are assigned based on objective criteria, and with which they do not necessarily identify.
People, however, can also associate with a group on a subjective basis, by feeling some psychological sense of attachment to the group. These subjective group identities have been found to generate more loyalty from group members than objective group memberships, and therefore to have greater effects on individual behavior and intergroup relations (Huddy 2001).
The survey items that generated the outcomes in figure 3.4 do not allow respondents to explain to what degree they _feel_ like they are connected to the other members of the group, how similar they are to other members of the group, in short, how strongly they identify with each group label. Fortunately, the ANES items used in figure 3.2 were also used to measure identification with other social groups. In comparison with a simple "Are you Catholic or Protestant?" prompt, these items ask respondents to specify who they feel close to, providing much more insightful evidence of a distinctly psychological partisan sorting.
Figure 3.5 includes these items, indicating the difference between the two parties in the percentage of partisans who felt psychologically identified with each group. The results, though limited in time, still provide a solid picture of the trend in social sorting in recent decades. The trend echoes that seen in figure 3.4, and additionally allows some insight into the group-based feelings in each party.
Figure 3.5. Identity-based social sorting
Note: Bars represent the difference between the percentage of Republicans who claimed to feel identified with each group minus the percentage of Democrats who did so (including partisan leaners). Data drawn from the ANES data files of 1972, 1992, and 2000.
The ideological-identity differences reiterate the data shown in figure 3.2 and indicate, again, a strengthening of both liberal identity among Democrats and conservative identity among Republicans, with a distinctly larger increase in ideological identity among Republicans. By 2000, Republicans overwhelmingly felt that conservatives were their kind of people. Democrats felt this way about liberals as well, but not to the same degree.
In addition to these very large differences in ideological identity, figure 3.5 indicates a steadily growing increase in racial identity differences between the two parties. Between 1972 and 2000, the percentage of people in each party who felt close to black people increased from an 8 percentage point difference to a 12 percentage point difference between the two parties. Likewise, respondents who claimed to feel particularly close to white people grew increasingly Republican, with an 8 percentage point difference between the parties in 1972 growing to a 13 percentage point difference in 2000.
There was also a growing income disparity, not only in objective income levels but also in the extent to which people identified with certain income levels. The difference between the parties in the percentage of people identified with poor people remained relatively steady between 1972 and 2000, with Democrats consistently more strongly identified with the poor. In the number of people who identified with "businessmen," the difference between the parties increased from 11 percentage points in 1972 to 18 percentage points in 2000. Democrats consistently identified with poor people, while Republicans felt more and more like businessmen.
Finally, in comparison with the objective measure of southern residence, the percentage of each party that _felt_ like a southerner was about the same in each party in 1972. In 1992, 5 percentage points more Democrats than Republicans felt connected to southerners as a group, suggesting that even as the Republican Party became more geographically southern than the Democratic Party, many Democrats still felt connected to the South. This is likely reflective of the southern provenance of the Democratic presidential candidate, Bill Clinton. But by 2000, 7 percentage points more Republicans than Democrats felt particularly close to southerners. The southern partisan reversal was firmly under way, and it was occurring on a psychological level as well as a purely geographical level.
These subjective feelings of attachment are important indicators in the trend of social sorting. It might have been possible for the parties to grow objectively sorted into observable but psychologically unimportant groups. This, however, did not happen. As the parties became more homogeneous in ideology, race, class, geography, and religion, partisans on both sides felt increasingly connected to the groups that divided them. As I explain in the following chapters, the psychological identifications with groups that divide along partisan lines have serious implications for the thinking, behavior, and emotions of American partisans.
## Sorting Made Easy
One important final element in the history of sorting is the role of non-policy-based phenomena that eased the way along the path toward American partisan division. Though a racial-policy divide may have started the trend in social sorting, it was not the sole culprit. Social sorting has its roots in a few separate developments.
First, the issue-based division of the Democratic Party was accompanied by a providential change in American civil society. As Bill Bishop argues in his book _The Big Sort_ (2009), the 1960s and 1970s saw a large decline in trust in government among both Democrats and Republicans—so large that it encouraged citizens to detach from their parties. Political scientists responded to this with declarations of the end of partisanship, writing articles with titles like "The End of American Party Politics" (Burnham 1969). This detachment, however, was part of a larger trend. Not only did Americans lose trust in their parties, they lost trust in all institutions, resulting in a decline in civic engagement. As Robert Putnam (2000) has observed, Americans had previously associated in bowling leagues, civic organizations such as the Grange, the Elks, and the Scouts, professional organizations, parent-teacher organizations, and politically diverse churches and neighborhood communities. During the 1970s and the 1980s, however, membership in all of these organizations declined sharply. Americans began to grow more isolated and independent, and their political ties loosened.
This social loosening freed Americans to rearrange their partisan, social, and civic affiliations. It also, however, led Americans to feel increasingly detached from their communities and country, and compelled them to seek comfort in increasingly homogeneous neighborhoods, towns, and churches, causing American citizens to sort themselves into geographically isolated groups that shared their culture, values, race, and politics. They disengaged from their old, community-centered groups and formed new affiliations, tailored exactly to meet their needs.
Unattached and increasingly mobile Democrats and Republicans moved into increasingly homogeneous communities. White urban residents moved to the suburbs while urban areas grew increasingly populated by black residents (Frey 1979). Roof and McKinney (1987) described a post-1960s religious environment that included "greater choice in religious affiliation" and a "heightened religious individualism" that allowed religious Americans to detach from old mainline churches and move to churches more precisely suited to their specific social requirements. The partisan segregation of US counties increased by 26 percentage points between 1980 and 2000 (Bishop 2009). Andrew Garner and Harvey Palmer in 2011 described American residential neighborhoods as "increasingly balkanized by political attitudes as well as . . . race, education, and income" (230). Much of this was due to Americans choosing to live in more homogeneous communities and surrounding themselves with people who felt familiar. As Gordon Allport explained in the 1950s, it simply takes less effort to surround yourself with people similar to you.
The second force to contribute to social sorting, however, was the relationship between citizens and their party leaders. While citizens had been disengaging and socially segregating, the Democratic and Republican parties had been changing to provide clearer partisan, ideological, and social cues to the electorate, particularly on the Republican side. Kyle Saunders and Alan Abramowitz looked at American political activism from 1972 to 2000 and found that, as the parties' ideological cues grew more distinct and more potent, more partisans were motivated to participate, particularly within the Republican Party (Saunders and Abramowitz 2004). The Republican Party did a better job of organizing sympathetic social groups behind it. So, although social sorting may have begun with a split in the Democratic Party, it was the solidification of the Republican Party into religious, middle-/upper-class, and white categories that increasingly led to a more socially sorted and divided electorate. Due to the clearer distinction between the parties, Americans had far more simple cues to follow. These cues helped citizens to understand that a highly religious Christian who is also wealthy and white will feel most at home among Republicans. Similarly, a secular, less-wealthy, black person will feel more comfortable surrounding herself with Democrats. The parties, by providing increasingly clear cues, have helped Americans to know which party is their own.
A third stimulus toward social sorting was a growing diversity of media sources. The increasingly clear partisan cues have been reinforced by an increasingly diverse set of media sources, many of which are overtly partisan and/or misleading. Partisans are now able to protect themselves from any exposure at all to the arguments and opinions of the other side. Already geographically and culturally isolated, these citizens are also informationally isolated. Americans are not only sorted into homogeneous parties, they have diminishing opportunities even to hear the arguments of their political opponents. The news media allow voters to listen only to the narratives of their own side, causing them to become increasingly consistent in understanding whose team they are on, and which other teams are on their side. Though the audience for this type of media represents a small portion of the American population, Matthew Levendusky (2013) has found that "partisan media have multiplier effects that allow a relatively limited medium that speaks to a narrow segment of the market to have an outsize influence on American politics" (7). The isolating effects of this segmented media environment make very clear where the partisan boundaries are between social groups. The news about which groups belong in each party spreads widely, allowing individual citizens to understand better which party is their home, and which party is their adversary.
All of these forces have worked to encourage social sorting far beyond the effects of simple policy positions. We have gone from two parties that are a little bit different in a lot of ways to two parties that are very different in a few powerful ways. These underlying social shifts have put the American population into a partisan team-based mindset, through which the country has split itself into us and them, the Rattlers and the Eagles. Democrats and Republicans have become different types of people, and not only in terms of the groups that can be measured by the ANES.
## Cultural Differences
Democrats and Republicans come from and create different kinds of families. The national decline in fertility and increase in the age of marriage that has occurred since the 1950s has been limited mostly to the "Blue" states. The "Red" states have comparatively higher levels of fertility and are married at younger ages (Cahn and Carbone 2010). Research has found a strong relationship between fertility rates among white voters and Republican voting that stands even when controlling for urbanization, wealth, female education, Evangelism, Mormonism, Catholicism, and geography (Lesthaeghe and Neidert 2006).
The parties are divided in what they watch on television. In 2012, TiVo Research and Analytics matched television viewing data with voter registration information from 186,000 American households (Carter 2012). They sorted television programs by how popular they were with members of each party, listing the top twenty shows for Democratic and Republican viewers. Not a single network show appeared on both lists. A 2016 _New York Times_ study found an urban/rural cultural divide in television watching that matched partisan voting patterns. The correlation between Trump votes and "fandom" for the show _Duck Dynasty_ (a Christian-value-based hunting show) was higher than for any other show. In fact, _Duck Dynasty_ viewing was more predictive of a Trump vote in 2016 than it was of a Bush vote in 2000 (Katz 2016). _Family Guy_ , an animated sitcom, was more correlated with 2016 Hillary Clinton support than any other show. According to Katz (2016), this pattern was consistent with most satirical comedy shows popular in cities, where Clinton tended to receive the most votes. What this means is that, even when we sit down to relax and watch TV, Democrats and Republicans are different kinds of people. Increasingly, we cannot even connect at the water cooler to discuss last night's shows.
This type of cultural difference is pervasive. Thomas Edsall (2012) explained the campaign strategy of "nanotargeting," a method that is only possible because Democrats and Republicans can be found purchasing and enjoying categorically different things. They receive news from different sources (Democrats like the _Washington Post_ , Republicans like the _Washington Times_ ); they eat at different restaurants (Democrats like Chuck E. Cheese's, Republicans like Macaroni Grill); they drive different cars (Democrats like hybrids, Republicans like Land Rovers); they drink different alcohol (Democrats like Cognac, Republicans like Amstel Light). These are cultural differences so notable that campaigns rely on them to target advertising at the voters they are most likely to attract.
The makeup of the two parties has changed a great deal in the past sixty years, increasing the social distance between them. Partisans have less and less in common. Fewer cross-cutting cleavages remain to link the parties together and allow the understanding, communication, and compromise necessary to fuel the American electorate, and, by extension, the American government. Democrats and Republicans have grown so different from each other that cooperation is receding as a perceived value. When two teams grow so distinct and isolated from each other, the status of the teams themselves grows in importance. The functional outcomes of governing matter less. The sorting of our identities into partisan camps has allowed these identities to increasingly drive polarized political behavior, thought, and emotion.
# FOUR
# Partisan Prejudice
Elections aren't just about policy choices. They're status competitions. When the polls swing your way, you feel a surge of righteous affirmation. Your views are obviously correct! Your team's virtues are widely recognized! You get to see the humiliation and pain afflicting your foes.
—David Brooks, "Poll Addict Confesses"
Social sorting affects political relations by a simple, powerful effect. Being a member of a group tends to change our perception of the world and bias our relationships with others. Any group member who feels connected to the group is powerfully motivated to evaluate other ingroup members more positively than nonmembers, to be more generous with ingroup members, in short, to show bias in favor of the ingroup. This effect is magnified when multiple group memberships coincide, which is why the sorting of multiple identities into two parties can exacerbate the general effect of a single identity. However, to explain the results of sorting, it is important to take a step back and begin by looking more closely at the effects of a single group identity.
Group identities, particularly partisan identities, have been well studied, and, before presenting my own analyses here, I provide a short summary of the work done to date.
## What Is Partisanship?
In political science, the understanding of partisanship has undergone a significant evolution. In the middle of the last century, political scientists and sociologists tended toward the idea of partisanship as a distinctly social phenomenon. Lazarsfeld, Berelson, and Gaudet (1944) wrote, "A person thinks, politically, as he is, socially" (27). In 1960, Angus Campbell and his colleagues at the University of Michigan published _The American Voter_ , which described partisan identification as a "psychological identification" and an "affective orientation." They believed that identifying with a party was not simply a record of past voting or an indicator of future vote choice. It was not only a list of issue positions that a voter attempted to match to one party or the other. Instead, they thought that the psychological and emotional sense of belonging to a party was capable of altering the thoughts, feelings, and actions of partisans. This "Michigan model" of partisanship was one of the first to discuss the real and apparent psychological effects of being part of a partisan group.
In later years, however, the Michigan model of partisanship began to meet challenges. A new wave of research suggested that party identity was more likely an endpoint—a reasoned decision based on a person's political opinions and rational evaluations of the performance of political leaders. In 1977, Morris Fiorina described candidate evaluations as being made up of two main elements: past political evaluations and current issue concerns. He explained, "Party ID combines additively with current issue concerns. But party ID at any given point is a function of issue concerns prior to that point" (611). Party identity, he explained in 1981, was a "running tally" of issue considerations. This literature viewed voters as information-based decision makers, more like bankers choosing an investment than like sports fans cheering for a team.
Since then, a more social approach to partisan identity has seen a resurgence. Between the 1970s and the 2000s, partisan voting markedly and significantly increased, as did the number of strong partisan identifiers (Bartels 2002). In 2002, Green, Palmquist, and Schickler likened partisan identity to religious identity, a social-group membership that is acquired early in life and acts as an organizing force in an individual's sense of identity and self, driving action and decision-making. Steven Greene, in three separate papers (1999, 2002, 2004), described and measured partisanship as a social identity in the classic, psychological sense of the term.
My own work with Leonie Huddy and Lene Aarøe in 2015 demonstrated the powerful effect of partisan social identity in driving emotional reactivity and political activism, independent of the effects of instrumental policy concerns. McConnell et al. (2016) have even found effects of partisanship on nonpolitical outcomes, such as willingness to engage in economic transactions with outgroup partisans.
The current climate in political science is one that generally accepts the social nature of partisan identity but also allows for the ability of individuals to understand some issues and apply this knowledge to their political choices. For example, Bullock (2011) found that among people who possess policy information (an admittedly small group), many are capable of applying their policy preferences to political decisions without being unduly influenced by partisan bias. Similarly, Carsey and Layman (2006) found that individual partisan identification can be influenced by particular issues that are salient to the voter and on which the voter is aware of party differences (again, a limited domain).
Other research attempts to combine the instrumental model of partisanship and the social, expressive models. The work on ambivalence by Lavine, Johnston, and Steenbergen (2012) allows for partisans to dislike their own party for a variety of reasons, weakening the bond between party and partisan. Groenendyk's 2013 book contributes to this synthesis of the instrumental and expressive models of partisanship, as he identifies the motivations that may lead partisans to avoid partisan bias.
Ultimately, however, partisanship itself is an undeniable psychological force in modern American politics. As Iyengar, Sood, and Lelkes observed in 2012, Democrats and Republicans have grown to "dislike, even loathe" each other, and this emotional partisan loathing is only minimally due to differences in policy opinions. There is a power that partisanship exerts on individual partisans in the way they see the world, and in the way they think about other citizens. Much of the current partisan loathing grows out of the enduring effects of partisan and other social identities.
## The Psychological Effects of Party Identity
Democrats and Republicans compete for the power to implement very different policy platforms, affecting the entire nation. Political victory provides power in government and increased freedom to enact real policy outcomes that often directly benefit the members of the winning party in the form of tax policies, welfare policies, business regulation, or social programs. However, as Tajfel and Turner (1979) explain, "It is nearly impossible in most natural social situations to distinguish between discriminatory intergroup behavior based on real or perceived conflict of 'objective' interests between the groups and discrimination based on attempts to establish a positively valued distinctiveness for one's own group" (46). In other words, though the parties are competing for real interests, they are also competing because it just feels good to win. Distinguishing between those motivations is not a simple matter, but it is important to remember that both motivations are separately present in any political competition.
One powerful example of these tangled motivations is visible in the government shutdown of 2013. In October of 2013, the federal government shut down for sixteen days. According to the Office of Management and Budget (2013), the shutdown was "the second longest since 1980 and the most significant on record, measured in terms of employee furlough days." Their most conservative estimates found that the shutdown lowered GDP growth by $2 billion to $6 billion, led to 120,000 fewer jobs created, hindered trade, disrupted private-sector and federal lending to businesses and individuals, cost Alaskan fisherman thousands of dollars per day, reduced small-business contracts with the government by a third, delayed approval of important drugs, and deprived businesses of information about the state of the economy. Federal employees lost $2.5 billion in compensation for their furloughed work time. The shutdown delayed tax refunds, prevented sick patients from enrolling in clinical trials, closed Head Start daycare centers to 6,300 children, delayed home loans for 8,000 rural families, and delayed food-safety inspections.
What was worth so much damage to the economy and the nation as a whole? House Republicans insisted that they would not pass a spending bill required for government operation unless the Affordable Care Act (ACA, also known as Obamacare) was defunded or otherwise derailed. The ACA was Democratic president Obama's signature achievement during his first term. It was a health-care reform bill that represented a massive victory for the Democratic Party and an infuriating loss for the Republicans. Republicans had challenged the act before the Supreme Court, and the act was upheld. Legislatively, judicially, and executively, the act was approved and was moving ahead. Logistically and by all rational measures, there was no way to stop it.
Prior to the government shutdown, Republicans in the House of Representatives had voted forty-one times to repeal the ACA, knowing each time that these bills would never pass the Senate. The forty-second vote to repeal the ACA was tied to the approval of the spending bill in October. Though Democrats in Congress and the president himself had assured them that this type of brinksmanship would have no effect on the ACA, Republicans in the House went ahead with their demands, costing the nation billions of dollars, thousands of jobs, and untold opportunity costs. By the time the shutdown ended, nothing had been gained for the Republican Party, and the American taxpayers had seen billions of dollars squandered. When asked why Republicans had gone ahead with a hopeless cause that did so much damage, Republican representative Jack Kingston from Georgia replied, "I think it was important to us to reestablish our brand as being against Obamacare" ( _All In with Chris Hayes_ , 2013). Members of the Republican Party were willing to significantly damage the greater good of the nation in order to improve their partisan "brand." It seems crazy, but it's not. Henri Tajfel could have predicted this kind of behavior decades ago. The Republicans had an identity to defend. As long as the ACA goes forward, Republicans will be reminded of a massive group loss. Standing against the ACA is like pushing back against defeat. Republicans, particularly strongly identified ones, must defend their group against losing, at nearly any cost.
The example of the 2013 government shutdown is not notable in the damage done to the nation in pursuit of partisan goals. Certainly partisans in government have supported or opposed legislation that would ultimately damage the nation, on behalf of strongly held values or beliefs, or for strategic purposes. What distinguishes the 2013 shutdown is the hopelessness of the cause, and the knowledge on the part of Republicans that their actions would not change any policy. The goal of the shutdown was not, in reality, to prevent the enactment of the ACA. It was, as many representatives suggested, to enhance their brand. As an identity grows increasingly central, and increasingly encompassing as an element of any individual's self-concept, the status of that identity grows more important. Even in the face of defeat, the shutdown was a way to say to supporters, "We are not entirely powerless, we still have status." It was an act of reassurance for partisans whose sense of self had been damaged. It was a way to repair the damaged egos of a team who had just suffered a massive loss. It was a way to hold the team together, and heal the wounds of defeat. The damage caused to the nation by shutting down the government was, from an identity standpoint, justified. Millions of strongly identified Republicans needed to know that they were not losers. Their party had to deliver that message. It cost a great deal, but it was necessary for protecting the group.
The work of Tajfel and Turner (1979) has grown into what is now known as social identity theory. They describe a social identity as "those _aspects of an individual's self-image_ that derive from the social categories to which he perceives himself as belonging" (40; emphasis added). Being part of a group informs each person's self-image. Part of the reason for the deep and enduring existence of ingroup bias and the quest for group victory is that people are compelled to think of their groups as better than others. Without that, they themselves feel inferior. According to social identity theory, group members, at a very primal level, are powerfully motivated to see outgroups as different from them and to view the world through a competitive lens, with importance placed on their own group's superiority. This is crucial to keep in mind in the pages that follow. There is something inherent in a group identity that causes group members to be biased against their opponents. All of the political arguments over taxes, welfare, abortion, compassion, responsibility, and the ACA are built on a base of automatic and primal feelings that compel partisans to believe that their group is right, regardless of the content of the discussion. A partisan prefers his or her own team partly for rational, policy-based reasons but also for irrational, automatic, self-defensive reasons. This can cause irrational behavior in the search for victory. It can also cause very deep feelings of prejudice toward other partisans.
The term _prejudice_ is used here interchangeably with the concept of ingroup bias. Ingroup bias is an essential element of any group identity, including a partisan identity, and it means essentially that a person prefers their own group to the outgroup, for no reason other than that they are part of the ingroup, as was so well demonstrated by the minimal group paradigm experiments. I call this prejudice because it is the element of the group identity that is the most visceral and tribal, and in this sense it is indistinguishable from the base motive that drives racial prejudice or religious prejudice. Partisan bias has been experimentally demonstrated in grading (Musgrave and Rom 2015), college admissions (Munro, Lasane, and Leary 2010), and among survey interviewers (Healy and Malhotra 2014)—with copartisans given preferential treatment and judged superior to outgroup partisans. This is similar to the racial bias found by Bertrand and Mullainathan (2003) in the evaluations of resumes in a job search. If racial bias in hiring could be called prejudice, then the motive driving partisans to prefer their own kind can also be understood to be a type of prejudice. It is something that Cass Sunstein (2015) has labeled "partyism." In defending the use of the term _prejudice_ , Sunstein writes, "A degree of antipathy—at least if it is not personal—may reflect principled disagreement, not prejudice at all. But there is a large difference between a degree of antipathy and the forms of partyism we are now observing" (10).
The analyses that follow provide evidence that current levels of partisan antipathy have moved beyond pure disagreements of principle. Partisans dislike each other to a degree that cannot be explained by policy disagreement alone.
### Warm Feelings
Every election year, the American National Election Studies (ANES) ask respondents to rate the two parties on a "feeling thermometer," essentially gauging how "warm" or "cool" the respondents feel toward Democrats and Republicans. Figure 4.1 shows mean levels of the difference between the two party-feeling thermometers, what I call "warmth bias," over time, compared against the average extremity of Americans' positions on six prominent policies. The partisan preference for one party over the other has been steadily increasing, adding a 10 percentage point difference in feelings toward the two parties since 1984. At the same time the overall extremity of American policy positions has not seen the same increase. In fact, between 2008 and 2012 the two trends significantly diverged—our feelings toward the parties grew more polarized while our average policy attitudes remained unchanged. Partisans maintained the extremity of their policy preferences, and liked each other less.
Figure 4.1. Warmth bias and policy extremity over time
Note: Warmth bias is the absolute difference between each respondent's thermometer rating of the Democratic and Republican parties. Policy extremity is an index of six issues including abortion policy, government services versus spending, government health insurance, government aid to minorities, government employment protections, and defense spending. Each issue is folded in half so that higher scores represent more extreme positions on both ends of the spectrum. The six folded issue scales are then combined into an index. Data are drawn from the ANES cumulative file through 2012 (fully weighted), using only observations for which answers are available for all six issues.
Where did this divergence come from? Part of the story comes from our partisan identities. In figures 4.2 and 4.3 I look at people's feelings of warmth toward the Democratic and Republican parties, respectively. In particular, I look at the predicted values of feelings among two different groups. First, I look at Democrats who hold conservative positions on the six policies mentioned above. Second, I look at Republicans who hold liberal positions on the same six policies. What I'm examining, then, is whether party identity or policy agreement is better at driving feelings of warmth toward the two parties.
Figure 4.2. Predicted feelings of warmth toward the Democratic Party
Note: Data drawn from the ANES cumulative file (fully weighted) and the ANES 2012 file (fully weighted). Predicted values are derived from OLS regressions, controlling for education, sex, race, age, southern residence, urban residence, and church attendance. For the full regressions, see appendix table A.1a.
Figure 4.3. Predicted feelings of warmth toward the Republican Party
Note: Data drawn from the ANES cumulative file (fully weighted) and the ANES 2012 file (fully weighted). Predicted values are derived from OLS regressions, controlling for education, sex, race, age, southern residence, urban residence, and church attendance. For the full regressions, see appendix table A.1b.
In every year, partisanship trumps policy positions in determining our feelings toward the two parties. Even if we agree with the opposing party on most policies, we still feel most warmly toward our own party. Democrats who hold consistently conservative policy positions generally report feelings between 60 and 80 degrees (out of 100) for the Democratic Party. However, Republicans who agree with Democrats on most issues rate the Democratic Party between 30 and 50 degrees, depending on the year. This difference has fluctuated since 1980, the earliest point available in the series. The smallest difference between these cross-pressured partisans occurred in 1994, when only 11 degrees separated their evaluations. The very next election year, in 1996, the two cross-pressured partisans rated the Democratic Party 50 degrees apart. The difference between these cross-pressured partisans, although slightly volatile, exists in every year going back to 1980.
The same basic pattern occurs in our feelings toward Republicans. Republicans whose policy attitudes rightly belong with the Democratic Party report feelings of warmth toward Republicans of around 60 to 70 degrees across time. At the same time, Democrats who agree with most of the Republican positions on policies feel between 30 and 50 degrees or warmth toward Republicans, mostly on the "cool" end of the thermometer spectrum. The difference between these cross-pressured partisans was smallest in 1984, when their ratings were about 16 points apart, and largest in 2008, when their ratings were about 42 points apart.
This is not a picture of a nation that is choosing its parties based entirely on its policy attitudes. Simply being a member of one party can cause significant differences in our preferences for the two parties, even when our policy positions conflict with those of our own party. These conflicted partisans are still very loyal to their partisan teams, despite a consistent disagreement with their goals.
Rather than holding parties responsible for their policy positions, partisans are inclined to cling to their own party, seeing it through rose-colored glasses. The real outcomes of government, and a person's opinions about those outcomes, take a back seat to the central importance of seeing the inparty as better than the outparty. As long as the inparty is winning, partisans will have little motivation to stray. As Tajfel found, and the House Republicans demonstrated in 2013, winning is often more important than the good of the population as a whole.
This is part of the reason that even when policy debates crack open and an opportunity for compromise appears—a chance to increase the greater good—partisans are psychologically motivated to look away from that possibility and instead to find a way for the team to win, even if it means that we all receive less than we could have won together. In the 2013 debate over expanding background checks for gun purchases, 83 percent of Democrats and 81 percent of Republicans personally supported a law expanding background checks in a Pew poll. But only 57 percent of Republicans supported the Senate passing a background check bill, an action that would have been a victory for Democrats (Pew 2013). On this issue, Republicans and Democrats in the electorate clearly and massively agreed on what they wanted as the outcome. However, when it came to the moment of public partisan competition, party victory trumped preferred policy for many Republicans. Party affiliation today means that a partisan cares a great deal about one party being the winner. Policy results come second.
### Friends and Neighbors
Sadly, this natural bias against the outparty does not end at the gates of policy and governance. According to the 2016 Pew poll, when asked about a hypothetical person moving into their neighborhood, 61 percent of both Democrats and Republicans thought it would be easier to get along with the new person if that person were a member of the same party. Data that I collected from a national sample in 2011 confirm this type of social discomfort between partisans in the electorate. I asked a sample of Americans how willing they would be to spend time with Democrats and Republicans at four different levels of social intimacy. These included spending occasional social time, being next-door neighbors, being close friends, and marrying a partisan from each side. The items were drawn from a social-distance scale originally used by the sociologist Emory Bogardus (1925) to gauge racial prejudice in the 1920s and 1930s. I suspected that these prompts might reveal a distinctly social difference between contemporary Democrats and Republicans.
Figure 4.4 shows mean levels of willingness to engage in social contact with ingroup partisans and outgroup partisans. The results follow what Gordon Allport would likely have predicted in 1954. Democrats and Republicans would much rather spend time with people from their own party. On the full range of willingness, Americans are 19 percentage points less willing to spend occasional social time with outgroup partisans than with ingroup partisans. They are 13 percentage points less willing to have an outgroup partisan as a next-door neighbor. They are 17 percentage points less willing to be close friends with a person from the opposing party, and they are 36 percentage points less willing to marry a political opponent than a political comrade. These results are nearly identical for Democrats and Republicans (though Democrats are slightly less willing to be close friends with Republicans than Republicans are with Democrats in this sample). These are all statistically significant differences. Partisans in America would prefer to spend time with their own kind. In 2011, 52 percent of American partisans said that they definitely or probably would not marry a member of the opposing party. As a point of comparison, when Bogardus asked these questions of white Protestant Americans in 1928, he found that 10 percent would not marry a Canadian or northern European, but 90 percent would not marry a southern or eastern European (Triandis and Triandis 1960). In their suitability for marriage, therefore, outgroup partisans today rank somewhere between Canadians and Italians in the early twentieth century. These results echo those found in 2012 by Shanto Iyengar and colleagues, in which nearly 50 percent of Republicans and 30 percent of Democrats in 2010 reported that they would feel somewhat or very unhappy if their child chose to marry an outgroup partisan.
Figure 4.4 reveals that American partisans are not simply unhappy with their political choices in government, they dislike their political opponents in the electorate as well. This is where the social element of social polarization becomes clear. Partisans prefer to spend time with members of their own party. In a purely rational view of partisanship, this could be explained by a general reluctance to encounter social disagreements. Perhaps Democrats and Republicans don't wish to spend time together simply because they want to avoid awkward political discussions. However, policy-based disagreements do not explain the entire effect seen in figure 4.4. One easy way to see this is to combine all four of these social domains into one measure and examine it at varying levels of partisanship and issue positions.
Figure 4.4. Social distance between Democrats and Republicans
Note: Data are drawn from an adult sample collected by YouGov Polimetrix, using funding from the National Science Foundation under grant no. SES-1065054. Bars represent mean levels of reported willingness among self-identified Democrats and Republicans, including independent leaners. Ninety-five percent confidence intervals around mean values shown to indicate significant differences.
In figure 4.5, I plot predicted values of the difference between the two parties' scores on this aggregated social-distance measure. I refer to this difference as social-distance bias—the relative willingness to spend time with members of the inparty versus members of the outparty in all four domains. Measuring social distance this way controls for people who simply don't like to spend time with any partisan. If this partisan bias against outparty friends and neighbors is due largely to an avoidance of policy arguments, a person's policy positions should drive most of the bias. However, if simple group identity drives a biased evaluation of the two parties, strong partisanship should play an independent role.
Figure 4.5. Predicted difference in social distance between ingroup and outgroup partisans
Note: Predicted values drawn from an OLS regression controlling for education, political knowledge, race, gender, income, age, and church attendance. Ninety-five percent confidence intervals shown. Originating regressions are shown in appendix table A.2. Data are drawn from the 2011 Polimetrix sample. Sample includes only those respondents who have indicated preference for one party ( _N_ = 774). Partisanship is a scale introduced and tested by Huddy, Mason, and Aarøe (2015), based on items often found in social-psychological identity scales (Luhtanen and Crocker 1992; Crocker et al. 1994). Issues are an index of five generally salient political issues, weighted by the rated importance of each one, as well as a measure of issue constraint. Social-distance scale ranges from 0 (no difference in willingness to interact socially with members of the two parties) to 1 (maximal difference in willingness to interact socially with members of the two parties).
Figure 4.5 reveals almost exactly what Tajfel would have predicted. Real conflicts and partisan identities are both driving ingroup bias. In this case, they are doing so to a relatively equal extent. The measures used here are superior to the ANES measures, with a four-item measure of social identity used to gauge partisan identity, and an issue-position measure that includes respondents' sense of whether each issue is important and how consistently liberal or conservative those issues are (see figure note for details).
The regression model predicting social-distance bias (SDB) is as follows:
SDB = _a_ \+ _B_ 1(PID) + _B_ 2(IE) + _B_ 3(IC) + _B_ 4(IE*IC) + _B_ i(controls) + _e_.
In this linear regression model, social-distance bias is a function of partisan social identity (PID), issue extremity (IE), issue constraint (IC), and the interaction of the two issue measures (as well as control variables listed under the figure). The reason for interacting the two issue measures is to account for two particular types of people who are often difficult to measure. First, those people who may hold a host of extreme issue positions but who are ideologically inconsistent (a mix of liberal and conservative extreme positions). The second type is a person who is consistently either liberal or conservative but never holds an extreme position (always just barely on one side of the central/moderate response). The interaction allows for extremity to be separated from constraint and for the power of the combination to be more precisely examined. In figure 4.5, I only examine the effects of both measures at low or high levels together.
In the first column of figure 4.5, the level of social-distance bias is estimated for a very weakly identified partisan with (a) the most moderate positions on immigration, health care, gay marriage, abortion, and the deficit, (b) very little sense that those issues are important, and (c) no particular tendency toward either the liberal or conservative end of the spectrum. This person is predicted to exhibit no social-distance bias whatsoever. That is, he or she does not care about the partisanship of their social contacts and may, in fact, prefer outgroup partisans (though this is not statistically distinguishable, as the confidence interval includes zero).
In the second column, the weakly identified partisan is given very strong positions on all five issues, a sense that all of those issues are highly important, and a consistently liberal or conservative answer to all questions. This person is about 21 percentage points more willing to spend time with ingroup partisans than outgroup partisans. This is the element of social-distance bias that comes from policy-based disagreement. It is not based in strong partisanship.
In the third column, however, social-distance bias is predicted for a very strongly identified partisan with the most moderate and unconstrained issue positions, and the sense that these issues are unimportant. This person is predicted to exhibit slightly more social-distance bias than the weak partisan who cares a great deal about issues. Even when policy does not matter at all, a strong partisan is still about 29 percentage points more willing to spend time with ingroup partisans than with outgroup partisans, an 8 percentage point increase over the weak partisan who feels strongly about issues (though this difference does not reach statistical significance). This is the portion of social-distance bias that is not entirely based in policy. A person who doesn't care at all about issues is unlikely to be worried about starting a political argument with an outgroup partisan. She or he simply does not want to spend time with someone outside the group.
The fourth column predicts the social-distance bias of a strong partisan with very strong, important, and constrained issue positions. This person is 55 percentage points more willing to spend time with ingroup partisans than with outgroup partisans. The combined effect of partisanship and issue positions is larger than the additive effect of each one taken separately. When strong rational disagreements are combined with a strong group-based psychological bias, people begin to demonstrate the "loathing" that Iyengar, Sood, and Lelkes (2012) have described. This social distance between Republicans and Democrats, however, is not entirely seated in policy disagreement. To a large extent, simply feeling part of a partisan group is driving people apart. The discriminatory intergroup behavior is partly based, as Tajfel predicted, on "real conflict of 'objective' interests between the groups" but also driven by "attempts to establish a positively valued distinctiveness for one's own group." Partisans prefer their own kind. It just feels right.
Of course, American politics wasn't always quite so reactive to these deeply rooted identities. There was a time when partisans found paths toward compromise, despite their natural inclinations toward ingroup triumph. Though partisan identity has always driven partisans to want to win, this victory hasn't always been such a powerful motivator of political behavior. And though partisans have always preferred their own party, they have not always felt as socially distant from outgroup partisans as they have in recent years. Iyengar, Sood, and Lelkes (2012) showed that less than 5 percent of partisans in 1960 were opposed to their children marrying outgroup partisans. Part of the reason for the subsequent increase is that the number of strongly affiliated partisans has risen. In 2012, 38 percent of Americans called themselves strong partisans versus 22 percent in 1974. But partisans today are not only more strongly affiliated. They are also, as I discussed in chapter 3, more socially similar to other members of their own party than they have been in decades. It is reasonable to expect that, as long as there are parties, partisans will want their party to win. The extent of that desire, and the extent to which it eclipses concrete policy outcomes, depends on not just party identities but the other social identities that have gathered around our two parties. The simple effects of partisan identity on our political perceptions are amplified when those party identities are joined by other social-group divisions.
## A Frightful Despotism
George Washington, in his farewell address of 1796, warned the new nation about "the expedients of party." He was concerned that if the nation divided itself into distinct parties, the priorities of the new government would focus on those parties, rather than on loyalty to the nation itself. He called this partisan loyalty "a frightful despotism." Washington's worry grew out of his experience observing human nature, and noting the natural inclination toward factionalism that seemed to be always close to the surface of political interaction. He warned that this factionalism, once set in motion, could cause citizens to misrepresent the opinions of other citizens, and could cause fellow citizens to consider each other as enemies, even as the nation itself was struggling to form.
The data presented in this chapter would do little to reassure Washington that partisanship has not done exactly what he predicted. However, while American partisanship has existed since Washington left office, the current brand differs in nature from what Washington may have expected. This is because the factionalism that Washington feared is not only applicable to partisan teams. People form factions in all sorts of dimensions. We have long known that religion, race, and even sports-team affiliations have driven people into factions, set against each other along a dividing line. Partisanship may be necessary for government to organize and assist its citizens in decision-making. The problem arises when partisanship implicitly evokes racial, religious, and other social identities. As the sorting of the previous chapter occurs, parties become increasingly socially homogeneous. It is this social dimension of the partisan divide that makes it far easier for individual partisans to dehumanize their political opponents.
Social contact and shared social identities are the things that allow individuals to understand each other and tolerate differences in opinion. As those connections grow scarce, the effects of party no longer affect parties alone. Partisan battles become social and cultural battles, as well as political ones. The social homogenization of parties reduces room for compromise and increases the importance of simple party victory. The brand matters more than the good of the nation. This is what George Washington was concerned about, and it is now increasingly visible as American social identities reinforce the partisan divide.
# FIVE
# Socially Sorted Parties
The recent increases in levels of partyism shown in the previous chapter are not rooted solely in increasing practical disagreements. Partisans do not need to hold wildly extreme political attitudes in order to grow increasingly biased against their opponents. Partisanship alone cannot tell the whole story either. It turns out that one essential factor driving partisan prejudice is a set of well-sorted political identities.
The social psychologist Marilynn Brewer and her colleagues noticed a few years ago that, while there was plenty of evidence of the psychological effects of a single social identity, little research existed that explained how exactly our social identities work _together_. After all, none of us has just one social identity. A strongly identified Democrat or Republican is also a member of any number of other social groups. Each partisan could also identify as a woman, a conservative, a Christian, a runner, a football fan, a graduate of their particular college or high school; the list is endless. Brewer and her colleagues therefore decided it was important to examine the psychological effects of holding multiple social identities (Roccas and Brewer 2002).
When they asked people to think about how much their various social identities overlapped, they found a large difference between the thinking of people with highly aligned identities and the thinking of people with very unaligned identities. Identities are aligned when a large portion of the members of one group are (or are believed to be) also members of the other group. When multiple identities align, Brewer and her colleagues found, people are less tolerant, more biased, and feel angrier at the people in their outgroups. As an example, people who are Irish and Catholic (highly aligned national and religious identities) are more likely to be intolerant of non-Irish people than are people who are Irish and Jewish (relatively unaligned national and religious identities).
This is because a person with two highly aligned social identities sees outsiders as very different from herself. Her understanding of who she is will be constrained, and the list of the identities that define her feels smaller. On the other hand, when a person holds two social identities that are unaligned, outside groups seem more approachable. A person with cross-cutting identities feels that she is defined by a broad range of groups, and this makes her more tolerant toward groups that aren't exactly like her. An Irish-Jewish person will feel closer to non-Irish people than an Irish-Catholic person will.
The intolerance generated by a set of aligned identities can also come simply from lack of exposure to people unlike oneself. According to Allport's intergroup contact hypothesis, interaction between members of different groups can, under the right circumstances, reduce prejudice against those outgroups. A homogeneous set of social identities reduces the chance for that outgroup exposure. In fact, Diana Mutz found in 2002 that cross-cutting political identities _do_ reduce intolerance toward outgroups by giving people the "capacity to see that there is more than one side to an issue, that a political conflict is, in fact, a _legitimate_ controversy with rationales on both sides" (Mutz 2002, 122). Without this exposure to members of the political outgroup, it becomes far easier to view opponents with prejudice and their values as illegitimate. Thus, even as increasing numbers of Americans call themselves political independents, they tend to maintain partisan allegiances, as the social identities connected to the parties remain intact (Klar and Krupnikov 2016).
A lack of exposure to other ideas and people can make other ideas seem extreme and other people seem totally foreign, even when they are not. This includes both an intolerance of the policy positions of the other side and, more basically, an intolerance of the increasing strangeness of the outsiders. It can make a relatively moderate person intolerant of other views. The response is based on the strength and alignment of the identities, not the content of the identity-linked issue positions. This type of intolerance does not require partisan identities to correspond to highly extreme policy opinions. Even when our policy opinions remain relatively moderate, the alignment between our partisan and other identities can drive us toward prejudice against our opponents.
## Magnified Ingroup Bias
The gradual sorting of partisans into the "correct" parties during the last fifty years has transformed a nation of cross-cutting partisan identities into a nation of increasingly aligned partisan identities. As Democrats and Republicans grow socially sorted, they have to contend not only with the natural bias that comes from being a partisan but also with their own growing intolerance, sharpened by the shrinking of their social world. A conservative Democrat will feel closer to Republicans than a liberal Democrat would. A secular Republican will feel closer to Democrats than an evangelical Republican would. The sorting of our parties into socially distinct groups intensifies the partisan bias that we've always had. This is the American identity crisis. Not that we have partisan identities, we've always had those. The crisis emerges when partisan identities fall into alignment with other social identities, stoking our intolerance of each other to levels that are unsupported by our degrees of political disagreement.
In the previous chapter, I looked at feelings toward the two parties as an example of the emergence of partisan prejudice. Partisanship alone, in those models, was capable of driving identity-based prejudice. Independent of policy positions, partisans like their own party better than the other party. This is not surprising, but it is also not an unbiased choice.
In this chapter, however, I look at the added effect of a well-sorted partisan identity. As other identities fall into alignment with party, partyism only grows stronger. The other party seems more distant to a partisan, and it becomes easier to dislike them. I use the same measures of partisan prejudice—warmth bias and social-distance bias—so that the contribution of sorting will be clear. Sorting can be thought of in two different ways. The traditional understanding in political science is that sorting is simply the alignment between party and ideology. I argue that a number of additional social identities can be involved as well, as was demonstrated in chapter 2. In order to provide a thorough picture of the effects of sorting, I include both types of sorting here—simple ideological sorting and the more complex social sorting.
## Warmer Feelings
In order to look at the added effect of sorting, I first examine only strong partisans. Although most people are not strong partisans, those who are most committed to their parties provide the strongest test for the effects of sorting. We know that strong partisans will be the most biased against the outgroup party, but if their ideological or other social identities are aligned with that strong partisan identity, can their partisan bias grow even larger? It turns out that it can.
In figure 5.1, I show the predicted difference in feeling-thermometer ratings between the two parties, controlling for political knowledge, education, race, gender, income, age, and church attendance. Importantly, the regressions used to generate these predicted values also control for the extremity and constraint of policy positions, so the intensity of policy attitudes about abortion, gay marriage, health care, race, immigration, defense spending, government spending in general, the importance of the deficit versus unemployment, and the degree to which they are all ideologically consistent is unchanging. I also constrain partisan identity to be as strong as it can be. The two bars in each cluster, therefore, are demonstrating the difference between a strong partisan with cross-cutting identities and a strong partisan with well-sorted identities. The only difference between the two bars is the level of sorting. This way, the added effect of sorting, above the regular effects of partisanship, is made clear. The two clusters of columns show the difference between sorted and unsorted partisans in two separate data sets.
Figure 5.1. Predicted values of partisan prejudice (warmth bias) among strong partisans across levels of sorting
Note: Bars represent predicted values of warmth bias at varying levels of partisan-ideological sorting in the ANES models and social sorting in the YouGov model, controlling for issue extremity and constraint, political knowledge, education, race, gender, income, age, and church attendance. Originating regressions are shown in appendix table A.4. Ninety-five percent confidence intervals are shown. ANES samples are fully weighted. The low sorting score in the ANES models is not zero but 0.0857, the lowest sorting score possible given a strong partisan identity. In the YouGov sample, the lowest possible value of the social-sorting scale among strong partisans is 0.4286, and this is therefore the low sorting value used.
### ANES Results
In the first column of figure 5.1, all of the American National Election Studies data from 1972 to 2012 are combined, showing a general picture of partisan prejudice averaged over the last few decades. In these data, sorting is measured as the alignment between party and ideology, both measured using the seven-point scale traditionally used in the ANES. The partisan-ideological sorting score is calculated by taking the difference between the party and ideology scores, reverse-coding that difference, then multiplying it by the rated strength of each identity (i.e., lean, weak, strong). The sorting variable is coded to range from 0 (weakest and least-aligned identities) to 1 (strongest and perfectly aligned identities). In the cumulative file sample, a strong partisan with a cross-cutting ideological identity is predicted to rate the two parties about 37 degrees apart. However, add a strong and matching ideological identity, and the two parties all of a sudden have 56 degrees of warmth separating them. On a scale of 0 to 100, this is a large difference, and it is also statistically significant (the 95 percent confidence intervals are shown). Once the predicted difference between the two parties' feeling thermometers rises above 50, the respondent must be, to some extent, feeling coolly toward one party and warmly toward another. Even if one party is given a thermometer rating of zero (coldest), a 51 degree difference would put the other party at 51 degrees (on the warm side of the scale). There is no room for analogous feelings when the two parties are separated by more than 50 degrees. The addition of a sorted ideological identity, even when nothing else changes, causes a strong partisan to prefer his or her own party by 20 more degrees than partisanship alone can account for.
### YouGov Results and the Social-Sorting Measure
The second cluster of bars is drawn from the 2011 YouGov study, which included the four-item social identity scale to measure partisan identity (described in chapter 4) and social-identity-based measures of six other identities, including liberal, conservative, secular, evangelical, black, and Tea Party. This is a far more powerful measure of sorting than the simple partisan-ideological sorting used in the ANES analyses and a far more powerful measure of partisan identity. It is created to assess the feeling described as early as 1961 by V. O. Key, who said, "A person may have so intimate an identification with the Republican Party that when it is assaulted he cringes as if he had been attacked personally" (219). This type of attachment is assessed by the social identity measure, and it is assessed for all of the identities listed.
The sociopartisan sorting scale is designed to (1) assess the objective alignment between a respondent's social identities, while (2) accounting for the subjective strength of those identities. This is done because the alignment between identities means nothing if a person does not identify with one or more groups. The objective alignment of these various identities is determined by linking each nonparty identity to one of the two parties according to connections found in prior research and verified by examining the mean level of each identity for each party separately in the data. Aligned identities are found to be, for the Democratic Party, liberal, secular, and black identities, and, for the Republican Party, conservative, evangelical, and Tea Party identities.
The social-sorting scale is constructed so that, for each party, aligned identities are coded with positive values while unaligned identities are coded negatively. The mean of the identity scores is then taken for each party, with aligned identities increasing the total value and unaligned identities decreasing the final score. The party-specific scores are gathered into one measure, recoded to range from 0 to 1, with 0 representing consistently weak or totally unaligned identities, and 1 representing the strongest, most consistently aligned identities. This is an additive scale, rather than an interactive model, because here I am not interested in what happens when one identity moves while the others are held constant. Instead, this measure is constructed to allow all of the identities to move in relation to each other, generating varying sorting scores.
Using the social identity scale generates stronger results than the seven-point scales available in the ANES. In figure 5.1, the strong partisan in the YouGov data whose ideological, racial, and/or religious identities are in conflict with his or her party still feels 58 degrees of difference between the inparty and the opposing party. This person already feels warmly toward one party and coolly toward the other. Even this powerful effect of partisanship, however, can be strengthened by adding a set of well-sorted social identities. Once this strong partisan is socially well sorted, the difference between the two parties rises to 78 degrees. This means that, even if the inparty is placed at the warmest possible location on the thermometer, the opposing party is nearly 30 degrees colder than a neutral evaluation.
The effect of sorting across both data sets is to increase the biasing effects of partisan identity, even when nothing else changes, including policy extremity and constraint. This means that as the country grows more sorted, our ability to judge each other fairly is diminished. Even if we can find realistic policy solutions that we could all agree on, the alignment of our social identities behind our parties can generate its own animosity and partisan bias.
### The Role of Issues
In figure 5.1, issue positions were held constant, but what role do issues play as our identities pull us apart? In figure 5.2, I look only at the YouGov data, due to their superior measurement possibilities, and I replicate the sorting effects versus the partisanship effects across three levels of issue-position extremity. At low issue extremity, people have consistently moderate policy attitudes, and they consider those issues to be unimportant. At mean issue extremity, people are generally representative of the average American's policy attitudes and their sense of the importance of those policies. At high issue extremity, people have very extreme policy attitudes that they consider to be highly important.
Figure 5.2. Predicted values of partisan prejudice (warmth bias) among strong partisans across levels of sorting and issue polarization
Note: Bars represent predicted values of warmth bias at varying levels of social sorting in the YouGov 2011 sample, across three levels of issue extremity, controlling for issue constraint, political knowledge, education, race, gender, income, age, and church attendance. Originating regression is shown in appendix table A.5. Ninety-five percent confidence intervals are shown. The lowest observed value of the social-sorting scale among strong partisans is 0.4286, and this is therefore the low sorting value used. The mean value of issue extremity is 0.65 (the median, not shown here, is 0.67).
Again, when partisanship is strong but combined with cross-cutting social identities, levels of bias are significantly lower than when a strong partisan identity is well aligned with other party-linked identities. But even more importantly, this is true across all three levels of issue-position extremity. At every level of issue extremity, a cross-cutting set of identities reduces the partisan bias of a strong partisan by about 20 degrees on the feeling thermometer. Though overall bias is lower when issue extremity is low compared to when issue extremity is high, the difference between sorted partisans and unsorted partisans is significant no matter whether issue positions are extreme or moderate. This means that even in the moderate center of the electorate, where partisans from both sides find common ground on issues, a sorted identity is capable of driving citizens to feel increasingly warmly toward their own party and coolly toward their partisan opponents. The moderation of their policy attitudes does not protect them from the biasing effects of social sorting. However, a cross-cutting set of identities combined with a moderate set of issue positions does appear to be the only condition in which a strong partisan might place both parties on the same end of the feeling thermometer.
### Matching
The previous tests looked at the effect of sorting among strong partisans. But this doesn't demonstrate the full effect of sorting in the electorate as a whole. Most citizens, after all, are not strong partisans. Another way to look at the effect of sorting on feelings of warmth is to account for the full range of sorting and to test it very strictly using a method called matching. Matching is a particularly strong test of the effect of sorting because it takes a large sample of people (here the full cumulative ANES file including 2012) and simulates the random assignment of sorting to the population. In essence, this method makes it possible to pretend that each respondent is part of an experiment in which they are randomly assigned to a level of sorting. It does this by matching as many respondents as possible so that they are nearly identical in their ideology, issue extremity, political knowledge, education, age, sex, race, geographical location, and religiosity. This group of matched respondents is then divided into two groups, with the only measured difference between them being the level of sorting, either low or high, depending on whether respondents score above or below the median value of sorting.
Once the matched sample is divided into low and high levels of sorting, I look at the differences between them in how warmly they feel toward the two parties. Because of the exact matching, a simple difference in means on the matched data can reveal whether sorting has a measurable effect on partisan prejudice. Little else can influence changes in partisan prejudice because the respondents are constrained to be essentially identical in all other aspects observed.
To find any effect of sorting at all on people who are identical in their education, political knowledge, age, sex, race, location, religiosity, issue-position extremity, and ideological identity would be a powerful outcome, particularly because people who are highly sorted are generally demographically _different_ from those who are unsorted. It would be unlikely to find, in the general population, people who are identical in these multiple demographic and social domains divided evenly between cross-cutting and sorted identities. More likely is that the types of people who hold well-aligned ideological and partisan identities are similar to each other in levels of education, knowledge, religiosity, and issue-position extremity and different from those with cross-cutting identities. These other social similarities would likely drive sorted individuals to hold even more partisan bias than what is observed here. If sorting has any effect on the partisan feelings of these matched and evenly divided individuals, it is likely an underestimate of the effect in the population as a whole.
In figure 5.3, the samples are matched on ideology (and the abovementioned variables), while the extent to which partisan identity is aligned with that ideological identity is varied. Ideologically identical people (in both identity and issue positions) are significantly more biased in their assessments of the two parties when their partisan identity is strong and in line with their ideological identity. The mean warmth bias score for a person with an inconsistent partisan identity is a difference of 27 degrees between the two parties, while an otherwise similar person with a consistent partisan identity rates the parties 48 degrees apart. Moving from unsorted to sorted increases the difference in feelings toward the two parties by 20 degrees, even among people who are otherwise identical in multiple domains.
Figure 5.3. Difference in warmth bias by sorting in matched samples
Note: Respondents are matched on ideology, issue extremity, political knowledge, education, age, sex, race, geographical location, and church attendance. The only observed difference between the two bars is the degree to which party identity aligns with ideological identity. Bars represent the mean level of warmth bias for each group. Ninety-five percent confidence intervals are shown.
As partisanship moves into alignment with ideological identity, even when little else changes, partisan prejudice increases. People who are identical in their demographics, knowledge, issue positions, and ideological identity become significantly more biased when their party is aligned with their ideology. As the partisan rift in the American electorate falls into line with an ideological rift, average citizens are finding the opposing party increasingly different, unlikeable. Even when these citizens have a great deal in common, sorting alone is able to drive their opinions of the two parties apart.
## Friends and Neighbors
It isn't only our feelings of warmth toward the political parties that are polarized by sorting. As the previous chapter showed, our feelings toward our fellow citizens are just as vulnerable to the prejudice that comes out of a highly sorted set of political identities. And again, just as in the case of warmth bias, our social bias against spending time with outgroup partisans is only strengthened by adding a set of sorted identities to our partisan loyalties. Figure 5.4 uses the YouGov data again to demonstrate the effects of social sorting, issue extremity, and constraint on individuals who are already strong partisans. Once again, though the level of policy-attitude extremity and constraint does make a difference in general levels of social-distance bias, it cannot eliminate the socially divisive effects of social sorting.
In figure 5.4, partisans with cross-cutting identities who care little about policy outcomes and have no ideological constraint (but are nonetheless strong partisans) are predicted to report a 21 percentage point difference in their willingness to be socially involved with members of the two parties (first set of bars). However, if that same strong partisan—the one who doesn't care much about any issues—also identifies with racial and religious social groups aligned with the party, that partisan is predicted to be 36 percentage points more willing to spend time with members of their own party than with members of the outgroup party. This is a person who holds moderate positions on abortion, gay marriage, immigration, health care, and the deficit and unemployment, considers all of these issues to be unimportant, and demonstrates no ideological bent one way or the other. Party-matched racial and religious social identities make a 15 percentage point difference in the social tolerance of ingroup and outgroup partisans. This social discomfort is unlikely to be due to worries about policy arguments, these individuals do not have strong opinions about any of the issues measured here. When issues don't matter, the alignment of racial and religious social identities behind a partisan identity can make the difference between a welcoming neighbor and a hostile one.
Figure 5.4. Predicted social-distance bias by social sorting
Note: Bars represent predicted values of social-distance bias at varying levels of social sorting in the YouGov 2011 sample, across three levels of issue positions, controlling for political knowledge, education, race, gender, income, age, and church attendance. Originating regression is shown in appendix table A.6. Ninety-five percent confidence intervals are shown. The lowest observed value of the social-sorting scale among strong partisans is 0.43, and this is therefore the low sorting value used. Social distance scale ranges from 0 (no difference in willingness to interact socially with members of the two parties) to 1 (maximal difference in willingness to interact socially with members of the two parties). There is a significant interaction between issue extremity and constraint in this model, so both variables are included, both set at their minimum, mean, and maximum values for each set of predicted values.
Even when a person cares somewhat about the issues measured, when they hold a set of well-aligned identities they are still less comfortable around outgroup partisans. In fact, those with mean levels of issue extremity and constraint are not statistically different from those who do not care about issues. These moderate partisans with cross-cutting identities (in the second set of bars) are about 15 percentage points more comfortable spending time with outparty (versus inparty) members than are the partisans with well-aligned social identities. A moderate involvement with policy outcomes generates the same social distance as nonengagement with these policies; the major effect is between those partisans who hold cross-cutting identities and those who are socially sorted. Almost regardless of policy attitudes, social sorting is associated with a social distancing of outgroup partisans.
Among those who hold consistently extreme and constrained issue positions that they consider to be extremely important (the third set of bars), the difference between an unsorted and a sorted set of identities has essentially the same effect that it has at other levels of issue engagement, but general levels of social distance are higher. The difference between cross-cutting and socially sorted identities increases social distance by about 15 percentage points, once again. This 15 percentage point difference, however, begins at a significantly higher level of social distance. Strong partisans with cross-cutting social identities but strong and constrained issue positions prefer to spend social time with the inparty rather than the outparty by about 41 percentage points. Add a set of well-sorted racial and religious social identities, and this preference difference increases to 57 percentage points. The effect of strong and constrained issue positions, then, is to increase the total level of social distance between partisans but not to alter the effect of social sorting at all.
All of this means that highly sorted partisans will be biased against their outparty friends, neighbors, and romantic interests no matter what they think about political issues. Not only that, but strong partisans with cross-cutting identities will demonstrate the most tolerance toward their political opponents. If any grounds can be found for political harmony in American politics, it will not be the common ground of shared policy opinions. One robust force for political harmony appears to be an increasingly rare set of cross-cutting political identities rather than a moderate set of issue positions. Even in a group of partisans who all hold moderate, conflicting issue positions, a set of sorted identities will drive them to dislike and avoid contact with their friends and neighbors from the outgroup party. And even among those strong partisans who care a great deal about issues, they are more likely to be socially tolerant of other partisans if their racial and religious identities do not match their party. The increasing levels of social sorting described in chapter 3 are encouraging Americans to avoid social contact with members of the opposing party. The more socially sorted American partisans become, the more they will want to pull away from one another. This outcome goes beyond the simple effect of partisan identity. Partisanship can drive significant levels of partisan prejudice, but, when our social identities line up behind our parties, our prejudices expand beyond what partisanship can do on its own. Social sorting, in other words, links our racial and religious prejudice directly to our partisan preferences and allows our political opinions to be driven by increasingly social divides.
This is one source of the political acrimony that characterizes so much of contemporary American politics. We are not dealing with normal partisan bickering. The sorting of American social identities into partisan teams has magnified the effects of ingroup bias and pulled American partisans apart. This social sorting has created, in essence, two megaparties, whose members dislike and avoid their political opponents, even when they live next door. While partisanship is not new in American politics, the social sorting that magnifies partisan prejudice is changing the power of partisan identity.
## Sorting and Policy Bias
The White House press starts from the premise: Is the President up or down today? Is this good politically or not good politically? There's far less interest in the substance of policy.
—Mike McCurry, White House Press Secretary, 2014
The primacy of social sorting as a driver of partisan prejudice is not consistent with a common view of politics. The folk theory of representative democracy, named by Achen and Bartels (2016a), assumes that individual citizens choose to vote for a party because it best represents their own interests and values. The classic Downsian view of voting assumes that parties compete, in their policy positions, for the approval of the median voter, who is always voting in his or her self-interest (Downs 1957). In this view, the party that comes closest to the median policy attitude among all voters will win the majority of the votes. This premise requires that voters make choices based mainly on the policy positions of parties and their proximity to voters' own positions. Davis, Hinich, and Ordeshook, in 1970, wrote that "the fundamental process of politics is the aggregation of citizens' preferences into a collective—a social—choice . . . in which the social choice is a policy package which the victorious candidate advocates" (426). The "social choice" is seen as policy-based. Policy positions are, in this traditional view, the fundamental basis of politics and of voter decision-making.
This view of the centrality of policy attitudes among the citizenry is a highly optimistic view of American political thought. A large body of literature has found Americans' understanding of political policy debates to be sorely lacking. Still, this is the way that many Americans tend to understand electoral contests and political battles. Democrats and Republicans are in a battle over health care, over abortion, over tax policy. The political fights in American politics are supposed to be _about_ something.
An abundance of evidence, however, contradicts this view. Geoffrey Cohen, in 2003, found issue positions to be highly dependent on group and party cues. In an experiment in which he varied the policies of the two parties, liberals expressed support for a harsh welfare program and conservatives expressed support for a lavish welfare program when they were told that their ingroup party supported the policy. Notably, these respondents did not believe that their position had been influenced by their party affiliation. They were capable of coming up with explanations for why they held these beliefs. This result, in particular, casts doubt on the general perception expressed among partisans and pundits that our political evaluations are drawn entirely from the conviction of our issue opinions. In fact, issue positions appear to be quite slippery.
A Pew poll from June 2013 found that, under Republican president George W. Bush, 38 percent more Republicans than Democrats believed that NSA surveillance programs were acceptable, while under Democratic president Barack Obama, Republicans were 12 percent _less_ supportive of NSA surveillance than Democrats. The question prompt was identical, the only difference was the party of the president. As in Cohen's experiment, it is likely that these voters, if asked, would have provided logical reasons for their change of heart. But, as Cohen experimentally demonstrated, the influence of party loyalty is capable of reversing a single person's well-argued issue position without them even realizing it.
Part of the reason that policy opinions are so vulnerable to partisan cues is that partisans tend to engage more in motivated reasoning when their social settings are more homogeneous (Klar 2014)—a condition that is more likely in more highly sorted groups (Mutz 2002). As we sort ourselves into socially uniform parties, we lose perspective on what we really believe and begin to simply defend the positions that our party takes. It is a self-defense mechanism that takes hold when our parties take up larger and larger parts of who we think we are. The more parts of our identities that are linked with our parties, the more the success of our parties becomes more important than any real policy outcomes.
This is why, when we judge the Democratic and Republican parties, our issue positions have become less consequential than our identities. In figure 5.5, predicted values of warmth bias are shown at varying levels of policy extremity and constraint and then, separately, at varying levels of sociopartisan sorting. The difference between the two panels is telling. The bars represent the predicted distance between the Democratic and Republican feeling thermometers. In the first panel, levels of issue extremity and constraint are varied while all other variables are held at their means or modes. Policy attitudes do have a significant effect on feelings toward the two parties. People with the most extreme and constrained policy attitudes place the two parties 60 degrees apart, while those with the most moderate and unconstrained policy attitudes place the parties 36 degrees apart. The difference in our feelings toward the two parties grows by 24 degrees when we move from a total moderate who cares little about issues to an extreme ideologue who cares a great deal about issues.
Figure 5.5. Predicted warmth bias by levels of social sorting and policy attitudes
Note: Bars represent predicted values of warmth bias (the difference between Democratic and Republican feeling thermometers) at varying levels of social sorting in the YouGov 2011 sample, controlling for political knowledge, education, race, gender, income, age, and church attendance. Originating regression is shown in appendix table A.7. Ninety-five percent confidence intervals are shown. Mean value of policy extremity is 0.66, of policy constraint is 0.56, and of social sorting is 0.71 out of 1.0.
This effect, however, is noticeably different from the effect of sorting shown in the second panel of figure 5.5. Those people with the most cross-cutting racial, religious, and partisan identities (and average policy attitudes) are predicted to place the two parties 14 degrees apart on the feeling thermometer. This is less than half the size of the partisan gap seen among those with the most moderate policy attitudes. A set of cross-cutting identities is much better than moderate issue positions at equalizing feelings toward the two parties. All else equal, the most moderate and unconstrained policy attitudes still allow a strong preference for one party over the other, while a set of cross-cutting identities is a truly moderating force in driving feelings toward the two parties. At the other end of the spectrum, a very well sorted person places the two parties 78 points apart. The difference in feelings toward the two parties grows by 64 degrees when we move from a set of cross-cutting identities to a set of well-sorted identities. The effect of sorting is nearly three times larger than the effect of issue extremity.
It shouldn't be much of a surprise that policy attitudes are less effective at changing feelings toward the two parties, considering that they seem to be relatively unreliable and vulnerable to identity-based influences. Since identity is at least partially responsible for the effect of policy attitudes on party evaluations, it makes sense that a direct measure of multiple party-linked identities would be a more direct way to determine the polarization of citizens' feelings toward the parties.
This does, however, fly in the face of the folk narrative about political polarization. Many media stories focus on the policy elements of partisan battles and present elections as referenda on public opinion about policies. Polls are examined to see what percent of America agrees with policies X, Y, and Z. Citizens assume that we have reasons for our opinions, and that parties win or lose based on our thoughtful choices. In fact, the data presented here reliably show that partisan identities and the social identities that line up behind them have a significant effect on our political judgments. Political scientists have long known that partisan identities can affect our policy attitudes and our feelings about political contests. What is new here is the idea that partisan identities are only part of the story. The sorting that has often been recognized as a simple realignment of identities has in fact been able to motivate substantially larger levels of partisan bias than partisanship alone could do.
When Americans decide how they feel about the Democratic and Republican parties, they only partially turn to an assessment of their own policy opinions. They are driven, substantially, by a need to maintain a positive distinctiveness for their own team. And as their team grows increasingly socially homogeneous, it becomes even more important for it to be the best. The opposing party becomes more distant and unfamiliar as our social identities line up behind our partisan identities. This makes it all the more important for partisans to see their own party as better than the other. Unfortunately, this is not normatively useful for democratic representation. The American system of democracy, as it grows increasingly socially polarized, will rely less on policy preferences and more on knee-jerk "evaluations" that should rightfully be called partisan prejudice.
## Is This Polarization?
In the study of partisan polarization, a debate continues between two camps of political scientists. On one side, scholars such as Alan Abramowitz argue that the American electorate is polarizing at the level of the mass electorate and that this polarization is defined by American policy attitudes. On the other side, scholars such as Morris Fiorina claim that the American electorate is not exceptionally polarized in their policy preferences and that American polarization is therefore limited to our highly policy-polarized elites. What this debate overlooks is the behavior, emotions, and actions of the electorate, aside from their policy attitudes. This chapter demonstrates the robust presence of an ingrained prejudice that grows out of the increasing alignment between our partisan, ideological, racial, and religious social identities. This is a distinctly social phenomenon, unbound by the extremity of policy attitudes, but undeniably a sign of a polarizing electorate. Political scientists can disagree until we are blue in the face over the extent of America's policy polarization, but are citizens prejudiced in their evaluations of political opponents? Absolutely. Even when they can agree with them.
Looking at partisan polarization in a less policy-focused way allows us to discover many areas of American political life, including our political judgment, emotion, and actions, that don't necessarily correspond to policy extremism or polarization but are nonetheless present. Policy attitudes are said to be polarized when they can demonstrate an extreme and bimodal distribution of policy positions, with Democrats clustered on the extreme liberal end of the spectrum and Republicans clustered on the extreme conservative end of the spectrum. Abramowitz and Fiorina have been going over this territory repeatedly in the last few decades, so I will not enter this particular debate here. However, it is possible for Americans to be socially polarized even when their policy positions are not, or when those policy attitudes are relatively less polarized. A distinctly social type of polarization that includes political prejudice, anger, enthusiasm, and activism does exist, and it is being driven by political and social identities. An electorate that increasingly treats its political opponents as enemies, with ever-growing levels of prejudice, offensive action, and anger, is a clear sign of partisan polarization occurring within the citizenry. If issue positions do not follow precisely this pattern of behavioral polarization, it does not make those increasingly tribal partisan interactions irrelevant.
# SIX
# The Outrage and Elation of Partisan Sorting
Trump has gotten voters who are so angry that they are willing to put their ideological concerns aside. We have never seen voters do that to this extent. They're saying, "We're so ticked off that that's the only message that matters."
—Patrick Murray, pollster, 2016 (quoted in Goldmacher 2016)
In April of 2014, the federal Bureau of Land Management (BLM) attempted to round up and repossess a herd of cows belonging to a man named Cliven Bundy. Bundy grazed his cattle on federal land in Nevada, for which he was legally required to pay grazing fees. He had refused to pay these fees since 1993, claiming ownership of the land. By 2014, the BLM estimated that Bundy owed the federal government $1 million. As members of the BLM began to round up Bundy's cattle, some members of Bundy's family began protesting and confronting federal officials. Within days, a protest camp formed at Bundy's farm with a sign at the entrance reading "MILITA SIGHN IN" (Fuller 2014). Hundreds of self-identified members of armed militias gathered on Bundy's land, preparing for a violent battle against the federal employees. They dressed in paramilitary gear, set up illegal checkpoints, aimed their weapons at law-enforcement officials and federal employees, and threatened to bomb and kill people at local businesses (MacNab 2014). The story exploded in the national media, with conservative news sources praising Bundy as a hero, and liberal news sources calling him a terrorist and a "big fat million dollar welfare dead beat" (Vyan 2014).
Conservatives were outraged at the federal government's treatment of Bundy. And, with guns drawn, hundreds of militiamen joined Bundy in expressing their anger, if not outright rebellion, against the government. An attorney named Larry Klayman wrote in support of Bundy:
> Before these government goons do come back, let this message go forth. Barack Hussein Obama, Harry Reid and the gutless Republican establishment leaders in Congress who roll over to and further this continued government tyranny, We the People have now risen up and we intend to remove you legally from office. This country belongs to us, not you. This land is our land! And, we will fight you will [ _sic_ ] all legal means, including exercising our legitimate Second Amendment rights of self-defense, to end your tyranny and restore freedom to our shores! (Klayman 2014)
The case of a local rancher who hadn't paid his taxes was quickly turned into a national fiasco, and a source of potent outrage among conservatives. How did Cliven Bundy so quickly become a national conservative icon? The answer—as Paul Waldman (2014) put it in the _Washington Post_ —was that "when conservatives looked at Bundy . . . everything about him told them he was their kind of guy."
Bundy checked off many of the boxes that make up the Republican Party. A strong conservative, a white man, a rural southerner, he represented the convergence of the social identities that hold the Republican Party together. This convergence of identities made it much easier for Republicans to get angry on his behalf, and for Democrats to get angry at him. Conservatives, as they defended Bundy, did focus on a policy aspect of the conflict—the overreach of the federal government. But for many conservatives, particularly under Democratic president Barack Obama, the federal government, as Larry Klayman decried, had become more of an enemy—a set of "goons"—than the foundation of a policy position.
This sort of intense anger is not rare in modern American politics. In 2009, when Congress was debating what would eventually become the Affordable Care Act, town hall meetings across America erupted with angry outbursts. _Politico_ reported, "Screaming constituents, protesters dragged out by the cops, congressmen fearful for their safety—welcome to the new town-hall-style meeting, the once-staid forum that is rapidly turning into a house of horrors for members of Congress" (Isenstadt 2009). At the same time, members of the Tea Party held angry protests in Washington. In 2011, in New York City, a liberal group calling themselves Occupy Wall Street protested against a number of economic, political, and social injustices. The New York protests spread to dozens of other cities and were described by the _New York Times_ as "Countless Grievances, One Thread: We're Angry" (Lacey 2011). Turn on almost any cable news station during the last ten years, and you can find a political pundit expressing anger at a new political development.
Perhaps the pinnacle of all of this anger has been the unexpected success of the 2016 presidential campaign of Donald Trump. According to a 2016 Pew poll, the Americans who expressed anger at the government tended strongly to be Trump supporters. Trump is a fascinating case because, as a candidate, his policy positions were well known to be quite flexible, if not nonexistent.
In a 2016 _Washington Post_ article, Philip Rucker and Dan Balz wrote, "Donald Trump fits no simple ideological framework. The presidential candidate collects thoughts from across the spectrum. Added together, however, his ideas represent a sharp departure from many of the Republican Party's values and priorities dating back half a century or more. . . . Trump's presidential candidacy has been described as a hostile takeover of the Republican Party. In reality it appears more a movement that threatens to subsume the GOP behind a menu of ideas and instincts that might best be described as 'America Wins.'" In this sense, the Trump candidacy distilled perfectly what Tajfel found in his minimal group paradigm experiments. Winning grows increasingly important as identities grow stronger. To this point, Trump's support was also strongest among those voters who shared multiple Republican-linked identities (Mason and Davis 2016). These particularly include white and Christian identities. Trump's campaign did not tear the Republican Party apart; he spoke directly to the social groups that have aligned with the Republican Party in recent years, and he did so with little real policy content.
The alignment of multiple social identities can directly affect the degree of anger with which individuals respond to identity threats. As identities have moved into alignment in recent years, levels of anger at outgroup candidates have also increased. Though these are simply correlational trends, they serve to set up the story to come.
One crude way to examine average levels of anger over time is to look at one question asked by the American National Election Studies every year beginning in 1980. The item asks respondents whether each presidential candidate "has—because of the kind of person he is, or because of something he has done—made you feel angry." I coded this item so that it refers only to people's feelings of anger toward the outgroup candidate. The numbers in figure 6.1 represent the percentage of people who have reported feeling angry at the outgroup candidate in each presidential election year.
Figure 6.1. Anger toward the outgroup presidential candidate
Note: Data drawn from the weighted ANES cumulative data file, 1948–2012. Numbers represent the percentage of people who reported feeling angry at the outgroup presidential candidate, coded as a presidential candidate of the party that the respondent does not belong to. Question was first asked in 1980. Pure independents are excluded.
This is a rough measure and fluctuates widely depending on the context of the election. For example, Barack Obama's 2008 Yes We Can campaign was generally oriented toward hope and change and was the first election in which an African American was elected president. Republicans (those for whom Obama was the outgroup candidate), had little to openly express anger about. Not only did social norms against racism briefly tamp down open partisan rancor, but it was, after all, a Republican president who had only months before presided over one of the greatest financial disasters in national history. Republicans may have been angry, but in that moment they were not angry at the relatively unknown Barack Obama. (That would come later.)
In the same election, John McCain, the Republican, had a reputation as a centrist, party-bucking politician who could be relied upon to make compromises. Democrats (those for whom McCain was the outgroup candidate) therefore had little to hold against him personally. Their ire was reserved for the sitting president, George W. Bush, who held some of the lowest approval ratings of all time.
An earlier drop in anger had registered in the election of 2000, when voters famously saw little difference between the two major party candidates. But these relatively low-anger elections did nothing to reduce anger in the following elections. If anything, the low-anger elections worked as slingshots, pulling levels of anger down, only to shoot them back up in the following elections to unprecedented levels. Voters in recent years, when they do feel angry, feel angrier. In both 2004 and 2012, levels of anger reached 60 percent of the partisan population for the first time since the measure was introduced in 1980.
In figure 6.1, the general trend over time is drawn as a straight line, which is moving upward, toward more anger. In 1980, 40 percent of partisans felt anger toward the opposing presidential candidate, and by 2012 that number had increased to 60 percent. Even accounting for fluctuations, the trend line indicates a 10 percentage point average increase in the proportion of people reporting angry feelings at their party's main opponent since 1980. The anecdotes of partisan rancor and vitriol don't seem to be simply isolated events. There has been a modest but increasing trend toward angrier American politics.
This is not the entire story, however. Americans are not only angrier at their political opponents, they are also happier with their own team's candidates. Figure 6.2 shows trends in the percentages of Americans who claim they have felt "proud" of their ingroup presidential candidate.
Figure 6.2. Enthusiasm for the ingroup presidential candidate
Note: Data drawn from the weighted ANES cumulative data file, 1948–2012. Numbers represent the percentage of people who reported feeling proud of the ingroup presidential candidate, coded as a presidential candidate of the party that the respondent does belong to. Pure independents are excluded.
Just as in the case of anger toward the outgroup candidate, pride for the ingroup candidate is steadily, if noisily, rising. The general trend from 1980 to 2012 is a mean increase in pride of 12 percentage points. In the three presidential elections since 2000, around 60 percent of partisans felt proud of their presidential candidate, compared to numbers hovering around 50 percent in the decades before. So, just as Americans are growing increasingly angry at their opponents' candidates, they are growing increasingly enthusiastic about their own.
Combine this anger and pride in every presidential election, and we see a picture of an electorate that is increasingly emotionally reactive. As time progresses, American partisans are more likely to feel angry at their opponents and proud of their own candidates. We are priming the pump for a very energetic battle.
## Why Are We So Emotional?
Where is all this emotion coming from? In fact, anger and enthusiasm can be understood as very natural reactions to the group-based competition and threats that partisans face on a regular basis. As elections grow longer—and political media coverage explains governing as a constant competition between Democrats and Republicans—partisans are inundated with messages that their group is in the midst of a fight for superiority over the outgroup. Every vote in Congress, then, has the potential to feel like a threat to an attentive partisan. These party threats are capable of motivating significant levels of both anger and enthusiasm in party identifiers, driven not simply by a dissatisfaction with potential policy outcomes or a potential policy victory, but by a much deeper, more primal psychological reaction to group competition.
Intergroup emotions theory (an outgrowth of social identity theory) has found that strongly identified group members react with stronger emotions, particularly anger and enthusiasm, to group threats (Mackie, Devos, and Smith 2000). According to this theory, group-based partisan bias leads strongly identified partisans to believe (correctly or not) that their party is the generally favored party—that Americans like them the best. The sense that the party is strong, enjoying collective support, increases their ability to feel anger and engage in confrontational behavior. This is because when the ingroup is perceived to be stronger than the outgroup, anger results from intergroup competition, while the perception of a weak ingroup leads to anxiety in the face of group competition. These are natural psychological reactions to group competition, driven not by practical thoughts about the concrete outcomes of an intergroup competition but by evolutionarily advantageous reactions to group competition and threat. A strong group is in a powerful position to react to threat with anger and offense, while a weak group is not. A weak group is expected to react to the same threat with anxiety. Partisan anger therefore is not only driven from a loss of tangible resources but also an outgrowth of natural offensive behavior that emerges from faith in the power of the ingroup and the aggressive tendencies that group allegiance allows.
Importantly, this emotional reaction depends on a threat to the status of the group. As identities grow stronger, anger only increases if the group is perceived to be under some kind of threat from the outgroup. Kevin Arceneaux and Martin Johnson in their 2013 book remind us that personalities on cable news shows "raise their voice in outraged frustration, badger hostile guests, and hurl insults at the other side. . . . The apparent goal is to steel and energize the in-partisans while taunting the out-partisans" (75). These types of partisan threats are present on cable news shows, in political commercials, in print media, and even, during election seasons, in simple polls. Status threats are potent emotional catalysts. Pierce, Rogers, and Snyder (2016) found that, in the week following the Democratic presidential victory in the 2012 election, Republicans felt significantly sadder than they had the previous week. But they weren't simply sad. They felt sadder than American parents felt in the week after hearing about the Newtown Shootings. They felt sadder than Bostonians in the week after the Boston Marathon bombing. Republicans, because their party had lost, reported feeling some extremely powerful negative emotions.
These negative reactions can, however, help to generate positive emotions as well. In 1994, Nyla Branscombe and Daniel Wann conducted a study in which they asked people to watch the movie _Rocky IV_. For some respondents, they altered the movie so that, in the end, Rocky is defeated by the Russian fighter, Ivan Drago. In this condition, those people who felt most closely identified with being American took severe hits to their own self-esteem. They felt very negatively about themselves after watching Rocky lose. But they were then given a chance to express their levels of distrust and dislike of Russians in general. Those who did this, who expressed many kinds of negative feelings about Russians, restored their self-esteem. These people felt better about themselves by making insulting judgments about their Russian outgroup. Imagine these effects, now, in terms of partisan competition. When partisans lose an election, they take a hit to their self-esteem, which is wrapped up in their partisan identity. One effective way of soothing this damage is to lash out at partisan opponents. It is this threat to self-esteem that drives partisan insults and rage, which lead to a consequent improvement in self-esteem. This is a cycle, in which threats to group status lead to angry and insulting reactions, which then lead to higher assessments of group status, which cause threats to have even larger effects. Our anger and enthusiasm are fueling each other.
Not only do strong identities push partisans to react to threats with anger and excitement but aligned identities add even more anger to every threat response. Sonia Roccas and Marilynn Brewer (2002) raised the possibility that those with highly aligned identities may be less psychologically equipped to cope with threats to group status. This is because a person with a highly sorted set of identities is more socially isolated and therefore less experienced in dealing with measured conflict. This can lead to higher levels of negative emotions when confronted with threat.
When multiple identities are strongly aligned, a threat to one identity affects the status of multiple other identities. The possible damage to a person's self-esteem grows as more identities are partnered with the damaged group. While stronger identities motivate increased anger and excitement in the face of group threat, more sorted identities have an even larger effect. We have more self-esteem real estate to protect as our identities are linked together.
Although anecdotal stories of political anger and fervor appear to be provoked largely by issues such as health-care reform, gay marriage, abortion, and taxation, social sorting can powerfully drive emotion, contrary to the popular perception that only practical disagreements trigger higher levels of political rancor. Because a highly sorted set of identities increases an individual's perceived differences between groups, the emotions that result from group conflict are likely to be heightened among well-sorted partisans, regardless of policy opinions.
## Why Do Emotions Matter?
Before delving into the evidence for the social roots of emotions, it is important to examine why emotions matter in the first place. Anger and enthusiasm seem like politically important emotions, but why? And why focus on these two emotions in particular?
The study of emotion in political science is relatively new, and only recently has it been studied using rigorous empirical methods. One of the better-known theories of emotion in politics is affective intelligence theory, introduced by Marcus, Neuman, and MacKuen (2000). This theory argued that it was not sufficient to study the simple difference between positive and negative (valence) emotions and that far more information could be obtained if researchers looked at different types of emotions within each category—particularly in the category of negative emotions. They determined that the difference between anger and anxiety was significant, especially when looking at political behavior. The two types of negative emotions, in fact, can have opposite effects on judgment and action. Anxiety was found to lead to more thoughtful processing of information, while anger led to more reliance on easily available cues such as social identities. More recent research on anxiety by Albertson and Gadarian (2015) has found that anxious citizens do in fact search out more information, but they do so in a biased way, looking especially for threatening information. In any case, while anxious citizens tend to look for new information, angry and enthusiastic citizens do not.
A related body of research has found anger and enthusiasm to be particularly good at driving action. In 2011, Valentino and colleagues found that those who were angry were more inclined to sign political petitions, register other people to vote, participate in political protests, volunteer for a campaign, and donate money. Van Zomeren, Spears, and Leach (2008) discovered that a strong group identity increased collective-action tendencies via group-based anger. That is, when members of a social group (students) were presented with a threat (rising student fees), they reacted with anger, and this anger precipitated collective political action. Furthermore, Groenendyk and Banks (2014) found that feelings of enthusiasm increased citizens' likelihood of urging others to vote for a particular candidate, wearing a campaign button, attending political rallies, and donating to a candidate or party. Banks (2016) has found fascinating evidence that feelings of anger in white Americans push them to think in more racial terms. In other words, the anger that Banks observes is directly linked to (and occurs prior to) racial divisions into ingroup and outgroup categories. Nonracial anger pushes racially conservative individuals to think about race.
Combined with intergroup emotions theory, all of this research points to the idea that strong group identities and intergroup divisions facilitate increasingly angry and enthusiastic responses to group threats. While political enthusiasm is not usually thought of as problematic, it, along with anger, leads to increased political activity based not on policy goals but on knee-jerk identity-defense responses. The key point, for the purposes of this book, is that anger and enthusiasm are the primary emotional drivers of political action, and they are not drivers of thoughtful processing of information. The following chapter addresses the direct effects of social sorting on activism, but these emotions are important to examine on their own, as they are capable of provoking much of the action and judgment that contributes to current levels of social polarization. The difference between anxious and angry responses (though they are highly correlated in any given event) helps to explain how it is that partisans can grow increasingly divided even when their policy positions do not diverge. Anger is a powerful emotion that can drive group identifiers apart, reflexively. It is therefore important to examine the group-based drivers of both anger and enthusiasm, the two emotions that lead to relatively thoughtless political action.
## Evidence from a Panel Study
Between 1992 and 1996, when partisan sorting was in flux, the ANES ran a panel study—interviewing the same people in 1996 that they had interviewed in 1992. In figure 6.3, I compare changes in anger at the outgroup candidate and issue intensity among three groups of citizens—those whose level of partisan-ideological sorting increased, those whose level of sorting did not change, and those whose level of sorting decreased between 1992 and 1996.
Figure 6.3. Change in anger and issue extremity between 1992 and 1996 by sorting
Note: Data drawn from the ANES 1992–1996 Panel Study. Demographic controls are not necessary as this is a reinterview of the same individuals. Sorting is limited to the partisan-ideological sorting measure, due to data limitations.
Among those people who became increasingly sorted between 1992 and 1996, reported anger at the outgroup candidate increased by 28 percentage points. In comparison, among those whose sorting did not change, anger increased by 22 percentage points, and those whose level of sorting decreased reported a 17 percentage point increase in feeling angry at the outgroup candidate. All three groups reported an increase in anger, which is at least partly contextual. In the intervening four years, Republicans had taken the majority of the House of Representatives for the first time in forty years, and a new partisan conflict between the Democratic president and the Republican House had heated up to the point of a government shutdown in 1995. There was a general feeling of mounting partisan discord. But, importantly, Americans didn't all get angry in the same way. The people whose partisan and ideological identities had moved into alignment were the ones whose anger increased the most. In comparison, when partisan and ideological identities had not moved toward alignment, people were less readily angered.
Was this anger all due to more potent policy attitudes among the increasingly sorted? Apparently not. Among the increasingly sorted, as their levels of anger increased by 28 percentage points, the intensity of their policy attitudes increased by only 1 percentage point. Not only did issue intensity remain essentially the same, its change was barely different from the issue-intensity changes of those whose sorting levels decreased. Those whose levels of sorting remained the same grew less intense by about 1 percentage point. The difference in anger between the sorted and reverse-sorted was 11 percentage points, while the difference in issue intensity between these two groups was less than 1 percentage point.
This pattern holds for both Democrats and Republicans. The only minor difference is that sorted Republicans grew more issue intense (increase of 0.12) than did Democrats (decrease of 0.05), but they also grew slightly less angered than Democrats did. Again, this does not support the idea that anger comes from issue intensity.
This is a picture of a nation whose partisan teams are raring to fight, despite an almost total lack of any substantive policy reasons to do so. It should be reiterated that the changes depicted here are changes among _the same people_ over time. When a single person went through a process of aligning their partisan and ideological identities, they came out the other end angrier than they entered. More so than other Americans. Partisan-ideological sorting, without affecting policy extremism, generated significant changes in anger.
## Sorting or Party Identity?
Is it possible that these effects of sorting on emotion can be explained by the effects of partisan identity alone? Not likely. Figure 6.4 looks once again at strong partisans in the cumulative ANES data, predicting their probability of feeling angry using logit models (see originating models in the appendix). Among these intense partisans, those who have cross-cutting ideological identities are certainly angry at the outgroup candidate. There is a 66 percent probability that a cross-pressured but strongly committed partisan will report feeling anger. However, once that strong partisanship is accompanied by a strong and well-aligned ideological identity, there is an 86 percent probability that they will report feeling anger. These are substantively and statistically significant differences.
Figure 6.4. Predicted probability of feeling anger at the outgroup candidate
Note: Predicted probabilities drawn from a logit model using weighted ANES data from the cumulative file through 2012. Controls are included for issue extremity and constraint (and their interaction), education, sex, race, age, southern location, urban location, and church attendance. Originating regression is shown in appendix table A.8. Ninety-five percent confidence intervals shown.
In an emotionally charged election, a simple change in the alignment of partisan and ideological identities has the power to increase the potential for anger by 20 percentage points. These models also are drawn from regressions in which race, gender, education, age, southern origin, urban origin, church attendance, and issue extremity and constraint are all held constant. Without any change in any of these characteristics, and even among the strongest partisans, simply moving from a cross-cutting ideological identity to a sorted ideological identity can drive a significant increase in feelings of anger. This is a psychological response to the feeling that the party makes up a larger part of a person's social world. Once an ideological identity lines up behind a partisan identity, it becomes harder to understand opponents as reasonable people and easier to feel threatened and angered by them.
The effects on pride are smaller but still significant. Figure 6.5 presents the same strong partisans but predicts their probability of feeling proud about their own candidate. Among intense partisans with cross-cutting ideological identities, there is a 77 percent probability that they will feel proud of their ingroup candidate. However, once an ideological identity is strong and well aligned with this strong partisan identity, that probability increases to 88 percent. This difference, again, is statistically significant. Therefore, even while holding issue positions constant, simply aligning an ideological identity with a strong partisan identity is capable of increasing the likelihood of feeling proud of your own candidate by 10 percentage points.
Figure 6.5. Predicted probability of feeling proud of ingroup candidate
Note: Predicted probabilities drawn from a logit model using weighted ANES data through 2012. Controls are included for issue extremity and constraint, education, sex, race, age, southern location, urban location, and church attendance. Originating regression is shown in appendix table A.8. Ninety-five percent confidence intervals shown.
Although the differences between cross-cut partisans and well-sorted partisans are relatively small in magnitude, they are significant, and they suggest something real about American politics. Even our strongest partisans have emotions that are kept slightly in check when their ideological identities are unaligned with their party. Once party and ideology move into alignment, as they have across large swaths of the American electorate, the likelihood that partisans are feeling angry and proud increases significantly. Sorting is pushing us into emotional territory that partisanship alone cannot.
## Matching
Another way to examine the effect of sorting on emotion is to go back to the matched sample used in chapter 5. Using the same sample—comprising members nearly identical in ideological identity, issue extremity, education, age, sex, race, geographical location, and religiosity—I again split the sample into low and high levels of sorting. This time I looked at the differences between the groups in their reported levels of anger at the outgroup candidate. This is a challenging test because, although the people are constrained to be matched, the political context in each year is drastically different, causing a large variance in anger across the cumulative ANES sample, which spans from 1972 to 2012. Despite the contextual variation, these highly similar individuals, when averaged across time, tend to be significantly angrier at the outgroup candidate when their ideological identities are aligned with their partisan identities. This is true despite large confidence intervals. Figure 6.6 presents the results of this matching test.
Figure 6.6. Percent angry at the outgroup candidate in matched sample
Note: Data drawn from ANES cumulative file through 2012. Ninety-five percent confidence intervals shown. Respondents matched on ideological identification, issue extremity, education, sex, race, age, southern location, and church attendance. Low sorting and high sorting are divided by cutting the sorting score at its median.
In this figure, ideologically matched people are significantly angrier at the outgroup candidate when their partisan identity is strong and aligned with their ideological identity. About 11 percent of people with cross-cutting partisan identities feel angry at the outgroup candidate, while 49 percent of the well-sorted sample reports feeling angry. This is a significant difference. Although they are quite similar in their characteristics, ideology, and political attitudes, those whose partisan identity is aligned with their matched ideologies are angrier, across the decades, than those whose partisan identities are unaligned with their matched ideologies. The alignment of these two identities is driving people to feel angrier, despite their agreement on policy outcomes and their similarity in every other measured way.
Looking at pride for the ingroup candidate offers a similar picture. In figure 6.7, the same matched sample is compared across levels of sorting. Once again, we see that across the years those individuals who are similar to each other in many ways, and differ in their level of partisan-ideological sorting, feel very differently toward their own party's candidates. Among those with cross-cutting partisan and ideological identities, only 14 percent report feeling proud of their own party's candidate. Move that party into alignment with ideology (again, an ideology matched across conditions), and 55 percent report feeling proud.
Figure 6.7. Percent proud of the ingroup candidate in matched sample
Note: Data drawn from ANES cumulative file through 2012. Ninety-five percent confidence intervals shown. Respondents matched on ideological identification, issue extremity, education, sex, race, age, southern location, and church attendance. Low sorting and high sorting are divided by cutting the sorting score at its median.
These differences are not only statistically and substantively significant, they are compelling because they occur among respondents that are as similar to one another as possible. Furthermore, though issue extremity is matched here, these models have been replicated using an issue-constraint measure instead, and the results support the same conclusions. Individuals who are similar on ideology and issue positions grow far more proud of their candidate when their party is well matched to their ideological identity, even though their beliefs do not differ. Partisan-ideological sorting is capable of encouraging an increasingly angry and enthusiastic electorate.
## Evidence from an Experiment
Up until now, the effects of sorting on anger have only been demonstrated in the case of partisan-ideological sorting. Furthermore, all of the models presented above have measured anger as a simple yes/no response to a question regarding anger at the outgroup candidate, which a respondent must recall from memory. According to social identity theory and intergroup emotions theory, a threat is necessary for group identities to activate anger. The outgroup candidate is a good representation of a threat to group status. He or she is, after all, the embodiment of the party whose victory will mean an inevitable defeat for a partisan's own party. However, more precision in measuring both anger and threat is possible.
In the 2011 YouGov survey, I included an experiment in which respondents were randomly assigned to one of five conditions. Some respondents were asked to read a message that threatened their party. They were told it was taken from a political blog, but in fact I fabricated it based on a number of blog comments I had collected, in order to make the messages as comparable as possible. For Republicans, this message read:
> 2012 is going to be a great election for Democrats. Obama will easily win re-election against whatever lunatic the Republicans run, we are raising more money than Republicans, our Congressional candidates are in safer seats, and Republicans have obviously lost Americans' trust. Our current Congress is proving to Americans that Republicans do not deserve to be in the majority, and Americans will make sure they're gone in 2012. Finally, we'll take the Congress back and won't have to worry about the Republicans shutting down government anymore! I'm glad that Americans have finally returned to their senses. Republicans should get used to being the minority for the foreseeable future. Democrats will hold our central place in the leadership of the country. Obama 2012!!
For Democrats, the message read:
> 2012 is going to be a great election for Republicans. We're going to defeat the hardcore socialist Obama, we are raising more money than Democrats, our Congressional candidates are in safer seats, and Democrats have obviously lost Americans' trust. Our current Congress is proving to Americans that Democrats do not deserve to be in the majority, and Americans will make sure they're gone in 2012. Finally, we'll take the government back, and we won't have to worry about Democrats blocking us at every turn! I am so glad that Americans have finally returned to their senses. Democrats should not get used to running the government. Republicans will take back our central place in the leadership of the country. Defeat Obama in 2012!!
Other respondents were asked to read a "blog message" that threatened their party's cherished policy outcomes. For Republicans, this message read:
> 2012 is going to be a great election for responsible political ideas. After this election we can finally fix the economy using wise tax increases to pay for our indispensable social programs and infrastructure, so that we can create jobs instead of blindly throwing money to corporations and giving tax cuts to the millionaires who caused this mess. After this election we'll be able to improve the health care bill by adding a public option, make sure every woman has clear access to abortions, every child has a chance to learn evolutionary theory in school, and make it easier for all adults to get married if they want to, no matter who they are. Finally, our country will be on the right path again!
For Democrats, this message read:
> 2012 is going to be a great election for responsible political ideas. After this election we can finally fix the economy by enforcing personal responsibility, using a true free-market system to make sure people aren't handed more than they've earned. We'll be able to shrink the government and get it off our backs, and lower taxes so that hard-working people have a reason to work. After this election we'll be able to stop socialized medicine, prevent the abortions of innocent babies all over the country, bring God back into the public sphere, and make sure that we are a country that respects that marriage is between a man and a woman. Finally, our country will be on the right path again!
A fifth group did not read any message at all. The four messages were randomly assigned, so some Democrats would read the Republican threat message and some Republicans would read the Democratic threat message. When this occurred, I coded this as a message of support for the party.
After reading one of the messages, respondents were asked how the message had made them feel. They could answer A great deal, Somewhat, Very little, or Not at all to the following emotion items: Angry, Hostile, Nervous, Disgusted, Anxious, Afraid, Hopeful, Proud, and Enthusiastic. I combined their responses to the Angry, Hostile, and Disgusted items to form a scale of anger (α = 0.91), and the Hopeful, Proud, and Enthusiastic responses to form a scale of enthusiasm (α = 0.93). In comparison to the yes/no anger responses measured above, this measure created a scale of emotion that ranges relatively continuously from 0 to 1, creating much more variation in the amount of anger or enthusiasm a person could report.
Figure 6.8 illustrates the main effects of each experimental treatment on emotion. As expected, in the threat conditions, anger is substantially stronger than enthusiasm, and in the support conditions enthusiasm is the main result. The party-threat conditions included language that had the potential to generate stronger emotions than the issue conditions, but, as the data show, emotional reactions to the issue threats are relatively similar to the main emotional effects of party threats.
Figure 6.8. Main effects of experimental treatment
Note: Ninety-five percent confidence intervals shown. Bars represent mean levels of each emotion in each treatment, across the entire sample.
As in chapter 5, I measured sorting using the full social-sorting scale, including partisan, ideological, black, secular, evangelical, and Tea Party identities, measuring each, if present, using the four-item social-identification scale. This creates a much fuller measure of sorting by including multiple social identities that may come into play in determining how angry or enthusiastic each partisan can be.
I expected the most socially sorted partisans to be the most emotionally volatile. I thought they would react to the party-based threats with the most anger and to messages of support with the most enthusiasm. I also expected to find somewhat smaller results for partisan identity alone, and much smaller effects among those with the most extreme issue positions. In other words, I expected to see that a conglomeration of identities is most emotionally responsive to threat (particularly group-based threat), that one identity is slightly less so, and that a set of extreme issue positions generates the smallest emotional response to threat.
In order to give the issue positions a fair test, however, I included the threats that were devoid of partisan labels and only threatened policy outcomes. If anything were to anger those with strong issue positions, it should be these issue-based threats. Furthermore, the issue measure used here accounts for not only issue extremity but also issue importance and issue constraint. I refer to this measure as issue intensity, due to its inclusion of multiple elements of issue attitudes.
### Experimental Results
What I found was relatively consistent with expectations but also slightly surprising. The results are presented in figures 6.9 and 6.10. In short, the intensity of issue positions does, indeed, generate significant emotional reactions to issue-focused messages. When issue positions are threatened (fig. 6.9a) or reassured (fig. 6.10a), those with the most extreme, consistent, and salient issue positions respond by growing angrier and happier, respectively. However, when party defeat (fig. 6.9b) or victory (fig. 6.10b) is promised, issue extremity has no significant emotional effects. Issue-focused citizens are different from their issue-moderate counterparts in the degree to which they are angered and excited by practical goals but not by status threats regarding their own parties.
Partisanship has a different influence on emotion. Strong partisans are significantly angrier than weak partisans when the party is threatened (fig. 6.9b) but not when policy success is threatened (fig. 6.9a). They also grow significantly more enthusiastic than weak partisans when party victory is discussed (fig. 6.10b) but not when policy victory is promised (fig. 6.10a). It doesn't really matter to partisans whether their policy positions are threatened. Strong partisans are emotionally engaged by messages of support regarding their party's status—but not by the actual policy outcomes of that status.
Figure 6.9. Predicted angry reactions to messages
Note: Bars represent the predicted values of anger at each level of issue extremity, partisan identity, or sorting. Originating regressions are shown in appendix table A.9. Ninety-five percent confidence intervals shown.
Figure 6.10. Predicted enthusiastic reactions to messages
Note: Bars represent the predicted values of enthusiasm at each level of issue extremity, partisan identity, or sorting. Originating regressions are shown in appendix table A.10. Ninety-five percent confidence intervals shown.
Sorting is the unique variable in this sequence, in that it is capable of affecting emotion no matter what kind of message is presented. But this occurs in an interesting way. In figure 6.9, social sorting does affect angry reactions to both issue-based and party-based messages of threat. Unlike either issue intensity or partisan identity alone, the difference between cross-cutting and well-sorted identities is apparent in response to both messages. However, one important point to note is that, in the issue-based threat condition, the highest levels of sorting do not generate anger that is significantly higher than the anger produced among the most issue intense. Similarly, in the party-based threat condition, the highest levels of sorting do not generate anger that is significantly higher than the anger produced among the strongest partisans. The main difference between the effects of sorting versus issue intensity and partisanship is found at the _low_ end of the scale.
The people with the most cross-cutting identities respond to both types of threat with significantly _less_ anger than either the least issue-intense or the least partisan individuals. In fact, for both types of threat, the cross-cut individuals respond with no anger in the case of party threat and with _negative_ anger in the case of issue threat. In other words, when these cross-cut individuals read a threatening political message, they remain impassive. These data suggest that Americans are not growing increasingly angry because the best-sorted identities drive the highest levels of anger. They are growing angrier because the people who tend to respond without anger (those with cross-cutting identities) are disappearing. As the sorting seen in chapter 3 continues, the people who have the best chance of remaining calm in the face of political conflict are shrinking as a proportion of the electorate.
A similar phenomenon is seen in the case of enthusiasm, shown in figure 6.10. In the presence of an issue-based message of victory (6.10a), the most socially sorted individuals are predicted to report no more enthusiasm than those who are the most issue intense. In this sense, sorting is not increasing enthusiasm beyond what it would already be among the most issue intense. However, at the low end of the spectrum, those with cross-cutting identities are significantly _less_ enthusiastic than the least issue intense. The 95 percent confidence interval for the least-sorted group in figure 6.10a crosses zero, suggesting that, once again, those with cross-cutting identities have no emotional response whatsoever, even to a positive message.
In the case of party-based messages of victory, the same basic pattern arises. People who are highly socially sorted are no more enthusiastic after hearing a victory message than are the strongest partisans. In this one case, those with cross-cutting identities are statistically indistinguishable from very weak partisans. So the dampening effect of cross-cutting cleavages does not go beyond the dampening effect of simple weak partisanship. However, one difference does exist. The confidence interval around the predicted level of enthusiasm for those with the most cross-cutting identities includes zero, which means it is statistically probable that these people do not respond to encouraging messages with any enthusiasm at all. In comparison, the confidence interval for the weakest partisan's level of enthusiasm does _not_ include zero (narrowly), and therefore, statistically speaking, a weak partisan is predicted to respond with some minimal level of enthusiasm.
Well-sorted citizens are broadly emotionally responsive. They get angry at any message of threat, and they get happy at any message of victory. Whether party-based or issue-based, highly sorted individuals react to political messages with emotional reactions that match those driven by the strongest partisans or the most issue-intense individuals. However, while the emotional reactions of highly sorted individuals match the maximum emotional reactions already found in the electorate, the reactions of cross-cut individuals are significantly less intense than the reactions of any other citizens measured here. Cross-cutting identities dampen emotional reactions to political messages, such that the most cross-cutting identities lead to a complete lack of emotional response. This lack of response exists only in the group of cross-cut citizens that are increasingly disappearing from the American electorate.
## Obstructive Anger
Emotional reactivity is obviously important when we are trying to understand why certain partisans react to politics with anger or excitement and others respond less emotionally. The more sorted we become, the more emotionally we react to normal political events, and the more cross-cutting our identities, the more calmly we respond. The anger on display at Cliven Bundy's ranch, at the 2010 town hall meetings over Obamacare, at the Occupy Wall Street protests, and at Donald Trump's 2016 rallies is fueled by our increasing social and partisan isolation. As Americans continue to sort into partisan teams, we should expect to see more of this emotional reaction, no matter how much we may truly agree on specific policies.
In examining intergroup conflict in other nations, Kahn et al. (2016) found that "hatred and anger, and the absence of positive intergroup sentiments and moral sentiments of guilt or shame, may be an important obstacle both to the type of interest-based agreements that would benefit all concerned and to the type of relationship-building programs that can humanize adversaries and create the trust necessary for more comprehensive agreements. Indeed, trying to produce such agreement through careful crafting of efficient trades of concessions, without attending to relational barriers may be an exercise in futility" (83).
In other words, the anger that is driven by intergroup conflict and the gradual reduction of cross-cutting identities in the electorate is actively harming our ability to reasonably discuss the important issues at hand. The more people who feel angry, the less capable we are as a nation of finding common ground on policies, or even of treating our opponents like human beings. Our emotional relationships with our opponents must be addressed before we can hope to make the important policy compromises that are required for governing.
The increasingly prevalent well-sorted partisans are not only more intransigent in governing but also more active in politics, to make their intransigent inclinations known. Their emotional reactions to sorting can lead to a distinctly emotional type of political participation, in which partisans participate not only to make their policy positions known but, largely, because they're feeling particularly angry or elated.
# SEVEN
# Activism for the Wrong Reasons
An important balance between action motivated by strong sentiments and action with little passion behind it is obtained by heterogeneity within the electorate. Balance of this sort is, in practice, met by a distribution of voters rather than by a homogenous collection of "ideal" citizens.
—Bernard Berelson, Paul Lazarsfeld, and William McPhee, _Voting_
In their 1954 book, Berelson and colleagues made a relatively controversial claim. They suggested that an electorate composed entirely of "deeply concerned" voters would be unresponsive at the aggregate level and unsuitable to the needs of a changing nation. They pointed out that citizens who were "subjected to conflicting social pressures," such as those with cross-cutting identities, were necessary for providing the "inconsistent" preferences that lead to flexible voting records. These voters may be "erratic" in their voting patterns, but they would provide a net benefit to society, as the government could not change directions without them.
This chapter looks at the relationship between social sorting and political activism. As I explain below, there is a positive relationship between the two, with social sorting increasing levels of political activism. While most political scientists agree that political participation is a necessary ingredient of a functioning democracy, Berelson and colleagues pointed not to the value of participation itself but instead to the makeup of the electorate, and this is my focus as well. I do not argue that all political participation is bad for democracy. What I do argue is that the makeup of the electorate matters. If social sorting drives increased participation, the increasingly sorted electorate will be more homogeneously made up of the "deeply concerned" citizens. Highly sorted citizens, as seen in prior chapters, care more for party victory and therefore will be more consistently partisan in their voting than the "erratic," cross-cut voters. Cross-cutting identities provide some flexibility in voter preferences. Changing cross-cut voters into well-sorted voters is not necessarily a net good for a functioning democracy, particularly if the well-sorted are more likely to vote. The well-sorted voters are more likely to be active on behalf of their identities and emotions, which drive them consistently in the direction of voting that is less responsive to changing conditions and events.
## A Brief History of Participation
Social identity theory would predict that an increase in political identity strength and sorting should increase political activism, but most of the stories we hear about the American electorate describe it as disengaged, uninformed, uninterested, and inactive. Robert Putnam, in his 2001 book _Bowling Alone_ , paints a clear picture of an American public that is growing increasingly disconnected from one another on a civic and community-based level. As Americans become more isolated from each other, our engagement with politics is expected to go the way of our engagement with churches and civic organizations. We simply disconnect. In fact, Bill Bishop's 2009 book, _The Big Sort_ , describing massive cultural and geographical partisan sorting relies on this process of disengagement to explain Americans' subsequent realignment along partisan lines.
So why isn't our narrative about activism the same as our narrative about polarization? Are we polarized and apathetic? Has our increasing sorting and polarization managed to keep us out of politics? The short answer is that, despite all of the lamentations about American political apathy, there has been an increase in activism to match our social sorting.
The traditional explanation for political participation in American politics is one of either socioeconomic status or resources. Brady, Verba, and Schlozman in 1995 elaborated a model of participation that relied on resources such as time, money, and civic skills. Those skills were learned in settings such as civic organizations and churches, exactly the type of organizations that Putnam saw Americans moving away from. As Americans abandoned these organizations and affiliations, they also avoided political participation. But Miller et al. (1981) pointed out that some groups in American politics have participated in politics at a higher rate than their socioeconomic resources alone could predict. They offered an early explanation that resonates today: people who are conscious of their group memberships are driven to participate in politics on behalf of their groups.
This is not always a normatively positive type of participation. A Pew study in 2016 found that "across most measures of political participation . . . individuals with no negative partisan stereotypes were less likely to participate in politics." This means that the active electorate is now more likely to include voters who hold negative partisan stereotypes, a result that matches an increasing identification with socially homogeneous parties.
Social psychology explains how political participation can be a direct outcome of our sense of feeling attached to others, not necessarily those that we see in distinctly community-based places but those with whom we simply feel socially connected. As Americans have rearranged their social affiliations, not around institutions but around socially similar others, they have laid the groundwork for getting involved in politics again. Work I did with Leonie Huddy and Lene Aarøe in 2015 explained how identifying with a political party in a social way can increase political participation, even more powerfully than partisans' issue positions. I therefore expect to find that, as levels of social sorting have increased, levels of activism have followed along.
Although the changes over time have been small, political action has increased significantly in recent years. The cumulative ANES data through 2012 show rising levels of participation. Figure 7.1 presents the percentage of the American population who report that they have voted in each year. This measure is famous for false claims of voting due to social desirability (Hanmer, Banks, and White 2014). However, even if we pretend that it is an entirely imaginary measure, always constructed from whole cloth by the respondents of the survey, it is clear that even the number of people who _wish to_ have voted has increased. These are small total values, but between 1972 and 2012, the percentage of people who report having voted increased by about 5 percentage points.
Figure 7.1. Percentage of US population reporting that they have voted
Note: Data drawn from the ANES cumulative file through 2012, weighted. Ninety-five percent confidence intervals shown. Linear trend line drawn for ease of interpretation.
Voting, of course, is not the only way to participate in politics. It is perhaps the most consequential, in terms of concrete electoral outcomes, but the ANES measures a series of other activities that certainly qualify as political activism, many of which require more commitment than simply casting a ballot (or reporting that you have done so, whether or not this is true). The ANES asks respondents whether they have tried to influence the vote of others, displayed a campaign button or bumper sticker or yard sign, attended a political rally, or done volunteer work for a political campaign.
Figure 7.2 lays out the trends in each of these types of political action between 1972 and 2012. By far the most popular, and the most rapidly increasing form of political activism is the simple act of speaking to another person and trying to convince them to vote for the candidate you support. Between 1972 and 2012, this very social form of activism increased by 11 percentage points. Levels have declined slightly since 2004, but even considering this decline, 2012 levels of political social contact are higher than they were in the 1970s, 1980s, and 1990s. It is also important to note that these measures do not account for Internet activism, which has been shown to have a mobilizing effect on voters (Settle et al. 2016).
Figure 7.2. Percentage of US population reporting having participated in each political activity
Note: Data drawn from the ANES cumulative file through 2012, weighted. Ninety-five percent confidence intervals shown.
Occupying a distant second place in popular political activism is wearing a button or putting a bumper sticker on your car or a sign in your yard. Though far fewer Americans engage in this type of public display of political affiliation, the number who have done so since 2000 has markedly increased. Between 2000 and 2004, the percentage of Americans who engaged in this public display increased by nearly 10 percentage points, with this increase declining to about 5 percentage points by 2012. However, since Bill Bishop, in his 2009 book, noted an increasing level of geographical polarization among American partisans, it is possible that, rather than indicating increasing partisanship, this increase simply indicates a more comfortable and nonconfrontational environment for public displays of partisanship.
To account for this possibility, I examined the button/sticker measure among those who never discuss politics with family and friends—indicating either an uncomfortable social environment for political engagement or a nonpolitical environment—and compared this against those who do discuss politics with family and friends. As figure 7.3 indicates, those who discuss politics with family and friends are significantly more likely to commit one of these public acts of partisanship. Furthermore, although both groups of people grew more likely to engage in these displays in 2004, those who do not discuss politics fell back almost completely to pre-2004 levels in the 2008 election. Those who were socially engaged did not experience the same drop off. These partisans remained far more likely to engage in a public display of partisanship in the following elections. Though the percentage of Americans engaged in these displays declined, they remained significantly more publicly partisan than they had been in 2000. The trend shown in figure 7.2, then, is driven largely by people who are _socially_ engaged in politics. The types of political engagement that are increasing are social engagements, and they occur most frequently among those who are in politically aware social networks.
Figure 7.3. Percentage of US population reporting displaying a button, sticker, or sign, by level of political discussion
Note: Data drawn from the ANES cumulative file through 2012, weighted. Ninety-five percent confidence intervals shown.
Going back to figure 7.2, it is clear that donating to campaigns, attendance at political rallies, and volunteer work have not significantly increased, or have increased only modestly, during the last few decades. Most of the increase in political activism has been in the persuasion element, and in the public display of partisan attachments. These are certainly the least costly forms of engaging in politics, but their increase does indicate a meaningful change in the American electorate. Americans feel more compelled to report that they voted, they are more often attempting to convince others to vote for their preferred candidates, and they are more often publicly displaying signals indicating which candidate they prefer. All of this points to an electorate that finds political preferences to be a more social and public phenomenon than they were thirty years ago.
## Identity-Driven Action
Social psychologists have already discovered that when people identify with a group of other people they are more likely to take political action on behalf of their group, particularly when that group is under threat. Kelly and Breinlinger (1996) found that identification as a woman increased the likelihood of participating in the women's movement. De Weerd and Klandermans (1999) discovered that social identification as a farmer increased action preparedness and farmers' protest participation. Ethier and Deaux (1994) observed that strongly identified Hispanic college students, when placed in an unfamiliar environment, engage more in group-based cultural activities than the less strongly identified. Huddy, Mason, and Aarøe (2015) demonstrated that a stronger partisan identity increased intentions to donate to and volunteer for a political campaign in the context of an election. Mackie, Devos, and Smith (2000) found that strong group identifiers were more likely to take action against a threatening outgroup.
It is tempting to argue that these actions are driven by self-interest (also called instrumental concerns) more than social identity. Group members take action because they will directly and individually benefit from the success of their group's objectives. But in most of these cases, the difference between the strong group identifier and the weak group identifier is simply the individual's psychological attachment to the group. A weakly identified farmer will reap the same benefits from political action as a strongly identified farmer, and yet only the strongly identified farmer takes action. The identification with the group drives the group member to take action to maintain positive group status, in line with the first imperative of a social identity—be victorious.
Partisans should be more likely to participate in politics not simply because the party holds sympathetic issue positions but also because the party is their team, it is under threat, and they are compelled to do something to maintain its status. As Klandermans (2003) explains, "People participate not so much because of the outcomes associated with participation but because they identify with the other participants . . . participation generated by the identity pathway is a form of automatic behavior, whereas participation brought forward by the instrumental pathway is a form of reasoned action" (687). Partisans are compelled to automatically participate in politics by social and psychological motivations. It should be noted that social and instrumental influences are never completely separate in a given individual. In most people, both types of motivations work simultaneously. The purpose here is to attempt to pull apart the effects of each so that the respective contributions of the two types of motivations can be more precisely seen.
In 2015, Leonie Huddy, Lene Aarøe, and I compared the effects of what we called an "expressive" partisan identity to the effects of an "instrumental" partisan identity. In other words, we measured whether social identification with a party drove more political action than issue-based identification with a party. We found that partisan identity was a significantly more powerful predictor of political action than an issue-based measure. We also found that the more social-identity-focused our measure was, the better the measure was at predicting activism. The traditional seven-point measure of partisan identification was less powerful than the four-item battery of social partisan identity. So, as we got closer to social identity, political action grew more likely. This is in line with what social psychology would have predicted. The same literature would predict that a set of highly aligned identities should have an even more powerful effect.
## Partisan-Ideological Sorting
One way to begin finding effects of sorting on activism is to look at the simple alignment of partisan and ideological identities. If I look only at partisan-ideological sorting, I can use the fairly strong test of the matched sample that I examined in prior chapters. Again, in this sample, the full ANES cumulative file through 2012 is used to match respondents on ideological identification, issue extremity, education, sex, race, age, geographical location, and church attendance. The only thing that moves in figure 7.4 is the extent to which partisanship lines up with ideological identification. When this very large set of data is combed to find people who are as similar as possible on many important traits, the effect of sorting on activism emerges as an important one. I put all of the five elements of activism from figure 7.2 into one combined measure, counting the number of activities that have been undertaken (and recoding this count to range from 0 to 1). The predicted levels of total activism in the presence of low and high levels of sorting are shown in figure 7.4. When a partisan identity is poorly aligned with an ideological identity, average levels of activism are significantly lower than when partisanship and ideology are well aligned. Again, the total effect size is small, as changes in activism over time also tend to be. However, there is a significant difference in political activism between the sorted and unsorted, even among people who are very similar, including in their political opinions.
Figure 7.4. Mean activism: Matching on ideology and issue extremity
Note: Ninety-five percent confidence intervals shown. Data drawn from the ANES cumulative file through 2012. Samples are matched on ideology, issue extremity, education, sex, race, age, south/nonsouth location, and church attendance.
These highly similar Americans are significantly more likely to participate in some form of political action when their partisan identities line up with their ideological ones. Even without the contribution of issue extremity—and in alternate models accounting for the constraint of issue positions—simple partisan-ideological sorting drives people to take political action. As seen above, this action is largely social in nature, like its motivations.
But do the effects of sorting surpass the effects of simple partisan identity? I look at this question by again examining partisan-ideological sorting while holding partisanship constant, looking only at strong partisans. As in prior chapters, this test again results in significant effects of sorting, this time on political action. Figure 7.5 uses the ANES cumulative sample through 2012 to present predicted values of the five-item activism scale, drawn from OLS regressions controlling for issue extremity, issue constraint, political knowledge, race, sex, income, age, and church attendance. While holding each of these variables constant, the predicted levels of activism among unsorted strong partisans is about 0.17, while the predicted level of activism among sorted partisans is 0.27, a significant 10 percentage point difference.
Figure 7.5. Predicted values of activism among strong partisans across levels of sorting
Note: Bars represent predicted values of activism at varying levels of partisan-ideological sorting controlling for a scale of issue extremity and issue constraint (and their interaction), political knowledge, race, sex, income, age, and church attendance. Originating regression is shown in appendix table A.11. Ninety-five percent confidence intervals are shown. This sample is fully weighted. The low sorting score is not zero, but 0.0857, the lowest sorting score possible given a strong partisan identity.
Once again, the difference between the sorted and unsorted partisans is significant, meaning that even among strong partisans a well-aligned ideological identity will generate more activism than a cross-cutting one. This model controls for issue extremity and constraint among other things. Therefore, the alignment of partisan and ideological identities is motivating increased political action even while issue positions are unchanging. American partisans are more willing to participate in politics when they are well sorted, no matter what their issue positions happen to be. This is political tribalism driving action. We are growing more politically engaged on behalf of our team spirit.
## Social Sorting and Activism
If the alignment of the seven-point scales of partisanship and ideological identity can predict increasing activism, a more social-identity-based measure of partisanship and other social identities should have even more powerful effects. When the concept of social sorting is measured using multiple social identities and measures expressly designed to assess social identification, stronger effects do indeed result.
When looking at prior political activism, including working for a political party or candidate, engaging in political protest, writing a letter or email to a political official, and donating money to a political party or candidate (a count recoded to range from 0 to 1), social sorting indicates a much more engaged partisan. Figure 7.6 presents the predicted values of this activism scale, according to levels of social sorting. Importantly, these predicted values are generated from an OLS model that controls for a measure of issue positions including the extremity, constraint, and rated importance of five issues, as described in chapter 4. In this figure, when all other variables are held at their means or medians, the most cross-pressured partisan is predicted to rate an activism score of about 0.19. A partisan with very high levels of social sorting, however, is predicted to report an activism score of about 0.52. This is an increase of about 30 percent of the full range of activism, which corresponds to more than one additional political activity out of the full range of 4. Again, this is the relationship between social sorting and activism, holding issue positions constant. This is a powerful effect of the teaming up of social identities that drives people to engage in politics, even when their issue positions are no different than those of their fellow citizens who hold cross-cutting identities.
Figure 7.6. Predicted values of a scale of prior activism, by level of social sorting
Note: Predicted values drawn from OLS models shown in the appendix, controlling for issue extremity, constraint and rated importance, race, sex, income, age, political knowledge, and church attendance. Originating regression is shown in appendix table A.12. Ninety-five percent confidence intervals shown.
Predicting future or intended activism is slightly more problematic because there is no guarantee that these promises are fulfilled. However, figure 7.7 presents the predicted values of an index of intended activism in which people indicate whether they intend to (1) donate to or (2) volunteer for (3) candidates or (4) parties in the upcoming 2012 election. Remember, these data were collected a full year before the 2012 election, suggesting that the results should be taken with a grain of salt, as people engage less with elections so far away.
Figure 7.7. Predicted values of a scale of intended future activism, by level of social sorting
Note: Predicted values drawn from OLS models shown in the appendix, controlling for issue extremity, constraint and rated importance, race, sex, income, age, political knowledge, and church attendance. Originating regression is shown in appendix table A.12. Ninety-five percent confidence intervals shown.
Still, these four possible activities are combined into a scale, again coded to range from 0 to 1. In figure 7.7, the promise of future political action is even more powerfully related to social sorting. Those with highly cross-cutting social identities are, in fact, predicted to participate in negative levels of activism, with a score of −0.07. The confidence interval is wide, however, and crosses zero, so it is fair to assume that these cross-cutting social identities essentially eliminate intentions to participate in politics. Those with very strong and well-aligned social identities are predicted to report an activism score of 0.36, a full 43 percentage points higher in the full range of intended activism than those with cross-cutting identities. With four activities in the scale, this increase in social sorting adds nearly two full activities. Even if this self-report of future behavior is inflated, the effect of social sorting is clearly large and powerful. People at least want to _say_ that they will participate in politics far more strongly when their social identities line up with their partisanship.
Even when a host of highly salient issue positions are held constant, these large effects emerge. Social sorting is capable of making people want to participate in politics.
However, controlling for an index of issue positions perhaps does not provide a fair test of the potential impact of a truly extreme and salient issue position. What if someone cares mostly about one issue, for example? The five-issue index wouldn't account for the unique power of the single issue. Also, can an issue drive action not only from an instrumental motivation but also by generating an identity around itself? If instrumental concerns about single issues can motivate action, the evidence that I presented above doesn't allow that effect to come through.
## Issue-Driven Action
One issue that tends to motivate single-issue voters is abortion. It is also a cause that has generated labels for the advocates of each side of the issue, potentially creating a social identity around issue advocacy. Pro-choice and pro-life opinions have produced potential political identities in a way that is distinct from the advocates of issues such as taxes or health-care reform.
### The Case of Abortion Attitudes
Craig McGarty and his colleagues explained in 2009, "Merely holding the same opinion as others is not sufficient for such a group to be said to exist, rather the shared opinion needs to become part of that social identity. In this way, people can come to perceive and define themselves in terms of their opinion group membership in the same way as they would with any other psychologically meaningful social category or group" (846). The main incentive that drove McGarty et al. to identify such an issue-based identity was the distinctly activist outcomes that result directly from these identities.
A single issue for a voter who cares a great deal about it may generate a social identity, with both the issue and the identity driving the individual toward greater action. Klandermans (2014) has laid out a compelling case for the collective-action effects of group identities. When groups form around a particularly salient issue, they tend to lead to political action. In this sense, issue-based identity is potentially a unique motivator for political action, as it involves both issue opinions and a sense of group loyalty. It is this identification with the issue-based group that I expect to drive significant levels of activism and partisan bias, more powerfully than simply an extreme issue position.
Abortion attitude identities in particular—a sense of social connection to others who call themselves pro-choice or pro-life—seem to be a powerful example of this issue-based identity. Pro-choice and pro-life groups have undeniably engaged in group-based activism since the 1973 ruling in _Roe v. Wade_ that legalized abortion. To this day, groups like Planned Parenthood and the National Right to Life Committee (NRLC) clash repeatedly over the legality and normative values at stake in abortion rights, with advocates on either side identifying with the issue-based labels "pro-choice" and "pro-life." I choose this topic mainly because of these preestablished and well-known names with which a person can identify to varying degrees. In fact, I specifically measured the social identity strength of affiliating with each side of the debate. I used the four-item scale of social identity and applied it to the terms _pro-choice_ and _pro-life_. The items for each label scaled together well, with both pro-life (α = 0.82) and pro-choice (α = 0.80) identities demonstrating a real identification with these labels. In fact, mean levels of identification with the pro-choice and pro-life labels was generally higher than partisan identification among partisans. Mean Democratic identity among Democrats (on a 0 to 1 scale) was 0.62 and Republican identity among Republicans was 0.60. But mean pro-choice identity among those who approved of abortion was 0.74, while mean pro-life identity among those who disapproved of abortion was 0.79.
In the sample used here, the issue of abortion was rated to be somewhat or extremely important by 87 percent of respondents, upholding its potential for representing issue-based identity. I also looked at abortion issue extremity, operationalized as the abortion-opinion scale including the following four options: "By law, abortion should never be permitted," "The law should permit abortion only in case of rape, incest, or when the woman's life is in danger," "The law should permit abortion for reasons other than rape, incest, or when the woman's life is in danger," and "By law, a woman should always be able to obtain an abortion as a matter of personal choice." This scale was folded in half, so that abortion extremity is limited to a two-point measure, which can indicate either position in the middle of the scale (coded 0) or either position on the ends of the scale (coded 1). Finally, I looked at abortion issue importance, which was simply the follow-up importance item to the abortion-opinion prompt, asking "How important is this issue to you?" with the potential responses of Very important, Somewhat important, Not very important, and Not at all important. This four-point scale was recoded to range from 0 (not at all important) to 1 (very important). These alternate measures were used to determine which element of this issue most motivated political action. Is it the sense of being connected to like-minded others? Or is it the actual extremity of belief? Or is it instead a sense that the issue is important? As V. O. Key says in his 1961 book, "The more concerned a person is about an issue, the greater is the probability that his opinion will be intense" (219). I therefore compare the identity behind abortion attitudes with a combined effect of abortion extremity and importance.
Key (1961) did not find a strong relationship between activism and issue intensity: "From our daily impressions of politics, we feel that persons who have opinions of high intensity are likely to seek energetically to achieve the ends in which they believe. . . . Little bands of dedicated souls leave their clear imprint on public policy" (229). However, in examining the data, he had to conclude that "under some circumstances persons have intense opinions that do not lead to heightened participation, although across the electorate there seems to be a slight tendency for participation to increase with intensity of opinion" (230). Key blames these weak findings on the "crudeness of our measures of both intensity of opinion and participation" (230). Perhaps then a better measure, like the one that expressly gauges the social identity associated with an issue position, could do a better job explaining the link between issue position and political action.
Figure 7.8 shows that when looking at general political engagement, measured as the political activities a person has done, the identity associated with the abortion issue is better at driving activism than is the extremity of that abortion opinion, or even the extremity interacted with the rated importance of the issue. Figure 7.8a shows the predicted levels of activism at each level of social identification with the pro-choice or pro-life label. This identity has a significant effect. Those with the weakest attachments to the group labels are predicted to engage in 24 percent fewer political activities than those with the strongest attachments. This effect is controlling for the effect of both abortion-opinion extremity and importance, so these identity effects are occurring independent of a person's actual position on or passion about abortion.
Figure 7.8. Predicted values of prior political engagement from abortion identity versus abortion extremity versus abortion extremity*importance
Note: Ninety-five percent confidence intervals shown. Identity measured from four items including (1) How important is being pro-choice/pro-life to you? (2) How well does the term _pro-choice_ / _pro-life_ describe you? (3) When talking about pro-choice/pro-life people how often do you use "we" instead of "they"? (4) To what extent do you think of yourself as pro-choice/pro-life? Prior political activism includes working for a political party or candidate, engaging in political protest, writing a letter or email to a political official, and donating money to a political party or candidate (a count recoded to range from 0 to 1). Panels A and C derive from a model including the interactive term between extremity and importance. Panel B only includes extremity and does not control for identity. All models control for race, sex, income, age, political knowledge, and church attendance. Originating regressions are shown in appendix table A.13.
In figure 7.8b, I look only at the effect of abortion-opinion extremity, not controlling for importance or the issue identity measured in 7.8a. As Key said, it is likely that importance and extremity go hand in hand, and this model is an attempt to see what power comes from allowing the extremity of the issue to account for all of the other issue-based influences. What I find, instead, is that the extremity of a person's position on abortion has some effect on general activism, but the effect is small—far smaller than the effect of abortion-opinion identity. The total effect of abortion-opinion extremity is less than half the size of the effect of identity, and this effect does not control for the effects of identifying with the abortion-opinion group. The best-case scenario for the motivating effects of extremism, therefore, is that it generates half the activism of the pro-choice or pro-life identity.
In figure 7.8c, I interact opinion extremity and importance to give issue intensity one last chance to prove its activating potential. Figure 7.8c looks at the marginal effects of extremity (the effect of moving from lowest to highest extremity) on activism at varying levels of issue importance. This effect is technically significant but does not reach an actual positive effect on activism until the rated importance of the issue passes its midpoint. This means that the extremity of the abortion position has no effect on general activism unless the issue is rated by a person to be at least somewhat important. Even then, the effect increases activism by about only 10 percent of the total range of activism. In comparison with the 24 percentage point increase seen in figure 7.8a, abortion attitudes are doing something, but they are far from the most powerful motivator of political action. The social attachment to the pro-choice and pro-life groups drives twice as much action as the intensity of the attitudes themselves.
The same result appears, if not even more powerfully, in figures 7.9a, 7.9b, and 7.9c, which examine intentions to participate in future activism. In these models I asked people if they intended to participate in the upcoming 2012 election. In figure 7.9a, those who felt strongly socially connected to the pro-choice or pro-life group labels showed essentially the same reaction to that identity as they did in the case of prior activity. The most strongly socially connected are predicted to participate in about 25 percent more political actions (or to intend to do so) than those who feel only a weak social connection to these groups. Again, this is controlling for an interaction between abortion-opinion extremity and importance. So, without any change in beliefs about abortion, the people who feel socially connected to the pro-choice or pro-life groups are promising significantly more political activism than those who don't feel socially connected.
Figure 7.9. Predicted values of intended future political engagement from abortion identity versus abortion extremity versus abortion extremity*importance
Note: Ninety-five percent confidence intervals shown. Identity measured from four items including (1) How important is being pro-choice/pro-life to you? (2) How well does the term _pro-choice_ / _pro-life_ describe you? (3) When talking about pro-choice/pro-life people how often do you use "we" instead of "they"? (4) To what extent do you think of yourself as pro-choice/pro-life? Intended activism includes intention to (1) donate to or (2) volunteer for (3) candidates or (4) parties in the upcoming 2012 election. Panels A and C derive from a model including the interactive term between extremity and importance. Panel B only includes extremity and does not control for identity. All models control for race, sex, income, age, political knowledge, and church attendance. Originating regressions are shown in appendix table A.14.
In comparison, in figure 7.9b, the effect of the extremity of the issue position alone (without even controlling for identity) is significant but quite small. The difference between an extreme opinion and a moderate opinion generates 6 percent more intention to participate in the upcoming election. In figure 7.9c, when the extremity and importance are combined, the effect of issue intensity on intentions to participate in 2012 only rises above zero at the very highest levels of issue importance. Even then, it predicts an increase in activist intentions of about only 6 percentage points, with a confidence interval that remains near zero.
When you ask people if they are planning to participate in the upcoming election, the intensity of their abortion opinion does very little to push them into action. However, their sense that they are socially connected to other people who call themselves pro-choice or pro-life is quite powerful. These single-issue activists are driven to act not by the intensity of their beliefs but by the sense that they are supported by like-minded others. Other people, not simple opinions, push activists into action.
Even in looking at the effects of singularly powerful issues on political activism, the biggest motivator for political engagement is not the issue itself but the community around the issue. The extremity and importance of an issue is not what drives people to take political actions—to urge others to vote, to wear a button, to promise to donate to a cause. The thing that drives people into action on behalf on an individual issue is the sense that there are other people on their side. Their opinion on that issue generates an identity—a team—and their team spirit, just as in the case of partisan identity, is what makes them act. We are not observing a strongly outcome-oriented electorate. Even when looking directly at one single issue that people want to see changed or supported, it is group feelings that drive political action. Without team spirit, the extreme believers barely differ from the moderate believers. Political activism, then, is being driven by social cues, even when it is in service of a single issue.
## Emotion-Driven Action
Injustice, wrong, injury excites the feeling of resentment, as naturally and necessarily as frost and ice excite the feeling of cold, as fire excites heat, and as both excite pain. A man may have the faculty of concealing his resentment, or suppressing it, but he must and ought to feel it. Nay he ought to indulge it, to cultivate it. It is a duty. His person, his property, his liberty, his reputation are not safe without it. He ought, for his own security and honour, and for the public good, to punish those who injure him. . . . It is the same with communities. They ought to resent and to punish.
—John Adams, diary entry of March 4, 1776, one month before Lexington and Concord (quoted in Philbrick 2013)
John Adams did not have a problem with action driven by resentment and anger. Be warned, however, that he was on the precipice of the Revolutionary War when he expressed this feeling.
The previous two chapters have demonstrated that social sorting and identity-based polarization have led not only to increased action but to prejudice and emotional volatility. If the American public was simply more biased and angry but did nothing about it, sorting would not necessarily have electoral effects. Unfortunately, sorting has driven American partisans to grow more biased and angry and to take more political action because of that bias and anger.
The motivating effects of emotion have been well studied. In chapter 6, social sorting was shown to be uniquely capable of driving angry and enthusiastic responses to political messages. These emotions are not only interesting in themselves. Emotions, particularly anger and enthusiasm, can change the way people think and analyze, and can drive them to action. In 2014, Eric Groenendyk and Antoine Banks found that "the emotions strong partisans experience help them to bypass individual-level utility calculations and take action on behalf of their party" (360). The emotions most often observed by Groenendyk and Banks among strong partisans were anger and enthusiasm. Psychologists have often discussed the particular power of these two emotions to drive action. They have been labeled "approach emotions" (Harmon-Jones, Harmon-Jones, and Price 2013). Both anger and enthusiasm tend to lead to more optimistic expectations for the future (Lerner and Keltner 2000). They are emotions that make you think that, if you get in the game, you are likely to win. These particular emotions are driving people toward action not because they have made a reasoned, utility-based calculation but because they are pushed by their feelings. Well-sorted people have more of these emotions, and they are more activist than those with cross-cutting identities.
When I look at the experimentally induced levels of anger and enthusiasm produced in the 2011 sample, these emotions have significant effects on people's intention to participate in the upcoming 2012 election. As a reminder, I had respondents read randomly assigned messages that were either threatening or supportive of groups (parties) or issues. For the purposes of simplification, the group- and issue-based messages are combined for this analysis. The activism questions were asked of the respondents after the experimental manipulations. As seen in chapter 6, the experimental messages generated significant anger and enthusiasm in the respondents, particularly among those with high levels of social sorting. These angry and enthusiastic responses significantly increased respondents' activism. Importantly, the results shown in figure 7.10 represent predicted values of activism drawn from an OLS regression that controls for issue-position extremity and constraint. So, just as Groenendyk and Banks (2014) found, these emotions are driving people toward action, holding issue positions constant.
Figure 7.10. Predicted activism at low and high anger and enthusiasm
Note: Ninety-five percent confidence intervals shown. Model only includes those who were presented with a threatening or reassuring message. OLS model with robust standard errors controls for issue extremity and constraint, race, sex, income, age, political knowledge, and church attendance. Originating regressions are shown in appendix table A.15.
This effect is not due to issue positions, which are controlled, and it is not entirely due to the correlation between emotions and sorting because the effect remains (though it shrinks) when sorting is controlled. These emotional reactions drive people to want to act. When they have read a political message that makes them feel highly angry or enthusiastic, they want to jump into the ring. They want to get involved.
The experimental manipulations themselves had a significant effect on intended activism but only for threat-based messages. This is consistent with other research that has found stronger results in motivating activism via anger rather than enthusiasm (see Valentino et al. 2011). In figure 7.11, individuals who receive a threat-based message are significantly more likely to report that they intend to participate in the upcoming election than are those who receive no threat (including the control group and messages of support). These are simply mean levels of intended activism without any controls, which are comparable due to the randomization of the treatments.
Figure 7.11. Mean levels of intended activism after threatening versus nonthreatening messages
Note: Ninety-five percent confidence intervals shown. Dots represent mean levels of intended activism in the presence of threatening or nonthreatening treatments. Data from 2011 YouGov sample.
A threatening message, in these data, does motivate increased intention to take political action. This is unique to the threatening message; the messages of support do not have the same power. This is, however, not surprising, considering the importance of threat to activating social identity and group defense. Without a threat to a social group, members are less likely to derogate outgroups, and they have less of a motivation to improve the status of the group. Exposure to a threatening message does, overall, increase activism.
In 2011, Valentino and colleagues made the point that a simple resources-based model cannot account for the changes in activism that occur within individuals from year to year. Something else changes in each election that either motivates participation or depresses it. In 2001, psychologist Barbara Fredrickson explained that emotions, particularly positive emotions, lead people to engage in behavior that is not necessarily goal-oriented but, instead, rewarding in and of itself. The emotions that emerged in chapter 6, due largely to social sorting, guide people toward action because that action feels like the right thing to do. They are participating in politics, like Klandermans's farmers, because they feel connected to their groups, and that connection makes them feel more emotionally driven toward action. The highly sorted individuals don't necessarily hold more intense issue positions or stand to gain more tangible resources from a party victory, they act because it feels good to act.
## But Don't We Want an Active Electorate?
"I used to spend ninety per cent of my constituent response time on people who call, e-mail, or send a letter, such as, 'I really like this bill, H.R. 123,' and they really believe in it because they heard about it through one of the groups that they belong to, but their view was based on actual legislation," Nunes said. "Ten per cent were about 'Chemtrails from airplanes are poisoning me' to every other conspiracy theory that's out there. And that has essentially flipped on its head." The overwhelming majority of his constituent mail is now about the far-out ideas, and only a small portion is "based on something that is mostly true." He added, "It's dramatically changed politics and politicians, and what they're doing."
—Devin Nunes, Republican congressman from California (quoted in Lizza 2015)
Activism may have increased over the last few decades, but this is not necessarily a responsible, outcome-based participation. As Republican congressman Devin Nunes told _New Yorker_ reporter Ryan Lizza in 2015, the types of people who reach out to him (a form of participation) are increasingly ignorant of the actual policies they wish to see enacted. They are participating, but they are doing so on the basis of misinformation and ill-formed ideas. The findings of this chapter have shown that the most-sorted partisans are the most likely to take political action, regardless of their policy attitudes, particularly when they feel threatened and angry. In combination with the findings of chapter 3, this suggests that an increasing portion of the American electorate is driven to action by identity-centric motivations, while a decreasing portion of the population (those with cross-cutting identities) is relatively inactive.
The fact that action occurs because it simply feels good to act is not a great shining light of our contemporary democracy. Our political identities—partisan, ideological, racial, religious—and the alignment between them move us toward action without necessarily informing us about policy outcomes. We get up off of our sofas and put on buttons, talk to neighbors, go and vote, because it feels righteous. The bigger and more socially homogeneous the parties are, the more we have to fight for. We are only partially motivated by issue-based outcomes.
We are acting because our emotions and the self-esteem that is driven by our identities compel us to do something. We must defend our groups, and the larger and wider the group, the more necessary it is to defend it. When a threat to any of these party-related groups is perceived, political activism in defense of party status grows more likely. Even when we are acting in the pursuit of a single important issue, a large portion of that action is driven by our membership in a social group to which we feel socially and emotionally connected. Our actual opinions—the intensity of our attitudes—can't compel the same sort of political activism that our simple sense of social connection can. We take political action, potentially making real political changes, because we feel close to particular groups of people and want them (and therefore ourselves) to be winners.
This is partially why we see strong partisans out campaigning and voting even when they are nearly certain their candidate will win. They feel compelled to take political action not to achieve change but to express support for their team. It feels right to get out there and defend the team, even when the team is a guaranteed victor—possibly even more so when victory is imminent. There is a gleeful joy in participating in your own team's victory. The team doesn't need all of our votes and participation, but partisans gladly provide it anyway.
This group-based activism is unlikely to abate any time soon. Dinas (2014) found that not only does a strong partisan identity drive political action, including voting, but the act of voting also drives increasingly strong identification with the party. Therefore, the more we feel partisan, the more we vote, and the more we vote, the more partisan we feel. It is a self-reinforcing cycle. As our partisanship joins forces with many other social identities, this effect grows stronger. Our sense of social connection to other people is what drives us to take political action, not simply the intensity of our issue opinions.
While activism is generally a desirable element of a functioning democracy, blind activism is not. These results demonstrate that American partisans are working hard to participate in politics, but the ones who are most active tend to be those who cannot be convinced to change their minds. They react to threat, anger, and the strength of a whole cohort of identities that are increasingly harmonized. When individuals participate in politics driven by team spirit or anger, the responsiveness of the electorate is impaired. If their own party—linked with their race and religion—does something undesirable, they are less likely to seriously consider changing their vote in the ballot booth. As people grow more sorted, they are less likely to split their ballots or to vote for outgroup partisans in addition to ingroup partisans (Davis and Mason 2015). Without voters who may be cross-pressured or otherwise likely to allow their vote to respond to events on the ground, government has the potential to be too rigid to respond to modern changing conditions.
# EIGHT
# Can We Fix It?
Identity is what gets the blood boiling, what makes people do unspeakable things to their neighbors. It is the fuel used by agitators to set whole countries on fire.
—Ian Buruma, _The Blood Lust of Identity_ (quoted in Kalyvas 2006)
How does American politics get back to the work of governing instead of focusing so much of our energy on partisan victory, conflict, and pride? Donald Trump won the presidential election of 2016. His ascent was bewildering to political scientists and pundits alike. One defining characteristic of his campaign was the effectiveness of his use of identity to anger and divide the electorate. By calling out distinct social groups, including women, Muslims, Mexicans, blacks, and a continuing list of others, Trump has made it clear that the American electorate can and should be divided into identity-based groups. While many conservatives lament his lack of consistently conservative policy positions, he remains remarkably popular with the base of the Republican Party to date, and this encourages establishment Republicans to fall in line behind him. This popularity does not come from his policy-based bona fides but from his use of the power of simple identity to rile up a significant portion of the American population.
Based on the research presented in this book, his success should not be surprising. Trump has taken advantage of a country whose parties have grown socially segregated. Threats to the status of any social group are linked to the status of other social groups as well, so that a single political threat has the ability to mobilize, anger, and bias large swaths of the electorate. The election of 2016 is the result of decades of social sorting, which has allowed a larger portion of the electorate to engage in politics out of defensiveness, judgment, anger, and a need to win. This is not a productive type of engagement, it is one that only deepens the divide between racial, religious, and partisan groups. The 2016 election is a perfect example of what happens to a nation that has seen its citizens gradually isolated from those who are socially unlike themselves.
## What Happens to a Sorted Nation?
In the late eighteenth century, the term _party_ was not meant to indicate the brittle, divided groups we have today. Robert Dahl describes the original concept of parties as "a current of political opinion, rather than an organized institution." As parties began to organize into more factional groups, Senator Hillhouse of Connecticut declared in 1808 that "party spirit is the demon which engendered the factions that have destroyed most free governments" (Dahl 1967, 207). In fact, it isn't only party spirit that can damage governments.
In political research outside of American politics, the alignment of multiple social identities has had some powerful and even dangerous outcomes. Selway (2011) finds that cross-cutting identities significantly reduce the risk of civil war in a country, while the alignment of ethnic and religious identities speeds the onset of civil war. He writes, "ethno-religious cross-cuttingness makes civil war less likely because it reduces the saliency of out-group differences and thus makes it harder for potential rebel leaders to recruit by appealing to ethnicity" (Selway 2011, 112). In a 2012 study, Gubler and Selway find that civil war is twelve times less probable in societies where ethnicity is cross-cut by another social identity such as class, geography, or religion. Kalyvas in his 2006 book writes, "The intuition is that if a population is clustered around a small number of distant but equally large poles, it is likely to undergo violent conflict. . . . The underlining mechanism is dislike so intense as to cancel even fraternal ties, imagined or real" (Kalyvas 2006, 64).
Even in the context of American politics, Dahl (1981) explained America's own civil war by stating, "Never before or since in American history has the pattern of moderate conflict with crosscutting divisions been so fully transformed into the pattern of severe conflict and polarization" (321). Fisher (1997) described the American civil war experienced in Tennessee as a distinctly social rupture: "Dedication to the cause and party consciousness broke former bonds of friendship and kinship; there was a tendency to greet and be friendly towards members of the same group, whilst systematically avoiding the others. Quarrels, rivalry and hatred developed out of these estrangements. Each group had its cafe, its meetings and even its feast days, religious on the one side and secular on the other" (Fisher 1997, 85).
The American electorate today is not engaged in a civil war. However, it is difficult to read Fisher's account and not feel echoes of contemporary American political culture. I do not argue here that the social sorting of the American electorate is inevitably leading to violent political conflict. After all, as Marilynn Brewer states, "In various contexts, groups have managed to live in a state of mutual contempt over long periods without going to war over their differences" (Brewer 2001a, 32). In fact, Scarcelli (2014) clarified that highly aligned identities don't always lead to civil war, and that catalysts such as economic decline or "adverse regime change" are often needed to exacerbate intergroup tensions to the point of organized violence. This is something to be wary of, but it is by no means unavoidable.
It should be emphasized, still, that increasing trends of social sorting are not simply benign, ignorable reorganizations. There can be real consequences when cross-cutting identities give way to orderly, segregated political teams. The question, then, becomes, is there a way to reduce or reverse the trends of social sorting or the sociological and emotional effects of this sorting?
Of course, some partisan conflict is necessary for any successful democracy. Parties should disagree about policy, and partisans should care which side wins. When the parties become socially isolated from one another, however, the conflict between them becomes less about governing and more about the conflict itself. This type of conflict is the one that needs to be addressed.
There are a number of ways to approach this problem, so I explore each strategy in turn, considering its usefulness for American polarization in particular. First, there is a broad literature that examines how to reduce intergroup conflict. This research was designed to reduce racial or religious group conflict, but in many cases it can be applied to partisan conflict as well. Second, some research in social psychology has begun looking at the effect of manipulating self-esteem and self-affirmation on political polarization, and this is another possible avenue for conflict reduction. Finally, there is a possibility for an unsorting or dealignment of our partisan and social identities. This could occur through demographic trends or by the reappearance of a major rift in one party.
In the end, none of these solutions may be effective or likely. However, each may provide some further insight into the depth and nature of the problem at hand.
## How Social Science Deals with Intergroup Conflict
We do not need to disagree to feel connected to our social groups. The sense of well-being that we receive simply from being in the groups is reason enough to join them. And we do not need to dislike outgroups simply because we like our own group. It is possible to feel connected to social ingroups and not feel antagonism toward outgroups. In an ideal world, we would enjoy our own social-group identities without wishing harm upon others. Unfortunately, this is not how American politics works today.
Luckily, social scientists are not new to the concept of intergroup conflict. Since before the Rattlers and the Eagles set up camp, people have been trying to understand what makes us fight the people who aren't us. They have studied it in order to find ways to stop it. Considering the continuing existence of prejudice, none of these methods are absolutely effective. However, in the second half of the twentieth century there has at least been a reduction in the acceptability of expressing racial and religious prejudice in public, particularly among elected officials (with the troubling exception of Donald Trump). Perhaps some of these methods could prove useful in addressing the identity-fueled partisan prejudice that is currently so evident in American politics.
### Contact Theory
See that man over there?
Yes.
Well, I hate him.
But you don't know him.
That's why I hate him.
—Gordon Allport, _The Nature of Prejudice_
The concept behind contact theory goes back to the work of Gordon Allport (1954) who specified that certain types of social contact can reduce prejudice between groups. This theory was so well accepted that it "provided the foundation for the Social Science Statement submitted to the U. S. Supreme Court in connection with the _Brown v. Board of Education_ decision of 1954," desegregating public schools (Brewer and Kramer 1985, 232). Optimally, according to Allport, this contact would meet four conditions. It would occur (1) among groups of equal status who (2) have common goals, (3) no competition between them, and (4) the support of relevant authorities. The contact between opposing partisans is blatantly lacking conditions two and three. Luckily, later research by Pettigrew et al. (2011) found that Allport's conditions do help to reduce prejudice, but they aren't all absolutely necessary. They found that prejudice can be reduced in the presence of intergroup friendships, in the absence of anxiety, and in the presence of empathy. Even better, indirect contact can also reduce prejudice, such as contact through mass media or simply having a friend of a friend in the opposing group.
In terms of reducing American partisan prejudice, contact theory would send Democrats and Republicans into the same social arenas and ask them to simply see each other with a calm and friendly set of eyes. One way to accomplish this could be via media. Duckitt (1992) found that "the manner in which the media present and portray social and intergroup 'realities' can markedly influence the perceived salience of (a) intergroup distinctions, roles, and inequities; (b) negative stereotyping; (c) the social acceptability of prejudice; and (d) norms that govern intergroup behavior" (255). This approach would compel our partisan news media to present opposing partisans in more sympathetic ways, but would also add sympathetic partisans of both sides to simple entertainment-based television shows, including shows consumed mainly by partisans of each party. Recent research has shown that the gender roles portrayed in sitcoms have an effect on viewers' attitudes toward gender-based policies (Swigger 2017). Perhaps partisan portrayals could affect viewers' perceptions of who Democrats and Republicans are. In fact, since Democrats and Republicans reliably watch different types of television in a wide range of arenas, contact theory would suggest implanting sympathetic partisans of the opposing team into each party's television shows. Although this would be a welcome development, American partisans generally live in different social arenas and partisan television tends to pander to its own fans, making this development unlikely.
While exposure to opposing political ideas and individuals can moderate intolerance and polarization, this exposure is growing far less frequent. Furthermore, as Brewer and Kramer (1985) explain, attempts to locate the prejudice-reducing effects of contact have generated mixed results. In particular, when a group member comes into contact with a member of the opposing group, if she or he considers that member to be "atypical" of the opposing group the effects do not last beyond the moment of contact. Jacoby-Senghor, Sinclair, and Smith (2015) found that the people with the most bias against outgroups are not only less likely to have direct contact with outgroup members "but will also be less likely to have friends with outgroup friends." Therefore our most biased partisans will have little direct or even indirect contact with political opponents. In short, the amount of contact between Democrats and Republicans is decreasing, and it is quite unlikely that this trend will reverse without some larger outside influence.
### Social Norms
Don't boo. Vote.
—President Barack Obama, 2016 Democratic National Convention
In 1994, Newt Gingrich sent members of the Republican Party a memo titled "Language: A Key Mechanism of Control." This memo came to be known as the GOPAC memo, and was meant to instruct Republicans on what types of words to use when describing their political opposition. As he wrote in the memo, "This list is prepared so that you might have a directory of words to use in writing literature and mail, in preparing speeches, and in producing electronic media. The words and phrases are powerful" (Gingrich 1994). Gingrich's list of recommended words to describe Democrats included betray, bizarre, decay, destroy, devour, greed, lie, pathetic, radical, selfish, shame, sick, steal, and traitors, among many others. To this day, this is the type of language used by party leaders to demonize opponents.
In 2015, Republican senator John McCain accused Democratic secretary of state John Kerry of being less trustworthy than the leaders of Iran, a known adversary of the United States. McCain and Kerry had once been friendly Senate colleagues. In response to McCain's comments, President Barack Obama announced, "That's an indication of the degree to which partisanship has crossed all boundaries. It needs to stop" (Coll 2015). Unfortunately, Obama was not the best person to make this argument. In fact, a prominent Republican would have presented the best chance for this message to have an effect, as the action that needed to be addressed came from within the Republican team.
One way that outright partisan prejudice may be addressed is for the parties themselves to establish new norms for partisan behavior. Putting aside for a moment how or why they would do this, scholars do know that "what group members think of what others are thinking may play a key role in influencing intergroup relations and perceptions" (Putra 2014). In other words, if group members believe that other ingroup members are tolerant of the outgroup, this can turn into a more broadly tolerant approach toward the opposing team.
In fact, Allport himself, back in 1954, argued that "the remedy for prejudiced opinion is not suppression, but rather a free-flowing counteraction by unprejudiced opinion" (469). If the parties themselves had any interest in reducing levels of partisan prejudice, they could likely do so simply by encouraging the prominent flag-bearers of the party to loudly and freely discuss partisan opponents in an unprejudiced way. According to Hogg (2001), group leaders have power to influence group members because group members "cognitively and behaviorally conform to the prototype" of the leader.
What if the leaders of the Democratic and Republican parties decided to take on a tolerant rhetoric toward the opposing team? What if party prototypes started discussing real differences rather than demonizing their opponents? What if party opinion leaders (of both parties) started talking about politics by commending compromise and acknowledging the humanity and validity of the opposing team? What if there were a new, opposite version of the GOPAC memo, in which the demonizing words were discouraged rather than encouraged?
There is no reason to believe that this will occur in the near term, particularly in the Republican Party. Trump supporters at the 2016 Republican National Convention repeatedly called for the imprisonment of Trump's Democratic opponent Hillary Clinton. Trump himself repeatedly encouraged bias and intolerance. Furthermore, party leaders are incentivized to maintain conflict and incivility. It draws attention and votes. But if for some reason both parties were to stand up for norms of civil partisan interaction, it could reduce partisan conflict and prejudice in American politics in general. This, however, is highly unlikely without some secondary intervention.
### Superordinate Goals
Back in 1954, once the Rattlers and Eagles had reached a point of such violent conflict that they had to be separated, the experimenters decided to try to bring them back into friendly relations. They knew that superordinate goals, or goals that go beyond group boundaries and include both groups, had been theorized to help groups mend rifts between them. The experimenters presented the Rattlers and the Eagles with a number of challenges that could only be solved if the teams worked together. In one case, the experimenters cut off the water supply to the camp, and as the boys grew more and more thirsty, they were compelled to cooperate to find and solve the problem with the water supply. In a second case, the teams were asked to contribute money to fund a movie night at the camp. They were forced to decide together how much money each team would contribute to obtain this precious treat. Finally, a precariously angled tree that threatened the camp was chopped down and pulled away by both teams of boys working together. These superordinate goals allowed the boys a chance to see each other as human beings, and though they still identified as Rattlers and Eagles, the animosity between them began to subside. After these exercises, the boys remained partial to their own teams, but they did agree to ride home in the same bus at the end of camp. Prior to the exercises, both teams had refused to share a bus with the others.
A modern political example of this can be seen in the aftermath of the terrorist attacks of September 11. For a short time afterward, Democrats and Republicans came together, at least in their approval of the president, George W. Bush. However, the activation of a superordinate American identity did not heal the rift between the parties. In fact, the attacks spurred increasing disputes between the parties over how best to respond, and a drawn-out war further divided the parties. As of this writing, news of Russian tampering in the 2016 election may be another influence that can unite Democrats and Republicans in service to the same goal of protecting America, but already Republican public approval of Russian president Vladimir Putin has increased (Nussbaum and Oreskes 2016).
In the best case, as Bert Klandermans wrote in 2014, "superordinate identity makes it possible for people to accept disadvantages done to their subgroup in the interest of the larger community. People trust authorities to make sure that next time their group will benefit. This implies that superordinate identity and trust in authorities are intimately related" (14). Unfortunately, lack of trust in outgroup authorities does appear to be preventing at least some of the potential benefit of the common American identity. As Hetherington (2015) explains, "Trust among those who identify with the party outside the White House is much lower than historical norms and, indeed, almost completely evaporated among Republicans during Barack Obama's presidency" (446). With the decline in trust of outgroup partisans, superordinate goals are possibly no longer powerful enough to bring the parties together. Brewer (2001a) points out that "when intergroup attitudes and relations have moved into the realm of outgroup hate or overt conflict, . . . the prospect of superordinate common group identity may constitute a threat rather than a solution . . . when intense distrust has already developed, common group identities are likely to be seen as threats (or opportunities) for domination and absorption. In this case, the prescription for conflict reduction may first require protection of intergroup boundaries and distinctive identities" (36). Brewer adds that the best way to protect distinctive identities is for a society to be divided along multiple lines, in a cross-cutting pattern. Social sorting is a barrier to the possible solution of superordinate goals.
The challenge, then, is to find any goal that could unify Democrats and Republicans and not simply cause more harm than good. The magnitude of this challenge is, in fact, what has been revealed in the multiple models described in this book. Despite a large number of common goals and essentially American problems, partisans have yet to find a way to unite in defense of outcomes rather than lashing out with partisan rancor regardless of the consequences. So far, Democrats and Republicans still won't board the same bus to go home.
## Self-Affirmation
It is the frustration of basic needs by instigating conditions that leads group members, whose individual identity is shaken, to turn to the group for identity, to focus more on their social identity, or to "give themselves over" to an identity group. This frustration also leads to scapegoating and the creation of destructive ideologies (which identify enemies), that turns the group against another group.
—Ervin Staub, "Individual and Group Identities in Genocide and Mass Killing"
A great deal of attention has been paid recently to uneducated, poor, white Americans, who are rising in prominence in American politics due to their increasingly visible racial prejudice, affiliation with Republican president Donald Trump, and their uniquely increasing rates of mortality due to drug and alcohol abuse (Case and Deaton 2015). Repeatedly, we see them described as feeling as if they have been left behind. As Trump reminded his supporters, "We got $18 trillion in debt. We got nothing but problems. . . . We're dying. We're dying. We need money. . . . We have losers. We have people that don't have it. We have people that are morally corrupt. We have people that are selling this country down the drain. . . . The American dream is dead" (Frum 2016). This is not an uplifting message. In fact, it is a message that explicitly reminds these Americans that they should have higher status but unfairly do not.
According to a theory from social psychology, there is good reason to believe that these voters are suffering from damaged self-esteem, driven by either a lack of economic opportunity, a fear of a culturally changing country, or some combination thereof. Those with damaged self-esteem normally look for a way to enhance their self-image. One powerful place to find such a thing is a person's group identity, which provides an alternate way for individuals to feel highly esteemed.
The study of self-esteem and self-uncertainty has been ongoing in the field of social psychology for some time. Recent research has examined the effects of self-esteem and self-affirmation on political variables. Self-esteem has been shown to affect ideological closed-mindedness (Cohen et al. 2007), American patriotism (Hohman and Hogg 2015), opinions about the Affordable Care Act (Bendersky 2014), evaluations of debate performances (Binning et al. 2010), and violence, extremism, and authoritarianism (Hogg, Kruglanksi, and van den Bos 2013), among other things.
Hogg (2014) has argued that uncertainty about a person's own status can lead them to identify with more extreme positions and groups, while Hogg, Adelman, and Blagg (2010) have shown that this same uncertainty can lead to stronger religious identification. Other work based in social identity theory finds that when self-esteem declines, people attempt to improve it by using the status of their social groups as a buttressing agent, privileging ingroups and derogating outgroups (Crocker et al. 1987). In essence, when people feel badly about themselves they turn to their groups for self-affirmation, becoming more strongly affiliated with the group.
Furthermore, when self-esteem is threatened, people tend to prefer their social groups to be increasingly homogeneous (Jetten, Hogg, and Mullin 2000). When people feel self-esteem declining, they not only cling more strongly to their group identities but they "circle the wagons" of social identity, a process very much like social sorting, in order to keep their multiple identities as aligned (and therefore impervious to outsiders) as possible.
In the realm of American politics, this leads to the social polarization that emerges out of strong and aligned political group identities, as well as the increasingly aggressive activism that arises from these strengthened identities.
The good news is that Cohen et al. (2007) have found that simply reminding a person of their own self-worth, a technique called self-affirmation, can significantly reduce extremism and ideological closed-mindedness. To date, very little research within political science has taken advantage of the self-esteem literature to explain political attachments and/or political polarization in American politics, but future research may be able to use this information in a way that can safeguard economically disadvantaged white Americans from using their group identities to soothe their own sense of inadequacy.
Alternatively, an economic upturn or change in economic status for these and other Americans could reduce the intensification of outgroup loathing that is currently occurring among American partisans.
## Demographic Trends
Although we do not have reliable voter data from before the 1950s, McCarty, Poole, and Rosenthal (2008) have documented polarization in Congress going back to the Civil War. From these data, it is possible to see that the current levels of polarization, at least in Congress, are relatively close to what they were directly after the Civil War. So, considering that these levels of polarization significantly decreased between the Civil War and the 1950s, it may be possible to use some of the information from that period to help guide us toward a less polarized nation. In 2012, Hetherington and Haidt summarized for the _New York Times_ multiple theories from political science that explain the depolarization period, including "shifts in the coalitions that composed each party, the shared experiences of war and economic calamity and very low levels of immigration, which allowed a stronger sense of national identity to form."
In all that this book has explored, it seems that the sense of national identity is one of the main victims of the social homogenization of the two parties. As the coalitions that make up Democrats and Republicans grow increasingly socially distant from one another, the superordinate identity of the nation grows less powerful and may even drive partisans apart.
In addition to this, we are witnessing a major change in the racial makeup of the American population. Ruy Teixeira, William Frey, and Robert Griffin predicted in February 2015 that the United States eligible voting population would reach majority-minority status in 2052. Some states, such as New Mexico, have already reached that status, while other states, like Colorado would have to wait until 2060. A number of Red states, however, would turn majority-minority even sooner. Texas is predicted to be majority-minority in 2019, Georgia in 2036, and Louisiana in 2048. Mississippi, Oklahoma, Virginia, and North Carolina are predicted to reach majority-minority status during the 2050 decade. Furthermore, Alabama, Arkansas, Kansas, Massachusetts, Michigan, Oregon, Pennsylvania, Rhode Island, South Carolina, and Utah are expected to reach at least 40 percent minority populations by 2060. They describe a "superdiversification" of American children, explaining, "In 1980, children were 25 percent minority; today, they are 46 percent minority. And diversification will not stop in the future: In 2040, children are projected to be 57 percent minority, and in 2060, children should be 65 percent minority" (Teixeira, Frey, and Griffin 2015, 11).
Unfortunately, decades of research, beginning with Allport, have found a positive relationship between the size of a minority group and levels of white prejudice against minorities. Craig and Richeson (2014) found that, when white respondents were presented with these statistics about the _projected_ population breakdown, they more strongly preferred to spend time with other whites than did respondents who were told about current population statistics. Craig and Richeson found that concern about the status of white Americans drives this effect. This means that the changing demographics of the American population are not leading toward racial tolerance, at least not soon. Furthermore, recent research that I have done with Leonie Huddy and Nechama Horwitz has found that Latinos are increasingly identifying with the Democratic Party and that this is partially driven by their sense of general discrimination against Latinos (Huddy, Mason, and Horwitz 2016).
In terms of social sorting, racial minorities tend to identify with the Democratic Party, while whites identify as Republicans. The changes in the makeup of the American electorate suggest that, should current alignments persist, the Democratic Party will gradually grow to win increasing numbers of local, state, and national elections, despite having lost the 2016 elections across the board. According to social identity theory, group status matters a great deal to group members. When a group's status is low, as would be the case if Republicans began to lose elections in a consistent manner, a group member has three choices. A group member can (1) exit the group, (2) grow increasingly creative about how to describe group status, or (3) fight to change the group's status in society. Currently, strongly identified Republicans have been fighting, via activism, to maintain their group's status, just as Democrats have. If, however, the status or coherence of the Republican Party declines in the next few years or decades, it may be the case that increasing numbers of Republicans will choose to exit the group (likely becoming independents). If that occurs, it is possible that American partisans will experience a new realignment, which would reduce party homogeneity and therefore reduce social polarization.
## Rift in One Party: An Unsorting
Should Mr. Trump clinch the presidential nomination, it would represent a rout of historic proportions for the institutional Republican Party, and could set off an internal rift unseen in either party for a half-century, since white Southerners abandoned the Democratic Party en masse during the civil rights movement.
—Alexander Burns, Maggie Haberman, and Jonathan Martin, "Inside the Republican Party's Desperate Mission to Stop Donald Trump"
It is reasonable to argue that one major reason for the period of partisan dealignment in the 1970s and 1980s had a lot to do with the flight of southern conservatives from the Democratic Party. This era of dealignment, while messy, was also full of cross-cutting cleavages, which held levels of partisan rancor and social polarization lower than they are today. The Donald Trump phenomenon in 2016 was predicted by many to generate similar rifts in the Republican Party, though as of this writing few have appeared. Fred Malek, the finance chairman of the Republican Governors Association, was quoted by the _New York Times_ : "There's no single leader and no single institution that can bring a diverse group called the Republican Party together, behind a single candidate. It just doesn't exist" (Burns, Haberman, and Martin 2016). Just as southern Democrats did not immediately join the Republican Party after 1964, it would take some time for any rifts in the Republican Party to realign into a new system of party coalitions. On the other hand, since the Republican victories in 2016, most party-infighting discussion has focused on the rifts within the Democratic Party.
Still, divisions in the Republican Party were emerging even before the appearance of Donald Trump. The rise of the Tea Party in 2010 and of the Freedom Caucus in the House of Representatives demonstrated genuine divisions between factions of the Republican Party. Ragusa and Gaspar (2016) found that the Tea Party and the Freedom Caucus have independently generated "party-like" effects in Congress, different from the Republican Party itself. Bode et al. (2015) analyzed Twitter posts and found that conservatives could be divided into three distinct groups. Since the widespread Republican victories in 2016, these various parts of the party must cooperate in governing. If the demographic trends toward racial diversity continue, and Republicans as a group begin to disagree on governing principles, it is distinctly possible that the party could reorganize itself into new sets of social groups. Social identities could be divided between factions of the party, which would generate the cross-cutting cleavages that suppress social polarization and social distance. This may be an unlikely scenario, but it is one way to imagine an unsorting of the American electorate, building, perhaps ironically, toward a more tolerant set of partisans on both sides.
## Where Does This Leave Us?
Nothing in politics is forever, and party alignments change and move over time. It just so happens that the current alignment of social identities within the two parties is promoting a greater focus on partisan victory than on the good of the nation. This may be, in the context of the 2016 election, happening among Republicans more than Democrats. However, the social homogenization of the parties has made it difficult for partisans to learn to like, or even humanize, their partisan opponents. They are stereotyped, vilified, and rejected out of hand. The unfortunate truth of this, however, is that these deep social divisions are allowing opportunities for policy compromise to go unnoticed. A 2015 study found that there are multiple points of agreement across party lines, even on a polarizing topic like abortion (Vavreck 2016). Dozens of other studies have found the same regarding other issues, including gun control. Partisans of the two parties are capable of coming to agreement on many issues. But today, they will change their positions rather than agree with the other side.
As parties grow more sorted, the incentives for party identifiers to compromise with the opposition decline. As Tajfel found so many decades ago, when people are socially identified with a group, "it is the winning that seems more important to them." The social makeup of the two parties has real consequences for American politics. A more socially homogeneous set of parties generates an electorate that is unresponsive to external challenges. It reduces the portion of the population that can listen to political messages impassively. It generates prejudice between citizens who identify with differing parties but may otherwise get along.
Identity is a crucial component of American democracy. As I explain in chapter 2 and demonstrate in chapter 4, partisan identity can be separated from issue preferences, and the identity element can be a powerful motivator of human judgment, emotion, and behavior. Partisans have natural incentives to see the world through a partisan lens, and to privilege their own party over their opponents. They are naturally inclined to prefer to spend social time with members of their own party, and to interpret the actions and characteristics of the other party with bias.
Though political science has long understood the social power of partisan identity, or, separately, racial or religious identity, this book has shown that these identities are far more informative when they are examined in relation to one another. No single person holds one single identity. The convergence or divergence of multiple social identities has real consequences for political behavior, particularly when partisan identities are involved. As the Democratic and Republican parties have grown increasingly socially distinct from one another, as I document in chapter 3, the potential for compromise and cooperation have declined. In chapter 5, I describe the increased bias and social distance that is induced when partisan identities are aligned with other types of social identities. More than partisan identity alone, socially sorted parties motivate a preference for ingroup partisans and prejudice in evaluating national figures and conditions.
Social sorting also does another very important thing. When parties are socially divided, their members react emotionally to political messaging, leading to behavioral consequences elaborated in chapter 6. But when partisans hold cross-cutting social identities, their emotional reactions are dampened. These emotional reactions are partial drivers of political activism, as chapter 7 shows. Therefore, when a nation changes from one made up of many cross-cutting identities to one built on socially segregated parties, the result is an electorate that is, on average, more angry, excitable, and active than it was before the social shift.
I have taken the position that the social sorting of the American electorate has been, on balance, normatively bad for American democracy. Many may disagree, thinking that an engaged and excited electorate is desirable. I maintain that an electorate that is emotionally engaged and politically activated on behalf of prejudice and misunderstanding is not an electorate that produces positive outcomes. The social sorting of American partisans has changed the electorate into a group of voters who are relatively unresponsive to changing information or real national problems. The voting booths are increasingly occupied by those who fiercely want their side to win and consider the other party to be disastrous. This effect exceeds that of bias based on partisanship alone. As long as a social divide is maintained between the parties, the electorate will behave more like a pair of warring tribes than like the people of a single nation, caring for their shared future.
# Appendix
Figure A.1. Alternate to figure 4.5, varying issue extremity and constraint
Figure A.2. Distribution of social-sorting score
Table A.1a Originating regressions for figure 4.2: Democratic feeling thermometer
---
| 1980 | 1982 | 1984 | 1986 | 1988 | 1990 | 1992 | 1994
Partisan identity (strong Democrat = high) | **0.37** | (.02) | **0.37** | (.01) | **0.35** | (.01) | **0.36** | (.01) | **0.37** | (.01) | **0.29** | (.01) | **0.35** | (.02) | **0.37** | (.02)
Issue scale (most liberal = high) | **0.09** | (.03) | **0.10** | (.03) | **0.10** | (.03) | **0.06** | (.03) | **0.10** | (.03) | **0.06** | (.03) | **0.16** | (.04) | **0.25** | (.04)
Education | **−0.12** | (.02) | **−0.10** | (.02) | **−0.08** | (.01) | **−0.08** | (.01) | **−0.08** | (.02) | **−0.09** | (.02) | **−0.12** | (.02) | **−0.05** | (.02)
Male | −0.01 | (.01) | **−0.03** | (.01) | **−0.02** | (.01) | **−0.03** | (.01) | **−0.03** | (.01) | **−0.04** | (.01) | **−0.03** | (.01) | −0.02 | (.01)
White | **−0.06** | (.01) | **−0.04** | (.01) | **−0.07** | (.01) | **−0.04** | (.01) | **−0.06** | (.01) | **−0.04** | (.01) | **−0.03** | (.01) | **−0.07** | (.02)
Age | **0.00** | (.00) | **0.00** | (.00) | **0.00** | (.00) | **0.00** | (.00) | **0.00** | (.00) | **0.00** | (.00) | **0.00** | (.00) | **0.00** | (.00)
South | **0.02** | (.01) | **0.03** | (.01) | 0.00 | (.01) | 0.01 | (.01) | 0.00 | (.01) | 0.02 | (.01) | **0.03** | (.01) | 0.02 | (.01)
Urban | 0.00 | (.01) | 0.01 | (.01) | 0.00 | (.01) | 0.00 | (.01) | 0.01 | (.01) | 0.01 | (.01) | **0.03** | (.01) | 0.01 | (.01)
Church attendance | **0.03** | (.01) | 0.01 | (.01) | 0.01 | (.01) | 0.02 | (.01) | 0.00 | (.01) | 0.01 | (.01) | −0.01 | (.01) | −0.02 | (.02)
Constant | **0.41** | (.03) | **0.42** | (.03) | **0.44** | (.02) | **0.45** | (.02) | **0.41** | (.03) | **0.47** | (.03) | **0.37** | (.04) | **0.27** | (.04)
_R_ 2 | 0.38
| |
0.46
| |
0.41
| |
0.38
| |
0.39
| |
0.29
| |
0.42
| |
0.51
|
_N_ | 1580
| |
1370
| |
2198
| |
2115
| |
1977
| |
1909
| |
1077
| |
991
|
| 1996 | 1998 | 2000 | 2004 | 2008 | 2012 | | | |
Partisan identity (strong Democrat = high) | **0.48** | (.03) | **0.34** | (.02) | **0.45** | (.02) | **0.42** | (.02) | **0.46** | (.02) | **0.49** | (.02)
| | | |
Issue scale (most liberal = high) | −0.02 | (.07) | **0.13** | (.03) | **0.05** | (.02) | **0.10** | (.04) | **0.11** | (.02) | **0.25** | (.03)
| | | |
Education | 0.01 | (.04) | **−0.07** | (.02) | −0.03 | (.02) | **−0.10** | (.02) | **−0.09** | (.02) | **−0.06** | (.02)
| | | |
Male | **−0.07** | (.02) | **−0.03** | (.01) | **−0.03** | (.01) | −0.02 | (.01) | **−0.03** | (.01) | −0.02 | (.01)
| | | |
White | **−0.08** | (.02) | **−0.09** | (.01) | **−0.05** | (.01) | **−0.05** | (.01) | **−0.05** | (.01) | **−0.07** | (.01)
| | | |
Age | 0.00 | (.00) | **0.00** | (.00) | 0.00 | (.00) | **0.00** | (.00) | 0.00 | (.00) | 0.01 | (.02)
| | | |
South | 0.03 | (.02) | 0.01 | (.01) | 0.01 | (.01) | 0.02 | (.01) | 0.00 | (.01) | 0.00 | (.01)
| | | |
Urban | 0.00 | (.02) | −0.02 | (.01) | **0.03** | (.01) | —
| |
—
| |
**0.03** | (.02)
| | | |
Church attendance | −0.03 | (.03) | 0.01 | (.01) | 0.02 | (.01) | −0.01 | (.01) | 0.01 | (.01) | **−0.03** | (.01)
| | | |
Constant | **0.37** | (.06) | **0.42** | (.03) | **0.34** | (.03) | **0.38** | (.03) | **0.38** | (.03) | **0.25** | (.02)
| | | |
_R_ 2 | 0.49
| |
0.41
| |
0.42
| |
0.48
| |
0.52
| |
0.63
| | | | |
_N_ | 395
| |
1258
| |
1754
| |
1186
| |
2030
| |
3437
| | | | |
Note: Coefficients indicate change in Democratic feeling thermometer. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
Table A.1b Originating regressions for figure 4.3: Republican feeling thermometer
---
| 1980 | 1982 | 1984 | 1986 | 1988 | 1990 | 1992 | 1994
Partisan identity (strong Democrat = high) | **−0.32** | (.02) | **−0.37** | (.02) | **−0.37** | (.01) | **−0.36** | (.01) | **−0.39** | (.01) | **−0.33** | (.02) | **−0.37** | (.02) | **−0.33** | (.02)
Issue scale (most liberal = high) | **−0.10** | (.03) | **−0.17** | (.03) | **−0.21** | (.03) | **−0.20** | (.03) | **−0.18** | (.03) | **−0.15** | (.03) | **−0.21** | (.04) | **−0.10** | (.05)
Education | **−0.08** | (.02) | −0.04 | (.02) | **−0.06** | (.02) | −0.01 | (.02) | **−0.04** | (.02) | **−0.06** | (.02) | **−0.12** | (.02) | −0.02 | (.03)
Male | 0.00 | (.01) | −0.01 | (.01) | −0.01 | (.01) | −0.01 | (.01) | 0.00 | (.01) | **−0.02** | (.01) | **−0.05** | (.01) | 0.00 | (.01)
White | −0.02 | (.02) | 0.03 | (.02) | **−0.03** | (.01) | −0.02 | (.01) | 0.00 | (.01) | −0.02 | (.01) | 0.00 | (.02) | −0.03 | (.02)
Age | **0.00** | (.00) | 0.00 | (.00) | **0.00** | (.00) | 0.00 | (.00) | **0.00** | (.00) | 0.00 | (.00) | 0.00 | (.00) | 0.00 | (.00)
South | 0.01 | (.01) | **0.03** | (.01) | **0.03** | (.01) | 0.02 | (.01) | **0.04** | (.01) | 0.00 | (.01) | **0.03** | (.01) | **0.05** | (.02)
Urban | −0.02 | (.01) | 0.01 | (.01) | **−0.03** | (.01) | −0.02 | (.01) | 0.01 | (.01) | 0.01 | (.01) | 0.00 | (.01) | −0.02 | (.02)
Church attendance | **0.04** | (.01) | **0.04** | (.01) | **0.04** | (.01) | **0.03** | (.01) | **0.04** | (.01) | **0.05** | (.01) | 0.02 | (.02) | 0.00 | (.02)
Constant | **0.83** | (.03) | **0.81** | (.03) | **0.90** | (.03) | **0.90** | (.03) | **0.85** | (.03) | **0.86** | (.03) | **0.90** | (.04) | **0.81** | (.04)
_R_ 2 | 0.24
| |
0.33
| |
0.35
| |
0.29
| |
0.33
| |
0.25
| |
0.33
| |
0.29
|
_N_ | 1580
| |
1370
| |
2198
| |
2115
| |
1977
| |
1909
| |
1077
| |
991
|
| 1996 | 1998 | 2000 | 2004 | 2008 | 2012 | | | |
Partisan identity (strong Democrat = high) | **−0.38** | (.03) | **−0.40** | (.02) | **−0.38** | (.02) | **−0.46** | (.02) | **−0.49** | (.02) | **−0.45** | (.02)
| | | |
Issue scale (most liberal = high) | −0.09 | (.07) | −0.04 | (.04) | **−0.05** | (.02) | **−0.20** | (.04) | **−0.07** | (.02) | **−0.25** | (.03)
| | | |
Education | **−0.14** | (.05) | **−0.09** | (.02) | **−0.07** | (.02) | **−0.09** | (.03) | **−0.08** | (.02) | **−0.05** | (.02)
| | | |
Male | −0.02 | (.02) | 0.00 | (.01) | 0.00 | (.01) | −0.02 | (.01) | **−0.04** | (.01) | **−0.02** | (.01)
| | | |
White | 0.02 | (.03) | −0.02 | (.02) | 0.00 | (.01) | **−0.04** | (.02) | **−0.03** | (.01) | 0.02 | (.01)
| | | |
Age | 0.00 | (.00) | 0.00 | (.00) | 0.00 | (.00) | 0.00 | (.00) | 0.00 | (.00) | **0.06** | (.02)
| | | |
South | 0.00 | (.02) | **0.04** | (.01) | 0.00 | (.01) | **0.03** | (.01) | **0.03** | (.01) | 0.01 | (.01)
| | | |
Urban | 0.01 | (.03) | −0.02 | (.01) | −0.01 | (.02) | —
| |
—
| |
**0.04** | (.02)
| | | |
Church attendance | **0.07** | (.03) | 0.02 | (.02) | **0.04** | (.01) | 0.00 | (.02) | **0.03** | (.01) | −0.01 | (.01)
| | | |
Constant | **0.86** | (.06) | **0.81** | (.03) | **0.78** | (.03) | **0.94** | (.04) | **0.83** | (.03) | **0.81** | (.02)
| | | |
_R_ 2 | 0.35
| |
0.33
| |
0.31
| |
0.44
| |
0.46
| |
0.54
| | | | |
_N_ | 395
| |
1258
| |
1754
| |
1186
| |
2030
| |
3435
| | | | |
Note: Coefficients indicate change in Republican feeling thermometer. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
Table A.2 Originating regression for figure 4.5
---
| Social-distance bias
Partisan identity strength | **0.34** | (.04)
Issue extremity | 0.07 | (.10)
Issue constraint | −0.21 | (.11)
Issue extremity*constraint | **0.40** | (.17)
Education | −0.01 | (.01)
Sophistication | **0.11** | (.04)
White | 0.02 | (.04)
Hispanic | 0.05 | (.04)
Black | 0.10 | (.05)
Male | −0.02 | (.02)
Income | **−0.09** | (.03)
Age (decades) | 0.00 | (.01)
Church attendance | **−0.05** | (.03)
Constant | −0.05 | (.08)
_R_ 2 | 0.20
|
_N_ | 774
|
Note: Coefficients indicate change in social-distance bias. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
## Explanation of the Social-Sorting Measure
The _social-sorting_ scale begins with assessing the subjective strength of various political identities. It includes the _partisan-identity_ and _ideological-identity_ four-item scales described in the previous chapter, as well as identity scales created to measure the strength of evangelical, secular, African American, and Tea Party identities. These additional identity scales are created in the same manner as the partisan and ideological-identity measures and likewise form reliable scales: evangelical (α = 0.88), secular (α = 0.80), black (α = 0.78), and Tea Party (α = 0.90).
For ease of explanation, table A.3 presents two potential Republican subjects, one who scores the highest possible score on the sociopartisan sorting scale and another who receives the lowest possible score.
Individual A in this example would score highest on Republican identity, conservative identity, evangelical identity, and Tea Party identity. This score can rise no higher but could fall were she to identify less strongly with her party or any of her party-consistent groups. It could also fall were she to identify at all with any of the party-inconsistent groups. Individual B, still a Republican (because she chose to answer the traditional seven-point party-identification scale on the Republican end of the scale), holds the weakest possible Republican social identity, the strongest possible liberal identity, the strongest secular identity, and the strongest black identity. Her score can fall no lower, but it could increase were she to identify less strongly with a cross-cutting identity, or to identify at all with any of the party-consistent identities, or to identify more strongly with Republicans.
The distribution of this variable, after recoding, is presented below. Average score for Republicans is .72 and for Democrats is .68.
Table A.3 Example calculation of sociopartisan sorting scores
---
Individual A: Highest-score Republican | Individual B: Lowest-score Republican
Republican identity = 1 | Republican identity = 0
Democratic identity = NA | Democratic identity = NA
Conservative identity = 1 | Conservative identity = NA
Liberal identity = NA | Liberal identity = −1
Evangelical identity = 1 | Evangelical identity = NA
Secular identity = NA | Secular identity = −1
Black identity = NA | Black identity = −1
Tea Party identity = 1 | Tea Party identity = NA
Sorting score = (1 + 1 + 1 + 1)/4 = 1 | Sorting score = (0 − 1 − 1 − 1)/4 = −0.75
Note: Sorting score rescaled to range from 0 to 1 by adding 0.75, then dividing by 1.75. No respondent rated the lowest sorting score. One Democrat scored −0.61 before recoding, and one Republican scored −0.27. The lowest recoded score is therefore 0.08. Nineteen respondents scored a perfect 1. All variables are coded to range between 0 and 1. Each identity score is generated from the four-item scale assessing social identification with the group. Party-inconsistent groups are reverse coded to range from –1 to 0. "NA" indicates a missing score.
Table A.4 Originating regressions for figure 5.1
---
| Cumulative ANES to 2012 | 2011 YouGov
Partisan strength | **0.26** | (.02) | **0.31** | (.05)
Partisan-ideological sorting | **0.21** | (.03)
| |
Sociopartisan sorting
| | |
**0.36** | (.08)
Issue extremity | 0.00 | (.03) | **0.32** | (.13)
Issue constraint | −0.07 | (.05) | 0.14 | (.13)
Issue extremity*constraint | **0.28** | (.08) | −0.21 | (.20)
Political knowledge | 0.00 | (.02) | **0.14** | (.04)
Education | −0.03 | (.02) | −0.01 | (.01)
White | 0.02 | (.02) | 0.04 | (.04)
Hispanic | 0.02 | (.03) | 0.03 | (.04)
Black | **0.08** | (.03) | 0.09 | (.05)
Male | −0.02 | (.01) | −0.03 | (.02)
Income | −0.04 | (.02) | 0.00 | (.00)
Age | 0.00 | (.00) | **0.01** | (.01)
Church attendance | 0.00 | (.00) | −0.04 | (.03)
Constant | **0.07** | (.04) | **−0.26** | (.10)
_R_ 2 | 0.26
| |
0.25
|
_N_ | 2147
| |
775
|
Note: Coefficients indicate change in partisan warmth bias. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1. Age is measured in ten-year increments. Coefficients should be interpreted with caution, as the partisan-identity and sorting measures are related by construction, so holding one constant while moving the other is not precisely possible. The predicted values take account of this problem by holding both variables constant at one value, but the regression coefficients do not.
Table A.5 Originating regression for figure 5.2
---
| Warmth bias
Partisan identity | **0.31** | (.05)
Sociopartisan sorting | **0.36** | (.08)
Issue extremity (including importance) | **0.21** | (.07)
Issue constraint | −0.01 | (.04)
Education | **−0.01** | (.01)
Sophistication | **0.14** | (.05)
White | 0.04 | (.04)
Hispanic | 0.03 | (.04)
Black | **0.09** | (.05)
Male | −0.03 | (.02)
Income | **0.00** | (.00)
Age (decades) | **0.01** | (.01)
Church attendance | −0.04 | (.03)
Constant | **−0.19** | (.07)
_R_ 2 | 0.25
|
_N_ | 775
|
Note: Coefficients indicate change in partisan warmth bias. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1. Coefficients should be interpreted with caution, as the partisan-identity and sorting measures are related by construction, so holding one constant while moving the other is not precisely possible. The predicted values take account of this problem by holding both variables constant at one value, but the regression coefficients do not.
Table A.6 Originating regression for figure 5.4
---
| Social-distance bias
Partisan identity | **0.25** | (.05)
Sociopartisan sorting | **0.27** | (.09)
Issue extremity | 0.05 | (.10)
Issue constraint | **−0.23** | (.11)
Issue extremity*constraint | **0.38** | (.16)
Education | −0.01 | (.01)
Sophistication | 0.06 | (.04)
White | 0.03 | (.04)
Hispanic | 0.05 | (.04)
Black | **0.10** | (.05)
Male | −0.02 | (.02)
Income | **−0.01** | (.00)
Age (decades) | 0.00 | (.01)
Church attendance | −0.05 | (.03)
Constant | −0.14 | (.09)
_R_ 2 | 0.21
|
_N_ | 774
|
Note: Coefficients indicate change in social-distance bias. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1. Coefficients should be interpreted with caution, as the partisan-identity and sorting measures are related by construction, so holding one constant while moving the other is not precisely possible. The predicted values take account of this problem by holding both variables constant at one value, but the regression coefficients do not.
Table A.7 Originating regression for figure 5.5
---
| Warmth bias
Partisan identity | **0.31** | (.05)
Sociopartisan sorting | **0.36** | (.08)
Issue extremity | **0.32** | (.13)
Issue constraint | 0.14 | (.13)
Issue extremity*constraint | −0.21 | (.20)
Education | −0.01 | (.01)
Sophistication | **0.14** | (.04)
White | 0.04 | (.04)
Hispanic | 0.03 | (.04)
Black | 0.09 | (.05)
Male | −0.03 | (.02)
Income | 0.00 | (.00)
Age (decades) | **0.01** | (.01)
Church attendance | −0.04 | (.03)
Constant | **−0.26** | (.10)
_R_ 2 | 0.25
|
_N_ | 775
|
Note: Coefficients indicate change in partisan warmth bias. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1. Coefficients should be interpreted with caution, as the partisan-identity and sorting measures are related by construction, so holding one constant while moving the other is not precisely possible. The predicted values take account of this problem by holding both variables constant at one value, but the regression coefficients do not.
Table A.8 Originating regressions for figures 6.4 and 6.5
---
| Anger ANES | Pride ANES
Sorting | **1.28** | (.24) | **0.85** | (.25)
Partisan strength | **1.54** | (.14) | **2.42** | (.14)
Issue extremity | −0.35 | (.31) | −0.17 | (.31)
Issue constraint | 0.98 | (.59) | 0.64 | (.60)
Issue extremity*constraint | 0.93 | (.90) | −0.42 | (.87)
Education | **0.74** | (.15) | 0.20 | (.15)
Male | −0.03 | (.08) | −0.08 | (.08)
White | 0.17 | (.10) | −0.18 | (.11)
Age | 0.01 | (.02) | 0.04 | (.02)
Southern location | **−0.20** | (.09) | **0.20** | (.09)
Urban | −0.19 | (.11) | **−0.64** | (.12)
Church attendance | **−0.06** | (.03) | 0.01 | (.03)
Constant | **−1.67** | (.25) | **−1.43** | (.24)
Pseudo _R_ 2 (from unweighted model) | 0.13
| |
0.15
|
_N_ | 4395
| |
4395
|
Note: Dependent variables are coded 1 for reporting feeling angry/proud, and 0 for no report of this emotion. All variables are coded to range from 0 to 1. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are logit models with sample weights.
Table A.9 Originating regressions for figure 6.9
---
| Issue polarization | Partisan identity | Sociopartisan sorting
**Angry responses to party-based threats**
Partisan identity
| | |
−0.08 | (.04)
| |
Sociopartisan sorting
| | | | |
**−0.30** | (.12)
Group threat | **0.25** | (.05) | 0.13 | (.07) | −0.12 | (.13)
Partisan identity × group threat
| | |
**0.24** | (.11)
| |
Sociopartisan sorting × group threat
| | | | |
**0.68** | (.23)
Issue polarization | 0.03 | (.05) | 0.04 | (.06) | 0.09 | (.06)
Issue polarization × group threat | 0.12 | (.10) | 0.09 | (.10) | −0.02 | (.11)
White | 0.01 | (.04) | 0.02 | (.04) | **0.09** | (.03)
Black | **−0.15** | (.05) | **−0.14** | (.05)
| |
Male | 0.00 | (.02) | 0.00 | (.02) | 0.01 | (.03)
Income | 0.00 | (.00) | 0.00 | (.00) | 0.00 | (.00)
Age (decades) | **0.02** | (.01) | **0.02** | (.01) | **0.02** | (.01)
Sophistication | **0.14** | (.05) | **0.15** | (.05) | **0.13** | (.05)
Church attendance | 0.04 | (.03) | 0.04 | (.03)
| |
Constant | **0.14** | (.06) | **0.16** | (.06) | **0.22** | (.08)
_R_ 2 | 0.16
| |
0.17
| |
0.18
|
_N_ | 859
| |
859
| |
753
|
**Angry responses to issue-based threats**
Partisan identity
| | |
−0.03 | (.04)
| |
Sociopartisan sorting
| | | | |
**−0.40** | 0.12
Issue threat | **0.14** | (.05) | 0.09 | (.06) | **−0.27** | 0.13
Partisan identity × issue threat
| | |
0.08 | (.09)
| |
Sociopartisan sorting × issue threat
| | | | |
**0.70** | 0.23
Issue polarization | −0.06 | (.06) | −0.05 | (.06) | 0.00 | 0.07
Issue polarization × issue threat | **0.39** | (.10) | **0.38** | (.10) | **0.26** | 0.12
White | 0.01 | (.04) | 0.01 | (.04) | 0.09 | 0.03
Black | **−0.16** | (.05) | **−0.16** | (.05) | 0.02 | 0.02
Male | 0.01 | (.02) | 0.01 | (.02)
| |
Income | 0.00 | (.00) | 0.00 | (.00) | 0.00 | 0.00
Age (decades) | 0.01 | (.01) | 0.01 | (.01) | 0.01 | 0.01
Sophistication | 0.06 | (.05) | 0.06 | (.05) | 0.06 | 0.05
Church attendance | 0.04 | (.03) | 0.04 | (.03)
| |
Constant | **0.25** | (.06) | **0.27** | (.06) | **0.39** | 0.08
_R_ 2 | 0.17
| |
0.17
| |
0.19
|
_N_ | 859
| |
859
| |
753
|
Note: Coefficients represent changes in reported levels of anger. Bold coefficients are significant at the _p_ < .05 level in a two-tailed test. Standard errors in parentheses. All models are OLS regressions with robust standard errors. All variables are coded to range from 0 to 1. Shaded cells represent marginal effects for ease of interpretation.
Table A.10 Originating regressions for figure 6.10
---
| Issue polarization | Partisan identity | Sociopartisan sorting
**Enthusiastic responses to party-based reassurances**
Partisan identity
| | |
**0.12** | (.04)
| |
Sociopartisan sorting
| | | | |
**0.30** | (.13)
Group support | **0.25** | 0.05 | 0.03 | (.07) | −0.06 | (.13)
Partisan identity × group support
| | |
**0.34** | (.09)
| |
Sociopartisan sorting × group support
| | | | |
**0.54** | (.21)
Issue polarization | 0.02 | 0.06 | −0.01 | (.06) | −0.04 | (.06)
Issue polarization × group support | 0.11 | 0.11 | 0.09 | (.10) | 0.01 | (.11)
White | −0.02 | 0.04 | −0.02 | (.04) | **−0.07** | (.03)
Black | 0.05 | 0.05 | 0.04 | (.05)
| |
Male | 0.02 | 0.02 | 0.03 | (.02) | 0.01 | (.03)
Income | 0.00 | 0.00 | 0.00 | (.00) | 0.00 | (.00)
Age (decades) | 0.00 | 0.01 | 0.00 | (.01) | 0.00 | (.01)
Sophistication | −0.06 | 0.05 | −0.08 | (.05) | −0.10 | (.06)
Church attendance | 0.05 | 0.03 | 0.03 | (.03)
| |
Constant | **0.36** | 0.07 | **0.33** | (.07) | **0.29** | (.08)
_R_ 2 | 0.12
| |
0.15
| |
0.16
|
_N_ | 859
| |
859
| |
753
|
**Enthusiastic responses to issue-based reassurances**
Partisan identity
| | |
**0.15** | (.04)
| |
Sociopartisan sorting
| | | | |
**0.28** | (.12)
Issue support | **0.17** | (.05) | 0.12 | (.07) | −0.09 | (.14)
Partisan identity × issue support
| | |
0.08 | (.10)
| |
Sociopartisan sorting × issue support
| | | | |
**0.51** | (.24)
Issue polarization | −0.04 | (.06) | −0.07 | (.06) | −0.10 | (.06)
Issue polarization × issue support | **0.42** | (.11) | **0.39** | (.11) | **0.27** | (.13)
White | −0.04 | (.04) | −0.04 | (.04) | **−0.07** | (.03)
Black | 0.03 | (.05) | 0.02 | (.05)
| |
Male | 0.01 | (.02) | 0.02 | (.02) | 0.01 | (.02)
Income | −0.01 | (.00) | 0.00 | (.00) | **−0.01** | (.00)
Age (decades) | 0.00 | (.01) | 0.00 | (.01) | 0.00 | (.01)
Sophistication | −0.07 | (.05) | −0.08 | (.05) | −0.09 | (.05)
Church attendance | 0.03 | (.03) | 0.02 | (.03)
| |
Constant | **0.41** | (.06) | **0.37** | (.06) | **0.31** | (.08)
_R_ 2 | 0.19
| |
0.20
| |
0.23
|
_N_ | 859
| |
859
| |
753
|
Note: Coefficients represent changes in reported levels of enthusiasm. Bold coefficients are significant at the _p_ < .05 level in a two-tailed test. Standard errors in parentheses. All models are OLS regressions with robust standard errors. All variables are coded to range from 0 to 1. Shaded cells represent marginal effects for ease of interpretation.
Table A.11 Originating regression for figure 7.5
---
| Activism
Partisan strength | **0.06** | (.01)
Partisan-ideological sorting | **0.10** | (.02)
Issue extremity | 0.03 | (.03)
Issue constraint | 0.09 | (.06)
Issue extremity*constraint | 0.00 | (.08)
Political knowledge | **0.08** | (.01)
White | 0.00 | (.01)
Black | **0.05** | (.02)
Male | 0.01 | (.01)
Income | **0.04** | (.02)
Age | **0.01** | (.00)
Church attendance | 0.00 | (.00)
Constant | −0.04 | (.02)
_R_ 2 | 0.11
|
_N_ | 3426
|
Note: Coefficients represent changes in levels of activism. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1. Coefficients should be interpreted with caution, as the partisan-identity and sorting measures are related by construction, so holding one constant while moving the other is not precisely possible. The predicted values take account of this problem by holding both variables constant at one value, but the regression coefficients do not. Data are weighted and drawn from the cumulative ANES file through 2012.
Table A.12 Originating regressions for figures 7.6 and 7.7
---
| Past activism | Intended activism
Social sorting | **0.10** | (.02) | **0.44** | (0.08)
Issue extremity | 0.03 | (.03) | 0.13 | (0.14)
Issue constraint | 0.09 | (.06) | 0.21 | (0.18)
Issue extremity*constraint | 0.00 | (.08) | −0.14 | (0.25)
White | 0.00 | (.01) | **−0.10** | (0.03)
Male | 0.01 | (.01) | **0.07** | (0.03)
Income | **0.04** | (.02) | **0.01** | (0.00)
Age | **0.01** | (.00) | 0.02 | (0.01)
Political knowledge | **0.08** | (.01) | **0.26** | (0.05)
Church attendance | 0.00 | (.00) | 0.05 | (0.04)
Constant | −0.04 | (.02) | **−0.50** | (0.11)
_R_ 2 | 0.19
| |
0.17
|
_N_ | 776
| |
776
|
Note: Coefficients represent changes in activism indices. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
Table A.13 Originating regressions for figure 7.8
---
| Prior activism
**Panels A and C**
Abortion social identity | **0.32** | (0.06)
Abortion issue extremity | −0.07 | (0.07)
Abortion issue importance | **−0.17** | (0.07)
Abortion extremity*importance | **0.18** | (0.09)
White | 0.00 | (0.03)
Male | **0.06** | (0.02)
Income | **0.01** | (0.00)
Age | **0.03** | (0.01)
Political knowledge | **0.38** | (0.05)
Church attendance | 0.01 | (0.03)
Constant | **−0.27** | (0.08)
_R_ 2 | **0.20**
|
_N_ | **869**
|
**Panel B**
Abortion issue extremity | **0.09** | (0.02)
White | −0.01 | (0.03)
Male | **0.05** | (0.02)
Income | **0.01** | (0.00)
Age | **0.03** | (0.01)
Political knowledge | **0.42** | (0.04)
Church attendance | 0.04 | (0.03)
Constant | **−0.19** | (0.05)
_R_ 2 | **0.18**
|
_N_ | **1041**
|
Note: Coefficients represent changes in the four-item index of prior activism. Bold coefficients are significant at the _p_ < .05 level. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
Table A.14 Originating regressions for figure 7.9
---
| Intended activism
**Panels A and C**
Abortion social identity | **0.28** | (0.07)
Abortion issue extremity | −0.08 | (0.08)
Abortion issue importance | **−0.17** | (0.08)
Abortion extremity*importance | 0.14 | (0.09)
White | **−0.07** | (0.03)
Male | **0.07** | (0.02)
Income | **0.01** | (0.00)
Age | **0.02** | (0.01)
Political knowledge | **0.30** | (0.05)
Church attendance | **0.07** | (0.03)
Constant | **−0.29** | (0.08)
_R_ 2 | **0.14**
|
_N_ | **868**
|
**Panel B**
Abortion issue extremity | **0.06** | −(0.02)
White | **−0.07** | −(0.03)
Male | **0.08** | −(0.02)
Income | **0.01** | (0.00)
Age | **0.02** | −(0.01)
Political knowledge | **0.34** | −(0.04)
Church attendance | **0.09** | −(0.03)
Constant | **−0.24** | −(0.05)
_R_ 2 | **0.18**
|
_N_ | **1041**
|
Note: Coefficients represent changes in the four-item index of intended activism. Bold coefficients are significant at the _p_ < .05 level. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
Table A.15 Originating regressions for figure 7.10
---
| Intended activism | Intended activism
Anger
| | |
**0.12** | (0.06)
Enthusiasm | **0.23** | (0.05)
| |
Issue extremity and constraint | **0.13** | (0.07) | **0.16** | (0.08)
White | −0.05 | (0.07) | −0.01 | (0.06)
Black | 0.01 | (0.09) | 0.13 | (0.08)
Male | 0.05 | (0.04) | **0.10** | (0.04)
Income | **0.01** | (0.00) | 0.01 | (0.00)
Age (decades) | 0.00 | (0.01) | **0.02** | (0.01)
Political knowledge | **0.36** | (0.08) | **0.21** | (0.08)
Church attendance | 0.04 | (0.05) | 0.01 | (0.05)
Constant | **−0.29** | (0.09) | **−0.25** | (0.10)
_R_ 2 | 0.17
| |
0.13
|
_N_ | 366
| |
376
|
Note: Coefficients represent changes in the four-item index of intended activism. Bold coefficients are significant at the _p_ < .05 level. Standard errors in parentheses. All models are OLS models with robust standard errors. All variables are coded to range from 0 to 1.
# Notes
## Chapter One
. According to the 2016 American National Election Study (ANES), with full sample weights, and independent "leaners" coded as partisans. The ANES is a series of election studies conducted by the ANES since 1948 to support analysis of public opinion and voting behavior in US presidential elections. The 2016 study features a dual-mode design with both face-to-face interviewing ( _n_ = 1,181) and surveys conducted on the Internet ( _n_ = 3,090), and a total sample size of 4,271. Respondents were interviewed in a pre-election survey between September 7 and November 7, 2016. The study reinterviewed as many as possible of the same respondents in a postelection survey between November 9 and January 8, 2017. The study was funded by the National Science Foundation via grants to the University of Michigan and Stanford University (grant nos. SES-1444721 and SES-1444910 respectively). The data were released by the ANES in April 2017. The data can be accessed at www.electionstudies.org.
. This concept makes up the origin of social categorization theory and social identity theory, pioneered by Henri Tajfel and John Turner (1979).
. For an early summary of these experiments, see Brewer (1979).
. See Levendusky (2010) for this argument.
. See Garner and Palmer (2011) for this argument.
## Chapter Two
. Data drawn from a study by the author, funded by the National Science Foundation under grant no. SES-1065054 and fielded by Polimetrix, who collect an online sample of Americans and use matching techniques to construct a sample that is as similar as possible to a nationally representative sample.
## Chapter Three
. South here is measured as what the ANES characterizes as the "political south," including only the eleven secession states: Alabama, Arkansas, Florida, Georgia, Louisiana, Mississippi, North Carolina, South Carolina, Tennessee, Texas, and Virginia.
## Chapter Four
. The issue extremity measure is an index of six issues: (1) The circumstances under which abortion should be allowed by law (four-point scale); (2) Whether to prioritize government services or spending (seven-point scale); (3) Whether government should have a role in health insurance (seven-point scale); (4) Whether government should provide aid to minorities/blacks (seven-point scale); (5) The extent to which defense spending should be increased or decreased (seven-point scale); and (6) Whether government should guarantee jobs to citizens (seven-point scale). Each issue is folded in half such that extreme liberal and conservative views are coded 1 and moderate views are coded 0. All of these folded issues are then combined into a scale. This is a well-differentiated measure—314 respondents or 3.16 percent of the sample score 0 on the scale, while 236 respondents or 2.37 percent of the sample score 1. The median score is 0.44 and the mean score is 0.45. The choice to measure issue extremity this way, without including any element of ideological constraint, was an intentional one. Broockman (2016) has found that "many Americans are not very ideologically consistent, [which] can lead summary measures that average individuals' positions across multiple issues to give the impression that these individuals support moderate policies 'on average' because they support some liberal policies and some conservative policies. An individual may well want policy to be very far left on one issue and very far right on another, but summarizing this individual's views as 'moderate' is misleading" (3). This extremity measure is examined here in order to account for an issue-based radicalization that can be seen to match the change in partisan feelings. An alternate measure that includes constraint (such as the one that Broockman argues can obscure issue extremism) does, in fact, increase somewhat over time, by nearly 10 points, while the warmth-bias measure increases by 15 points. An increase in constraint, however, does not necessarily mean that partisans are growing more radical in their issue positions, simply less conflicted. Extremity and constraint will be measured separately in later chapters.
. Originating regressions can be found in the appendix.
. The 1996 ANES, however, had a reduced sample size. For a comparable sample, the largest partisan difference was in 2000, in which the two cross-pressured partisans placed the Democratic Party 39 points apart.
. 1980 is the first year that all six issues were included together in the ANES.
. Eleven hundred respondents answered a web-based survey conducted by Polimetrix during November of 2011. Polimetrix maintains a panel of respondents, which it recruits through their polling website in return for incentives. Since recruitment into the panel is voluntary, the sample may be unrepresentative of the national population. However, sample matching was employed to draw a nearly nationally representative sample from the larger, nonrepresentative sample. The matching resulted in a sample with characteristics as similar as possible to the national population. This sample was balanced between Democrats and Republicans.
. Responses included "I absolutely would do this," "I probably would do this," "I probably would not do this," and "I absolutely would not do this."
. Items include the following: (1) How important is being a [identity] to you? [ _Extremely important_ , _Very important_ , _Not very important_ , _Not important at all_ ] (2) How well does the term [identity] describe you? [ _Extremely well_ , _Very well_ , _Not very well_ , _Not at all_ ] (3) When talking about [identity]s how often do you use "we" instead of "they"? [ _All of the time_ , _Most of the time_ , _Some of the time_ , _Rarely_ , _Never_ ] (4) To what extent do you think of yourself as being a [identity]? [ _A great deal_ , _Somewhat_ , _Very little_ , _Not at all_ ]
. The issues include the following: (1) Should the number of legally permitted immigrants be increased or decreased? (2) Do you support or oppose health care reforms passed by Congress in 2010? (3) Should abortion be permitted? (4) Do you support or oppose same-sex marriage? (5) Which is more important—reducing the deficit or reducing unemployment? These items form a reliable scale (α = 0.78). Each issue position is followed by an _issue-importance_ item that asks, "How important is this issue to you?" Issue positions are measured using two separate scales, which are then interacted. First, each issue item is folded in half and coded to range from 0 (weakest issue position) to 1 (most extreme issue positions on either end of the spectrum). These folded items are then weighted by the issue importance items and combined to form a scale of _issue extremity_. The full weighted index is coded to range from 0 (weakest, least important issue positions) to 1 (strongest, most important issue positions) on both ends of the spectrum. A second measure assesses _issue constraint_ , or how well these issue positions align on the liberal or conservative end of the ideological spectrum. This is measured by counting the percentage of answers on the liberal side of the issues and the percentage of answers on the conservative side of the issues. The difference between these two numbers is used to represent constraint.
In OLS models, these two measures, extremity and constraint, are interacted in order to account for separate variations in each type of issue polarization. The predicted values set the two measures of issue positions at either their minimum or maximum values.
. For the curious reader, various combinations are presented in appendix figure A.1. The general picture is that no combination of extremity and constraint does anything to increase social-identity bias without the addition of partisan identity, with the exception of the strongest levels of both extremity and constraint.
## Chapter Five
. These predicted values are based on regressions that can be found in the appendix.
. This is the interaction of the extremity and constraint measures explained in chapter 4.
. I also examined an interaction between partisan identity and sorting and found no significant results. For example, in looking at the predicted values of the YouGov data in figure 5.1, weak partisans with cross-cutting identities are still significantly less biased than weak partisans with more highly sorted identities.
. Also controlled for here is political knowledge, education, race, sex, income, age, and church attendance.
. Tea Party identity is significantly correlated (in pairwise correlations) with Republican Party identity (r = 0.34), conservative social identity (r = 0.65), evangelical identity (r = 0.34), issue polarization (r = 0.59), and black identity (r = −0.17). Although it is strongly related to conservative social identity and conservative issue positions, I treat it here as a separate social identity as it does appear to be an identity of its own, related to evangelical, Republican, and white social identities, and not simply an ideology.
. By including religion, race, and political-movement identities in addition to ideological identities (all measured using a four-item scale of social identification), this measure of sorting should drive higher levels of partisan prejudice. It accounts for the powerful combination of all of these social identities with partisanship. This data set, however, also includes a far more powerful measure of partisan identity (a four-item scale of social identification), which could make a more difficult test for the added effect of sorting.
. For a practical example of coding the social-sorting scale, see appendix table A.3.
. I manipulate only issue extremity, rather than constraint, as the extremity variable was the only one to have a significant effect in the originating regression. The interaction with constraint was not significant, so these models are run simply controlling for constraint rather than including the interaction, which would complicate predicted values.
. Only partisan-ideological sorting is examined here, because it is the only sorting measure available in the ANES cumulative data file.
. Because a number of the key variables are continuous, coarsened exact matching is used to make the matches more feasible (Iacus et al. 2012). The univariate imbalance in means is below 0.00001 for all covariates. This indicates that the samples are very well balanced and thus require no statistical model to account for any remaining imbalance. The ANES cumulative data file (through 2012) is used to provide as large a sample as possible for the matching process, with standard errors clustered by year. Matching on year was not feasible as the sample size was too severely restricted.
. In an alternate model not shown here, issue constraint rather than extremity is matched (matching on both too severely restricted the sample size). In this model, almost identical results were obtained.
. See, to begin, Converse (1964).
. Extremity and constraint are interacted in this model, though the interaction term is not significant. However, both variables are set to minimum, mean, and maximum levels in producing predicted values.
## Chapter Six
. In other analyses not shown, levels of anger toward the ingroup candidate have been decreasing over the same period. For more information on these trends, see Mason (2016).
. For the purposes of this figure, issue extremity and constraint are combined into one measure that I am calling issue intensity. This measure is coded by creating a scale of issues ranging from most extremely and consistently liberal to most extremely and consistently conservative (by averaging the six issue positions). The entire scale is then folded in half so that 0 represents the most moderate/conflicted positions and 1 represents the most extreme/constrained positions.
. In a separate model, in which an interaction between partisanship and sorting is included, this interaction is, in fact, significant. Predicted probabilities taken from this model still reveal significant differences between strong partisans with cross-cutting identities (67 percent probability) and strong partisans with well-aligned identities (76 percent probability). The interactive model is, however, very difficult to interpret, as the partisanship and sorting measures are related by construction.
. I included each type of threat separately in a regression, controlling for demographic variables (race, gender, political knowledge, age, income, and church attendance), and included interactions between each variable of interest (sorting, partisan identity, issue extremity) and either party-based or issue-based threats. The interactions showed the effect of each of these variables only when the respondent had read a threatening message. In the sorting and partisan-identity models, issue extremity is also interacted with threat in order to control for the effects of issue extremity more fairly. All other variables are held at their means or modes. Modes in this sample for the dichotomous variables are female and white. See originating regressions in the appendix.
. The full set of issue positions is combined into a scale ranging from most liberal on all issues to most conservative on all issues. Each issue in the scale is weighted by its importance, as rated by the respondent. This full weighted scale is then folded in half, to range from the most conflicted/moderate/unimportant issues to the most consistent/extreme/important issue positions on either end of the ideological spectrum. This measure proved to generate the most powerful emotional results for issue positions and is therefore used here so as not to undersell the impact of issue positions. As noted in chapter 4, Broockman (2016) has argued that including constraint in the measure obscures important heterogeneity in extreme responses across ideological boundaries. This is true. However, the current measure is theoretically appropriate in the case of this particular experiment, as the models here are run under the assumption that issue positions are constrained. The issue threats are full of party-consistent issue positions. Furthermore, parties constrain issues, and partisanship is a key element of the models. An unconstrained set of issues, for the purposes of this particular model, should be expected to reduce emotional reactivity due to their cross-cutting effects. Including constraint in the issue-polarization measure is therefore a decision that is intentional.
. It should also be noted that in all the following models, those that examine the effects of sorting or partisan identity control for issue intensity. The issue-intensity models do not control for partisanship or sorting. These models therefore provide a particularly generous test of the effects of issue polarization. Also controlled for in all models are white race, sex, income, age, political knowledge, and church attendance. In the issue-intensity and partisanship models, black race is also controlled for, but, as it is included in the sociopartisan sorting measure, it is not separately controlled in the social-sorting models.
## Chapter Seven
. Political engagement is obviously not entirely driven by sorting, but sorting is one contributor. This section establishes the context of political engagement; subsequent sections look more directly at causality.
. An alternate model with samples matched on issue constraint provides nearly identical results. Including both extremity and constraint in the matching severely reduced the sample size.
. The items, again, are (1) How important is being a [insert identity] to you? (2) How well does the term [insert identity] describe you? (3) When talking about [insert identity]s how often do you use "we" instead of "they"? (4) To what extent do you think of yourself as being a [insert identity]?
. The issue that was rated as somewhat or extremely important by the most respondents was whether we should focus on reducing the deficit or unemployment (98.3 percent of respondents). In future research, it may be interesting to examine the potential for issue-based identities around these economic concerns, but the group labels on either side do not naturally come to mind.
. The extremity and importance were interacted, and the interactive effect is presented.
. Models including only abortion importance provided weaker results than those including opinion extremity.
. Models including only abortion importance provided weaker results than those including opinion extremity.
. For those who are wondering what type of people are moderate on the issue but identify strongly with the group, a few answers exist. The whole sample includes only fifty-four of them (with the absolute strongest social connection to the abortion-group label and a moderate position on abortion). Of these fifty-four, about 91 percent are pro-life, 80 percent are Republicans, 90 percent are white, half are male, they earn $50,000 to $70,000 per year, they are about fifty-seven years old, and they go to church almost weekly. So, at least in this sample, the respondents who identify strongly with the pro-life social identity but can bend on the actual issue are middle-aged, middle-class, churchgoing, white Republicans.
## Chapter Eight
. It should be noted that, as social groups, racial minorities are not monolithic. This partially explains the higher prevalence of sorting-related behavior among Republicans, who do not have multiple racial groups associated with the party.
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# Index
Aarøe, Lene, , , 107–8
abortion, attitudes toward, 114–16, 118–19,
Abramowitz, Alan, , , , 76–77
Achen, Christopher, ,
Adams, John,
Adelman, Janice R.,
Affordable Care Act (ACA), , , ; repeal, attempts to, 48–49. _See also_ Obamacare
Aglioti, Salvatore M.,
Alabama, , 159n1
Albertson, Bethany,
aligned identities, , ; and intolerance,
Allport, Gordon, , , , , , 130–33, 137–38
American electorate: as emotionally reactive, 83–85; homogeneous parties, sorting into, 19–20; identity-centric motivations of, ; ideological rift, ; ideological sorting, increase in, ; makeup of, ; partisan lines, division among, , ; partisan polarization, 76–77; partisan prejudice, ; policy debate knowledge, as lacking, 73–74; political preferences, public display of, ; social sorting, and violent political conflict,
American National Election Studies (ANES), 27–28, , , , , , , 64–66, , , , , , 104–5, 109–10, 159n1 (chap. 1), 159n1 (chap. 2)
American Political Science Association (APSA), 4–5, ,
American political system: cross-cutting, 24–25; "Normal System" of,
_American Voter, The_ (Campbell),
anger, 3–4, 6–7, , , , 77–85, 87–91, , 95–97, 99–101, , 122–23, 126–27; political action, as drivers of,
Arkansas, , 159n1
Avenanti, Alessio,
Balz, Dan,
Banks, Antoine J., , 122–23
Bartels, Larry, ,
Berelson, Bernard, 7–8,
Bertrand, Marianne,
Bianco, William T.,
_Big Sort, The_ (Bishop), , , ,
Billig, Michael,
Bishop, Bill, , , ,
Blagg, Robert D.,
_Blood Lust of Identity, The_ (Buruma),
"Blue" states,
Bode, Leticia,
Bogardus, Emory, social-distance scale, 54–55
Boston Marathon bombing,
_Bowling Alone_ (Putnam),
Brady, Henry E.,
Branscombe, Nyla,
Breinlinger, Sara,
Brewer, Marilynn, , , , , ,
Broockman, David, 159–60n1, 162–63n5
Brooks, David,
_Brown v. Board of Education_ , ,
Bullock, John G.,
Bundy, Cliven, 78–79,
Bureau of Land Management (BLM),
Burns, Alexander,
Buruma, Ian,
Bush, George W., , , ,
Campbell, Angus, ,
Carmines, Edward G., 32–33
Carsey, Thomas M.,
Christian Coalition,
civic engagement, decline in,
civil rights, ,
Civil Rights Act, ,
Civil War, ,
Clinton, Bill,
Clinton, Hillary, 43–44,
Cohen, Geoffrey, ,
collective identity, political action,
Colorado,
Committee on Political Parties,
Conover, Pamela Johnston, ,
contact theory, 130–31
Contract with the American Family,
Converse, Philip,
Craig, Maureen A.,
cross-cutting identities, , , , , , 89–90, 99–103, 111–13, , , ,
Dahl, Robert, ,
Davis, Nicholas T.,
Deaux, Kay,
democracy, , 6–7, , , , ; blind activism, ; cross-pressured voters, need for, , 24–25; as defined, ; electorate, makeup of, ; folk theory of, , ; and identity, ; as identity-based, ; and partisanship, ; party loyalty, ; political activism, ; political parties, ; well-sorted voters,
Democrats, 2–5, 12–15, 17–18, , 43–44, , , , 74–76, , 83–84, , 93–95, 132–33, , , ; abortion issue, ; Affordable Care Act (ACA), ; alignments with, ; background checks, for gun purchases, ; blacks, loyal to, ; changes in, 31–32; contact theory, ; homogeneous communities, preference for, ; issue-based division of, ; liberal identity, , , , , 36–39, ; national identity, ; as neighbors, ; poor, identification with, ; racial minorities, identification with, ; Republicans, increasing ideological divide between, , ; Republicans, social contact with, ; Republicans, social distance with, 54–55, 58–59; self-reported ideology of, ; social distance measure, ; social sorting in, , 62–63; and South, 25–26, , ; southern conservatives, defection of, , ; warmth bias, 51–53
depolarization,
Derks, Belle, 12–13
Devos, T.,
de Weerd, Marga,
Dinas, Elias,
Downs, Anthony,
_Duck Dynasty_ (television show),
Duckitt, John H.,
Eagles and Rattlers, 1–2, , 12–13, , 19–20, , , , 133–34. _See also_ Robbers Cave experiment
Edsall, Thomas,
Ellis, Christopher,
emotion-driven action, , 123–24; approach emotions,
Ethier, Kathleen A.,
factionalism, 59–60
_Family Guy_ (television show), 43–44
Feldman, Stanley,
Fiorina, Morris, , , 76–77
Fisher, Noel C., 128–29
Florida, 159n1
Fredrickson, Barbara,
Freedom Caucus, ,
Frey, William,
Gadarian, Shana Kushner,
Garner, Andrew,
Gaspar, Anthony,
Gaudet, Hazel,
Georgia, , 159n1
Gingrich, Newt, ,
government: decline in trust of, ; partisan goals, ; shutdown of,
Green, Donald, ,
Greene, Steven, ,
Griffin, Robert,
Groenendyk, Eric W., , , , 122–23
group conflict,
group identity, , , , ; exclusivity of, 9–10; group experiments, winning, importance of, ; groups, discrimination between, ; group victory, ; ingroup bias, ; multiple identities, working in concert, ; and partisanship, ; physical effects of, ; and saliva, ; self-esteem, 135–36
Gubler, Joshua R.,
Haberman, Maggie,
Haidt, Jonathan,
Hetherington, Marc J., ,
Hillhouse, James,
Hinich, Melvin J.,
Hobson, Nicholas M.,
Hogg, Michael A., ,
homogeneity, ; homogenous parties, ; and intolerance, 6–7
Horwitz, Nechama,
Huddy, Leonie, , , 107–8,
identity, 8–9, , ; cross-cutting, and reduction, of partisan bias, ; and democracy, ; identity-based democracy, ; identity-based groups, ; identity-based social sorting, 38–40; identity-driven action, 107–8; identity politics, 18–19; and ideology, 21–23; intergroup competition, ; issue extremity, and cross-cutting, effects of, ; issues, role of, ; policy attitudes, ; self-concept,
identity-based ideological sorting,
identity-based ideology, . _See also_ symbolic ideology
ideological identity, , ; ideological-identity scale, 149–50; partisan identity, 90–91; and partisanship, ; policy preferences,
ideological sorting, , ; increase in,
ideology, , , , ; identity-based, ; ideological identity, and policy positions, ; issue positions, set of, 28–29; and party, ; as social identity,
ingroup bias, , , , , 58–59; outgroup, hostility toward, , ; partisan identities, ; partisan teams, ; and prejudice, ; and pride,
intergroup conflict, ,
intergroup emotions theory, , ,
Inzlicht, Michael,
Iran,
issue-based identity,
issue-based ideology, . _See also_ operational ideology
issue-based polarization, , 17–18
issue-based sorting, ,
issue constraint, , 57–58, 63–64, , 70–71, 74–75, , , , 110–11, 159–60n1, 160–61n8, 161n9, 162n2, 162n8, 162n11, 162n13, 162–63n5
issue-driven action,
issue extremity, , , , 162n2; issue-extremity measure, 159–60n1, 160–61n8
issue intensity, 162n2, 163n6
issue positions, as dependent on group and party cues,
Iyengar, Shanto, , ,
Jacoby-Senghor, Drew S.,
Johnson, Lyndon B.,
Johnston, Christopher D., ,
Kagan, Robert,
Kahn, Dennis T.,
Kalyvas, Stathis,
Kansas,
Katz, Josh,
Kelly, Caroline,
Kennedy, John F.,
Kerry, John,
Key, V. O., , , 115–16,
Kingston, Jack,
Klandermans, Bert, 107–8
Klandermans, P. G., , , ,
Klar, Samara,
Klayman, Larry, 78–79
Kramer, R. M.,
Lavine, Howard G., ,
Layman, Geoffrey C.,
Lazarsfeld, Paul F., , ,
Leach, Colin Wayne,
Lee, Frances,
Lelkes, Yphtach, ,
Levendusky, Matthew, ,
Lipset, Seymour, 24–25
Lizza, Ryan,
Lodge, Milton,
Louisiana, , 159n1
Mackie, D. M.,
MacKuen, Michael,
Malek, Fred,
Marcus, George E.,
Martin, Jonathan,
Masket, Seth,
Mason, Lilliana, 107–8
Massachusetts,
matching, 68–69, 91–93
McCain, John, ,
McCarthyism,
McCarty, Nolan,
McConnell, Christopher,
McCurry, Mike,
McGarty, Craig,
McKinney, William,
McPhee, William N., ,
megaparties, , ,
Michigan,
Miller, Arthur H.,
Mississippi, , 159n1
Mullainathan, Sendhil,
Mulvaney, Mick,
Murray, Patrick,
Mutz, Diana, ,
"nanotargeting,"
National Right to Life Committee (NRLC),
_Nature of Prejudice, The_ (Allport),
Neuman, W. Russell,
Nevada,
New Mexico,
Newtown shooting,
Niemi, Richard G.,
North Carolina, , 159n1
Nunes, Devin,
Obama, Barack, , , , , 93–94, , ; Yes We Can campaign,
Obamacare, 48–49, . _See also_ Affordable Care Act (ACA)
Occupy Wall Street, ,
Office of Management and Budget (OMB),
Oklahoma,
Oklahoma City,
operational ideology, . _See also_ issue-based ideology
Ordeshook, Peter C.,
Oregon,
outgroup bias, 10–14, , , , 49–50, , 58–59, 61–63, 70–72, 83–84, , , , , 130–32, , ; group status, as threat to, ; outgroup candidates, 80–82, 87–89, 91–93
Palmer, Harvey,
Palmquist, Bradley, ,
partisan identity, , , , , , 63–64, , , , 140–41; biasing effects of, ; as expressive, ; ideological identity, aligned with, , , 109–10; ingroup bias, driving of, ; as instrumental, ; and intolerance, ; issue extremity, ; partisan identity scale, 149–50, 161n6; party identities, ; and polarization, ; political action, 108–10; social elements of, ; social identities, lining up of, ; social sorting, 72–73; and sorting, , ; and voting,
partisan-ideological sorting, 87–88, , 109–11
partisan prejudice, , , , , 161n6; increase in, ; and partisanship, 72–73; political identities, ; social norms, 132–33
partisans, , , , ; and compromise, ; with cross-cutting identities, 70–72; as intolerant, 22–23; partisan ambivalence, , ; partisan anger, 83–84; partisan bias, , , ; partisan brand, , ; partisan divide, dehumanizing aspect of, ; partisan homogenization, ; partisan loathing, ; partisan loyalty, as "frightful despotism," ; partisan polarization, , 76–77; partisan politics, and winning, 11–12; partisan preference, increase in, 51–52; partisan stereotyping, ; rise of,
partisanship, 60–61, ; and activism, ; cross-cutting cleavages, , , 25–26, ; emotion, influence on, ; expressive model of, ; group identity, ; idea of, 45–46; ideological identity, ; instrumental model of, ; as mega-identity, ; "Michigan model" of, ; partisan prejudice, 72–73; policy opinion, effect on, ; policy positions, trumping of, 52–54; political action, ; public displays of, 106–7; social contact, 54–55; social sorting, 72–73; and sorting, ; voting behavior, ; winning, importance of, , , , ,
partisan sorting, ,
party identity, , , ; cultural differences, 43–44; ideological differences, ; income differences, ; income disparity, ; policy extremism, ; racial differences, ; racial identity, , ; religious divide,
partyism, , ,
party loyalty, , ; issue positions, ; policy preferences, back seat to,
Pennsylvania,
Pettigrew, Thomas F.,
Pierce, Lamar,
Planned Parenthood,
polarization, , 17–18, , , 76–77, ; as geographical, ; and sorting,
policy opinions, vulnerability to partisan cues,
Polimetrix, 160n5
political activism, 105–7, 110–13, , , 121–24, ; blind activism, as undesirable, ; group-based activism, ; and misinformation, ; and partisanship, ; and polarization,
political participation, ; political activism, ; and voting, 104–5
political parties: as homogeneous, 6–7, ; as polarized,
political science, ; study of emotion in, 85–86
Poole, Keith,
prejudice, , ; partisan bias,
_Principles of Sociology, The_ (Ross),
Putin, Vladimir,
Putnam, Robert, ,
race, ; racial bias, ; racial minorities, 164n1
racial sorting,
Ragusa, Jordan M.,
red scare,
"Red" states, ,
Reid, Harry,
Republican Governors Association,
Republicans, 2–5, 12–15, 17–18, 24–25, , 43–44, , , 65–66, 74–76, , , 83–84, , 93–95, , , , 149–50, 164n1; abortion issue, ; Affordable Care Act (ACA), 48–49; alignments with, ; background checks, for gun purchases, ; conservative identity of, , , , ; contact theory, ; Democrats, increasing ideological divide between, , ; Democrats, social contact with, ; Democrats, social distance with, 58–59; divisions in, ; GOPAC memo, 132–33; homogeneous communities, preference for, ; national identity, ; as neighbors, ; as partisan "brand," ; religious right, common cause with, ; self-reported ideology of, ; social distance measure, ; social sorting in, , 62–63; southern identity, ; Tea Party, 161n5; Trump, hostile takeover, ; warmth bias, 51–53; wealthy, as party of, ; whites, identification with, 32–33,
Rhode Island,
Richeson, Jennifer A.,
Robbers Cave experiment, , , , 19–20, ; group membership, ; intergroup conflict, . _See also_ Eagles and Rattlers
Robertson, Pat,
Roccas, Sonia,
_Rocky IV_ (movie),
_Roe v. Wade_ , ,
Rogers, Todd,
Roof, Wade Clark,
Roosevelt, Franklin,
Rosenthal, Howard,
Ross, Edward Alsworth,
Rucker, Philip,
Sampasivam, Sinthujaa,
Saunders, Kyle,
Scarcelli, Marc,
Schattschneider, E. E., ,
Scheepers, Daan, 12–13
Schickler, Eric, ,
Schlozman, Kay Lehman,
self-esteem, ; as threatened,
Selway, Joel Sawat,
September 11 attacks,
Sherif, Muzafer, ,
Sinclair, Betsy, ,
Sirigu, Angela,
Smith, Colin Tucker,
Smith, E. R., , ,
Snyder, Jason A.,
social categorization theory, 159n2
social cohesion: and differentiation, ; and inclusion,
social-distance bias (SDB), 56–59, ,
social-group identity,
social identity, , , 61–62, , 114–15; cross-cutting identities, , , ; intolerance, increase in, ; partisan divide, reinforcing of, ; party evaluations, effect on, ; and polarization, ; policy preference, ; self-esteem, as threatened, ; social identity scale, 65–66
social identity theory, , , , , , 159n2; ingroup bias of, ; outgroups, as different from, 49–50
social polarization, , 15–17, ; cross-pressured voters, as buffer against, ; emotional reactivity, ; and megaparties, ; partisan prejudice, ; political action,
social psychology, , , 135–36
social sorting, , , 32–34, 36–38, , , , , , , ; activism, effects on, , , 111–13; attachment, subjective feelings of, ; citizens, and party leaders, ; civic engagement, decline in, ; and homogeneity, ; identity-based polarization, ; institutions, loss of trust in, ; media sources, diversity of, 42–43; megaparties, creation of, ; opposing party, avoidance of, ; participation, driving of, ; partisan identity, ; and partisanship, 72–73; political activism, ; political relations, effect on, ; roots of, 41–43; social polarization, effect on, ; social-sorting scale, 149–50; superordinate goals, ; 2016 election, ; violent political conflict, wariness of,
sociopartisan sorting, ; sociopartisan sorting scale, 65–66
Sood, Gaurav, ,
sorting, 17–18, , , , , , 95–96, , 163n1; effects of, , , , , , ; and emotions, , ; ideological sorting, ; issue extremity, ; and matching, 68–69, 91–93; partisan bias, ; partisan identity, , ; and partisanship, ; and polarization, ; social sorting, ; theories of, 31–32. _See also_ ideological sorting; issue-based sorting; partisan-ideological sorting; partisan sorting; racial sorting; social sorting; sociopartisan sorting
South Carolina, , 159n1
Spears, Russell,
Stanley, Harold W.,
Staub, Ervin,
Steenbergen, Marco R., ,
Stimson, James A., , 32–33
Sundquist, James, 31–32
Sunstein, Cass,
superordinate goals, 133–34
Sykes, Charles,
symbolic ideology, . _See also_ identity-based ideology
Taber, Charles,
Tajfel, Henri, 10–11, , , , , , , , , 159n2
Tea Party, , , 95–96, , , 161n5
Teixeira, Ruy,
Tennessee, , 159n1
Texas, , 159n1
Theriault, Sean,
Tocqueville, Alexis de,
Truman, Harry S.,
Trump, Donald, 2–4, , , , , , 138–39; Republican Party, hostile takeover by, ; Republican Party base, popularity with,
Turner, John, , , , 159n2
unaligned identities, 61–62,
United States, ; demographic trends in, 137–38; identity crisis in, ; majority-minority status, ; partisan segregation of, 41–42; Russian tampering, in 2016 election,
Utah,
Valentino, Nicholas A., ,
Van Zomeren, Martijn,
Verba, Sidney,
Virginia, , 159n1
voters: as angry, 81–83; behavior, sources of, ; motivated reasoning, 13–14; party loyalty, ; policy opinion, 20–21; poor, white Americans, ; self-esteem, ; well-sorted,
_Voting_ (Berelson, Lazarsfeld, and McPhee), 7–8,
Waldman, Paul,
Wann, Daniel,
warmth bias, , 69–70,
Washington, George: factionalism, warnings against, 59–60; farewell address of,
_Washington Post_ (newspaper),
_Washington Times_ (newspaper),
well-sorted identities, , , 63–64, , , , , , 99–101,
YouGov study, , , ,
# Contents
1. Cover
2. Title Page
3. Copyright Page
4. Contents
5. Acknowledgments
6. ONE / Identity-Based Democracy
7. TWO / Using Old Words in New Ways
8. THREE / A Brief History of Social Sorting
9. FOUR / Partisan Prejudice
10. FIVE / Socially Sorted Parties
11. SIX / The Outrage and Elation of Partisan Sorting
12. SEVEN / Activism for the Wrong Reasons
13. EIGHT / Can We Fix It?
14. Appendix
15. Notes
16. References
17. Index
## Guide
1. Cover
2. Title Page
3. Copyright Page
4. Contents
1. i
2. ii
3. iii
4. iv
5. v
6. vi
7. vii
8. viii
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| {
"redpajama_set_name": "RedPajamaBook"
} | 1,156 |
using System;
using XamarinForms.QbChat.Android;
using Android.Provider;
using Xamarin.Forms;
using QbChat.Pcl.Interfaces;
[assembly: Xamarin.Forms.Dependency(typeof(AndroidDeviceUid))]
namespace XamarinForms.QbChat.Android
{
public class AndroidDeviceUid : IDeviceIdentifier
{
#region IDeviceIdentifier implementation
public string GetIdentifier ()
{
return Settings.Secure.GetString (Forms.Context.ContentResolver, Settings.Secure.AndroidId);
}
#endregion
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,154 |
Discovery Bay ist der Name folgender Orte:
Discovery Bay (Jamaica), Ort im Saint Ann Parish auf Jamaika
Discovery Bay (Kalifornien), Ort im Contra Costa County in Kalifornien, Vereinigte Staaten
Discovery Bay (Ort in Washington), Ort im Jefferson County in Washington, Vereinigte Staaten
Discovery Bay (Hongkong), Ort im Islands District, Hongkong
Discovery Bay, ein früherer Name von Milne Bay Province, Provinz von Papua-Neuguinea
Discovery Bay ist der Name folgender Buchten:
Discovery Bay (Australien), Bucht an der Küste Australiens
Discovery Bay (O'Sullivan Lake), Bucht im O'Sullivan Lake in der Provinz Ontario, Kanada
Discovery Bay (Shebandowan Lakes), Bucht in den Shebandowan Lakes in der Provinz Ontario, Kanada
Discovery Bay (Südliche Shetlandinseln), Bucht auf Greenwich Island
Discovery Bay (Bucht in Washington), Bucht im Jefferson County in Washington, Vereinigte Staaten
Discovery Bay, anderer Name von Undine Harbor, Bucht an der Küste Südgeorgiens
Discovery Bay, anderer Name von Snug Harbor (Bucht), Bucht in Alaska, Vereinigte Staaten | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,375 |
PwC's Worldtrade Management Services (WMS) advises clients in every industry on customs and international trade regulatory issues. The international trade environment, with its varying trade rules, regulations and compliance requirements presents an exciting and challenging career opportunity that offers great cross-border exposure.
As a Senior Manager, you will be leading teams and working closely with Directors and Partners to deliver both local and regional projects, and to maintain and grow our client and intra-firm relationships.
Management and development of our team of dedicated customs and trade specialists.
In-depth knowledge of customs valuation & classification, Free Trade Agreements, import / export compliance, Japan-specific procedures and protocols, and trade strategy development.
At least 8 to 10 years of related customs and trade professional experience, preferably in a consulting firm or equivalent senior position in industry managing multiple projects simultaneously. A proven track record of successful client engagements with definable contributions to business. Strong supply chain / logistics experience is a plus.
Japan tax advisory experience would be a plus..
Bilingual language ability in both written and spoken Japanese and English.
A desire to pursue a career in the field of customs and international trade.
Please indicate the reference code of position applied for. We regret that only shortlisted candidates will be notified. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,710 |
Pride Flashback: Do you remember 1996? Were you There or were You on the Sidelines?
Pride Parade Seattle, photo by Heidi S.
Last Sunday, I managed to navigate the crowds and spend a few hours at the Seattle Pride parade. The crowd was massive, in fact a little overwhelming. And the parade didn't disappoint. It was festive, beautiful, fun, and full of community groups, nonprofits, government departments and elected officials, businesses, religious groups, etc. It felt like everyone was there. In fact, it is really starting to feel like the Pride parade is the place everyone and every organization wants to be seen.
What I couldn't help but wonder is if all the straight ally community groups, businesses, politicians, and religious groups were there 20+ years ago. Did your organization support gay rights before the American Psychiatric Association removed homosexuality from the list of mental illnesses, before sexual orientation become part of the protected class from discrimination, before "Don't ask, Don't Tell" got repealed, before legal same-sex civil unions or gay marriage, before the Supreme Court ruled in favor of gay marriage nationwide? Did your organization stand up for LGBTQIA (Lesbian, Gay, Bisexual, Transgender, Queer, Intersex, Asexual) rights when it was uncomfortable, when you really risked losing funding, getting targeted or boycotted, when no other organizations around you were standing up?
Don't get me wrong the current support feels great! It feels like the warm societal hug that I wanted and needed when I struggled with coming out. It feels like the affirmation I needed when I wished my life could just be more "normal" (read: straight). It feels like the support I wish I had when I couldn't and wouldn't talk about my "partner" and ended up referring to her as my "roommate or best friend." It feels like the support I really wished I had 20 years ago.
Remembering 1996
Let's go back 20 years. It's a nice round number, and also happens to be when I graduated from college so I have clearer memories of that time. For those of you whose memories are worse than mine, or those of you who were too young to remember 1996, here are some mainstream pop culture highlights. The Macarena was the number one music single (you're doing the dance I hope). The number one grossing movie was Independence Day. It was also the year Jerry Maguire came out, "Show me the money." Nintendo released its first gaming console. A postage stamp cost 32 cents. And, Prince Charles and Diana, Princesses of Whales got divorced. I still had bangs, wore my clothes way too baggy, and was contemplating putting a corporate logo on my body. I am happy to report I was wise enough then to not get the tattoo.
The LGBT context in 1996 was not as festive as the atmosphere at the Pride parade on Sunday. The United States Court of Appeals for the Second Circuit upheld "Don't Ask, Don't Tell." The United States Senate passed the Defense of Marriage Act (defining marriage as the union between one man and one woman) in an 85 to 14 vote, and rejected prohibiting discrimination based on sexual orientation in the private sector in a 49 to 50 vote. And, based on a Gallup poll, 68% of the public opposed same-sex marriage. Contract that with the now 61% of people who believe in same-sex marriage in 2016. I think it's fair to say that if you or your organization were standing with gay people in 1996, you were taking a risk. You were brave at a moment in time when it was uncomfortable to be brave. You were an ally or accomplice when it wasn't trendy or easy to stand beside us. We also hope your organization was changing policies and behaviors to model inclusiveness and acceptance.
A Fakequity blog post wouldn't be complete without a racial equity analysis. So here is a little taste of what societal racism looked like and sounded like in 1996. The O.J. Simpson trial had just ended in an acquittal a few months earlier in October 1995. Hilary Clinton's infamous speech where she talked about "certain kids as super-predators" happened in 1996. She made the comment in reference to Black urban kids to justify "three strikes." A referendum to end affirmative action passed in California. Over 30 Black Churches were burned down in nine different states. It was also the year of the "Hollywood Blackout" at the Academy Awards. People magazine featured a story announcing of the 166 Oscar nominees, only one was black. Yes, this happened in 1996 too (and most years before and after), even before Twitter and 2016 trending #OscarsSoWhite.
As for looking at the intersection of sexual orientation and racism in 1996, I couldn't find much reference to gay people of color, except for a few academic articles. It's as if we didn't exist. Which is strange because now I know plenty of gay people of color who were around in 1996. The invisibility of our experiences in how and what we remember about homophobia and racism speaks loudly.
The 1996 LGBTQIA Context Still Exists in 2016
I am aware that Seattle's Pride parade is a bubble, although an expanding bubble. I know that we still have work to do gaining acceptance and full rights in our society. This is particularly true in places, communities, and organizations that still feel like that 1996 context. In fact, that 1996 context is everywhere. Here are a few places that come to mind:
In white dominated LGBTIA space – Too many LGBTQIA spaces are full of whiteness. Too many spaces are disproportionately (and often, unintentionally) focused on addressing LGBTQIA issues in white communities and for white people. The voices, perspectives, and experiences of LGBTQIA people of color are missing, silenced, or ignored. We need White individual and organizational allies to no longer tolerate these spaces. We need you to direct resources to the amazing things that are happening in Queer POC spaces.
In many communities of color – As a queer person of color, I know talking about relationships (besides persistent questions about marriage to someone of the opposite sex), much less sexual orientation can be taboo in many communities. There are also cultural issues upholding silence around homosexuality. This is often complicated by societal pressures to assimilate to white norms and give up aspects of our culture. But we need to explicitly address the messages that are putting our cultural identities at odds with being gay. We need to be talking about the mass shootings in Orlando at a gay club, and the disproportionate impact the Latinx and Muslim communities with our families. We can't avoid addressing the fact that in the words of Alan Palaez Lopex, "It's not safe to be a queer person of color in America." This is especially true for transgender people of color.
In the anti-transgender movement – This is happening everywhere. From the recent attackon a transgender man on Capitol Hill in Seattle, to the waste of resources and energy on anti-transgender initiatives such as Washington's I-1515. As a community and country we should really be directing our resources and energy to fighting the fact that according to a 2013 national report "More than two thirds of the homicide victims were transgender women, while 67% of victims of homicide were transgender women of color… This data follows a multi-year trend where the victims of fatal hate violence are overwhelmingly transgender women, and in particular transgender women of color"
What are you going to do today? Do you want to be a leader or a follower?
Ten and twenty years from now what do you want to say you were doing to advance social justice causes. Do you want to say you were marching and working with communities of color or do you want to say you were on the sidelines?
Recently Erin told me a story about how Starbucks failed #RaceTogether campaign makes other Fortune 500 companies leery about boldly leading on racial equity. Starbucks took a risk and it didn't work, but doesn't mean they and others shouldn't continue to push for social justice causes and examine and speak out about racism. Do they want to be on the sidelines or do they want to be leaders? The same conversation people were having in 1996 about being seen in the Pride Parade.
Posted by Heidi Schillinger
Why we Need to Stop Using the Word Minority
Accommodations versus Doing What is Right — Fix the System | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,198 |
import contextlib
import errno
import io
import logging
import os
import os.path
import subprocess
import tempfile
from scality_manila_utils.exceptions import EnvironmentException
log = logging.getLogger(__name__)
@contextlib.contextmanager
def elevated_privileges():
"""
Obtain temporary root privileges.
"""
previous_uid = os.geteuid()
previous_gid = os.getegid()
# Become root
log.debug("Elevating privileges")
os.seteuid(0)
try:
os.setegid(0)
except OSError:
os.setegid(previous_gid)
raise
try:
yield
finally:
# Drop root privileges
log.debug("Dropping elevated privileges")
try:
os.setegid(previous_gid)
finally:
os.seteuid(previous_uid)
def find_pids(process):
"""
Find pids by inspection of procfs.
:param process: process name
:type process: string
:returns: list of pids
"""
process_pids = []
pids = [f for f in os.listdir('/proc') if f.isdigit()]
for pid in pids:
status_path = os.path.join('/proc', pid, 'status')
try:
with io.open(status_path, 'rt') as f:
line = f.readline()
_, process_name = line.split()
if process_name == process:
process_pids.append(int(pid))
except IOError as e:
# Pass on processes that no longer exist
if e.errno != errno.ENOENT:
raise
log.debug("PIDs for '%s': %r", process, process_pids)
return process_pids
def binary_check(binary, paths):
"""
Check if a binary exists on the given paths.
:param binary: name of binary
:type binary: string
:param paths: list of paths to inspect for binary
:type paths: list of strings
:raises: :py:class:`scality_manila_utils.exceptions.EnvironmentException`
if the binary couldn't be found
"""
for path in paths:
if os.path.exists(os.path.join(path, binary)):
return
log.error("No '%s' found in PATH (%s)", binary, ', '.join(paths))
raise EnvironmentException("Unable to find '{0:s}', make sure it "
"is installed".format(binary))
def process_check(process):
"""
Check if a process is running.
:param process: process name
:type process: string
:raises: :py:class:`scality_manila_utils.exceptions.EnvironmentException`
if the process isn't running
"""
process_pids = find_pids(process)
if not process_pids:
log.error("'%s' is not running", process)
raise EnvironmentException("Could not find '{0:s}' running, "
"make sure it is "
"started".format(process))
def fsync_path(path):
"""
Fsync a directory.
:param path: path to directory to fsync
:type path: string (unicode)
"""
fd = None
try:
fd = os.open(path, os.O_RDONLY | os.O_DIRECTORY)
os.fsync(fd)
finally:
if fd is not None:
os.close(fd)
def safe_write(text, path, permissions=0o644):
"""
Write contents to file in a safe manner.
Write contents to a tempfile and then move it in place. This is guarenteed
to be atomic on a POSIX filesystem.
:param text: the content to write to file
:type text: string
:param path: path to write to
:type path: string
:param permissions: file permissions
:type permissions: int (octal)
"""
# Make sure that the temporary file lives on the same fs
log.debug("Writing '%s'", path)
target_dir, _ = os.path.split(path)
with tempfile.NamedTemporaryFile(mode='wt', dir=target_dir,
delete=False) as f:
os.chmod(f.name, permissions)
f.write(text)
f.flush()
os.fsync(f.fileno())
os.rename(f.name, path)
# fsync the directory holding the file just written and moved
fsync_path(target_dir)
@contextlib.contextmanager
def nfs_mount(export_path):
"""
Mount an NFS filesystem, and keep it mounted while in context.
:param export_path: exported filesystem to mount, eg. `127.0.0.1:/`
:type export_path: string
:returns: path to where the filesystem was mounted
"""
try:
mount_point = tempfile.mkdtemp()
subprocess.check_call(['mount', export_path, mount_point])
log.debug("Mounted nfs root '%s' at '%s'", export_path, mount_point)
except (OSError, subprocess.CalledProcessError):
log.exception('Unable to mount NFS root')
raise
try:
yield mount_point
finally:
try:
subprocess.check_call(['umount', mount_point])
except subprocess.CalledProcessError:
log.exception('Unable to umount NFS root')
raise
log.debug('Unmounted nfs root')
try:
os.rmdir(mount_point)
except OSError as e:
log.warning("Unable to clean up temporary NFS root: %s", e)
def is_stored_on_sofs(path):
"""
Check if the given location is stored on a SOFS filesystem.
:param path: an absolute path, e.g `/ring/fs/samba_shares`
:type path: string
:rtype: boolean
"""
with elevated_privileges():
try:
output = subprocess.check_output(['df', '-P', path])
except subprocess.CalledProcessError:
log.exception("Unable to get fs type of '{0:s}'".format(path))
raise
# df will output a header, followed by the filesystem capacity usage
fsline = output.splitlines()[-1]
fstype = fsline.split()[0]
return fstype == '/dev/fuse'
def execute(cmd, error_msg):
"""
Utility function to execute a command
:param cmd: the command with arguments to execute
:type cmd: iterable of `str` or a single `str`
:param error_msg: the exception message in case something went wrong.
`error_msg` must include the placeholders `{stdout}` and `{stderr}`
:type error_msg: `str`
:rtype: (`unicode`, `unicode`)
"""
process = subprocess.Popen(cmd, stdout=subprocess.PIPE,
stderr=subprocess.PIPE)
# `subprocess.communicate` returns a byte stream
stdout, stderr = process.communicate()
stdout, stderr = stdout.decode(), stderr.decode()
if process.returncode != 0:
raise EnvironmentError(error_msg.format(stdout=stdout, stderr=stderr))
return stdout, stderr
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,360 |
Home of the former Formula One and sports car constructor.
Portsmouth Road, Send, GU23 7JY
Founded in 1945, and based at Send in Surrey, Continental Cars (later renamed Continental Engineering) produced their first road car in 1949, using a 1.7-litre Lea Francis engine and chassis.
Although early cars were built for road use, the company's main emphasis was on performance, and the cars soon became a familiar site at race meetings.
Three models of the aluminium-bodied road car were offered in all, the L1, 2 and 3. Graham Robson, in A-Z British Cars 1945-1980, indicates that a total of only 27 cars were sold. Production of the L series officially ended in 1953, as the company focussed its energy on the development of single seaters.
One of Connaught's most enthusiastic supporters was race driver Kenneth McAlpine (whose family established the famous construction company) who both raced Connaught cars and gave financial backing for their development.
Connaught's first Formula 2 car had its first competitive outing in 1951 in the Rufforth Stakes at the Gamston circuit in Nottinghamshire. Racing success developed swiftly, and by 1952, Connaught cars were appearing in international F2 and F1 events in the hands of drivers that included Stirling Moss, Prince Bira and Peter Collins.
In 1955, the team gained the first win for a British car in an overseas Grand Prix race since 1924 when Tony Brooks, then aged 23, won the Syracuse Grand Prix.
Although there were further successes, the team was forced to withdraw from motor sport in early 1957. Insufficient funds made it impossible to continue the programme of racing. Later in the year everything relating to racing and car manufacture was sold – including two cars that were bought by a young team owner, Bernie Ecclestone.
• British Sports Cars, Gregor Grant, Foulis, 1958
Surrey locations
Byfleet
Dunsfold
Thames Ditton | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,250 |
package Tapper::MCP::State::Details;
use 5.010;
use strict;
use warnings;
use Moose;
use List::Util qw/max min/;
use Tapper::Model 'model';
use YAML qw/Dump Load/;
has state_details => (is => 'rw',
default => sub { {current_state => 'invalid'} },
);
has persist => (is => 'rw',);
sub BUILD
{
my ($self, $args) = @_;
my $testrun_id = $args->{testrun_id};
my $result = model('TestrunDB')->resultset('State')->find_or_create({testrun_id => $testrun_id});
$self->persist($result);
$self->state_details($result->state);
}
=head2 db_update
Update database entry.
@return success - 0
@return error - error string
=cut
sub db_update
{
my ($self) = @_;
$self->persist->state($self->state_details);
$self->persist->update;
return 0;
}
=head1 NAME
Tapper::MCP::State::Details - Encapsulate state_details attribute of MCP::State
=head1 SYNOPSIS
use Tapper::MCP::State::Details;
my $state_details = Tapper::MCP::State::Details->new();
$state_details->prc_results(0, {success => 0, mg => 'No success'});
=head1 FUNCTIONS
=head2 results
Getter and setter for results array for whole test. Setter adds given
parameter instead of substituting.
@param hash ref - containing success(bool) and msg(string)
=cut
sub results
{
my ($self, $result) = @_;
if ($result) {
push @{$self->state_details->{results}}, $result;
$self->db_update();
}
return $self->state_details->{results};
}
=head2 state_init
Initialize the state or read it back from database.
@return success - 0
@return error - error string
=cut
sub state_init
{
my ($self, $data) = @_;
$self->state_details($data);
$self->state_details->{current_state} = 'started';
$self->state_details->{results} = [];
$self->state_details->{prcs} ||= [];
$self->state_details->{keep_alive}{timeout_date} = $self->state_details->{keep_alive}{timeout_span} + time if defined $self->state_details->{keep_alive}{timeout_span};
foreach my $this_prc (@{$self->state_details->{prcs}}) {
$this_prc->{results} ||= [];
}
$self->db_update();
return 0;
}
=head2 takeoff
The reboot call was successfully executed, now update the state for
waiting for the first message.
@return int - new timeout
=cut
sub takeoff
{
my ($self, $skip_install) = @_;
my $timeout_current_date;
if ($skip_install) {
$self->current_state('reboot_test');
my $prc = $self->state_details->{prcs}->[0];
$timeout_current_date = $prc->{timeout_current_date} = $prc->{timeout_boot_span} + time();
} else {
$self->current_state('reboot_install');
my $install = $self->state_details->{install};
$timeout_current_date = $install->{timeout_current_date} = $install->{timeout_boot_span} + time();
}
$self->db_update();
return ($timeout_current_date);
}
=head2 current_state
Getter and setter for current state name.
@param string - state name (optional)
@return string - state name
=cut
sub current_state
{
my ($self, $state) = @_;
if (defined $state) {
$self->state_details->{current_state} = $state;
$self->db_update;
}
return $self->state_details->{current_state};
}
=head2 set_all_prcs_current_state
Set current_state of all PRCs to given state.
@param string - state name
=cut
sub set_all_prcs_current_state
{
my ($self, $state) = @_;
if (defined $state) {
for ( my $prc_num = 0; $prc_num < @{$self->state_details->{prcs}}; $prc_num++) {
$self->state_details->{prcs}[$prc_num]{current_state} = $state;
}
$self->db_update;
}
}
=head2 keep_alive_timeout_date
Getter and setter for keep_alive_timeout_date
@optparam int - new timeout_date for keep_alive
@return int - timeout date for keep_alive
=cut
sub keep_alive_timeout_date
{
my ($self, $timeout_date) = @_;
$self->state_details->{keep_alive}{timeout_date} = $timeout_date if defined $timeout_date;
$self->state_details->{keep_alive}{timeout_date};
}
=head2 set_keep_alive_timeout_span
Getter for keep_alive_timeout_date
@param int - new timeout date for keep_alive
@return int - new timeout date for keep_alive
=cut
sub set_keep_alive_timeout_span
{
my ($self, $timeout_span) = @_;
$self->state_details->{keep_alive}{timeout_date} = $timeout_span;
}
=head2 keep_alive_timeout_span
Getter and setter for keep_alive_timeout_span.
Note: This function can not set the timeout to undef.
@optparam int - new timeout_span
@return int - timeout date for keep_alive
=cut
sub keep_alive_timeout_span
{
my ($self) = @_;
return $self->state_details->{keep_alive}{timeout_span};
}
=head2 installer_timeout_current_date
Getter and setter for installer timeout date.
@param int - new installer timeout date
@return string - installer timeout date
=cut
sub installer_timeout_current_date
{
my ($self, $timeout_date) = @_;
if (defined $timeout_date) {
$self->state_details->{install}{timeout_current_date} = $timeout_date;
$self->db_update;
}
return $self->state_details->{install}{timeout_current_date};
}
=head2 start_install
Update timeouts for "installation started".
@return int - new timeout span
=cut
sub start_install
{
my ($self) = @_;
$self->state_details->{install}->{timeout_current_date} =
time + $self->state_details->{install}->{timeout_install_span};
$self->db_update;
return $self->state_details->{install}->{timeout_install_span};
}
=head2 prc_boot_start
Sets timeouts for given PRC to the ones associated with booting of this
PRC started.
@param int - PRC number
@return int - boot timeout span
=cut
sub prc_boot_start
{
my ($self, $num) = @_;
$self->state_details->{prcs}->[$num]->{timeout_current_date} =
time + $self->state_details->{prcs}->[$num]->{timeout_boot_span};
$self->db_update;
return $self->state_details->{prcs}->[$num]->{timeout_boot_span};
}
=head2 prc_timeout_current_span
Get the current timeout date for given PRC
@param int - PRC number
@return int - timeout date
=cut
sub prc_timeout_current_date
{
my ($self, $num) = @_;
return $self->state_details->{prcs}->[$num]->{timeout_current_date};
}
=head2 prc_results
Getter and setter for results array for of one PRC. Setter adds given
parameter instead of substituting. If no argument is given, all PRC
results are returned.
@param int - PRC number (optional)
@param hash ref - containing success(bool) and msg(string) (optional)
=cut
sub prc_results
{
my ($self, $num, $msg) = @_;
if (not defined $num) {
my @results;
for ( my $prc_num=0; $prc_num < @{$self->state_details->{prcs}}; $prc_num++) {
push @results, $self->state_details->{prcs}->[$prc_num]->{results};
}
return \@results;
}
if ($msg) {
push @{$self->state_details->{prcs}->[$num]->{results}}, $msg;
$self->db_update;
}
return $self->state_details->{prcs}->[$num]->{results};
}
=head2 prc_count
Return number of PRCs
@return int - number of PRCs
=cut
sub prc_count
{
return int @{shift->state_details->{prcs}};
}
=head2 prc_state
Getter and setter for current state of given PRC.
@param int - PRC number
@param string - state name (optional)
@return string - state name
=cut
sub prc_state
{
my ($self, $num, $state) = @_;
return {} if $num >= $self->prc_count;
if (defined $state) {
$self->state_details->{prcs}->[$num]{current_state} = $state;
$self->db_update;
}
return $self->state_details->{prcs}->[$num]{current_state};
}
=head2 is_all_prcs_finished
Check whether all PRCs have finished already.
@param all PRCs finished - 1
@param not all PRCs finished - 0
=cut
sub is_all_prcs_finished
{
my ($self) = @_;
# check whether this is the last PRC we are waiting for
my $all_finished = 1;
for ( my $prc_num=0; $prc_num < @{$self->state_details->{prcs}}; $prc_num++) {
if ($self->state_details->{prcs}->[$prc_num]->{current_state} ne 'finished') {
$all_finished = 0;
last;
}
}
return $all_finished;
}
=head2 prc_next_timeout
Set next PRC timeout as current and return it as timeout span.
@param int - PRC number
@return int - next timeout span
=cut
sub prc_next_timeout
{
my ($self, $num) = @_;
my $prc = $self->state_details->{prcs}->[$num];
my $default_timeout = 60 + 60; # (time between SIGTERM and SIGKILL in PRC) + (grace period for sending the message)
my $next_timeout = $default_timeout;
given ($prc->{current_state}){
when('preload') { $next_timeout = $prc->{timeout_boot_span}}
when('boot') {
if (ref $prc->{timeout_testprograms_span} eq 'ARRAY' and
@{$prc->{timeout_testprograms_span}}) {
$next_timeout = $prc->{timeout_testprograms_span}->[0];
} else {
$next_timeout = $default_timeout;
}
}
when('test') {
my $testprogram_number = $prc->{number_current_test};
++$testprogram_number;
if (ref $prc->{timeout_testprograms_span} eq 'ARRAY' and
exists $prc->{timeout_testprograms_span}[$testprogram_number]){
$prc->{number_current_test} = $testprogram_number;
$next_timeout = $prc->{timeout_testprograms_span}[$testprogram_number];
} else {
$prc->{current_state} = 'lasttest';
$next_timeout = $default_timeout;
}
}
when('lasttest') {
my $result = { error => 1,
msg => "prc_next_timeout called in state testfin. This is a bug. Please report it!"};
$self->prc_results($num, $result);
}
when('finished') {
return;
}
}
$self->state_details->{prcs}->[$num]->{timeout_current_date} = time() + $next_timeout;
$self->db_update;
return $next_timeout;
}
=head2 prc_current_test_number
Get or set the number of the testprogram currently running in given PRC.
@param int - PRC number
@param int - test number (optional)
@return test running - test number starting from 0
@return no test running - undef
=cut
sub prc_current_test_number
{
my ($self, $num, $test_number) = @_;
if (defined $test_number) {
$self->state_details->{prcs}->[$num]{number_current_test} = $test_number;
$self->db_update;
}
return $self->state_details->{prcs}->[$num]{number_current_test};
}
=head2 get_min_prc_timeout
Check all PRCs and return the minimum of their upcoming timeouts in
seconds.
@return timeout span for the next state change during testing
=cut
sub get_min_prc_timeout
{
my ($self) = @_;
my $now = time();
my $timeout = $self->state_details->{prcs}->[0]->{timeout_current_date} - $now;
for ( my $prc_num=1; $prc_num < @{$self->state_details->{prcs}}; $prc_num++) {
next unless $self->state_details->{prcs}->[$prc_num]->{timeout_current_date};
$timeout = min($timeout, $self->state_details->{prcs}->[$prc_num]->{timeout_current_date} - $now);
}
return $timeout;
}
1;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,168 |
Salk technology leads to gene therapy success
Using a gene-therapy delivery system developed in the laboratory of Inder Verma at the Salk Institute for Biological Studies, an international research team successfully treated two boys from Spain suffering from adrenoleukodystrophy (ALD), the rare, inherited disease that was the focus of the Hollywood film "Lorenzo's Oil." The genetic disorder, in which the fatty insulation of nerve cells degenerates, leads to progressive brain damage and results in death. Two years after mixed gene-therapy and bone marrow transplants, the boys remain disease-free.
Verma pioneered the use of stripped-down versions of HIV, the virus that causes AIDS, to ferry intact versions of genes that are defective or missing to cells throughout the body.
Before this innovation, viruses used in gene delivery could only infect actively dividing cells, drastically limiting its utility. Verma's modified HIV virus is capable of infecting nondividing cells and delivering genes efficiently into a wide variety of cells.
The virus was produced by CellGenesys under a license agreement with the Salk Institute and the treatment administered by a French research team. The findings appear in the journal Science (https://bit.ly/2Yk9aC).
Monitoring oil spills
One day, swarms of inexpensive, miniature underwater robotic devices could predict where ocean currents will carry oil spills. Such "drifter" devices are being designed, built and deployed by researchers at Scripps Institution of Oceanography and the Department of Mechanical and Aerospace Engineering at UCSD.
Critical to the technology's success is the development of robotic control systems. This research area has been given a boost by a $1.5 million National Science Foundation grant awarded to engineers at the UCSD Jacobs School of Engineering (https://bit.ly/2v64N5).
Underwater ocean currents are poorly characterized despite their importance for understanding marine protected areas, algal blooms, oil spills and the path sewage takes after it is pumped into the ocean. Autonomous underwater robotic explorers will provide insights in fundamental oceanographic mechanisms that can be used to determine underwater ocean currents on the order of a few kilometers.
Potential to slow ALS?
In animal studies, researchers in the Department of Cellular and Molecular Medicine at UCSD have shown that a chemical cousin of the drug-activated protein C (APC) — currently used to treat sepsis — can dramatically slow the progression of a form of amyotrophic lateral sclerosis (ALS), better known as Lou Gehrig's disease. Not only were researchers able to significantly extend the lifespan of mice with an aggressive form of ALS, the compound reduced the pace of muscle wasting, thereby extending the length of time mice were able to function well despite showing some symptoms of the disease.
While more research must be done before being tested in people, researchers are encouraged that the work involves a compound that has already been proven to be safe and is currently given to patients for another condition. The team hopes to test the enzyme as an ALS treatment in patients within five years. The finding appears in the online edition of Journal of Clinical Investigation (https://bit.ly/20DtAv).
Lynne Friedmann is a science writer based in Solana Beach. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,094 |
L'ancienne Abbaye des Prés de Douai était un monastère de moniales cisterciennes sis dans la commune de Douai dans le nord de la France. Une communauté religieuse fondée au début du devient cistercienne en 1220. L'abbaye est fermée et les religieuses chassées par le pouvoir révolutionnaire à la fin du .
Situation
L'abbaye est située dans la ville de Douai, non loin de la Scarpe, dans l'actuelle Rue de l'abbaye des Prés.
Historique
Fondation
Deux poèmes, l'un en latin et l'autre en français, rédigés au Moyen Âge, permettent de connaître les débuts de l'abbaye. Six jeunes femmes décident, à une date inconnue (plusieurs estimations se réfèrent à 1212), de fonder ensemble une communauté de prière. Trois d'entre elles sont sœurs de sang et filles de Raoul le Roux ou de le Hale : Sainte, Rosselle et Foukeut. Les trois autres sont Marie, Fressent et une servante anonyme. La donation d'un bourgeois de Douai, Werin Mulet, leur permet d'acquérir quelques cabanes et un « mes » au lieu-dit « Les Prés de Saint-Albin », jusque-là lieu à la réputation sulfureuse, liée aux pratiques de la jeunesse locale. Entre cette date de fondation et le premier document public attestant de l'existence de la communauté (, quand le chevalier Gossuin de Saint-Albin cède une rente aux religieuses), la vie de la jeune communauté est mal connue. On suppose une vie de béguines, alliant prière, travail manuel et eucharistie.
En 1217, les religieuses cherchent à se doter d'une règle. Mais le chapitre de Saint-Amé voit d'un mauvais œil la structuration de cette petite communauté, qui risque, si elle est érigée en abbaye, de priver la collégiale d'une partie de ses revenus. Néanmoins, le soutien actif d'un des membres de ce chapitre, le chanoine Jean Picquette, pousse l'une des religieuses à faire le voyage jusqu'à Rome pour obtenir le soutien pontifical. Les deux premiers voyages se soldent par des échecs, faute d'argent pour arriver à destination. Le troisième permet une rencontre avec Honorius III. Celui-ci, le , ordonne à Raoul de Neuville, évêque d'Arras, de faciliter la constitution de la petite communauté en abbaye cistercienne, et nomme trois clercs de l'archidiocèse de Cambrai pour en vérifier la réalisation ; le , l'évêque accepte — à contrecœur, semble-t-il — la constitution d'un prieuré. Côté cistercien, la réponse est plus rapide, et l'abbé de Vaucelles prend en la nouvelle fondation sous sa protection. Parallèlement, au printemps 1218, le petit groupe choisit de quitter le béguinage initial pour un site plus éloigné de la ville, et portant plus à l'isolement et à la prière.
Moyen Âge
Dès , une bulle d'Honorius III acte la transformation du prieuré en abbaye, signifiant, suivant les règles en vigueur chez les cisterciens, que l'abbesse est entourée d'au moins douze moniales. La première abbesse, Élissende, est issue de l'abbaye de la Brayelle, près de Lens ; les documents médiévaux suggèrent qu'elle n'a pas été choisie par les religieuses, mais imposée par l'Ordre, peut-être par l'abbaye de Vaucelles, pour faire correspondre la règle en usage aux Prés aux usages cisterciens (La Carta Caritatis). En revanche, la prieure, élue en 1218, est une des trois sœurs fondatrices, Sainte.
Lui succède dès 1220 une nommée Marie. En , le pape, toujours Honorius III, accorde à l'abbaye nouvellement fondée l'indépendance financière, le droit d'élire son abbesse et la protection du Saint-Siège. Il est possible que cette période ait été marquée par des malveillances externes, car une nouvelle bulle pontificale dénonce en les difficultés causées à l'abbaye.
Le martyrologe
L'abbaye médiévale est particulièrement connue par le martyrologe enluminé — aujourd'hui manuscrit 838 de la bibliothèque municipale de Valenciennes. Il s'agit probablement du seul martyrologe cistercien enluminé qui nous soit parvenu, sur seulement quatre ouvrages comparables et antérieurs au concernant les cisterciennes (les trois autres étant celle de Fontenelle, près de Valenciennes, celle du Jardin, près de Sézanne dans la Marne et celle de Maubuisson près de Pontoise). C'est un ouvrage composé de 131 feuillets de format 46×34 centimètres, pesant six kilogrammes.
Il a été offert à la communauté cistercienne par la famille Lenfant, riche famille de Douai dont la fille Marguerite avait rejoint la communauté ; l'historiographie traditionnelle a vu dans cette Marguerite l'enlumineuse du manuscrit ; les recherches récentes tendent à établir qu'il s'agirait plutôt d'un artiste laïc du début du , particulièrement lettré et créatif. Ce document est exceptionnel dans la littérature cistercienne, mais permet, par sa richesse iconographique (près de quatre cents images), d'illustrer la vie quotidienne des cisterciennes du Moyen Âge.
L'usage de cet ouvrage était quotidien. Chaque jour, après l'office de prime, lecture en était faite dans la salle capitulaire, avant que ne soit commnenté un chapitre de la règle de saint Benoît ; ce temps communautaire s'achevait également par la lecture de l'ouvrage, mentionnant les bienfaiteurs ainsi que les saints honorés dans la liturgie du jour.
Possessions et travail agricole
Sur le plan matériel, l'abbaye est bien dotée ; malgré sa fondation relativement tardive, elle n'est nullement lésée dans l'attribution de terres. Ses possessions s'étendent en effet dans 117 localités, en Flandre, Artois, Ostrevent, Cambrésis et Picardie. Ces possessions étaient pour partie arrentées, pour parties soumises à un cens particulier au Nord de la France, enfin pour partie exploitées directement par les religieuses autour le la grange de la Bouverie. Les comptes des exploitations agricoles sont tenus avec une extrême minutie : le registre allant de 1329 à 1380 nous est parvenu émoigne d'un soin comptable très poussé, ce qui est d'autant plus méritoire que ces années se caractérisent par la quasi-continuité de la guerre de Cent Ans et le passage de la peste noire. En revanche, l'analyse des productions révèle des rendements très faibles, et une prédominance des légumineuses et des fourrages servant à l'alimentation du bétail. L'analyse des volumes et des prix montre aussi que les cours sont très volatils, liés au contexte politique troublé, et donc que le blé était exporté du haut pays jusqu'à la Flandre et même au-delà. Enfin cette analyse montre que l'agriculture s'est moins contractée dans le Nord, en tout cas dans les terres exploitées par les sœurs, que ce que l'historiographie paysanne française a estimé pour les années 1330-1370 ; d'autre part, elle montre que les religieuses étaient capables d'une grande souplesse de gestion pour s'adapter à la conjoncture.
La commende
La Révolution
Sous la Révolution, l'abbaye est fermée de force et les religieuses chassées. La dernière abbesse, Henriette de Maes, s'enfuit en Angleterre avec la plus grande partie de ses religieuses. Parmi les religieuses qui sont chassées, figure sœur Hippolyte Lecouvreur (1747-1828). En 1799, elle retrouve en exil sa sœur de sang Hombeline Lecouvreur (1750-1829) , moniale à l'abbaye de la Brayelle ainsi que sœur Hyacinthe Dewismes (1760-1840), venant de la Woestyne. Ensemble, elles sont au début du à l'origine de la fondation des Cisterciennes bernardines d'Esquermes.
Abbesses
Le registre des abbesses, long de deux pages, recense les vingt-trois premières responsables de l'abbaye jusqu'en 1458 ; il a été établi sous l'abbatiat de Catherine du Bus (1458-1495) ; néanmoins, les diverses analyses de ce document, effectuées au fil des siècles, apportent des successions qui concordent, à l'exception de l'abbatiat d'Eustachia de Prats ou Praets, que la Gallia Christiana place en septième position à la fin du . Or, la même abbesse dite « Marie » est attestée par les sources du en 1270, avant l'arrivée possible d'Eustachia aux Prés, et en 1293, après la mort de cette dernière.
Les treize premières entrées du registre ne mentionnent que le nom de l'abbesse, et, pour cinq d'entre elles, le nom de leur père ; les dix dernières entrées sont complétées par une notice biographique plus fournie. La durée moyenne de l'abbatiat durant ces environ deux cent quarante années est donc de treize ans, ce qui correspond à la moyenne de cet exercice.
Notes et références
Notes
Références
Voir aussi
Articles connexes
Bibliographie
Abbaye dans le département du Nord
Douai
Douai
Fondation en 1219
Abbaye fondée au XIIIe siècle
Abbaye dédiée à Notre-Dame
Abbaye désaffectée | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,660 |
\section{INTRODUCTION}
To assist people in their daily activities, a robot would need to manipulate objects in specific ways, dependent on its current task, \emph{e.g.} it should not handle a knife in the same way for cutting vegetables as for handing it to a person with reduced mobility.
A human could teach such specificities to a robot by demonstrating the task to reproduce.
In this work, we focus on the task of grasping rigid objects, as illustrated in Figure~\ref{fig:introduction}.
Reproducing exactly a human grasp is impossible for a robot, because a robotic gripper is usually quite different from a human hand: different size, number of fingers, actuation, \emph{etc.} (see Figure~\ref{fig:comparison} for a comparison).
Instead, the robot should grasp objects \emph{like} the human did.
Most existing grasp retargeting approaches~\cite{dexpilot,dexmv,contacttransfer} rely on handcrafted correspondences between the human hand and the robotic gripper, either in terms of joints, surfaces, or key vectors, and they do not consider the object.
Some other methods such as ContactGrasp~\cite{contactgrasp} refine the grasps produced by GraspIt!~\cite{graspit} by optimizing the contact surface and reranking them. However, such approach is slow as it takes tremendous time and effort to generate and refine grasp candidates. It can moreover lead to significant differences with the human grasp.
The fundamental problem is that the exact meaning of ``grasping like a human'' is not well defined.
In this study we nonetheless introduce some generic proxies for grasp similarity, namely the \emph{contact} surface and the grasp \emph{orientation}.
Indeed, the affordance~\cite{gibson1979ecological} of a grasped object is typically dependent on the open space surrounding this object. Grasping an object from a similar orientation, with a similar contact surface on the object as in the human demonstration should therefore in general enable to perform with this object similar actions as the human.
In this paper, we investigate how well the use of these proxies enable to produce robotic grasps similar to human demonstrations.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{image/intro.png}
\vspace{-0.7cm}
\caption{\textbf{Grasping like humans.} Given an input human grasp (left), our method outputs a configuration of a multi-fingered gripper grasping the same object %
\emph{like the human} demonstration. We experiment with the Allegro (top) and BarrettHand (bottom) grippers.
}
\label{fig:introduction}
\end{figure}
To do so, we propose a multi-step optimization-based method that takes a human grasp demonstration as input -- represented by a 3D mesh of the object and a parametric MANO model of the hand pose~\cite{mano} -- and returns the configuration of the corresponding robotic grasp.
We define an objective function that encourages similar contact surfaces and global orientation for the human and the robotic grasps, while penalizing interpenetrations of the gripper and the object.
To avoid local minima, we perform a multi-stage optimization where the gripper global position and orientation are initialized similarly to the human demonstration.
Fingers are then closed by minimizing the distance between the fingertips and contact areas on the object, before optimizing for our full objective function in a last step.
To validate the genericity of our approach, we experiment with two off-the-shelf robotic hands: the \emph{Allegro}~\cite{allegrohand} and the \emph{BarrettHand}~\cite{barretthand} grippers (see Figure~\ref{fig:comparison}).
We evaluate our method using human grasps from the YCB-affordance dataset~\cite{ganhand} with various quality metrics, and we also perform a user study to compare qualitatively our approach with related methods. Both evaluations show that our approach allows to predict reasonable grasps that are better -- and more similar to the human demonstration -- than existing state-of-the-art grasp retargeting methods.
In the end, we validate the applicability of the approach in the real world on a Panda robotic arm.
In summary, the main contributions of this work are: (1) A novel objective function consisting of four losses which encourages a valid grasp while capturing the similarity between the human hand grasp and robot gripper grasp. (2) A novel multi-step optimization-based pipeline to transfer a human grasp demonstration to any multi-fingered gripper. (3) An extensive quantitative and qualitative evaluation and comparison between our approach and other related methods.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{image/comparison.png}
\vspace{-0.5cm}
\caption{\textbf{Comparison between different grippers} at the same scale with a human hand (left), Allegro (middle) and BarrettHand (right). Note that the size of the gripper and in particular the fingers are significantly different. The \textcolor{blue}{blue} vector represents the normal vectors of the human hand and robot hands, the \textcolor{green}{green} vector represents the forward vector (best seen in color).}
\label{fig:comparison}
\end{figure}
\section{Related work}
In this section, we review various works related to grasping like humans with multi-fingered grippers.
\paragraph*{Grasp prediction}
Predicting potential grasps for a given object is a classical research topic, illustrated by the seminal \emph{GraspIt!} simulator~\cite{graspit}.
Most recent approaches~\cite{multifingan,wu2020generative,varley2015generating,lu2020multi,liu2019generating} focus on learning-based techniques,
with some approaches~\cite{contactgraspnet,wu2020generative} modeling reachability constraints in the scene.
\paragraph*{Grasping from demonstration}
Learning from demonstration is also an important paradigm in robotics~\cite{pomerleau1991efficient,billard2008survey,argall2009survey,rajeswaran2017learning}.
It aims at teaching a particular task to a robot from a few examples of a human performing a similar task.
Most current approaches for learning from demonstration in the context of object manipulation focus on complex manipulation tasks with simple parallel-jaw grippers~\cite{zhang2018deep,schmeckpeper2020reinforcement}.
On the contrary, we focus in this study on simpler manipulation tasks %
(static grasping)
but with more complex multi-fingered grippers -- that could allow more advanced grasps and manipulations.
\paragraph*{Pose retargeting}
A solution to transfer a human grasp demonstration to a robotic gripper it to define some fixed correspondences between the human hand and the robotic gripper. DexPilot~\cite{dexpilot} and DexMV~\cite{dexmv} use some handcrafted motion retargeting techniques to do so.
Similarly, ContactTransfer~\cite{contacttransfer} relies on fixed correspondences between the surface of the human hand and the robotic gripper.
The applicability of such approaches is arguably limited however because the human hand and the gripper may have significantly different characteristics in practice.
For instance, the Allegro gripper has only 4 fingers and is about 1.6 times larger than a typical human hand (see Figure~\ref{fig:comparison}).
Moreover, these approaches do not consider the object being grasped.
\paragraph*{Contact-based retargeting}
More related to our work are approaches trying to predict robotic grasps sharing similar contact areas with the object as in the human demonstration, without requiring explicit correspondences between fingers of the human and the robot.
In~\cite{zhu2021toward}, Zhu~\emph{et al.} propose to annotate functional parts of the objects -- \emph{i.e.}, where humans would grasp the object or not -- to generate potential grasps for these objects.
Recently, ContactGrasp~\cite{contactgrasp} was proposed and uses GraspIt!~\cite{graspit} to generate a set of grasps that are iteratively refined and reranked such that the contact areas of the gripper on the object become closer to the ones of the human grasp. This approach has several drawbacks however.
First, it is about 40 times slower than our method as it has to generate and refine hundreds of grasp candidates each time.
Second, \emph{GraspIt!} mainly generates power grasps, and thus the refined grasps have similar properties.
Third, by focusing only on contact areas, ContactGrasp can produce grasps in which the gripper is occluding some important parts for the affordance of the grasped object.
In comparison, our proposed optimization approach is faster
and leads to grasps more similar to human ones, thanks to a simple yet effective initialization and thanks to additionally taking into account the grasp orientation. %
\section{GRASPING LIKE HUMANS}
\label{sec:optim}
In this section, we describe our optimization-based approach to generate a robot grasp `similar' to a given human grasp. %
After formalizing the problem and notations (Section~\ref{sub:problem}), we introduce the optimized objective function in Section~\ref{sub:loss} and detail all the steps of our approach in Section~\ref{sub:pipeline}.
\subsection{Problem and Notations}
\label{sub:problem}
We consider as input a rigid object grasped by a human hand. We represent the object by a 3D mesh $\mathcal{M}_{object}$, and we adopt the MANO~\cite{mano} model to represent the pose of the hand by a global rigid transformation $(R_{hand}, t_{hand}) \in SO(3) \times \mathbb{R}^3$ relative to the object and by its local joints configuration $\theta_{hand} \in SO(3)^{20}$.
Similarly, we assume that a kinematic model of the robotic gripper is available.
We aim to predict a global pose $(R_{robot}, t_{robot}) \in SO(3) \times \mathbb{R}^3$ relative to the object and some joints configuration $\theta_{robot} \in \mathbb{R}^n$ describing a static grasp with this gripper similar to the human demonstration ($n=16$ for Allegro, $n=7$ for Barrett).
We formulate this as an optimization problem and minimize an objective function $\mathcal{L}(R_{robot}, t_{robot}, \theta_{robot})$ representing the \emph{dissimilarity} of the robotic grasp with the human demonstration.
\begin{figure}
\center
\includegraphics[width=0.9\linewidth]{image/heatmap_example_2.png}
\vspace{-0.3cm}
\caption{\textbf{Contact heatmaps} on the object mesh corresponding to a human (top) and robotic (bottom) grasp. Our optimization-based approach tries to minimize the discrepancy between these heatmaps.
Red color denotes regions close to the hand/gripper while blue color denotes regions far from the hand/gripper.
}
\label{fig:contact-region-heatmap}
\end{figure}
\subsection{Objective Function}
\label{sub:loss}
Our objective function $\mathcal{L}$ is composed of a contact-heatmap loss $\mathcal{L}_{C}$ that incites contacts on the object to be similar, a hand orientation loss $\mathcal{L}_{O}$, as well as losses that penalize interpenetration with the object $\mathcal{L}_{I}$ and self-penetration of the gripper $\mathcal{L}_{S}$, \emph{i.e.}:
\begin{equation}
\label{eq:objective_function}
\mathcal{L} = \lambda_{C} \mathcal{L}_{C} + \lambda_{O} \mathcal{L}_{O} + \lambda_{I} \mathcal{L}_{I} + \lambda_{S} \mathcal{L}_{S}
\end{equation}
with weights experimentally set to $\lambda_{C}=10, \lambda_{O}=10, \lambda_{I}=0.5$ and $\lambda_{S}=1$. We detail these terms in the following paragraphs.
\paragraph{Contact Heatmap Loss $\mathcal{L}_C$}
Intuitively, grasps are similar if their contact regions on the target object are similar. Based on this observation, we propose an object-centric contact heatmap loss term, which encourages the contact regions of the input human hand and the robotic gripper on the object to be similar.
Specifically, we represent the contact regions of the human hand and the robot gripper by scalar contact heatmaps $H$ on the object.
At each vertex $o_i \in \mathcal{M}_{object}$ of the object mesh, we define the values of these heatmaps as
\begin{equation}
\left\lbrace
\begin{aligned}
H^{hand}(o_i) &= \exp(-d(o_i, \mathcal{M}_{hand})/\tau) \\
H^{robot}(o_i) &= \exp(-d(o_i, \mathcal{M}_{robot})/\tau) \>
\end{aligned}
\right.
\end{equation}
where $d(o_i, \mathcal{M})$ denotes the $L_2$-distance of $o_i$ to the set of vertices of the mesh $\mathcal{M}$, and where $\tau$ is a constant used to define contacts in a soft manner (\emph{i.e.} $H(o_i)=1$ when $d(o_i, \mathcal{M})=0$, and $H(o_i) \approx 0$ when $d(o_i, \mathcal{M}) >> \tau$). In our experiments, we use uniformly sampled meshes and choose $\tau=0.01m$.
Figure~\ref{fig:contact-region-heatmap} shows examples of contact heatmaps for different human grasps.
We define our object-centric contact heatmap loss as the $L_1$-distance between these generated heatmaps:
\begin{equation}
\mathcal{L}_{C} = \sum_{ o_i \in \mathcal{M}_{object}} |H^{hand}(o_i) - H^{robot}(o_i)|
\label{eq:contact_loss}
\end{equation}
\paragraph{Hand Orientation Loss $\mathcal{L}_{O}$} %
Grasps have high similarity if the hands are oriented similarly towards the object, thus resulting in a similar free space around the object, and thus potentially to a similar affordance. Therefore, we introduce a loss to encourage the orientation of the hand and the gripper to be similar. To this end, for each human/robot hand model, we define two unit vectors which are inherent to the model, the forward vector $f$ and the normal vector $n$.
Examples of these two vectors for different models are shown in Figure~\ref{fig:comparison}.
The normal vector $n$ is defined as the unit normal vector of the palm surface. The forward vector $f$ is defined as the unit vector that is parallel to the palm surface and pointing to the `pushing' direction.
We define the hand orientation loss as the $L_1$-distance between these two unit vectors:
\begin{equation}
\mathcal{L}_{O} = \vert n_{robot} - n_{hand} \vert + \vert f_{robot} - f_{hand} \vert \>
\label{eq:label_loss}
\end{equation}
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{image/pipeline.png}
\vspace{-0.2cm}
\caption{\label{fig:grasp-pairs-generation} \textbf{Overview of our pipeline for transferring human hand grasp to robot gripper grasp.} We first initialize the gripper with open fingers at the location of the hand. We then initialize the fingers position on the object surface by minimizing the distance between the fingertips and the contact regions of the human demonstration. At last, we refine the grasp by minimizing the overall objective function.}
\end{figure*}
\paragraph{Gripper-Object Interpenetration Loss $\mathcal{L}_{I}$}
To avoid interpenetration while ensuring realistic contacts between the robotic gripper and the object, we take inspiration from Müller~\emph{et al.}~\cite{selfcontact} and add a loss
\begin{equation}
\mathcal{L}_{I} = \alpha_1 \mathcal{L}_{push} + \beta_1 \mathcal{L}_{pull} + \gamma_1 \mathcal{L}_{normal} \>
\label{eq:interpenetration_loss}
\end{equation}
to our objective function.
It consists of three weighted terms.
\newline\noindent $\bullet$ The first term $\mathcal{L}_{push}$ aims at avoiding interpenetration by pushing the penetrated parts of the robotic gripper towards their nearest surface on the object mesh. To do so, we consider $\mathcal{U}_{robot} \subset \mathcal{M}_{robot}$ the set of vertices on the robotic gripper mesh that are inside the object mesh, and $\mathcal{U}_{object} \subset \mathcal{M}_{object}$ the set of vertices on the object mesh that are inside the robotic gripper mesh. In practice, we detect these two sets of vertices using the generalized winding numbers~\cite{winding_numbers}.
We define
{
\small
\begin{equation}
\mathcal{L}_{push} =
\hspace{-1em}
\sum_{o_i \in
\mathcal{U}_{object}}
\hspace{-1em}
\text{tanh} \left( \frac{d(o_i, \mathcal{M}_{robot})}{\alpha_2} \right) +
\hspace{-1em}
\sum_{r_k \in \mathcal{U}_{robot}}
\hspace{-1em}
\text{tanh} \left( \frac{d(r_k, \mathcal{M}_{object})}{\alpha_2} \right)
\end{equation}
}
to penalize interpenetration.
\newline\noindent $\bullet$
The second term $\mathcal{L}_{pull}$ encourages contacts for points of the gripper closer than a threshold $\delta$ to the object, while being constant for points farther away:
\begin{equation}
\mathcal{L}_{pull} = \sum_{ r_k \in \mathcal{M}_{robot}} \text{tanh} \left( \frac{\min(d(r_k, \mathcal{M}_{object}), \delta)}{\beta_2} \right) %
\end{equation}
We use $\delta=2mm$ in practice.
\newline\noindent $\bullet$
To further ensure realistic contacts, a third term is added that encourages normals of both meshes to be opposite at contact locations $\mathcal{V} = \{ r_k \in \mathcal{M}_{robot} \vert d(r_k, \mathcal{M}_{object}) < \delta \}$:
\begin{equation}
\mathcal{L}_{normal} = \sum_{r_k \in \mathcal{V}} 1 + \langle N(r_k), N( o^k_i ) \rangle
\end{equation}
where $N(\cdot)$ denotes the unit normal vector at a given vertex, and $o^{k}_i = arg\,min_{o_i \in \mathcal{M}_{object}} d(r_k, o_i)$ denotes the closest point on the object for any vertex $r_k \in \mathcal{V}$.
Hyperparameters values are experimentally set to $\alpha_1=2.4$, $\beta_1=7$, $\gamma_1=0.001$, $\alpha_2=4cm$, $\beta_2=6cm$.
\paragraph{Gripper Self-penetration Loss $\mathcal{L}_{S}$}
$\mathcal{L}_I$ considers griper-object penetration, but some
configurations of the gripper could also lead to self-penetration between different gripper components such as its fingers. We thus add a loss to avoid self-penetration. To this end, we use the exact same loss as $\mathcal{L}_{push}$
but apply it between the gripper mesh and itself, resulting in a loss $\mathcal{L}_{S}$.
\subsection{Optimization Pipeline}
\label{sub:pipeline}
Our objective function of Equation~\eqref{eq:objective_function} admits many local minima, and several optimization terms admit zero gradient when the gripper is far from the object. Having a good initialization is therefore important, and we thus propose a multi-step optimization pipeline whose overview is shown in Figure~\ref{fig:grasp-pairs-generation}.
It consists of 3 steps: (a) initializing the robotic gripper with open fingers around the same location as the human hand, (b) closing the fingers until contact with object, (c) refining all degrees of freedom.
\paragraph*{(a) Open Gripper Initialization}
Because of the hand orientation loss $\mathcal{L}_O$,
the optimal global position and orientation of the gripper ($R_{robot}, t_{robot}$) is likely to be close to the global position and orientation of the human hand ($R_{hand}, t_{hand}$) in the object coordinate system. This is why we initialize the gripper position and orientation at the same position and orientation as the human hand. At this stage, we assume that the rest of the parameters, \emph{i.e.}, the angle of the finger joints correspond to a fully-open position and we thus refer to this stage as `open gripper initialization'.
\paragraph*{(b) Fingers Initialization}
To initialize the fingers and make the fingers touch the object at the right place, we first detect the contact region of the human grasp, then we minimize the distance between the fingertips and their nearest contact region using the gripper-object interpenetration loss $\mathcal{L}_{I}$ defined in Equation (\ref{eq:interpenetration_loss}) with the self-penetration loss $\mathcal{L}_{S}$.
In this way, we can put the fingers of the robot hand to their closest region of contact and at the same time avoid gripper-object interpenetration and self-penetration.
\paragraph*{(c) Refining the Results}
We finally run the full optimization from this initialization. We use AdamW~\cite{adamw} as our optimizer, the initial learning rate is set to 0.001 for the translation $T_{robot}$ and 0.01 for rotation $R_{robot}$ and pose parameters $\theta_{robot}$. Each grasp is optimized for 100 iterations. The learning rate decreases by 10 at iteration \#50. During the optimization, we use the rotation parametrization introduced in~\cite{zhou2019continuity}.
\begin{table*}
\centering
\caption{\label{tab:sota} \textbf{Comparison of our approach with state-of-the-art methods.}
$^\dagger$ indicates methods that use different hyperparameters for different grippers.
For GraspIt!, we generated 100 grasps per human demonstration and selected the one with the lowest orientation difference and contact heatmap difference, \emph{i.e.}, the lowest $\mathcal{L}_{C}+\mathcal{L}_{O}$ loss.
}
\small
\vspace{-0.2cm}
\begin{tabular}{ll @{\hskip 1cm} ccccc}
\toprule
& & Grasp & Max Penetration & Penetration & Orientation & Contact Heatmap \\
& & $\epsilon$-quality $\uparrow$ & Depth ($cm$)$\downarrow$ & Volume ($cm^3$)$\downarrow$ & Difference $\downarrow$ & Difference $\downarrow$ \\
\midrule
\multirow{4}{*}{\rotatebox[origin=c]{90}{\parbox{1cm}{\centering Allegro \\ Hand}}}
& DexPilot$^\dagger$~\cite{dexpilot} & {\bf 0.535} & 2.91 & 5.04 & 0.011 & 0.176 \\
& ContactGrasp$^\dagger$~\cite{contactgrasp} & 0.460 & 3.53 & 6.94 & 1.818 & 0.195 \\
& GraspIt! (best $\mathcal{L}_{C}+\mathcal{L}_{O}$)~\cite{graspit} & 0.345 & 2.76 & {\bf 1.27} & 0.420 & 0.254
\\
& \textbf{Ours} & 0.466 & {\bf 2.57} & 4.89 & {\bf 0.001} & {\bf 0.153} \\
\midrule
\multirow{3}{*}{\rotatebox[origin=c]{90}{\parbox{0.8cm}{\centering Barrett \\ Hand}}}
& ContactGrasp$^\dagger$~\cite{contactgrasp} & 0.523 & 4.65 & 6.28 & 2.003 & 0.225 \\
& GraspIt! (best $\mathcal{L}_{C}+\mathcal{L}_{O}$)~\cite{graspit} & 0.354 & 4.52 & {\bf 0.88} & 0.714 & 0.258
\\
& \textbf{Ours} & {\bf 0.566} & {\bf 4.09} & 2.91 & {\bf 0.001} & {\bf 0.166} \\
\bottomrule
\end{tabular}
\end{table*}
\section{EXPERIMENTS}
\label{sec:xp}
In this section, after presenting datasets and metrics (Section~\ref{sub:xpdata}), we provide the results of an ablation study in Section~\ref{sub:ablations} and a comparison to the state of the art in Section~\ref{sub:sota}. We then describe a user study (Section~\ref{sub:userstudy}) that validates that our approach leads to grasps more similar to the human demonstrations than the state of the art.
\subsection{Datasets and Metrics}
\label{sub:xpdata}
To measure performance, we consider the human grasps from the YCB-Affordance dataset~\cite{ganhand}. For the 52 objects of the YCB-Objects~\cite{ycbobjects}, different types of human grasps are manually annotated and refined using GraspIt!~\cite{graspit}. This leads to a diversity in terms of grasps, including not only power grasps but also pinch grasps, \emph{etc.}, as shown by the annotations of the grasp categories provided with the dataset~\cite{ganhand} and illustrated in the top row of Figure~\ref{fig:results_examples}.
For evaluation, we measure both the grasps quality using the Grasp $\epsilon$-quality metric, which corresponds to the radius of the largest ball centered at the origin which can be enclosed by the convex hull of the wrench space~\cite{miller1999examples}, as commonly used in, \emph{e.g.}, \cite{graspit_score, multifingan, ycbobjects}. We also report the \emph{Max Penetration Depth}, \emph{i.e.}, the maximum distance between a vertex of the gripper that is inside the object and its closest vertex on the object, and the \emph{Penetration Volume}, \emph{i.e.}, the estimated volume of penetration between the two meshes.
In addition to these grasp quality metrics, we propose two metrics that are used to measure the similarity between the human hand grasp and the robot gripper grasp: the \emph{Contact Heatmap Difference} that measures contact similarity and the \emph{Orientation Difference} that measures similarity in terms of contact angle. We use the metric $\mathcal{L}_{C}$ introduced in Section~\ref{sub:loss} to evaluate numerically the \emph{Contact Heatmap Difference} and $\mathcal{L}_{O}$ for the \emph{Orientation Difference}. %
\subsection{Ablations}
\label{sub:ablations}
We first ablate our approach in Table~\ref{tab:losses} for the Allegro gripper.
To start with, we replace the second step of our optimization pipeline by another finger closing strategy inspired by~\cite{ganhand}:
starting from an \emph{open} configuration of the gripper, we discretize the gripper configuration space and pick iteratively for each finger joint the bin corresponding to the most \emph{closed} configuration that does not penetrate the object.
We observe that this significantly degrades the grasp $\epsilon$-quality, the penetration volume and the contact heatmap similarity.
We then ablate the losses of the final optimization step of our approach by removing them one by one. Removing the contact similarity loss $\mathcal{L}_{C}$ significantly degrades the contact heatmap difference from 0.153 to 0.189, while also impacting negatively the grasp $\epsilon$-quality metric and the interpenetration.
Additionally removing the loss on the angle similarity $\mathcal{L}_{O}$ leads to grasps that are even less similar and leads to higher penetration volume.
Furthermore, removing the self-penetration loss $\mathcal{L}_{I}$ degrades the grasp $\epsilon$-quality even more.
We finally evaluate the performance of our approach without the global optimization (the third step of our pipeline) in the last row, to show its importance for achieving good grasps.
\begin{table*}
\centering
\small
\caption{\label{tab:losses} \textbf{Ablation study} of our approach with the Allegro gripper. In the first row, we replace the second step of our pipeline with contact optimization for the fingers' initialization by a discrete closing strategy. In the rows below, we remove the losses one by one. The last row without any loss corresponds to the absence of Step 3 of our optimization pipeline. }
\vspace{-0.2cm}
\begin{tabular}{c @{\hskip 1em} cccc @{\hskip 1em} ccccc}
\toprule
Fingers & \multirow{2}{*}{$\mathcal{L}_{I}$} & \multirow{2}{*}{$\mathcal{L}_{S}$} & \multirow{2}{*}{$\mathcal{L}_{O}$} & \multirow{2}{*}{$\mathcal{L}_{C}$} &
Grasp & Max Penetration & Penetration & Orientation & Contact Heatmap \\
Init. (step 2) & & & & &
$\epsilon$-quality $\uparrow$ & Depth ($cm$)$\downarrow$ & Volume ($cm^3$)$\downarrow$ & Difference $\downarrow$ & Difference $\downarrow$ \\
\midrule
Discrete init. & \checkmark & \checkmark & \checkmark & \checkmark & 0.294 & 2.59 & 5.83 & 0.011 & 0.213 \\
\midrule
\multirow{5}{*}{Contact optim.} &
\checkmark & \checkmark & \checkmark & \checkmark & \bf{0.466} & 2.57 & \bf{4.89} & \bf{0.001} & \bf{0.153} \\
& \checkmark & \checkmark & \checkmark & & 0.438 & 2.70 & 6.32 & \bf{0.001} & 0.189 \\
& \checkmark & \checkmark & & & 0.467 & \bf{2.55} & 6.60 & 0.041 & 0.190 \\
& \checkmark & & & & 0.411 & 2.69 & 6.39 & 0.031 & 0.187 \\
\cmidrule{2-10}
& & & & & 0.387 & 2.96 & 8.52 & 0.011 & 0.170 \\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Comparison with the state of the art}
\label{sub:sota}
We compare our approach to the state of the art and report performances for the Allegro gripper as well as for the BarrettHand gripper in Table~\ref{tab:sota}. We also show various examples in Figure~\ref{fig:results_examples}.
As a first approach, we compare to a manually-defined mapping between the human hand and the Allegro gripper
using a re-implementation of DexPilot~\cite{dexpilot}. Note that this approach would require new manual annotations for another type of robotic gripper.
As a second approach, we use ContactGrasp~\cite{contactgrasp} that proposes to refine and rerank the grasps generated by GraspIt! by exploiting contact information. For each object, GraspIt! generates grasps from different directions around the object (we consider 100 grasps in practice). The generated grasps are fed to the ContactGrasp pipeline with the contact region heatmaps generated using the code provided by the authors.
In the end, we consider the best-ranked robotic grasp for each reference human grasp.
As a third approach, we use GraspIt!~\cite{graspit} to generate 100 different grasps for an object and select the one that minimizes the sum of \emph{Orientation Difference} and \emph{Contact Heatmap Difference}.
While DexPilot obtains a higher $\epsilon$-quality metric than our approach with the Allegro gripper, the grasps are actually less realistic as there are larger interpenetrations with the object, as illustrated in the left example of Figure~\ref{fig:results_examples}. Additionally, the grasp similarity in terms of both orientation and contact heatmap is lower than our approach. Note also that DexPilot
is not generic in that it has been handcrafted for the Allegro gripper.
We also compare our method to the best grasp from ContactGrasp~\cite{contactgrasp}, which uses different hyperparameters for the two grippers. We observe that our approach leads to higher quality grasps, with less interpenetrations, and with higher similarity with the input human grasps. This is particularly true for the orientation similarity as illustrated in the examples of Figure~\ref{fig:results_examples}: while ContactGrasp optimizes for similar contact regions, it does not enforce the gripper to approach the object from a similar direction, which can lead to grasps with significantly different properties than the human grasps in terms of free space around the object. Lastly, our method outperforms GraspIt! on every metric except for the penetration volume.
\begin{figure}
\centering
\small
\includegraphics[height=0.7in]{image/real_grasp_mustard_human1.jpg}
\includegraphics[height=0.7in]{image/real_grasp_mustard_human2.jpg}
\hfill{}
\vline{}
\hfill{}
\includegraphics[height=0.7in]{image/real_grasp_mustard_allegro1.jpg}
\includegraphics[height=0.7in]{image/real_grasp_mustard_allegro2.jpg}
\hfill{}
\includegraphics[height=0.74in]{image/real_grasp_mustard.jpg}
\vspace{-0.3cm}
\caption{\label{fig:realworld} \textbf{Real-world experiments.} \textit{Left:} input human demonstration. \textit{Middle:} corresponding Allegro grasp prediction. \textit{Right:} execution.}
\end{figure}
\paragraph*{Running time.}
We evaluated all the methods on a machine with 20 Intel(R) Core(TM) i9-9900X CPUs and one NVidia GeForce RTX 2080Ti card. Our approach takes about 1 minute for a given grasp.
For comparison, our implementation of DexPilot takes about 1 second but does not consider the geometry of the object, and ContactGrasp takes on average 43 minutes for one input human grasp as it first requires to generate 100 robotic gripper grasps using GraspIt! before refining all of them.
\begin{figure*}
\centering
\small
\includegraphics[width=\linewidth]{image/visual_example.png}
\vspace{-0.4cm}
\caption{\label{fig:results_examples} \textbf{Example of generated grasp transfers} for our approach, DexPilot\cite{dexpilot} and ContactGrasp\cite{contactgrasp}}
\end{figure*}
\subsection{User Study}
\label{sub:userstudy}
We aim at enabling robots to \emph{grasp like humans}, but the metrics above do not necessarily express this notion well. We therefore conducted a user study to better evaluate the similarity between human and robotic grasps.
It is difficult for people to quantitatively evaluate this similarity, thus we resorted to a comparative evaluation.
We randomly selected 120 human grasps from the YCB-Affordance~\cite{ganhand} dataset.
For each human grasp, we generated corresponding robotic grasps using different methods and asked participants to select the one which -- in their opinion -- is the most similar to the human demonstration.
For the Allegro gripper, we compared our method with ContactGrasp~\cite{contactgrasp} and DexPilot~\cite{dexpilot}. For BarrettHand, we compared our method with ContactGrasp~\cite{contactgrasp}. We also included an additional grasp generated randomly using GraspIt!~\cite{graspit} as baseline.
We received in total 1,392 votes from 58 participants -- each participant sharing its preference regarding 24 human grasps. Results are summarized in Table~\ref{tab:user_study}.
Overall, the participants favored grasps produced by our method in 51\% of the cases for the Allegro gripper, and 73\% of the cases for BarrettHand. These scores are way above random chance (25\% for Allegro, 33\% for BarrettHand), and they suggest that our generated grasps are considered significantly more similar to the human demonstrations than the grasps generated using the other evaluated methods.
Further analysis of the results showed that the preference for our method could be explained in all cases by the smaller difference of global orientation between the robot and human hands when using our method.
\begin{table}
\centering
\small
\caption{\label{tab:user_study} \textbf{User-Study:}
``Which grasp is the most similar to the human demonstration?''}
\begin{tabular}{lcc}
\toprule{}
\multirow{2}{*}{Grasp generation method} & \multicolumn{2}{c}{Number of votes (total: 1392)} \\
\cmidrule(lr){2-3}
& Allegro & BarrettHand \\
\midrule
GraspIt! (random)~\cite{graspit} & 49 & 38 \\
ContactGrasp~\cite{contactgrasp} & 117 & 101 \\
DexPilot~\cite{dexpilot} & 265 & - \\
\textbf{Ours} & \textbf{440} & \textbf{383} \\
\bottomrule{}
\end{tabular}
\vspace{-0.4cm}
\end{table}
\subsection{Real World Experiments}
\label{sub:video}
We focus in this work on predicting static grasps that describe the pose and joints configuration of a robotic gripper with respect to an object.
Grasping however is fundamentally a dynamic process, involving robot motion and contact forces.
To demonstrate the usability of our approach in real scenarios, we performed grasping experiments using an \emph{Allegro} gripper mounted on a \emph{Panda} robotic arm, from \emph{Franka Emika} (see one grasp example in Figure~\ref{fig:realworld}, and the attached video with 5 grasping examples on 5 different objects).
These experiments allowed to check that grasps produced by our method are physically feasible, while being similar to the human demonstrations.
Robot perception is out of the scope of this study, therefore we used as input some human demonstrations from the YCB-Affordance~\cite{ycbobjects} dataset, and we manually placed the objects in known poses before attempting the grasps with the robot. We did not conduct any quantitative evaluations because of this manual step.
Note that state-of-the-art methods for object~\cite{labbe2020} and hand+object pose estimation~\cite{obman,honnotate,egohands,dexycb,grab,hasson20_handobjectconsist} could be used to overcome this limitation.
\section{CONCLUSIONS}
We propose a multi-step optimization-based approach for transferring grasps from a human demonstration to a multi-fingered robotic gripper, so as to enable a robot to \emph{grasp like a human}.
The proposed approach is generic and can be applied to arbitrary multi-fingered gripper, as shown by our experimental evaluation with both Allegro and BarrettHand grippers.
Our results -- based on quantitative metrics and a qualitative user study -- suggest that it produces grasps significantly more similar to the human demonstrations than state-of-the-art methods, and we validated its applicability in the real world using an Allegro gripper mounted on a Panda arm.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,767 |
Q: Javascript epoch in c# I search the forum for my problem but found nothing. :(
This DateTime conversion drives me mad.
I try to convert a millisecond epoch to DateTime.
I found this Methode in the Internet:
private DateTime TimeFromUnixTimestamp(int unixTimestamp)
{
DateTime unixYear0 = new DateTime(1970, 1, 1, 0, 0, 0, DateTimeKind.Utc);
long unixTimeStampInTicks = unixTimestamp * TimeSpan.TicksPerSecond;
DateTime dtUnix = new DateTime(unixYear0.Ticks + unixTimeStampInTicks);
return dtUnix;
}
private DateTime TimeFromJavaTimestamp(long javaTimestamp)
{
return TimeFromUnixTimestamp((int)(javaTimestamp / 1000));
}
Now to test the method I run this code in JavaScript:
Date.UTC(2014,05,06,0,0,0,0);
You can test it here (jsfiddler)
The result is 1402012800000.
So far so good. Now I test my c# methode:
var test = TimeFromJavaTimestamp(1402012800000L);
and as result I get {06.06.2014 00:00:00}!
One month offset to what I do expected??
Can somebody explain this to me???
Regards Steffen
A: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Date/UTC
month An integer between 0 and 11 representing the month.
So, yeah, the month 05 is June. Looks like your code is working.
A: This is how I would do it.
In JavaScript:
var timestamp = new Date().getTime();
Then in C# to convert a JavaScript timestamp to a DateTime object:
public static DateTime ToDateTime(long timestamp)
{
var dateTime = new DateTime(1970,1,1,0,0,0,0, DateTimeKind.Utc);
return dateTime.AddSeconds(timestamp / 1000).ToLocalTime();
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,226 |
\section{Introduction}
Since the first numerical investigations of four-dimensional
non-abelian gauge theories
with dynamical Wilson fermions were performed in the early 80's
\cite{an,hamber},
the understanding of the phase and vacuum structure of lattice QCD with
Wilson fermions at non-zero lattice spacing, and of the way in which chiral
symmetry is recovered in the continuum limit, has been a goal of lattice
field theorists. The complexity of the phase structure of
this model was known a long time ago. The existence of a phase with parity
and flavor symmetry breaking was conjectured for this model by Aoki in the
middle 80's \cite{a1,a2}. From that time on, much work has been done in order
to confirm this conjecture, to establish a quantitative phase diagram
for lattice QCD with Wilson fermions, and to delimit the parameters region
where numerical calculations of physical quantities should be performed.
References \cite{a1}-\cite{ilg3} are a representative but incomplete list
of the work done on this subject.
In this paper we analyze the vacuum structure of lattice QCD with
Wilson fermions at non-zero lattice spacing, with the help of the probability
distribution function of the parity-flavor fermion bilinear order parameters
\cite{pdc}\cite{vw2}. We find that if the spectral
density $\rho_U(\lambda,\kappa)$ of the overlap hamiltonian, or Hermitian
Dirac-Wilson operator, in a fixed
background gauge field $U$ is not symmetric in $\lambda$ in the
thermodynamical limit for the relevant gauge configurations, Hermiticity of
$i\bar\psi_u\gamma_5\psi_u+i\bar\psi_d\gamma_5\psi_d$ will
be violated at finite $\beta$. Assuming that the Aoki phase ends at finite
$\beta$, this would imply the lost of any physical interpretation of this
phase in terms of particle excitations. In addition, the lost of Hermiticity
of $i\bar\psi_u\gamma_5\psi_u+i\bar\psi_d\gamma_5\psi_d$ at finite $\beta$
also suggests that a reliable determination of the $\eta$ mass would require
to be near enough the continuum limit. If on the contrary the spectral
density $\rho_U(\lambda,\kappa)$ of the Hermitian Dirac-Wilson operator
in a fixed background gauge field $U$, is symmetric in $\lambda$, we explain how, under certain assumptions, the
existence of the Aoki phase implies also the appearance of other phases, in
the same parameters region, which can be characterized by a non-vanishing vacuum expectation value of
$i\bar\psi_u\gamma_5\psi_u+i\bar\psi_d\gamma_5\psi_d$,
and with vacuum states that can not be connected with the Aoki vacua by parity-flavor symmetry
transformations.
The outline of this paper is as follows. In Sec. II we derive the $p.d.f.$
of $i\bar\psi\gamma_5\tau_3\psi$ and $i\bar\psi\gamma_5\psi$ and analyze the
conditions to have an Aoki phase with spontaneous parity-flavor symmetry
breaking. Section III contains our derivation of the p.d.f. of the same
fermion bilinears of Sec. II but in presence of a twisted mass term in
the action. We discuss also in this Sec. how if the spectral
density $\rho_U(\lambda,\kappa)$ of the Hermitian Dirac-Wilson operator
in a fixed background gauge field $U$ is not symmetric in $\lambda$,
a violation of Hermiticity manifests in a negative vacuum expectation value
for the square of the Hermitian operator $i\bar\psi\gamma_5\psi$ in the
infinite lattice limit. In Sec. IV we discuss the other possible scenario
i.e., we assume a symmetric spectral density $\rho_U(\lambda,\kappa)$ and
then derive, assuming that the Aoki vacuum does exists, that new vacuum
states should appear. These new vacua, which are not
connected with the Aoki vacua by parity-flavor symmetry
transformations, are analyzed in Sec. V.
Section VI contains some numerical results obtained by diagonalizing
quenched configurations in $4^4, 6^4, 8^4$ lattices. The goal of this
analysis was to distinguish between the two possible scenarios. Section
VII contains our conclusions.
\section{The Probability Distribution Function in the Gibbs State}
The model we are interested in is QCD with two degenerate Wilson quarks.
The fermionic part of the Euclidean action is
\begin{equation}
S_F=\bar\psi W(\kappa) \psi
\label{acfer}
\end{equation}
\noindent
where $W(\kappa)$ is the Dirac-Wilson operator, $\kappa$ is the hopping
parameter, which is
related to the bare fermion mass $m_0$ by $\kappa = 1/(8+2m_0)$, and
flavor indices are implicit ($W(\kappa)$ is a two-block diagonal matrix).
The standard wisdom on the phase diagram of this model in the gauge
coupling $\beta, \kappa$ plane is the one shown in Fig. 1. The two different
regions observed in this phase diagram, A and B, can be characterized as
follows; in region A parity and flavor symmetries are realized in the vacuum,
which is supposed to be unique. The Gibbs state is then very simple in this
region, and continuum QCD should be obtained by taking the $g^2\rightarrow 0$,
$\kappa\rightarrow 1/8$ limit from within region A. We will call region A
the QCD region. In region B, on the contrary, parity and flavor symmetries are
spontaneously broken, there are many degenerate vacua connected by
parity-flavor transformations in this region, and the Gibbs state is
therefore made up from many equilibrium states. In what follows we will call
region B as the Aoki region.
\begin{figure}[h]
\resizebox{8 cm}{!}{\includegraphics{Fig1.eps}}
\caption{Aoki (B) and physical (A) region in the $(\beta,\kappa)$ plane. Adapted from \cite{ilg} with courtesy of the authors.}
\end{figure}
The order parameters to distinguish the Aoki region from the QCD region are
$i\bar\psi\gamma_5\tau_j\psi$ and $i\bar\psi\gamma_5\psi$, with $\tau_j$
the three Pauli matrices. For the more standard election, $j=3$, they
can be written in function of the up and down quark fields as follows
$$
i\bar\psi\gamma_5\tau_3\psi=i\bar\psi_u\gamma_5\psi_u-i\bar\psi_d\gamma_5\psi_d
$$
\begin{equation}
i\bar\psi\gamma_5\psi=i\bar\psi_u\gamma_5\psi_u+i\bar\psi_d\gamma_5\psi_d
\label{orpa}
\end{equation}
The Aoki phases are characterized by \cite{a2}
$$
\langle i\bar\psi\gamma_5\tau_3\psi \rangle \neq 0
$$
\begin{equation}
\langle i\bar\psi\gamma_5\psi \rangle = 0
\label{orpaao}
\end{equation}
The first of the two condensates breaks both parity and flavor symmetries.
The non-vanishing vacuum expectation value of this condensate signals the
spontaneous breaking of the $SU(2)$ flavor symmetry down to $U(1)$, with
two Goldstone pions. Notwithstanding parity is spontaneously broken if
the first equation in \eqref{orpaao} holds, the vacuum expectation value of the
flavor diagonal condensate $\langle i\bar\psi\gamma_5\psi\rangle$ vanishes because it
is also order parameter for a discrete symmetry, composition of parity and
discrete flavor rotations, which is assumed to be realized \cite{a2}.
Following the lines developed in \cite{pdc}, we wish to write the $p.d.f.$ of the two fermion bilinear order parameters \eqref{orpa}. The motivation to develop
this formalism was precisely the study of
the vacuum invariance (non-invariance), in quantum theories regularized on a
space time lattice, under symmetry transformations which, as chiral, flavor
or baryon symmetries, involve fermion fields. Notwithstanding that Grassmann
variables cannot be simulated in a
computer, it was shown in \cite{pdc} that an analysis of spontaneous
symmetry breaking based on the use of the $p.d.f.$ of fermion local
operators can also be done in $QFT$ with fermion degrees of freedom.
The starting point is to choose an order
parameter for the desired symmetry $O(\psi,\bar\psi)$ (typically a
fermion bilinear) and characterize each vacuum state $\alpha$ by the
expectation value $c_\alpha$ of the order parameter in the $\alpha$ state
\begin{equation}
c_\alpha= \frac{1}{V} \int \langle O(x)\rangle_\alpha d^4 x
\label{exval}
\end{equation}
The $p.d.f.$ $P(c)$ of the order parameter will be given by
\begin{equation}
P(c)=\sum_\alpha w_\alpha \> \delta (c-c_\alpha)
\label{pdcferm}
\end{equation}
\noindent
which can also be written as \cite{pdc}
\begin{equation}
P(c)=\left\langle \delta(\frac{1}{V} \int O(x) d^4x -c) \right\rangle
\label{pdcferm2}
\end{equation}
\noindent
the mean value computed with the integration measure of the path
integral formulation of the Quantum Theory.
The Fourier transform $P(q)=\int e^{iqc} P(c) dc$ can be written, for the
model we are interested in, as
$$
P(q) =
$$
\begin{equation}
{\frac{\int [dU] [d\bar\psi d\psi]
\exp \{-S_{YM} +\bar\psi W(\kappa)\psi+ {\frac{iq}{V}} \int d^4x
\>\> O(x)\}}{\int [dU] [d\bar\psi d\psi] \exp
\{-S_{YM} +\bar\psi W(\kappa)\psi \}} }
\label{pdiqferm}
\end{equation}
In the particular case in which $O$ is a fermion bilinear of $\bar\psi$
and $\psi$
\begin{equation}
O(x)=\bar\psi(x) \tilde O \psi(x)
\label{bili}
\end{equation}
\noindent
with $\tilde O$ any matrix with Dirac, color and flavor indices, equation
\reff{pdiqferm} becomes
\begin{equation}
P(q)= {\frac{\int [dU] [d\bar\psi d\psi]
\exp \{-S_{YM} +\bar\psi (W(\kappa)+{\frac{iq}{V}} \tilde O)\psi \}}{\int [dU] [d\bar\psi d\psi] \exp
\{-S_{YM} +\bar\psi W(\kappa)\psi \}} }
\label{pdiqbili}
\end{equation}
Integrating out the fermion fields in \reff{pdiqbili} one gets
\begin{equation}
P(q)= {\frac{\int [dU] e^{-S_{YM}} \det(W(\kappa)+{\frac{iq}{V}} \tilde O) }
{\int [dU] e^{-S_{YM}} \det W(\kappa) } }
\label{pdiqbi}
\end{equation}
\noindent
which can also be expressed as the following mean value
\begin{equation}
P(q)=\left\langle {\frac{\det (W(\kappa)+{\frac{iq}{V}} \tilde O)}{\det W(\kappa)}} \right\rangle
\label{pdiqb}
\end{equation}
\noindent
computed in the effective gauge theory with the integration measure
$$
[dU] e^{-S_{YM}} \det W(\kappa)
$$
\noindent
Notice that zero modes of the Dirac-Wilson operator, which would produce a
singularity of the operator in \reff{pdiqb}, are suppressed by the fermion
determinant in the integration measure (zero mode configurations have on the
other hand vanishing measure).
The particular form expected for the $p.d.f.$ $P(c)$, $P(q)$,
depends on the realization of the corresponding symmetry in the
vacuum. A symmetric vacuum will give
\begin{equation}
P(c)=\delta(c)
\label{delta}
\end{equation}
$$
P(q)=1
$$
\noindent
whereas, if we have for instance a $U(1)$ symmetry which is
spontaneously broken, the expected values for $P(c)$ and $P(q)$
are \cite{pdc}
$$
P(c)=[ \pi (c_0^2-c^2)^{1/2} ]^{-1} \qquad\qquad -c_0 < c < c_0
$$
$$
{\rm otherwise} \qquad P(c)=0
$$
\begin{equation}
P(q)={\frac{1}{2\pi}} \int_{-\pi}^\pi d\theta e^{iqc_0\cos\theta}
\label{pdfu1}
\end{equation}
\noindent
the last being the well known zeroth order Bessel function of the
first kind, $J_0(qc_0)$.
In the simpler case in which a discrete $Z(2)$ symmetry is
spontaneously broken, the expected form is
$$
P(c)= {\frac{1}{2}} \delta(c-c_0) + {\frac{1}{2}} \delta(c+c_0)
$$
\begin{equation}
P(q)=\cos(q c_0)
\label{pdfz2}
\end{equation}
\noindent
or a sum of symmetric delta functions ($P(c)$) and a sum of
cosines ($P(q)$) if there is an extra vacuum degeneracy.
If we call $P_j(q), P_0(q)$ the p.d.f. of $i\bar\psi\gamma_5\tau_j\psi$ and
$i\bar\psi\gamma_5\psi$ in momentum space, we have
$$
P_1(q) = P_2(q) = P_3(q) = \left\langle \prod_j \left ( - {\frac{q^2}{V^2
\lambda_j^2}} +1 \right ) \right \rangle
$$
\begin{equation}
P_0(q) = \left\langle \prod_j \left ( {\frac{q}{V \lambda_j}} +1 \right )^2
\right \rangle
\label{pdfmom}
\end{equation}
\noindent
where $V$ is the number of degrees of freedom (including color and Dirac but
not flavor d.o.f.) and $\lambda_j$ are the real eigenvalues of the
Hermitian Dirac-Wilson operator $\bar W(\kappa) = \gamma_5 W(\kappa)$. Notice
that whereas $P_3(q)$ has not a definite sign, $P_0(q)$ is always positive
definite.
The q-derivatives of $P(q)$ give us the moments of the distribution $P(c)$.
In particular we have
\begin{equation}
{ \frac{d^n{P(q)}}{dq^n}} \bigg\rvert_{q=0} = i^n \langle c^n \rangle
\label{qder}
\end{equation}
Since $c$ is order parameter for the symmetries of the lattice action
and we integrate over all the Gibbs state, the first moment will vanish
always, independently of the realization of the symmetries.
The first non-vanishing moment, if the symmetry is spontaneously broken,
will be the second. Thus, for the particular case
$$
c_0= {\frac{1}{V}} \sum_x i\bar\psi(x)\gamma_5 \psi(x)
$$
\begin{equation}
c_3= {\frac{1}{V}} \sum_x i\bar\psi(x)\gamma_5 \tau_3 \psi(x)
\label{c0c3}
\end{equation}
we get
$$
\left\langle c_0^2 \right\rangle = 2 \left\langle {\frac{1}{V^2}}
\sum_j {\frac{1}{\lambda_j^2}}\right \rangle - 4 \left\langle
\left ( {\frac{1}{V}} \sum_j {\frac{1}{\lambda_j}} \right )^2 \right\rangle
$$
\begin{equation}
\left\langle c_3^2 \right\rangle = 2 \left\langle {\frac{1}{V^2}}
\sum_j {\frac{1}{\lambda_j^2}}\right \rangle
\label{vasp}
\end{equation}
In the QCD region flavor symmetry is realized. The p.d.f. of $c_3$ will
be then $\delta(c_3)$ and $\langle c_3^2\rangle = 0$. We get then
$$
\left\langle c_0^2 \right\rangle = - 4 \left\langle
\left ( {\frac{1}{V}} \sum_j {\frac{1}{\lambda_j}} \right )^2 \right\rangle,
$$
\noindent
which should vanish since parity is also realized in this region. Furthermore
a negative value of $\langle c_0^2\rangle$ would violate Hermiticity of
$i\bar\psi\gamma_5\psi$. We will come back to this point in the next section.
In the Aoki region \cite{a2} there are vacuum states in which the condensate
$c_3$ \eqref{c0c3} takes a non-vanishing vacuum expectation value. This implies
that the p.d.f. $P(c_3)$ will not be $\delta(c_3)$ and therefore $\langle c_3^2\rangle$
\eqref{vasp} will not vanish. Indeed expression \eqref{vasp} for $\langle c_3^2\rangle$ seems to
be consistent with the Banks and Casher formula \cite{BC} which relates
the spectral density of the Hermitian Dirac-Wilson operator at the origin
with the vacuum expectation value of $c_3$ \cite{sharpe}.
If, on the other hand, $\langle i\bar\psi\gamma_5\psi\rangle=0$
in one of the Aoki vacua, as conjectured in
\cite{a2}, $\langle i\bar\psi\gamma_5\psi\rangle=0$
in all the other vacua which are connected with the standard
Aoki vacuum by a parity-flavor transformation, since
$i\bar\psi\gamma_5\psi$ is invariant under
flavor transformations and change sign under parity. Therefore if we assume
that these are all the degenerate vacua, we conclude that
$P(c_0)= \delta(c_0)$
and $\langle c_0^2\rangle=0$, which would imply the following non-trivial relation
\begin{equation}
\left\langle {\frac{1}{V^2}}
\sum_j {\frac{1}{\lambda_j^2}}\right \rangle = 2 \left\langle
\left ( \frac{1}{V} \sum_j {\frac{1}{\lambda_j}} \right )^2 \right\rangle
\neq 0
\label{rela}
\end{equation}
\section{QCD with a Twisted Mass Term: the Non Symmetric Case}
In this section we will consider lattice QCD with Wilson fermions with
the standard action of previous section, plus a source term
\begin{equation}
\sum_x i m_t \bar\psi(x)\gamma_5\tau_3\bar\psi(x)
\label{source}
\end{equation}
\noindent
that explicitly breaks flavor and parity. The flavor symmetry is thus
broken from $SU(2)$ to $U(1)$. This is the standard way to
analyze spontaneous symmetry breaking. First one takes the thermodynamic
limit and then the vanishing source term limit.
We can calculate again the p.d.f. $\bar P_0(q)$ and $\bar P_3(q)$ of
$i\bar\psi\gamma_5\psi$ and $i\bar\psi\gamma_5\tau_3\psi$
with this modified action. Simple algebra give us the following
expressions
$$
\bar P_0(q) = \left\langle \prod_j \left ( \frac{\frac{q^2}{V^2} + {\frac{2q}{V} \lambda_j}}{m_t^2+\lambda_j^2} + 1 \right ) \right \rangle
$$
\begin{equation}
\bar P_3(q) = \left\langle \prod_j \left ( {\frac{{\frac{q^2}{V^2}} + {\frac{2q}{V}} i m_t}{m_t^2+\lambda_j^2}} - 1 \right ) \right \rangle
\label{pdfsou}
\end{equation}
\noindent
where again $\lambda_j$ are the real eigenvalues of the Hermitian
Dirac-Wilson operator and the mean values are computed now with the
integration measure of the lattice QCD action modified with the
symmetry breaking source term.
By taking the q-derivatives at the origin of $\bar P_0(q)$ and $\bar P_3(q)$
we obtain
$$
\langle c_0 \rangle = {\frac{2 i}{V}} \left\langle \sum_j {\frac{\lambda_j}{m_t^2+\lambda_j^2}} \right\rangle
$$
$$
\langle c_0^2 \rangle = {\frac{4}{V^2}} \left\langle\sum_j {\frac{\lambda_j^2}{(m_t^2+\lambda_j^2)^2}}
\right\rangle -
{\frac{2}{V^2}} \left\langle \sum_j {\frac{1}{m_t^2+\lambda_j^2}}\right\rangle -
$$
\begin{equation}
-4 \left\langle \left ( \frac{1}{V} \sum_j {\frac{\lambda_j}{m_t^2+\lambda_j^2}} \right )^2 \right\rangle
\label{paraor}
\end{equation}
\noindent
and
\begin{equation}
\langle c_3 \rangle = {\frac{2}{V}} m \left\langle \sum_j {\frac{1}{m_t^2+\lambda_j^2}} \right\rangle
\label{baca}
\end{equation}
Equation \eqref{baca} is well known. If we take the infinite volume limit and then
the $m\rightarrow 0$ limit we get the Banks and Casher result
\begin{equation}
\langle c_3 \rangle = 2\pi \rho(0)
\label{dens0}
\end{equation}
\noindent
which relates a non-vanishing spectral mean density of the Hermitian Wilson
operator at the origin with the spontaneous breaking of parity and flavor
symmetries.
The first equation in \eqref{paraor} is actually unpleasant since it predicts an
imaginary number for the vacuum expectation value of a Hermitian operator.
However it is easy to see that $\langle c_0\rangle=0$ because it is order parameter for
a symmetry of the modified lattice action, the composition
of parity with discrete flavor rotations around the x or y axis.
Concerning the second equation in \eqref{paraor}, one can see that the first and
second contributions to $\langle c_0^2\rangle$ vanish in the infinite volume limit for
every non-vanishing value of $m$. The third contribution however, which is
negative, will vanish only if the spectral density of eigenvalues of the
Hermitian Wilson operator $\rho_U(\lambda)$ for any background gauge field
$U$ is an even function of $\lambda$. This is actually not true at finite
values of $V$, and some authors \cite{sharpe,heller} suggest that the
symmetry of the eigenvalues will be recovered not in the thermodynamic limit,
but only in the zero lattice spacing or continuum limit. If we take this last
statement as true, we should conclude:
i. The Aoki phase, which seems not to be connected with the critical
continuum limit point $(g^2=0, \kappa=1/8)$ \cite{ilg,ster} is unphysical
since the $\langle c_0^2\rangle$ would be negative in this phase and this result breaks
Hermiticity.
ii. In the standard QCD phase, where parity and flavor symmetries are
realized in the vacuum, we should have however negative values for the
vacuum expectation value of the square of the Hermitian operator
$i\bar\psi\gamma_5\psi$, except
very near to the continuum limit. Since this operator is related to the
$\eta$-meson, one can expect in such a case important finite lattice spacing
effects in the numerical determinations of the $\eta$-meson mass.
This is the first of the two possible scenarios mentioned in the first section
of this article. In the next section we will assume a symmetric spectral
density of eigenvalues of the
Hermitian Wilson operator $\rho_U(\lambda)$ for any background gauge field
$U$ in the thermodynamic limit, and will derive the second scenario.
\section{QCD with a Twisted Mass Term: Symmetric Spectral Density of
Eigenvalues}
In this section we will assume that the spectral
density of eigenvalues of the
Hermitian Wilson operator $\rho_U(\lambda)$ for any background gauge field
$U$ is an even function of $\lambda$. In such a case equation \eqref{paraor} will
give a vanishing value for $\langle c_0^2\rangle$ at any value of $m_t$
\begin{equation}
\langle c_0^2 \rangle = 0
\label{clust}
\end{equation}
\noindent
Therefore the $p.d.f.$ of $c_0$ is $\delta(c_0)$ and
\begin{equation}
\langle i \bar\psi\gamma_5\psi \rangle = 0
\label{inva}
\end{equation}
\noindent
for any value of $m_t$, and also in the $m_t\rightarrow 0$ limit. Thus
we can confirm that under the assumed condition,
$\langle i\bar\psi\gamma_5\psi\rangle=0$ in the Aoki vacuum selected by the external
source \eqref{source}, as stated in \cite{a2}; but since $i\bar\psi\gamma_5\psi$
is flavor invariant and change sign under parity, we can conclude that
$\langle i\bar\psi\gamma_5\psi\rangle=0$, not only in the vacuum selected by the
external source \eqref{source}, but also in all the Aoki vacua which can be obtained
from the previous one by parity-flavor transformations. In order to see
the fact that, if there is an Aoki phase with parity-flavor symmetry spontaneously
broken, the previous vacua are not all the possible vacua, we will assume
that is false and will get a contradiction.
If all the vacua are the one
selected by the twisted mass term and those obtained from it by
parity-flavor transformations, the spectral density of the Hermitian Wilson
operator will be always an even function of $\lambda$, since the eigenvalues
of this operator change sign under parity and are invariant under flavor
transformations. Then the symmetry of
$\rho_U(\lambda)$ will be realized also at $m_t=0$ in the
Gibbs state. Now let us come
back to expression \eqref{vasp} which give us the vacuum expectation values
of the square of $i\bar\psi\gamma_5\psi$ and $i\bar\psi\gamma_5\tau_3\psi$
as a function of the spectrum of the Hermitian Wilson operator, but
averaged over all the Gibbs state (without the external symmetry breaking
source \eqref{source}). By subtracting the two equations in \eqref{vasp} we get
\begin{equation}
\langle c_3^2 \rangle - \langle c_0^2 \rangle =
4 \left\langle \left ( \frac{1}{V} \sum_j {\frac{1}{\lambda_j}} \right )^2
\right \rangle
\label{diffe}
\end{equation}
This equation would \emph{naively} vanish, if the spectral density of eigenvalues of the Hermitian Wilson operator
were an even function of $\lambda$. Therefore we would reach the following conclusion for the Gibbs state
\begin{equation}
\langle c_3^2 \rangle = \langle c_0^2 \rangle
\label{gibsta}
\end{equation}
Nevertheless, S. Sharpe put into evidence in a private communication (developed deeply in \cite{sharpe2}) an aspect that we, somewhat, overlooked: A sub-leading contribution to the spectral density may affect \eqref{gibsta} in the Gibbs state ($\epsilon$-regime, in $\chi$PT terminology), in such a way that, not only $\langle c_0^2 \rangle$, but every even moment of $i\bar\psi\gamma_5\psi$ would vanish, restoring the standard Aoki picture. The thesis of Sharpe, although possible, would enforce an infinite series of \emph{sum rules} to be complied, similar to those found by Leutwyler and Smilga in the continuum \cite{Leut}. We agree that such a possibility is open, at
least from a purely mathematical point of view: In fact sub-leading
contributions to the spectral density may exist, and conspire to
enforce the vanishing of all the even moments of the $p.d.f.$ of
$i\bar\psi\gamma_5\psi$. However
we believe such possibility not to be very realistic, and indeed we have
physical arguments, which will be the basis for subsequent work on the
topic, suggesting that the Aoki scenario is incomplete. Therefore, we will reasonably assume in the following \eqref{gibsta} to be true, in the case of a symmetric $\rho_U(\lambda)$, assumption that leads us to the conclusion that the \emph{Chiral Perturbation Theory may be incomplete}, for the new vacua derived from \eqref{gibsta} are not predicted in $\chi$PT.
If as conjectured by Aoki and verified by numerical simulations, a
phase with a non-vanishing vacuum expectation value of
$i\bar\psi\gamma_5\tau_3\psi$ does exists,
the mean value in the Gibbs state $\langle c_3^2\rangle$ inside this phase will be non-zero.
Then equation \eqref{gibsta} tell us that also $\langle c_0^2\rangle$ will be non-zero inside
this phase. But since in the Aoki vacua $\langle c_0^2\rangle=0$, this is in contradiction
with the assumption that the Aoki vacua are all possible vacua.
This is the second possible scenario mentioned in the Introduction of this
article.
\section{The New Vacua}
To understand the physical properties of these new vacuum states we will
assume, inspired by the numerical results reported in the next section,
that the spectral density of eigenvalues $\rho_U(\lambda)$ is an even
function of $\lambda$ in the Gibbs state of the Aoki region ($m_t=0$). Then
equation \eqref{gibsta} holds (taking into account the aforementioned discussion raised by S. Sharpe), and hence the $p.d.f.$ of the flavor singlet
$i\bar\psi\gamma_5\psi$ order parameter can not
be $\delta(c_0)$ inside the Aoki phase, and therefore new vacuum states
characterized by a non-vanishing vacuum expectation value of
$i\bar\psi\gamma_5\psi$ should appear.
These new vacua can not be connected, by mean of parity-flavor
transformations, to the Aoki vacua, as previously discussed.
In order to better characterize these new vacua, we have added to the lattice
QCD action the source term
\begin{equation}
i m_t \bar\psi\gamma_5\tau_3\psi + i \theta\bar\psi\gamma_5\psi
\label{2sour}
\end{equation}
\noindent
which breaks more symmetries than \eqref{source}, but still preserves the $U(1)$
subgroup of the $SU(2)$ flavor. By computing again the first moment of the
p.d.f. of $i\bar\psi\gamma_5\psi$ and $i\bar\psi\gamma_5\tau_3\psi$ and
taking into account that the mean value of the first of these operators is
an odd function of $\theta$ whereas the second one is an even function of
$\theta$, we get
$$
\langle i\bar\psi\gamma_5\psi \rangle = -{\frac{2\theta}{V}} \left\langle
\sum_j {\frac{-\lambda_j^2+m_t^2-\theta^2} {(\lambda_j^2+m_t^2-\theta^2)^2
+4\theta^2\lambda_j^2}}\right\rangle
$$
\begin{equation}
\langle i\bar\psi\gamma_5\tau_3\psi \rangle = {\frac{2m_t}{V}} \left\langle
\sum_j {\frac{\lambda_j^2+m_t^2-\theta^2}{(\lambda_j^2+m_t^2-\theta^2)^2
+4\theta^2\lambda_j^2}}\right\rangle
\label{fimo}
\end{equation}
\noindent
where $\lambda_j$ are again the eigenvalues of the Hermitian Wilson operator
and the mean values are computed using the full integration measure of
lattice QCD with the extra external source \eqref{2sour}. This
integration measure is not positive definite due to the presence of the
$i\bar\psi\gamma_5\psi$ term in the action.
By choosing $\theta=rm_t$ in the action and taking the thermodynamic limit
we get for the two order parameters the following expressions
$$
\langle i\bar\psi\gamma_5\psi \rangle = \int {\frac{2rm_t\lambda^2-2rm_t^3
(1-r^2)}{\left( m_t^2(1-r^2)+\lambda^2\right)^2+4r^2m_t^2\lambda^2}}
\rho(\lambda) d\lambda
$$
\begin{equation}
\langle i\bar\psi\gamma_5\tau_3\psi \rangle = \int {\frac{2m_t^3(1-r^2)+
2m_t\lambda^2}{\left( m_t^2(1-r^2)+\lambda^2\right)^2
+4r^2m_t^2\lambda^2}}
\rho(\lambda) d\lambda
\label{inte}
\end{equation}
\noindent
where $\rho(\lambda)$ is the mean spectral density of the Hermitian
Wilson operator averaged with the full integration measure.
Taking now the $m_t\rightarrow 0$ limit i.e., approaching the vanishing
external
source \eqref{2sour} point in the $\theta, m_t$ plane on a line crossing the origin and
with slope r, we obtain
$$
\langle i\bar\psi\gamma_5\psi \rangle = 2\rho(0)\int_{-\infty}^{+\infty}
{\frac{rt^2-r(1-r^2)}{\left( 1-r^2+t^2\right)^2+4r^2t^2}}dt
$$
\begin{equation}
\langle i\bar\psi\gamma_5\tau_3\psi \rangle = 2\rho(0)\int_{-\infty}^{+\infty}
{\frac{1-r^2+t^2}{\left( 1-r^2+t^2\right)^2+4r^2t^2}}dt
\label{intem0}
\end{equation}
In the particular case of $r=0$ ($\theta=0$) we get the Banks and Casher
formula
\begin{equation}
\langle i\bar\psi\gamma_5\tau_3\psi \rangle = 2\pi\rho(0)
\label{baca2}
\end{equation}
We see how, if $\rho(0)$ does not vanish, we can get many vacua
characterized by a non-vanishing value of the two order parameters
$i\bar\psi\gamma_5\psi$ and $i\bar\psi\gamma_5\tau_3\psi$. We
should point out that the value of $\rho(0)$ could depend on the slope
$r$ of the straight line along which we approach the origin in the
$\theta, m$ plane, and therefore, even if results of numerical simulations
suggest that
$\rho(0)\ne 0$ when we approach the origin along the line of vanishing slope,
this does not guarantee that the same holds for other slopes. However the
discussion in the first half of this section tell us that if $\rho(0)\ne 0$
at $r=0$, $\rho(0)$ should be non-vanishing for other values of $r$.
\section{Quenched Numerical Simulations}
In order to distinguish what of the two possible scenarios derived in the
previous sections is realized, we have performed quenched simulations of
lattice QCD with Wilson fermions in $4^4, 6^4$ and $8^4$ lattices.
We have generated an ensemble of well uncorrelated configurations
for each volume and then a complete diagonalization of the Hermitian
Wilson matrix, for each configuration, gives us the respective eigenvalues.
We measured the volume dependence of the asymmetries in the eigenvalue
distribution of the Hermitian Wilson operator, both inside and outside
the Aoki phase.
We want to notice that because of kinematic reasons (properties of the Dirac
matrices), the trace of all odd p-powers of the Hermitian Wilson operator
$\bar W(\kappa) = \gamma_5 W(\kappa)$
vanish until $p=7$, this included. This means that the asymmetries in the
eigenvalue distribution of
$\bar W(\kappa)$ start to manifest with a non-vanishing
value of the ninth moment of the distribution. We have found that these
asymmetries, even if small, are clearly visible in the numerical simulations.
In Figs. from 2 to 6 we plot the quenched mean value
\begin{equation}
A(\beta,\kappa,m_t) = \left \langle \left ( \frac{1}{V} \sum_j { \frac{\lambda_j}{m_t^2+\lambda_j^2}} \right )^2 \right \rangle_Q
\label{qmv}
\end{equation}
multiplied by the volume for three different volumes, in order to see the scaling of the asymmetries in the eigenvalue distribution of $\bar W(\kappa)$. As previously discussed, $A(\beta,\kappa,m_t)$ give us a quantitative measure of these asymmetries. We have added an extra $V$ factor to make the plots for the three different volumes distinguishable. Since we found that the value of $A(\beta,\kappa,m_t)$ decreased as the volume increased, the plots of the larger volumes were negligible with respect to the plot of the smaller volume $4^4$. Multiplying by $V$ all the plots are of the same magnitude order.
The $m_t$
term in the denominator of \eqref{qmv} acts also as a regulator in the quenched
approximation, where configurations with zero or near-zero modes are not
suppressed by the fermion determinant. This is very likely the origin of the
large fluctuations observed in the numerical measurements of \eqref{qmv} near $m_t=0$
in the quenched case. That is why our plots are cut below $m_t=0.05$; in the physical phase, this cutoff is not really needed, but in the Aoki phase it is more likely to find zero modes which spoil the distribution.
Figs. 2 and 3 contain our numerical results in
$4^4, 6^4$ and $8^4$ lattices at $\beta=0.001, \kappa=0.17$ and
$\beta=5.0, \kappa=0.15$. These first two points are
outside the Aoki phase, the first one in the strong coupling region. The second one intends to be a point where typically QCD simulations are performed.
\begin{figure}[h]
\resizebox{8.3 cm}{!}{\includegraphics{Fig2.eps}}
\caption{Point outside of the Aoki phase ($\beta = 0.001$, $\kappa = 0.17$) and in the strong coupling regime. The superposition of plots clearly states that the asymmetry of the eigenvalue distribution decreases as $\frac{1}{V}$. Statistics: 240 configurations ($4^4$), 2998 conf. ($6^4$) and 806 conf. ($8^4$)}
\end{figure}
\begin{figure}[h]
\resizebox{8.3 cm}{!}{\includegraphics{Fig3.eps}}
\caption{Another point outside of the Aoki phase ($\beta = 5.0$, $\kappa = 0.15$) in a region in which QCD simulations are commonly performed. The conclusion is the same as in Fig. 2. Statistics: 400 conf. ($4^4$), 900 conf. ($6^4$) and 200 conf. ($8^4$)}
\end{figure}
Figs. 4, 5 and 6 represent our numerical results in
$4^4, 6^4$ and $8^4$ lattices at $\beta=0.001, \kappa=0.30$, $\beta=3.0, \kappa=0.30$ and
$\beta=4.0, \kappa=0.24$. These points are well inside the Aoki phase, and the structure of the distribution is different from the structure observed in the previous plots of the physical phase. Nevertheless, the qualitative behaviour as the volume increases is the same.
\begin{figure}[h]
\resizebox{8.3 cm}{!}{\includegraphics{Fig4.eps}}
\caption{Point inside the Aoki phase ($\beta = 0.001$, $\kappa = 0.30$) and in the strong coupling regime. Although there is no clear superposition of plots, it is evident that the asymmetry goes to zero as the volume increases. Statistics: 368 conf. ($4^4$), 1579 conf. ($6^4$) and 490 conf. ($8^4$)}
\end{figure}
\begin{figure}[h]
\resizebox{8.3 cm}{!}{\includegraphics{Fig5.eps}}
\caption{Point inside the Aoki phase ($\beta = 3.0$, $\kappa = 0.30$). The asymmetry disappears as the volume increases. Statistics: 400 conf. ($4^4$), 1174 conf. ($6^4$) and 107 conf. ($8^4$)}
\end{figure}
\begin{figure}[h]
\resizebox{8.3 cm}{!}{\includegraphics{Fig6.eps}}
\caption{Point inside the Aoki phase ($\beta = 4.0$, $\kappa = 0.24$). Same conclusions as in the other Aoki plots. Statistics: 398 conf. ($4^4$), 1539 conf. ($6^4$) and 247 conf. ($8^4$)}
\end{figure}
We observe large fluctuations in the plotted quantity
near $m_t=0$, specially inside the Aoki phase. However the behavior with the lattice volume may suggests a
vanishing value of $A(\beta,\kappa,m_t)$ in the infinite volume limit in both regions, inside and outside the Aoki phase.
If this is actually the case in the unquenched model, the second scenario
discussed in this article would eventually be realized.
\section{Conclusions}
We have analyzed the vacuum structure of lattice QCD with two degenerate
Wilson flavors at non-zero lattice spacing, with the help of the probability
distribution function of the parity-flavor fermion bilinear order parameters.
From this analysis two possible scenarios emerge.
In the first scenario we assume a spectral
density $\rho_U(\lambda,\kappa)$ of the Hermitian
Dirac-Wilson operator, in a fixed
background gauge field $U$, not symmetric in $\lambda$. This property is
realized at finite $V$ for the single gauge configurations, even if a
symmetric distribution of eigenvalues is recovered if we average over
parity conjugate configurations. We find that under
such an assumption, Hermiticity of
$i\bar\psi_u\gamma_5\psi_u+i\bar\psi_d\gamma_5\psi_d$ will
be violated at finite $\beta$. This lost of Hermiticity for the pseudoscalar
flavor singlet operator suggests that a reliable determination of the $\eta$
mass would require to be near enough the continuum limit where the symmetry
of the spectral density $\rho_U(\lambda,\kappa)$ should be recovered.
Furthermore assuming that the Aoki phase ends at finite $\beta$, the
violation of Hermiticity obtained in this scenario implies the lost
of any physical interpretation of this phase in terms of particle excitations.
In the second scenario a symmetric spectral
density $\rho_U(\lambda,\kappa)$ of the Hermitian Dirac-Wilson operator
in the infinite volume limit is assumed, and then we show that the
existence of the Aoki phase implies also the appearance of other phases, in
the same parameters region, which can be
characterized by a non-vanishing vacuum expectation value of
$i\bar\psi_u\gamma_5\psi_u+i\bar\psi_d\gamma_5\psi_d$,
and with vacuum states that can not be connected with the Aoki vacua by parity-flavor symmetry
transformations. These phases, however, are not related to those mentioned in \cite{GM}, for we keep the twisted mass parameter $m_5$ equal to zero, whereas the phases studied by G. M\"unster appear at large $m_5$.
Sharpe and Singleton \cite{sharpe,shasi} performed an analysis of lattice QCD with two flavors of Wilson
fermions near the continuum limit, by mean of the chiral effective Lagrangians. In their analysis, they
found essentially two possible realizations, depending on the sign of a coefficient $c_2$, which appears
in the potential energy, expanded up to second order in the quark mass term. If $c_2$ is positive, a
phase with spontaneous flavor symmetry breaking and an Aoki vacuum can be identified.
If, on the contrary, $c_2<0$, flavor symmetry is realized in the vacuum and a first order
transition should appear. Either case may be realized in different regions of parameter space.
Indeed numerical simulations with dynamical fermions, performed at small lattice spacing \cite{jansen},
give evidences of metastability that can be related with the existence of a first order phase
transition and hence $c_2<0$, while the Aoki phase found at smaller $\beta$ values \cite{ilg} supports
$c_2>0$. We want to emphasize that the new vacua we find are coexisting with the standard Aoki vacuum.
These new vacua do exist if, and only if, the Aoki vacuum exists. But these new vacua are not explained in any way in the chiral effective
Lagrangian approximation of Sharpe and Singleton. On the other hand, in order to recover the standard Aoki picture via $\chi$PT, an infinite set of sum rules for the eigenvalues of the Hermitian Wilson operator must be imposed. As we stated previously, this possibility does not appeal us, in the sense that it seems unphysical (we would call it a `mathematical' possibility). Nevertheless we have not definite proof of the new vacua.
From our position, these conclusions lead us to cast a doubt into the completeness of the Chiral Perturbation theory. In any case we believe that, in order to definitely clarify this issue,
a careful investigation of the spectral properties of Hermitian Dirac-Wilson
operator for actual gauge field configuration in the full unquenched
theory is mandatory.
In order to distinguish what of the two possible scenarios, as mentioned at the beginning
of this section, is realized, we performed quenched simulations of
lattice QCD with Wilson fermions in $4^4, 6^4$ and $8^4$ lattices, and
measured the volume dependence of the asymmetries in the eigenvalue
distribution of the Hermitian Wilson operator, both inside and outside
the Aoki phase. To measure this asymmetries we diagonalized exactly
the Hermitian Wilson operator for all the gauge configurations generated
with the quenched measure, and measured the quenched average
$A(\beta,\kappa,m)$ \eqref{qmv}. Due to the fact that configurations with zero or
near-zero modes are not suppressed by the fermion determinant in the quenched
case, this quantity fluctuates violently near $m=0$, specially for the
points in the Aoki phase. Notwithstanding that,
the observed behavior with the lattice volume seems to suggest a
vanishing value of $A(\beta,\kappa,m)$ in the infinite volume limit, both
inside and outside the Aoki phase.
If this were actually the case in the unquenched model, the second scenario
discussed in this article would be realized. However a verification of
these results in the unquenched case would be very relevant in order to
discard the first scenario.
\acknowledgments
It is a pleasure to thank Fabrizio Palumbo for useful discussions. We also thank Steve Sharpe for his sharp comments and remarks.
This work has been partially supported by an INFN-MEC collaboration, CICYT (grant FPA2006-02315)
and DGIID-DGA (grant2007-E24/2).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 461 |
package org.apache.camel.component.urlrewrite.http4;
import org.apache.camel.builder.RouteBuilder;
import org.apache.camel.component.urlrewrite.BaseUrlRewriteTest;
import org.apache.camel.component.urlrewrite.HttpUrlRewrite;
import org.apache.camel.impl.JndiRegistry;
import org.junit.Test;
/**
*
*/
public class Http4UrlRewriteLoadBalanceRoundRobinTest extends BaseUrlRewriteTest {
@Override
protected JndiRegistry createRegistry() throws Exception {
JndiRegistry jndi = super.createRegistry();
HttpUrlRewrite myRewrite = new HttpUrlRewrite();
myRewrite.setConfigFile("example/urlrewrite2.xml");
jndi.bind("myRewrite", myRewrite);
return jndi;
}
@Test
public void testHttpUriRewrite() throws Exception {
// we should round robin between app2 and app3
String out = template.requestBody("http4://localhost:{{port}}/myapp/products/1234", null, String.class);
assertEquals("http://localhost:" + getPort2() + "/myapp2/products/index.jsp?product_id=1234", out);
out = template.requestBody("http4://localhost:{{port}}/myapp/products/5678", null, String.class);
assertEquals("http://localhost:" + getPort2() + "/myapp3/products/index.jsp?product_id=5678", out);
out = template.requestBody("http4://localhost:{{port}}/myapp/products/3333", null, String.class);
assertEquals("http://localhost:" + getPort2() + "/myapp2/products/index.jsp?product_id=3333", out);
}
@Override
protected RouteBuilder createRouteBuilder() throws Exception {
return new RouteBuilder() {
@Override
public void configure() throws Exception {
from("jetty:http://localhost:{{port}}/myapp?matchOnUriPrefix=true")
.loadBalance().roundRobin()
.to("http4://localhost:{{port2}}/myapp2?bridgeEndpoint=true&throwExceptionOnFailure=false&urlRewrite=#myRewrite")
.to("http4://localhost:{{port2}}/myapp3?bridgeEndpoint=true&throwExceptionOnFailure=false&urlRewrite=#myRewrite");
from("jetty:http://localhost:{{port2}}/myapp2?matchOnUriPrefix=true")
.transform().simple("${header.CamelHttpUrl}?${header.CamelHttpQuery}");
from("jetty:http://localhost:{{port2}}/myapp3?matchOnUriPrefix=true")
.transform().simple("${header.CamelHttpUrl}?${header.CamelHttpQuery}");
}
};
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,299 |
Q: javascript: unintentional feedback loop I am trying to create an animation in javascript that is triggered by a mouseover event and then returns to the initial state on mouse out. When the user runs their cursor over an image on the page, another div with an initial height of 0px gradually rises in height to 50px over the bottom portion of the image.
The problem I am facing is that when they move the cursor from the image to the div which now covers the bottom portion of the image, it triggers the mouseout (as it is a separate element from the image) and then a new mouseover event in quick succession because the div disappears when it detects that the cursor is no longer over the image (meaning the div appears and disappears quickly, over and over again).
I am wondering how I would go about breaking such a loop so that the div does not disappear when the cursor runs over it from the image (i.e. prevent the onmouseout event from triggering unless the mouse moves to some other element that is not the newly created div).
Here's an image to hopefully better illustrate the problem:
http://i.imgur.com/qcE64.jpg
A: I think for this situation you'd want to use a wrapper div around both the image and the div you're animating. Attach the mouseover/mouseout events to the wrapper and they will trigger when you're expecting them to. Here's a jsfiddle example
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,535 |
\section{Introduction}
The magnetic susceptibility at $H=0$ of the two dimensional Ising model on a
square lattice was shown in 1999 by Nickel \cite{nickel1,nickel2}
to have the remarkable (and unexpected) property that as a function of a complex
temperature variable there is a dense set of singularities\footnote{The emergence of an
accumulation of singularities had already been seen on resummed series
expansions of {\em anisotropic} Ising models~\cite{ge}. Here we restrict
our study to the {\em isotropic} Ising model in a magnetic field.} at the locus of the
zeros of the $H=0$ partition function of the finite size lattice.
On the other hand in 2003 Fonseca and Zamolodchikov \cite{fz} presented a compelling
scenario, since supported by extensive numerical studies \cite{man1,man2},
for the behavior of the Ising model in a magnetic field in the scaling field theory limit
which assumes analyticity at the locus of singularities.
The compatibility of these two approaches is an open question which
needs to be understood.
In this paper we investigate this compatibility by means of studying
the dependence on the magnetic field of the temperature zeros
of the finite size partition
function and of the equimodular
curves of the corresponding transfer matrix. This will use and extend
the work of \cite{mccoy1}.
It would be highly desirable to treat these questions of analyticity by
rigorous mathematical methods but, somewhat surprisingly, we will see
that the needed tools do not seem to exist.
In section 2 we give a precise formulation of the problem. The
partition function zeros are studied in section 3 and the transfer
matrix eigenvalues in section 4. In section 5 we use
these studies to formulate an interpretation which reconciles the
singularities of Nickel with the analyticity of Fonseca and
Zamolodchikov. Our conclusions are summarized in section 6.
\section{Formulation}
The isotropic two dimensional Ising model on a square lattice in
the presence of a magnetic field is defined by the
interaction energy
\begin{equation}
{\mathcal E}=-\sum_{j,k}(E\sigma_{j,k}\sigma_{j+1,k}
+E\sigma_{j,k}\sigma_{j,k+1}+H\sigma_{j,k})
\label{interaction}
\end{equation}
where $\sigma_{j,k}=\pm 1$ is the spin at row $j$ and column $k$ and
the sum is over all spins in a
lattice of $L_v$ rows and
$L_h$ columns with either cylindrical or toroidal boundary
conditions or the boundary conditions of Brascamp-Kunz \cite{bk}
where on a finite cylinder (with periodic boundary conditions in the
$L_h$ direction) one end interacts with a fixed row of up spins and the
other end interacts with an alternating row of up and down spins
with $L_h$ is even.
The partition function on the $L_v\times L_h$ lattice at
temperature $T$ is defined as
\begin{equation}
Z_{L_v,L_h}=\sum_{\sigma=\pm 1} e^{-\beta {\mathcal E}}
\end{equation}
where $\beta=1/k_BT$ (with $k_B$ being Boltzmann's constant).
$Z_{L_v,L_h}$ is a polynomial in the variables
$u=e^{-2E/k_BT}~~{\rm and}~~x=e^{-2H/k_BT}$.
However, we note that for appropriate boundary conditions
including Brascamp-Kunz \cite{bk} and toriodal (but not cylindrical) the
dependence is only on $u^2$.
The thermodynamic limit is the limit where $L_v,L_h\rightarrow
\infty$ with $L_v/L_h$ fixed away from zero and infinity. The free energy is
defined in the thermodynamic limit as
\begin{equation}
-F/k_BT=\lim_{L_v,L_h\rightarrow \infty}\frac{1}{L_vL_h}\ln Z_{L_v,L_h}.
\end{equation}
At $H=0$ the free energy of the Ising model is \cite{ons}
\begin{equation}
\hspace{-.2in}-F/k_BT=\frac{1}{2}\ln (2s)+
\frac{1}{8\pi^2}\int_{-\pi}^{\pi}d\theta_1\int_{-\pi}^{\pi}d\theta_2
\ln(s+s^{-1}-\cos\theta_1-\cos \theta_2)
\label{free}
\end{equation}
where
\begin{equation}
s=\sinh (2E/k_BT)=\frac{u^{-1}-u}{2}.
\end{equation}
This integral has a singularity at a temperature $T_c$ such that $s_c=\pm
1$, where negative $s$ implies that $E$ is negative and hence that
the system is antiferromagnetic.
For $H=0$ the zeros of the partition function accumulate in the
thermodynamic limit
on the circle
\begin{equation}
|s|=1
\end{equation}
which in terms of the variable $u$ becomes the two
circles
\begin{equation}
u=\pm 1+2^{1/2}e^{i\theta}~~~{\rm with }~0\leq \theta< 2\pi
\end{equation}
and the ferromagnetic (antiferromagnetic) critical temperatures are
given by
\begin{equation}
u_c=\sqrt{2}-1~~{\rm ferromagnetic},~~~u_c=\sqrt{2}+1~~{\rm
antiferromagnetic}.
\end{equation}
For Brascamp-Kunz boundary conditions all the zeros of the
partition function for $H=0$ are
exactly on the unit circle at the positions
\begin{equation}
s+s^{-1}=\cos\frac{(2n-1)\pi}{L_h}+\cos\frac{m\pi}{L_v+1}
\label{bkzeros}
\end{equation}
with $1\leq n\leq L_h/2,~1\leq m\leq L_v,$ and $L_h$ even.
The magnetic susceptibility is given as the second derivative of
the free energy with respect to $H$ as
\begin{equation}
\chi=\frac{\partial M(H)}{\partial H}=k_BT\frac{\partial^2 \ln
Z}{\partial H^2}.
\end{equation}
In 1999/2000 Nickel \cite{nickel1,nickel2} discovered that in
the thermodynamic limit for
both $T<T_c$ and $T>T_c$ the susceptibility has an infinite number of
singularities on the circle $|s|=1$ at
\begin{equation}
s_j+s_j^{-1}=\cos (2\pi m/j)+\cos(2\pi n/j)
\label{nickel}
\end{equation}
where
\begin{equation}
0\leq m,n\leq j-1~{\rm with}~m=n=0~{\rm excluded}.
\end{equation}
Here $j$ is a positive integer which is odd for $T>T_c$ and the
singularity at $s_j$ is proportional to
\begin{equation}
\epsilon^{2j(j-1)-1}\ln \epsilon
\end{equation}
where $\epsilon=s-s_j$.
For $T<T_c$ the integer $j$ is even and the singularity at $s_j$ is
proportional to
\begin{equation}
\epsilon^{2j^2-3/2}.
\end{equation}
\section{Partition function zeros}
The partition function depends on the two variables $x$ and $u$ and in
principle should be considered as a polynomial in two
variables. However, here we will consider the dependence on $x$ and $u$
separately and not jointly.
\subsection{Dependence on $x$}
The earliest study of partition function zeros is for zeros in the
plane of $x=e^{-2H/k_BT}$ for fixed values of $u=e^{-2E/k_BT}$
where for ferromagnetic interactions $E>0$ and for free, toroidal or
cylindrical boundary conditions Lee and Yang \cite{ly} proved that the
zeros all lie on the unit circle $|x|=1$
\begin{equation}
Z_{L_v,L_h}(x)=x^{-N/2}\prod_{n=1}^N(x-e^{i\theta_n^{(N)}})
\end{equation}
where $N=L_vL_h$ and $\theta_n^{(N)}$ is real and satisfies
\begin{equation}
\theta^{(N)}_n=-\theta^{(N)}_{N-n}
\end{equation}
and we note that $Z_{L_v,L_h}(x)=Z_{L_v,L_h}(x^{-1})$.
For $T<T_c$, where $0\leq
u<{\sqrt 2}-1,$ the zeros lie on the entire circle $|x|=1$ and for
$T>T_c$, where ${\sqrt 2}-1<u\leq 1,$ the zeros lie on an arc
$x=e^{i\theta}$ where $0<\theta_{LY}\leq \theta \leq
2\pi-\theta_{LY}$.
There have been several numerical studies \cite{kg}-\cite{kim2} of
these zeros and these
studies are all consistent with the limiting statement that, numbering the
zeros as an increasing sequence $\theta^{(N)}_n$ for $1\leq n\leq N$
the limit
\begin{equation}
\lim_{N\rightarrow \infty}N(\theta^{(N)}_{n+1}-\theta^{(N)}_n)
\end{equation}
exists and is non zero. This allows us to define a density for
${\bar\theta^{(N)}_n}=(\theta_n^{(N)}+\theta^{(N)}_{n+1})/2$ as
\begin{equation}
D({\bar\theta_n})
=\lim_{N\rightarrow \infty}\frac{1}{N(\theta^{(N)}_{n+1}-\theta^{(N)}_n)}
\label{lydensity}
\end{equation}
and for $T>T_c$ this density diverges as $\theta\rightarrow
\theta_{LY}$ and $\theta \rightarrow 2\pi-\theta_{LY}$.
Unfortunately, there are no mathematical proofs for these empirical statements.
For example there is no proof that the density defined by
(\ref{lydensity}) exists and even if it does exist
the only thing we know about its properties are the values at
$\theta=0$ \cite{yang} and $\pi$ \cite{ly,mccoy3} where
for all $0\leq T<\infty$
\begin{equation}
D(\pi)=\left[
\frac{(1+u^2)^2}{1-u^2}(1+6u^2+u^4)^{-1/2}\right]^{1/4}
\end{equation}
and
\begin{equation}
\hspace{-.8in}D(0)=0~~ {\rm for}~~T>T_c,
~~~D(0)=\left[\frac{1+u^2}{(1-u^2)^2}(1-6u^2+u^4)^{1/2}\right]^{1/4}
{\rm for}~~T<T_c.
\end{equation}
It is very tempting to write the free energy as
an integral over the density $D(\theta)$ using
\begin{equation}
Z_{L_v,L_h}(x)
=x^{-N/2}\prod_{n=1}^N(x-e^{i\theta^{(N)}_n})
=x^{-N/2}\exp\sum_{n=1}^N\ln(x-e^{i\theta^{(N)}_n})
\end{equation}
so that
\begin{eqnarray}
\hspace{-.5in}
F/k_BT&=&-\lim_{L_vL_h\rightarrow \infty}\frac{1}{L_vL_h}\ln Z_{L_v,L_h}(x)
\nonumber\\
&=&\frac{1}{2}\ln x-\frac{1}{2\pi}\int_{\theta_{LY}}^{2\pi-\theta_{LY}}d\theta
D(\theta)\ln(x-e^{i\theta})
\end{eqnarray}
where $(2\pi)^{-1}\int_{\theta_{LY}}^{2\pi-\theta_{LY}}d\theta D(\theta)=1$.
This expression for the free energy is analytic for $|x|\neq 1$. Furthermore
it is universally assumed that on $|x|=1$ the only singularities are at
$x=e^{i\theta_{LY}},e^{i(2\pi -\theta_{LY})}$ for $T>T_c$ and at $x=1$
for $T<T_c$ \cite{langer} and the free energy can be analytically
continued through the arc of zeros on $|x|=1$. This is called the
``standard analyticity assumptions'' in \cite{fz}. However, there is
absolutely no proof of these assumptions of analyticity.
\subsection{Dependence on $u$ at $H=0~(x=1)$}
The dependence of the partition function on $u$ for arbitrary fixed $x$
is far more complicated than the dependence on $x$ for fixed $u$. In
particular the zeros in the $u$ plane will not in general lie on
curves but can fill up areas. The one exceptional case where the zeros
for the finite lattice do lie on curves is when for $H=0$ the lattice has
Brascamp-Kunz boundary conditions. We plot these zeros using
(\ref{bkzeros}) in Figure~\ref{fig1} for the $20\times 20$ lattice in both
the $s$ and the $u$ variable.
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(350,160)
\put(0,0){\epsfig{file=sq_BK_w20_roots.pdf,width=12cm,angle=0}}
\end{picture}
}
\end{center}
\caption{Zeros of the isotropic Ising
model partition function at $H=0~(x=1)$ with
Brascamp-Kunz boundary conditions for the $20\times 20$ lattice. The
full $s$ plane is plotted on the left. On the right the zeros are
plotted in the $u$ plane; the zeros are on the two circles
$u=\pm 1+2^{1/2}e^{i\theta}$ and only the first quadrant is shown.
}
\label{fig1}
\end{figure}
Unlike the case of the Lee-Yang zeros in the variable $x$ the zeros in
neither the $s$ nor the $u$ plane have the regular $1/N$ spacing
such that a limiting density defined
like (\ref{lydensity}) exists. Nevertheless Lu and Wu \cite{luwu}
write the free energy at $H=0$ in the form
\begin{equation}
-F/kT=\frac{1}{2}\ln(4s)+\int_0^{2\pi}d\alpha
g(\alpha)\ln(s-e^{i\alpha})
\end{equation}
where they ``define'' the density $g(\alpha)$
by saying that the
number of zeros in the interval
$[\alpha,\alpha+d\alpha]$
is $L_vL_hg(\alpha)d\alpha$
with
$\int_{0}^{2\pi}d\alpha g(\alpha)=1$.
This is, of course, a vague statement and is certainly not the same as
(\ref{lydensity}). Then from the two dimensional integral (\ref{free})
Lu and Wu
(and not from the formula for zeros) find
\begin{equation}
g(\alpha)=\frac{|\sin \alpha|}{\pi^2}K(\sin\alpha)
\label{lwdensity}
\end{equation}
where
\begin{equation}
K(k)=\int_0^{\pi/2}dt (1-k^2\sin^2t)^{-1/2}
\end{equation}
is the complete elliptic integral of the first kind.
We plot this density in Figure~\ref{fig2}.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.28]{LuWu_Density.pdf}
\end{center}
\vspace{-.1in}
\caption{The density $g(\alpha)$ of Lu and Wu \cite{luwu}.}
\label{fig2}
\end{figure}
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(320,440)
\put(0,0){\epsfig{file=sq_BK_density.pdf,width=11cm,angle=0}}
\end{picture}
}
\end{center}
\caption{Plots of the scale dependent density $g(\alpha;a)_N$
for the Brascamp-Kunz
zeros as a function of the angle $\alpha/\pi$ for the $20\times 20$
lattice on the left and the $100\times 100$ lattice on the right. In
the first row $a=1,$ in the second row $a=[L^{1/2}]$ and in the third
row $a=L=N^{1/2}$. This limiting density (\ref{lwdensity}) of
\cite{luwu} is shown in red. }
\label{fig3}
\end{figure}
\subsection{Definitions of the density of zeros}
In order to recover the result (\ref{lwdensity}) of \cite{luwu}
for $g(\alpha)$ from the partition function zeros of (\ref{bkzeros}) we need
to be more precise in the definition of density of zeros.
There are two slightly different ways to proceed. We can either divide
the circle $s=e^{i\alpha}$ into a set of intervals of equal size and count the
number of zeros in each interval or we can compute the size of an
interval needed to contain exactly a fixed number of zeros. We here
adopt the second method which generalizes (\ref{lydensity}) by defining
\begin{equation}
g(\alpha;a)=\lim_{N\rightarrow \infty}g(\alpha^{(N)}_j;a)_N
\end{equation}
where
\begin{equation}
g(\alpha^{(N)};a)_N=\frac{a}{N(\alpha^{(N)}_{j+a}-\alpha^{(N)}_j)}~~{\rm
with}
~~a=[cN^p].
\end{equation}
where $[x]$ denotes the integer part of $x$.
If $p=0$ and $c=1$ we recover the density definition
(\ref{lydensity}). If the limit exists for some $p_0<1$ it will
continue to exist for $p>p_0$. The quantity $p_0$ can be called the
scale for which the density exists.
We examine the existence of these limits for the Brascamp-Kunz zeros
on the $L\times L $ lattice where $N$ is proportional to $L^2$. In
Figure~\ref{fig3} we compare for the $20\times 20$ and $100\times
100$ lattices the scale dependent densities for $a=1,~a=[L^{1/2}]$ and
$a=L=N^{1/2}$. We see for $a=1$ and $a=[L^{1/2}]$ that the limit
does not appear to exist but the limit does seem to exist
for $a=L=N^{1/2}.$ Further
studies reveal that the limit does not exist for $0\leq p<1/2$
but does exist for $1/2<p<1$. However, we have no analytic proof
of these numerical observations.
\subsection{Dependence on u for $H>0$}
When $H>0$ the free energy is no longer invariant under
$E\rightarrow -E$ (ie. ferromagnetic $\rightarrow$ antiferromagnetic).
However, for Brascamp-Kunz boundary conditions the partition function
does remain symmetric under $u\rightarrow -u$ and hence
is a polynomial in $u^2$.
In addition, as the magnetic field $H$ increases the zeros in the $u^2$
plane move to infinity as $x=e^{-2H/k_BT}\rightarrow 0$ so instead of
$u^2$ we consider the rescaled variable
\begin{equation}
y=u^2x^{1/2}.
\end{equation}
We plot the zeros of the Ising partition function with Brascamp-Kunz
boundary conditions on the $22\times 22$ lattice for several values of
$x$\footnote{The partition function for a given value of $x$
is after multiplication by an appropriate constant a polynomial in $u$ with integer coefficients. The zeros of the partition
function can then be calculated numerically (to any desired accuracy) using
root finders such as {\tt MPSolve} \cite{Bini00} or {\tt Eigensolve} \cite{Fortune02}.} in Figure~\ref{fig4}.
These extend the earlier work of Matveev and
Shrock \cite{shrock1} on $7\times 8$ lattices with helical boundary
conditions and Kim \cite{kim} on $14\times 14$ lattices with
cylindrical boundary conditions.
\begin{figure}[h]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(320,440)
\put(0,0){\epsfig{file=sq_BK_w22_roots_field.pdf,width=11cm,angle=0}}
\end{picture}
}
\end{center}
\caption{Brascamp-Kunz zeros in the plane $y=u^2x^{1/2}$ on the
$22\times 22$ lattice for values
of $x=0.99,~0.90,~0.50,~0.10,~0.01,~0.0001$.}
\label{fig4}
\end{figure}
It is quite clear from these plots that as $H\rightarrow
\infty~(x\rightarrow 0)$ the zeros become symmetric under
$y\rightarrow -y$. This limiting case of the Ising model on
the isotropic square lattice is the hard
square system at fugacity
\begin{equation}
z=y^2
\end{equation}
which has been studied in \cite{mccoy1} for cylindrical boundary
conditions on the $40\times 40 $ lattice. We
plot these zeros in Figure~\ref{fig5} along with the similar plot
for hard hexagons on the $39\times 39$ lattice for comparison.
\clearpage
\begin{figure}[h!]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(360,180)
\put(0,0){\includegraphics[scale=0.3]{HS_CF_w40_roots.pdf}}
\put(180,0){\includegraphics[scale=0.30]{HH_CF_w39_roots.pdf}}
\end{picture}
}
\end{center}
\caption{Comparison in the complex fugacity plane $z$ of the
zeros of the partition function with
cylindrical boundary of hard squares on the $40\times 40$ lattice
to hard hexagons on the $39\times 39$ lattice taken from Figure 2 of
ref. \cite{mccoy1}.}
\label{fig5}
\end{figure}
It is strikingly obvious that as $H$ increases from zero that the
inner and outer loops in Figure~\ref{fig4} behave in drastically
different ways. The inner loop in Figure~\ref{fig4}
which separates the disordered from the ferromagnetic
ordered phase smoothly becomes the line $-1\leq z \leq z_d$ of hard squares
whereas the outer loop does not remain a curve and spreads out into a
two dimensional area. These two regions must be treated separately.
\subsection{The inner loop zeros}
To study the inner loop zeros in more detail we plot them on an
expanded scale in Figure~\ref{fig6} for a $22\times 22$ lattice.
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(300,450)
\put(0,0){\epsfig{file=sq_BK_w22_inner.pdf,width=11cm,angle=0}}
\end{picture}
}
\end{center}
\caption{Partition function zeros for the $22\times 22$ lattice with
Brascamp-Kunz boundary conditions on the inner loop in the plane
$y=u^2x^{1/2}$ for $x=1.0,~0.99,~0.98,~0.95.~0.90,~0.80$}
\label{fig6}
\end{figure}
These plots make it abundantly clear that there is a sharp change in
behavior which sets in as soon as $H$ is increased from zero and that
this transition has been completed for $x<0.95$. In the region
$0.95\leq x <1$ the deviations from a smooth curve become sufficiently
large that a one dimensional density formula becomes
inappropriate. Furthermore it is likely that the structure in this
region will change with increasing lattice size. However, for $x<0.95$
the locus of zeros has become quite smooth and we can consider a
density function
\begin{equation}
D(y_j)=\frac{1}{N|y_{j+1}-y_j|}
\label{bkdensity}
\end{equation}
where $y_j$ is the position of the $j^{th}$ zero as measured from the
endpoint on the right and $N$ is the number of zeros on the inner loop.
We plot this density in Figure~\ref{fig7} versus
the index $j$.
\begin{figure}[h]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(300,450)
\put(0,0){\epsfig{file=sq_BK_w22_inner_dens.pdf,width=11cm,angle=0}}
\end{picture}
}
\end{center}
\caption{The nearest neighbor density of zeros (\ref{bkdensity}) of
the $22\times 22$ lattice with Brascamp-Kunz boundary conditions
in the plane $y=u^2x^{1/2}$ for
$x=0.94,~0.90,~0.80,~0.50,~0.10,~0.01$ versus the
the index $j$ .}
\label{fig7}
\end{figure}
For $x>0.90$ it is clear from Figure~\ref{fig7} that the nearest neighbor
density is not smooth for $L=22$. This connects with the behavior
already seen for $H=0$. However, for $x\leq 0.8$ the nearest neighbor
density is very smooth except at the rightmost end and the
spacing of zeros behaves for large $N$ as $1/N$ which is what was
observed for hard squares and hexagons in \cite{mccoy1}.
Universality suggests that for sufficiently large $N$ the density at
the right-hand endpoint should diverge for all $x<1$. This is more or less seen
qualitatively in Figure~\ref{fig7} for $x<0.5$ and in the hard square limit
an exponent of $1/6$ was estimated in \cite{mccoy1} from the data of
the $40\times 40$ lattice. However, it is not possible to extract an
accurate exponent of divergence from the data shown in Figure~\ref{fig7}.
\subsection{Outer loop zeros}
The zeros on the outer loop behave very differently from the inner
loop zeros. Instead of the zeros of $H=0$ changing their spacing to
the density function (\ref{bkdensity}) the zeros have spread out into
an area which grows as $H$ increases. It may be conjectured that this
spreading into an area happens for the entire outer loop but for any
finite size lattice there will always be a region near the real axis
where this effect cannot be resolved.
\subsection{Toroidal and cylindrical boundary conditions}
In order to better understand the role on boundary conditions we plot
the zeros as a function of $H$ in the $y=ux^{1/4}$ plane for toroidal
boundary conditions on the $16\times 17$ lattice in Figure~\ref{fig8a} and for
cylindrical boundary conditions of the $20\times 20$ lattice in
Figure~\ref{fig8}.
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(260,400)
\put(0,0){\epsfig{file=sq_PB_w16_roots.pdf,width=9.5cm,angle=0}}
\end{picture}
}
\end{center}
\caption{The zeros in the plane of $y=ux^{1/4}$ for the $16\times 17$
lattice with toroidal boundary conditions for $x=1.0,~0.9,~0.5,~0.1,~0.01,~0.001.$}
\label{fig8a}
\end{figure}
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(280,500)
\put(0,0){\epsfig{file=sq_CF_w20_roots.pdf,width=9.5cm,angle=0}}
\end{picture}
}
\end{center}
\caption{The zeros in the $y=ux^{1/4}$ plane for the $20\times 20$
lattice with cylindrical boundary conditions for $x=1.0,~0.5,~0.1,
~0.01,~0.001,~0.0001,~0.00001,~0.000001.$}
\label{fig8}
\end{figure}
For cylindrical boundary conditions the exact partition function on
the finite lattice was computed in 1967 \cite{mccoy2}. In contrast
with Brascamp-Kunz boundary conditions the zeros are not symmetric
under $u\rightarrow -u$ and at $u=-1$ the $L\times L$ lattice has an
$L$ fold zero. The total number of zeros is $2L^2-L$.
As $H$ increases from $H=0$ the $L$ fold zero at $u=-1$ of the $L\times L$
lattice becomes $L$ zeros on the negative axis which for $L$ even
are in closely
spaced pairs. As $H$ is increased the pairs coalesce and become
complex conjugate pairs.
For sufficiently large $H$ they are all complex. However, the
imaginary part is sufficiently small that in the plots they appear
to be on the negative axis.
When $x$ is sufficiently small the three groups of $L$ zeros each tend to
infinity at angles $\pi,~\pm \pi/3$. This has previously been seen
in \cite{kim}. We
have no explanation for this phenomenon. The remaining $2L^2-L-3L$
zeros have a 4-fold symmetry (for L even) at $x\rightarrow 0$.
\section{Transfer matrix eigenvalues}
An alternative method to compute partition functions is to define a
(row to row) transfer matrix on the $L_v\times L_h$ lattice of
size $2^{L_h}\times 2^{L_h}$. We denote by
$T_C(L_h)$ the transfer matrix with periodic boundary conditions
in the $L_h$ direction and by
$T_F(L_h)$ the transfer matrix with free boundary conditions
in the $L_h$ direction.
In 1949 Kaufman \cite{kauf} computed all eigenvalues of $T_C(L_h)$ and
found that there are two sets
\begin{equation}
\lambda_{+}=\prod_{n=0}^{L_h-1}e^{\pm \gamma_{2n+1}}\hspace{.2in}
\lambda_{-}=\prod_{n=0}^{L_h-1}e^{\pm \gamma_{2n}}
\label{kauf1}
\end{equation}
with
\begin{equation}
e^{\pm\gamma_m}=s+s^{-1}-\cos \phi_m
\pm \left( (s+s^{-1}-\cos \phi_m)^2-1\right)^{1/2}
\label{kauf2}
\end{equation}
where $\phi_m=\pi m/L_h$
and there must be an even number of minus signs.
Each set of eigenvalues contains $2^{L_h-1}$ eigenvalues.
For all $\gamma_{m}$ for $m\neq 0$ the square roots are defined
as positive for $0<T<T_c~~(1<s<\infty)$.
For $|s|=1$ and all $\phi_m$ such that $(s+s^{-1}-\cos \phi_m)^2<1$
the modulus of
$e^{\pm \gamma_m}$ is unity and thus many eigenvalues on the circle
$|s|=1$ will have the same modulus.
For $\gamma_0$ a
factorization occurs under the square root and
\begin{equation}
e^{\gamma_0}=s+s^{-1}-1+(s-1)(s^{-2}+1)^{1/2}
\end{equation}
So $\gamma_0$ is positive for $s>1$ and
negative for $s<1$. For $T=T_c$ we have $s=1$ and $\gamma_0=0$.
There are four constructions of partition functions from these
transfer matrices.
\begin{itemize}
\item $L_v$ periodic, $L_h$ periodic
\begin{equation}
Z^{CC}_{L_v,L_h}={\rm Tr}T_C(L_h)^{L_v}=\sum_k\lambda^{L_v}_{C;k}(L_h),
\end{equation}
\item $L_v$ periodic, $L_h$ free
\begin{equation}
Z^{C,F}_{L_v,L_h}={\rm Tr}T_F(L_h)^{L_v}=\sum_k\lambda^{L_v}_{F;k}(L_h)
\end{equation}
\item $L_v$ free, $L_h$ periodic
\begin{equation}
Z^{FC}_{L_v,L_h}={\bf v}\cdot T_C^{L_v-1}(L_h){\bf v'}
=\sum_k{\bf v\cdot v_k}\lambda_{C;k}^{L_v-1}{\bf v_k\cdot v'}
\end{equation}
\item $L_v$ free, $L_h$ free
\begin{equation}
Z^{FF}_{L_v,L_h}={\bf v}\cdot T_F^{L_v-1}(L_h){\bf v'}
=\sum_k{\bf v\cdot v_k}\lambda_{F;k}^{L_v-1}{\bf v_k\cdot v'}
\end{equation}
\end{itemize}
where $\lambda_{C;k}$ and $\lambda_{F;k}$ are eigenvalues,
${\bf v}$ and ${\bf v'}$ are suitable boundary vectors and
${\bf v_k}$ are the eigenvectors.
It is obvious by symmetry that
$Z^{CF}_{L_h,L_v}=Z^{FC}_{L_v,L_h}$ and thus the explicit results of
1967 for $Z^{FC}_{L_v,L_h}$ must be obtainable from the eigenvalues
of $T_F(L_h)$ but the eigenvalues of $T_F(L_h)$ have never been
computed. Clearly something is missing.
\subsection{Equimodular curves}
The Ising model at $H=0$ and $H/k_bT=i\pi/2$ are the only models
where the finite size partition
function (at arbitrary size) has ever been computed from the transfer matrix eigenvalues.
For all other models when there is one eigenvalue $\lambda_{\rm max}$
that is dominant (i.e. of maximum modulus) on the finite lattice the free energy per site in
the thermodynamic limit is computed as
\begin{equation}
-F/kT=\lim_{L_h\rightarrow \infty}\lim_{L_v\rightarrow
\infty}\frac{1}{L_vL_h}\ln \lambda^{L_v}_{\rm max}(L_h).
\end{equation}
However an eigenvalue which is dominant in one portion of the
$u=e^{-2E/kT}$ plane will not, in general, be dominant in all parts of
the plane. The places where two or more eigenvalues have the same modulus form
equimodular curves and can separate the complex $u$ plane into
many distinct regions.
When there are only two equimodular eigenvalues $\lambda_1(L_h)$ and
$\lambda_2(L_h)$ on the equimodular curve and there are periodic
boundary conditions in the $L_v$ direction we can approximate the
partition function near the curve as
\begin{equation}
Z_{L_v,L_h} \sim \lambda_1(L_h)^{L_v}+\lambda_2(L_h)^{L_v}
\end{equation}
and thus for fixed $L_h$ as $L_v\rightarrow \infty$ there will
be a smooth distribution of zeros with a spacing of $1/L_v$ and a
density determined by the phase difference between the two
eigenvalues \cite{mccoy1}.
For free boundary conditions we have
\begin{equation}
Z_{L_v,L_h} \sim c_1\lambda_1(L_h)^{L_v}+c_2\lambda_2(L_h)^{L_v}
\end{equation}
where $c_j={\bf(v\cdot v_j)(v_j\cdot v')}$
When there are only two equimodular eigenvalues this relation for
zeros is sufficient for partition functions computed
by first taking $L_v\rightarrow\infty$ and then taking $L_h\rightarrow
\infty$ so that the aspect ratio $L_h/L_v$ vanishes.
For thermodynamics to be valid the free energy must be independent of
aspect ratio as long as $0<L_h/L_v<\infty$.
\subsection{Equimodular curves for $T_C(L_h)$ at $H=0$}
For the Ising model at $H=0$ the equimodular curves of the transfer
matrix $T_C(L_h)$ can be numerically computed from the eigenvalues
(\ref{kauf1}),(\ref{kauf2})
of Kaufman \cite{kauf} where we note that the corresponding momentum
is
\begin{equation}
P=\sum_{m}\phi_m~~({\rm mod}~2\pi).
\end{equation}
We plot these curves in the complex $u$ plane
in Figures~\ref{fig9} and \ref{fig10} for $L_h=8,10,12$.
\clearpage
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(360,180)
\put(0,0){\epsfig{file=equmod_L8.pdf,width=6cm,angle=0}}
\put(180,0){\epsfig{file=equmod_P0_L8.pdf,width=6.2cm,angle=0}}
\end{picture}
}
\end{center}
\caption{The equimodular curves in the $u$ plane for $T_C(L_h)$
for $L_h=8$. On the
left all eigenvalues are considered and on the right the restriction
to the momentum sector $P=0$ is made. The sectors where $\lambda_{+}$ is
dominant is marked by $+$ and the sector where $\lambda_{-}$ is
dominant is marked by a circle. The multiplicity of the crossings on
the curves are indicated by colors. On left panel:red=2,
green=3, black=4, blue=8, yellow=16, purple=32, brown=64
On right panel: red=2, green=4, blue=8,
brown =3, black=9.}
\label{fig9}
\end{figure}
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(360,180)
\put(0,0){\epsfig{file=equmod_P0_L10.pdf,width=6cm,angle=0}}
\put(180,0){\epsfig{file=equmod_P0_L12.pdf,width=6cm,angle=0}}
\end{picture}
}
\end{center}
\caption{The equimodular curves in the $u$ plane for $T_C(L_h)$
at $P=0$ for $L_h=10$ on
the left and 12 on the right. Red indicates a multiplicity of 2,
green of 4 and blue of 8.
For $L=10$ the sequence of multiplicities on the upper
(antiferromagnetic) sequence (increasing towards $u=i$) is 2,4,8,4,18,24
and the lower (ferromagnetic) sequence 2,2,4,4,8,8,18,28.
For $L=12$ the upper sequence 2,4,8,2,18,18,52,84 and the
lower sequence is 2,2,4,4,8,8,18,26,52,88}
\label{fig10}
\end{figure}
\clearpage
These curves have the following striking properties:
\begin{enumerate}
\item All eigenvalues are equimodular at $u=\pm i$.
\item The equimodular curves in the $u$ plane of the eigenvalues
$\lambda_{+}$ and the eigenvalues $\lambda_{-}$ are segments of the
two circles $u=\pm 1+2^{1/2}e^{i\theta}$
which is the curve on which there are Brascamp-Kunz zeros.
\item On most of the segments of this curve there are more than two
equimodular eigenvalues.
\item The equimodular curves formed by one eigenvalue $\lambda_{+}$ and
one $\lambda_{-}$ do not lie on the curve of Brascamp-Kunz zeros.
\end{enumerate}
The multiple degeneracies on the equimodular curves destroy the
mechanism for a smooth density of zeros of the $L_v=L_h=L$ lattice
with a $1/L^2$ spacing. The mechanism which changes the scale of
smooth zeros from $1/L^2$ to $1/L$ seen in section 3.3 is not understood.
\subsection{$u$ plane eigenvalues for $x=0.99$}
When $H$ is increased from $H=0$ the transfer matrix eigenvalues have been
computed numerically. In Figure~\ref{fig11} we plot the equimodular
curves for all eigenvalues for $x=0.99$.
(We note that the curves extending from the upper
branch to infinity are also present for $H=0$ but are not seen in
Figure~\ref{fig9} because in that figure the imaginary part of $u$ is
restricted to $0\leq \mathrm{Im}(u)\leq 1$.)
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(400,120)
\put(0,125){\epsfig{file=6bis-no-titles.pdf,width=4.5cm,angle=-90}}
\put(200,125){\epsfig{file=8bis-no-titles.pdf,width=4.5cm,angle=-90}}
\end{picture}
}
\end{center}
\caption{Equimodular curves in the $u$ plane for $x=0.99$ of
$T_C(L_h)$ for $L_h=6$ on the left and $L_h=8$ on the right.
Red is for singlet-singlet crossings,
green is for singlet-doublet and
blue is for doublet-doublet}
\label{fig11}
\end{figure}
By comparing Figure~\ref{fig11} with Figures~\ref{fig9} and \ref{fig10}
we see that several dramatic phenomena occur for $H>0$.
\begin{enumerate}
\item For $H>0$ the rays to the imaginary axis very rapidly retreat into
the curve of the Brascamp-Kunz zeros. This is caused by the
lifting of the near degeneracy of
eigenvalues in the $\lambda_+$ and $\lambda_{-}$ subspaces of
$H=0$. The larger $L_h$ the more rapid the retreat.
\item The rays to infinity separate regions of $P=0$ and $P=\pi$ and are
virtually unchanged for $H>0$.
\item The multiple degeneracies disappear. For momenta $P=0,\pi$
the eigenvalues are singlets for $P\neq 0,~\pi$ the momenta $\pm P$
are doubly degenerate. In Figure~\ref{fig11} all singlet-doublet
and doublet-doublet curves enclose regions
where the dominant eigenvalue has $P\neq 0,\pi$ but for $x=0.99$ some of the
regions are too small to be observed as areas.
\end{enumerate}
In Figure~\ref{fig12} we plot for $L_h=8$ the region near $u=i$ in more detail.
Thus far eigenvalues for $L_h\geq 10$ have not been computed for the
case $H\neq 0$.
\begin{figure}[ht]
\begin{center}
\hspace{0cm} \mbox{
\begin{picture}(200,180)
\put(0,180){\epsfig{file=6bis_zoom.pdf,width=6.5cm,angle=-90}}
\end{picture}
}
\end{center}
\caption{Equimodular curves in the $u$ plane for $x=0.99$ expanded near
$u=i$ for $T_c(L_h)$ with $L_h=8$. Red is for singlet-singlet crossings,
green is for singlet-doublet and blue is for doublet-doublet}
\label{fig12}
\end{figure}
\section{An interpretation}
It is very clear, both from the behavior of the partition function
zeros and the degeneracy of the equimodular curves, that there is a
drastic qualitative difference between $H=0$ and $H\neq 0$. We
conjecture here an interpretation of the
singularities (\ref{nickel}) found by
Nickel \cite{nickel1,nickel2} based on
this behavior. The argument is substantially different for the inner
(ferromagnetic) and outer (antiferromagnetic) loops in the $u$ plane.
Naturally conjectures concerning analyticity based solely on finite
size computations can only be suggestive.
\subsection{Scenario on the ferromagnetic loop}
We conjecture that on the ferromagnetic loop for $H>0$ the zeros
approach a curve as $L_hL_v=N\rightarrow \infty$ and that for
sufficiently large $N$ and fixed $H\neq 0$ the limit
\begin{equation}
\lim_{N\rightarrow \infty}N(u_{j+1}-u_j)<\infty
\end{equation}
exists. However, this cannot be uniform in $H$ and thus the limits
$H\rightarrow 0$ and $N\rightarrow \infty$ will not commute.
For both $H=0$ and $H\neq 0$ the free energy is analytic at the locus
of zeros. However, for $H\neq 0$ the analytic continuation beyond the
zero locus encounters many singularities which accumulate in the limit
$H\rightarrow 0$ to the singularities of Nickel (\ref{nickel}). The
location (and nature) of these singularities is different if the
continuation is from the interior (low temperature) or exterior (high
temperature) of the loop. The amplitude of the singularities vanishes
as $H^2$ at $H\rightarrow 0$ and hence the analyticity of the free
energy at $H=0$ is maintained.
In this scenario the singularities in the susceptibility at $|s|=1$ occur
because taking two derivatives with respect to $H$ kills the $H^2$
in the amplitude of the singularities but does not move the locations.
It can be argued that the non-integrability of the Ising model at
$H\neq 0$ is caused by these singularities in the analytic
continuation beyond the locus of zeros. Nevertheless, there are no
singularities on the locus of zeros except at the endpoints. The singularity
at the endpoint is expected \cite{mccoy1} to have the
same behavior as the endpoint
behavior of hard squares, hard hexagons and the Lee-Yang edge.
We may now make contact with the scenario of Fonseca and
Zamolodchikov \cite{fz} who assume that in the field theory limit the
free energy may be continued far beyond the locus of zeros.
The field theory limit is defined by $T\rightarrow T_c$ and
$H\rightarrow 0$ such that
\begin{equation}
\tau=(T-T_c)H^{-8/15}
\end{equation}
is fixed of order one. In terms of this scaled variable Fonseca and
Zamolodchikov posit that there is analyticity across the locus of
zeros and that there is an extensive region of analyticity in the
analytically continued free energy which sees none of the
singularities which, in this interpretation, produce the singularities
of Nickel. The analyticity of \cite{fz} will be consistent with our scenario
if the singularities which approach the point
$u={\sqrt 2}-1$ as $H\rightarrow 0$ is slower than the scaling $H^{8/15}$.
If this is indeed the case then there is no contradiction between the
field theory computations of \cite{fz} and the singularities
of \cite{nickel1,nickel2}.
\subsection{Scenario on the antiferromagnetic loop}
The behavior on the antiferromagnetic loop is quite different from
the behavior on the ferromagnetic loop because
now the zeros spread out into areas for $H\neq 0$. Moreover the pinching of the
zeros at the antiferromagnetic singularity at $u={\sqrt 2}+1$ remains
a pinch for all values of $x$ and furthermore the singularity in the
free energy in the hard square limit is numerically estimated from
high density series expansions
\cite{baxter2,kb} to be the same as the logarithmic singularity at
$T_c$ of the antiferromagnetic Ising model at $H=0$.
The zeros in
Figures~\ref{fig4} and \ref{fig5} do appear to be smoothly spaced in a two
dimensional region so from this point of view the distribution of zeros
which for $H=0$ was studied in section 3.3 has moved smoothly from
the circle to an area in the plane. There is, unfortunately, not sufficient
data to conjecture the behavior where the zeros in the $N\rightarrow
\infty$ limit pinch the positive $u$ axis. Even in the hard square
limit it cannot be concluded from Figure~\ref{fig5} if the zeros pinch
as a curve, as a cusp with an opening angle of zero or as a wedge with
a nonzero opening angle. The field theory argument of
\cite{fz} does not extend to the hard square limit and it is not
obvious how to consider analytic continuation into an area of zeros.
The second feature which needs an explanation is the
approach of the zeros to the hard square limit
in both Figure~\ref{fig4} for Brascamp-Kunz
boundary condition in the $y=u^2x^{1/2}$ plane and in
Figures~\ref{fig8} and \ref{fig8a}
for cylindrical and toroidal boundary conditions in the $y=ux^{1/4}$
plane. Namely
the emergence of the 2 fold symmetry for Brascamp-Kunz and the 4 fold
symmetry for cylindrical and toroidal boundary conditions. For all boundary
conditions new points of singularity are created in the complex $y$
plane as $H$ is increased, which in the hard square limit become identical
with the singularity on the positive $y$ axis. The mechanism for
the creation of these new points of singularity is completely unknown.
\subsection{The bifurcation points}
However, perhaps the most striking feature of the zeros is the existence
of the special points where the
one dimensional locus bifurcates into the two dimensional area.
It is the existence of these points which allows us to use the terms
ferromagnetic and antiferromagnetic branch.
At $H=0$ these points are at $u=\pm i$ where all eigenvalues are
equimodular and the free energy is singular \cite{shrock1}. In the hard square
limit this point is at $z=-1$ where
all eigenvalues are also equimodular \cite{fen}. It is natural to
conjecture that for all values of $H$ the free energy fails to be
analytic at these points.
\section{Conclusion}
In this paper we have presented the results of extensive numerical
computations of the zeros of the partition function of the Ising model
in a magnetic field $H$ and a companion study of the dominant
eigenvalues of the transfer matrix as
$H$ goes from $H=0$ to the hard square limit $H\rightarrow \infty$.
This reveals that in the ferromagnetic region the distribution of
zeros changes radically when $H$ is infinitesimally increased from $H=0$
and this feature is used to give an interpretation of the natural
boundary in the magnetic susceptibility conjectured by Nickel
\cite{nickel1,nickel2} which is consistent with the analyticity of
the scaling limit assumed by Fonseca and Zamolodchikov
\cite{fz}. However, an analytic argument for this scenario remains to
be found and further data is needed in order to reliably
understand the approach to the hard square limit.
\section*{Acknowledgments}
We are pleased to thank for their hospitality the organizers of
the conference ``Exactly solved models and beyond'' held at
Palm Cove, Australia July 19-25, 2015 in honor of the 75th birthday
of Prof. Rodney Baxter where much of this
material was first presented. One of us (JLJ) was supported by the
Agence Nationale de la Recherche (grant ANR-10-BLAN-0414), the
Institut Univeritaire de France, and the European Research Council
(advanced grant NuQFT). Two of us (MA and IJ) were supported by
funding under the Australian Research Council's Discovery Projects
scheme by the grant DP140101110. The work of IJ was also supported by
an award under the Merit Allocation Scheme of the NCI National Facility.
\newpage
\section*{ References}
\vspace{.1in}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,897 |
The Village of Lumby and surrounding area is policed by a five member RCMP Detachment.
Lumby Detachment is located at 2208 Shuswap Avenue at the corner of Shuswap Avenue and Glencaird Street.
Consisting of 24-30 volunteer members the Lumby & District Volunteer Fire Department serves the community under the direction of the Regional District of North Okanagan.
The BC Ambulance Service services the Village, with a service directly out of Lumby.
Learn more about Emergency Management BC click here.
Prepared BC your one-stop shop for disaster readiness information. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,475 |
Navajo Nation sues U.S. government for $160 million over toxic waste damage to their community
Friday, December 09, 2016 by: Vicki Batts
Tags: EPA, Gold King Mine Spill, Heavy metals, Navajo Nation
(Natural News) The Navajo Nation recently filed suit against the United States government – to the tune of nearly $160 million – for damages and ongoing injuries resulting from a mine spill and the toxic waste that was released into the environment near the tribe's territory.
The Environmental Protection Agency (EPA) has taken responsibility for the devastating spill, and the agency is also the lawsuit's primary target. The lawsuit is seeking a cool $159 million in damages, as well as an additional $3.2 million to cover expenses that have already been submitted to the EPA but have not yet been reimbursed.
The catastrophic spill took place in August 2015. An EPA clean-up crew was tasked with pumping out and decontaminating sludge from the mine, but things took a turn for the worse when the team destabilized a dam of loose rocks. Consequently, this led to 3 million gallons of mine waste water and tailings – containing heavy metals like cadmium and lead, and other toxic elements like arsenic and beryllium – into Cement Creek, a tributary of the Animas River.
Naturally, the EPA has maintained that this incident was just an accident. However, some have proposed that this was no accident. It was predicted that perhaps the agency would intentionally sabotage the mine as a means to secure Superfund money. And sure enough, just over a year after the accident, the Gold King Mine was declared a Superfund site. On September 7, 2016, the EPA declared that the area would be on its list of contaminated areas due for federally funded clean-up.
Why weren't they more careful the first time around? Who knows. NBC News even reported that the federal agency was well aware of the risks involved with the Gold King Mine. Their own internal documents have shown that they knew there was a serious potential for a disastrous "blowout" at an abandoned mine that could release "large volumes" of wastewater laced with heavy metals.
NBC News quotes the report as stating, "In addition, other collapses within the workings may have occurred creating additional water impounding conditions. Conditions may exist that could result in a blowout of the blockages and cause a release of large volumes of contaminated mine waters and sediment from inside the mine, which contain concentrated heavy metals." A subsequent report from May of 2015 referenced similar concerns.
The catastrophe at the mine contaminated rivers in Colorado, New Mexico and Utah. Just days after the spill, NBC News reported that the EPA was claiming that the contaminant levels in the water had already returned to pre-spill levels, but experts warned that the toxic heavy metals had likely just sunk down into the sediment – just waiting to be stirred back up some day. You would think that the expert folks who work at the EPA would have reached the same conclusion, but then again, it is a federal agency.
The Navajo Nation's lawsuit claims that the Gold King Mine spill has negatively impacted communities along the San Juan River in the tribe's territory. In a press release, the Navajo Nation Attorney General Ethel Branch stated that the spill converted the river from a "life-giver and protector," into a "threat" to the Navajo people, crops and animals.
The request for additional damages will cover long-term ecological and groundwater monitoring, assessments for livestock and agriculture, an on-site laboratory, additional water treatments, alternative water supply reservoirs, cultural preservation and the development of a plan to assess the damages to natural resources.
A letter signed by both the attorney general and attorney John C. Hueston, makes note of all the additional expenditures the Navajo Nation will have to make in order to keep their people safe – and it also accuses the EPA of failing to notify the tribe of the spill for "nearly two days." The letter also accuses the agency of ignoring the potential dangers, stating that the agency had "insufficient emergency protocols in place," and was "entirely unprepared to deal with the colossal damage it had unleashed."
One can only imagine the hurt and anger the people of the Navajo Nation must feel. After all, the EPA has admitted that they knew there were serious risks involved with the Gold King Mine, and yet, the agency proved to be unable to deal with said consequences, or even take the proper precautions to prevent such a tragedy.
TheDailySheeple.com
TheHill.com
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https://www.naturalnews.com/2016-12-09-160-million-lawsuit-against-us-governement-filed-by-navajo-nation-for-toxic-waste-damages-to-their-community.html
<a href="https://www.naturalnews.com/2016-12-09-160-million-lawsuit-against-us-governement-filed-by-navajo-nation-for-toxic-waste-damages-to-their-community.html">Navajo Nation sues U.S. government for $160 million over toxic waste damage to their community</a> | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,777 |
La vieille brasserie (en hongrois : régi sörgyár) est un monument industriel situé dans le de Budapest.
Monument historique dans le 10e arrondissement de Budapest
Architecture industrielle
Édifice construit en 1912 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,149 |
package org.apache.ignite.examples.ml.preprocessing.encoding;
import java.io.FileNotFoundException;
import org.apache.ignite.Ignite;
import org.apache.ignite.IgniteCache;
import org.apache.ignite.Ignition;
import org.apache.ignite.ml.dataset.feature.extractor.Vectorizer;
import org.apache.ignite.ml.dataset.feature.extractor.impl.ObjectArrayVectorizer;
import org.apache.ignite.ml.preprocessing.Preprocessor;
import org.apache.ignite.ml.preprocessing.encoding.EncoderTrainer;
import org.apache.ignite.ml.preprocessing.encoding.EncoderType;
import org.apache.ignite.ml.selection.scoring.evaluator.Evaluator;
import org.apache.ignite.ml.selection.scoring.metric.classification.Accuracy;
import org.apache.ignite.ml.tree.DecisionTreeClassificationTrainer;
import org.apache.ignite.ml.tree.DecisionTreeNode;
import org.apache.ignite.ml.util.MLSandboxDatasets;
import org.apache.ignite.ml.util.SandboxMLCache;
/**
* Example that shows how to use Label Encoder preprocessor to encode labels presented as a strings.
* <p>
* Code in this example launches Ignite grid and fills the cache with test data (based on mushrooms dataset).</p>
* <p>
* After that it defines preprocessors that extract features from an upstream data and encode string values (categories)
* to double values in specified range.</p>
* <p>
* Then, it trains the model based on the processed data using decision tree classification.</p>
* <p>
* Finally, this example uses {@link Evaluator} functionality to compute metrics from predictions.</p>
*/
public class LabelEncoderExample {
/**
* Run example.
*/
public static void main(String[] args) {
System.out.println();
System.out.println(">>> Train Decision Tree model on mushrooms.csv dataset.");
try (Ignite ignite = Ignition.start("examples/config/example-ignite.xml")) {
try {
IgniteCache<Integer, Object[]> dataCache = new SandboxMLCache(ignite)
.fillObjectCacheWithCategoricalData(MLSandboxDatasets.MUSHROOMS);
final Vectorizer<Integer, Object[], Integer, Object> vectorizer = new ObjectArrayVectorizer<Integer>(1, 2).labeled(0);
Preprocessor<Integer, Object[]> strEncoderPreprocessor = new EncoderTrainer<Integer, Object[]>()
.withEncoderType(EncoderType.STRING_ENCODER)
.withEncodedFeature(0)
.withEncodedFeature(1)
.fit(ignite,
dataCache,
vectorizer
);
Preprocessor<Integer, Object[]> lbEncoderPreprocessor = new EncoderTrainer<Integer, Object[]>()
.withEncoderType(EncoderType.LABEL_ENCODER)
.fit(ignite,
dataCache,
strEncoderPreprocessor
);
DecisionTreeClassificationTrainer trainer = new DecisionTreeClassificationTrainer(5, 0);
// Train decision tree model.
DecisionTreeNode mdl = trainer.fit(
ignite,
dataCache,
lbEncoderPreprocessor
);
System.out.println("\n>>> Trained model: " + mdl);
double accuracy = Evaluator.evaluate(
dataCache,
mdl,
lbEncoderPreprocessor,
new Accuracy()
);
System.out.println("\n>>> Accuracy " + accuracy);
System.out.println("\n>>> Test Error " + (1 - accuracy));
System.out.println(">>> Train Decision Tree model on mushrooms.csv dataset.");
}
catch (FileNotFoundException e) {
e.printStackTrace();
}
}
finally {
System.out.flush();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,630 |
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}
\markboth{}{Introduction}
The Gell-Mann--Okubo (GMO) mass relations following from $\text{SU}(3)$-flavor symmetry breaking in the strong sector have been proven to be of great success for describing hadron masses and classifying them into multiplets. The GMO mass relations of the decuplet, for instance, allowed Gell-Mann to predict the mass of the $\Omega^-$-particle, before it was discovered (cf. \cite{Zee2016} and \cite{Langacker2017}). Nowadays, it is widely believed that baryons being fermions have to obey GMO relations which are linear in the baryon masses, whilst mesons as bosons follow quadratic GMO relations (cf. \cite{Langacker2017}). This distinction -- allegedly first introduced by R.P. Feynman (cf. \cite{DeSwart1963}) -- is commonly justified with the observation that the mass enters linearly in a fermionic Lagrangian, but quadratically in a bosonic Lagrangian. In a supersymmetrical world, however, one expects a symmetry between fermionic and bosonic multiplets in a supermultiplet. This symmetry implies that every mass relation satisfied by a fermionic multiplet has to be satisfied by its bosonic supersymmetrical counterpart and vice versa. As the argument for distinguishing fermionic and bosonic mass relations should apply to all quantum field theories (QFTs) independent of the question whether the theory is realized in Nature, it should also apply to a supersymmetrical theory.
Clearly, the symmetry between fermion and boson masses conflicts with Feynman's distinction\footnote{``Feynman's distinction'' is used throughout this thesis as a phrase for the distinction of baryons and mesons into linear and quadratic GMO relations (or linear and quadratic hadronic mass relations, in general), respectively.} in this case. Since this reasoning is independent of the question whether Nature is supersymmetrical, the problem with Feynman's distinction must already arise on a theoretical level rendering the distinction itself questionable. Therefore, I challenge Feynman's distinction in this master's thesis. In the course of the thesis, I will further support this claim by presenting an in-depth analysis of the GMO relations on a theoretical and experimental level.\par
The theoretical side of this thesis concerns the derivation of the mass relations. In the course of this work, we will develop two approaches to the GMO relations: The first is the description of hadron masses within the framework of a hadronic effective field theory (EFT). An example for this EFT approach is chiral perturbation theory (cf. \cite{Scherer2011}). The second approach which we will call state formalism describes the hadrons and their masses as eigenstates and -values of the Hamilton operator, respectively. In both the EFT approach and the state formalism, we will obtain the GMO mass relations as a first order result of the $\text{SU}(3)$-flavor symmetry breaking to the isospin symmetry group $\text{SU}(2)\times\text{U}(1)$.\par
One difference between the EFT approach and the state formalism is central for our investigations: While Feynman's distinction seems to arise naturally in the EFT approach, the state formalism does not exhibit this distinction. At first glance, that seems to indicate that the EFT approach and the state formalism are incompatible, nevertheless, we will see that the discrepancy can easily be explained: The state formalism is only applicable, if the flavor symmetry breaking is small and can be treated as a perturbation. In that case, both linear and quadratic GMO mass relations are actually equivalent and Feynman's distinction is artificial.\par
Additional to the $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking, we will also incorporate isospin symmetry breaking, electromagnetic contributions, and heavy quark symmetry into the model of hadron masses. These effects give rise to additional mass relations in the hadronic sector which we will use for further investigations of Feynman's distinction.\par
The part of this work related to experimental data services as a ``proof of concept'' for the theoretical considerations: I will support my claims about Feynman's distinction by analyzing experimental data on hadron masses. In particular, we will see that the experimental hadron masses indicate that both linear and quadratic mass relations are applicable in most cases rendering Feynman's distinction artificial. Even if we need to distinguish between linear and quadratic mass relations, the distinction does not originate from the difference between baryons and mesons.\\
The thesis is organized as follows:\par
\autoref{chap:hadron_masses} provides an instructive introduction to the basic concepts of this work. For the sake of clarity, the mathematical details are postponed to \autoref{sec:GMO_formula}. At the beginning of the chapter (\autoref{sec:mass_matrix}), we will explore an easily understandable access to the GMO relations by considering the \text{SU}(3)-flavor transformation properties of mass matrices corresponding to singlets, triplets, sextets, octets, and decuplets. As a next step (\autoref{sec:Trafo_QCD}), we will consider the transformation properties of a Lagrangian involving three light quark flavors with a flavor symmetric interaction of the quarks exemplified by quantum chromodynamics (QCD). In particular, we will discuss which terms in the Lagrangian may spoil the \text{SU}(3)-flavor symmetry between the quarks. At the end of the chapter (\autoref{sec:EFT+H_Pert}), we will see how the transformation behavior of the Lagrangian from \autoref{sec:Trafo_QCD} dictates the symmetry structure of the hadron masses. We will find that the transition from the quark Lagrangian to the hadron masses can be achieved in two ways, namely the EFT approach and the state formalism. In the context of these two approaches, Feynman's distinction will be addressed.\par
In \autoref{chap:GMO_formula}, the description of hadron masses will be elevated to a higher mathematical level (\autoref{sec:GMO_formula}) and expanded to isospin symmetry breaking, electromagnetic interaction (\autoref{sec:add_con}), and heavy quark symmetry (\autoref{sec:heavy_quark}).\par
\autoref{chap:mass_relations} gives an overview over the relations that follow from the mass formulae of \autoref{chap:GMO_formula} for all multiplets realized in Nature. Additionally, the order of magnitude of the dominant correction(s) is calculated for each mass relation.\par
In \autoref{chap:data}, we check both linear and quadratic mass relations against experimentally determined hadron masses to verify our interpretation of the mass relations. Moreover, we use current scientific results and phenomenological observations to classify unassigned resonances into multiplets and utilize the data of yet incomplete multiplets to predict the masses of the missing particles.
\newpage
\chapter{Hadron Masses under \text{SU}(3)-Flavor Transformations}
\label{chap:hadron_masses}
The derivation of the GMO mass relations features many complex and noteworthy steps with regard to both mathematics and physics. A bottom-up derivation, starting with a Lagrangian and presenting every point in detail, may easily be convoluted and fails to be instructive. Therefore, in this chapter, we only point out the main features that lead to the GMO mass relations. In particular, we cover how the \text{SU}(3)-flavor transformations of hadrons in a multiplet, namely singlet, triplet, sextet, octet, and decuplet, define the transformation of the hadron mass matrices and how \text{SU}(3)-flavor symmetry breaking leads to mass relations in those multiplets. Furthermore, we explain how \text{SU}(3)-flavor symmetry breaking enters a Lagrangian with three quark flavors and how this symmetry breaking can be linked to the hadron masses. The last point is demonstrated in two different ways to show how Feynman's distinction may arise and how it can be understood. Throughout this chapter, many assumptions and mathematical claims are made without deeper reasoning for the sake of clarity. A detailed discussion of those points is postponed to \autoref{sec:GMO_formula}.
\section{Decomposition of Mass Matrices and GMO Mass Relations}\label{sec:mass_matrix}
In the early 1960s, when people began to realize that hadrons show a symmetry structure related to the Lie group \text{SU}(3) (cf. \cite{Gell-Mann1961}), it was observed that hadrons form multiplets of \text{SU}(3). The mathematical definition of the term ``multiplet'' is connected to the term ``representation''. A real or complex \textit{representation} $D^{(\rho)}$ -- sometimes simply written as $\rho$ or denoted by its vector space $V$ -- of a topological group $G$, for instance \text{SU}(3), on a real or complex Hilbert space $V$ is a group homomorphism
\begin{gather*}
D^{(\rho)}:G\rightarrow\text{GL}(V)
\end{gather*}
with $\text{GL}(V)\coloneqq \{A:V\rightarrow V\text{ linear and bounded}\mid A^{-1}\text{ exists and is bounded}\}$ such that the map $\Phi^{(\rho)}:G\times V\rightarrow V, (g,v)\mapsto D^{(\rho)}(g)(v)$ is continuous (cf. \cite{Knapp2001}). A subspace $W\subset V$ is called \textit{invariant}, if $D^{(\rho)}(g)(W)\subset W\ \forall g\in G$, and the representation $D^{(\rho)}$ is called \textit{irreducible}, if the only closed invariant subspaces of $V$ are $\{0\}$ and $V$ (cf. \cite{Knapp2001}). With these definitions in mind, we define a \textit{multiplet} of a group (mostly \text{SU}(3) in this work) to be an irreducible representation of that group.\par
The expression ``hadrons form multiplets of \text{SU}(3)" can now be understood in multiple ways: In the sense of a QFT, we may assume that each hadron which we want to group into one multiplet is described by a field and that the fields transform as the components of a vector in the vector space $V$ of the multiplet $D^{(\rho)}$. For example, if $V$ is the complex vector space $\mathbb{C}^n$ and the hadrons at hand are spin-$\frac{1}{2}$ fermions described by the fields $\psi_a$, then the fields $\psi_a$ transform under \text{SU}(3) as:
\begin{align*}
\begin{pmatrix}\psi_1\\ \vdots\\ \psi_n\end{pmatrix}&\xrightarrow{A\in\,\text{SU}(3)}\begin{pmatrix}\psi^\prime_1\\ \vdots\\ \psi^\prime_n\end{pmatrix} = D^{(\rho)}(A)\begin{pmatrix}\psi_1\\ \vdots\\ \psi_n\end{pmatrix}\\
\Leftrightarrow \psi_a &\xrightarrow{A\in\,\text{SU}(3)}\psi^\prime_a = \sum\limits^n_{b=1}\left(D^{(\rho)}(A)\right)_{ab}\psi_b.
\end{align*}
Note that if the fields $\psi_a$ can be decomposed in terms of creation and annihilation operators, the transformed fields $\psi^\prime_a$ cannot, in general, be decomposed in the same way. The reason for this is that we cannot, in general, pass down the transformation of the fields to the creation and annihilation operators (cf. \autoref{app:stateform}).\par
Alternatively, we may think of the hadrons in one multiplet as quantum mechanical states $\Ket{a}$ and assume that these states transform under {${A\in\text{SU}(3)}$} as:
\begin{gather*}
\Ket{a}\xrightarrow{A\in\,\text{SU}(3)}\Ket{a^\prime} = \sum\limits_{b} \left(D^{(\rho)}(A)\right)^\ast_{ab}\Ket{b}.
\end{gather*}
Note that the transformation coefficients of $\Ket{b}$ are complex conjugated in regard to the transformation of $\psi_a$. This is due to the fact that the states $\Ket{a}$ are related to $\overline{\psi}_a$.\pa
In both cases, we may now identify mass matrices, i.e., self-adjoint matrices whose eigenvalues are the hadron masses. The definition of the mass corresponding to a particle or resonance and its relation to operators and parameters of the Lagrangian are by no means trivial and require an in-depth discussion (cf. \autoref{sec:polemass} and \autoref{app:stateform}). However, we skip this discussion for now and use very simplistic pictures for both cases. In the case of fields, we identify the mass matrix (neglecting loop corrections) with the coefficients of the quadratic fields terms in a(n) (effective) Lagrangian:
\begin{gather*}
\mathcal{L}_M = -\sum\limits_{a,b}\overline{\psi}_a M_{ab}\psi_b.
\end{gather*}
Note that the mass matrix defined by the equation above has to be replaced by the squared mass matrix in the case of bosonic fields. We will discuss this point in more detail in \autoref{sec:EFT+H_Pert}.\par
In the case of states, the mass, as a physical observable, is given by a self-adjoint operator $\hat M$ and the mass matrix can be defined via the matrix elements of $\hat M$:
\begin{gather*}
M_{ab}\coloneqq \Bra{a}\hat M\Ket{b}.
\end{gather*}
If we now assume that $M_{ab} = 0$ if $a$ is a hadron in the multiplet and $b$ is not, only the mass matrix elements of the multiplet contribute to the mass of a hadron in that multiplet. Thus, it is sufficient to only consider the mass matrix elements of the multiplet for the transformation of the hadron masses in that multiplet and we are able to define the transformation of the mass matrix under \text{SU}(3) for both cases:
\begin{align*}
\sum\limits_{a,b}\overline{\psi}_a M_{ab}\psi_b&\xrightarrow{A\in\, \text{SU}(3)}\sum\limits_{a,b}\overline{\psi^\prime}_a M^\prime_{ab}\psi^\prime_b\coloneqq \sum\limits_{a,b}\overline{\psi}_a M_{ab}\psi_b;\\
M_{ab}&\xrightarrow{A\in\, \text{SU}(3)}M^\prime_{ab}\coloneqq \Bra{a^\prime}\hat M\Ket{b^\prime}.
\end{align*}
A quick calculation shows:
\begin{align}
\sum\limits_{a,b}\overline{\psi^\prime}_a M^\prime_{ab}\psi^\prime_b &= \sum\limits_{a,b}\left(\sum\limits_{c}\overline{\psi}_c \left(D^{(\rho)}(A)\right)_{ac}^\ast \right) M^\prime_{ab} \left(\sum\limits_{d}\left(D^{(\rho)}(A)\right)_{bd}\psi_d\right)\nonumber\\
& = \sum\limits_{c,d}\overline{\psi}_c\left(\sum\limits_{a,b}\left(D^{(\rho)}(A)\right)_{ac}^\ast M^\prime_{ab} \left(D^{(\rho)}(A)\right)_{bd}\right)\psi_d\nonumber\\
& = \sum\limits_{c,d}\overline{\psi}_c M_{cd}\psi_d\nonumber\\
\Rightarrow M^\prime_{ab} &= \sum\limits_{c,d}\left(D^{(\rho)}(A^{-1})\right)_{ca}^\ast M_{cd} \left(D^{(\rho)}(A^{-1})\right)_{db};\label{eq:M_compon_field}\\
M^\prime_{ab} &= \Bra{a^\prime}\hat M\Ket{b^\prime}\nonumber\\
&= \left(\sum\limits_{c}\Bra{c}\left(D^{(\rho)}(A)\right)_{ac}\right)\hat M\left(\sum\limits_{d}\left(D^{(\rho)}(A)\right)^\ast_{bd}\Ket{d}\right)\nonumber\\
&= \sum\limits_{c,d}\left(D^{(\rho)}(A)\right)_{ac} M_{cd} \left(D^{(\rho)}(A)\right)^\ast_{bd}\nonumber\\
&= \sum\limits_{c,d}\left(D^{(\rho)}(A)^\dagger\right)^\ast_{ca} M_{cd} \left(D^{(\rho)}(A)^\dagger\right)_{db}\label{eq:M_compon_state}.
\end{align}
In \autoref{eq:M_compon_field}, we have used that $D^{(\rho)}$ is a group homomorphism:\linebreak {${D^{(\rho)}(A)^{-1} = D^{(\rho)}\left(A^{-1}\right)}$}. Without loss of generality, we can assume that the matrix $D^{(\rho)}(A)$ is always unitary, as for every representation of the compact Lie group \text{SU}(3), there exists an Hermitian scalar product for which the representation is unitary (cf. \cite{Hamilton2017}). With this property, we can see that the mass matrix in both cases transforms in the same way, even though $\psi_a$ and $\Ket{a}$ transform differently. If we rewrite the transformation of $M_{ab}$ in matrix notation, we find:
\begin{gather*}
M^\prime = D^{(\rho)}(A)\cdot M\cdot D^{(\rho)}(A)^\dagger.
\end{gather*}
Note that $M^\prime$ is self-adjoint, as $M$ is a mass matrix and, therefore, self-adjoint.\par
Until now, the discussion of the transformation behavior of $M$ was rather abstract. To make the flavor transformation properties of hadronic mass matrices more concrete, it is instructive to consider examples of multiplets. In addition to clarity, examples of multiplets also provide an easy access to group theoretical mass parameterizations and relations. The assumptions and considerations we need to make in order to find the GMO mass formula and relations in \autoref{chap:GMO_formula} and \autoref{chap:mass_relations} are easily applied to singlets, triplets, sextets, octets, and decuplets. The structures these multiplets exhibit guide our expectation of the flavor transformation behavior of general multiplets and provide us with a list of (mathematical) observations we will prove for all \text{SU}(3)-multiplets in \autoref{sec:GMO_formula}.
\subsection*{Singlet}
First, take the multiplet $D^{(\rho)}$ to be the trivial representation or, equivalently, a singlet of \text{SU}(3), denoted by $\rho = 1$. The singlet operates on an one-dimensional vector space and is simply given by:
\begin{gather*}
D^{(1)}(A) = \mathbb{1}\quad\forall\, A\in\text{SU}(3).
\end{gather*}
This implies that $M$ does not transform at all under \text{SU}(3), meaning that $M$ itself transforms as $1$ under \text{SU}(3), and is just given by one component. Since $M$ is self-adjoint, this component is real and corresponds to the mass of the hadron in the singlet. In this case, the transformation behavior of $M$ does not reveal any information about the mass of the hadron.
\subsection*{Triplet}
A more interesting transformation behavior arises when we take $D^{(\rho)}$ to be a triplet or, equivalently, the fundamental representation of \text{SU}(3), denoted by $\rho = 3$. It operates on the three-dimensional vector space $\mathbb{C}^3$ and is given by:
\begin{gather*}
D^{(3)}(A) = A\quad\forall\, A\in\text{SU}(3).
\end{gather*}
The triplet describes three hadrons which we can identify with the vectors
\begin{gather*}
\begin{pmatrix}1\\0\\0\end{pmatrix},\ \begin{pmatrix}0\\1\\0\end{pmatrix},\ \begin{pmatrix}0\\0\\1\end{pmatrix}.
\end{gather*}
Hence, the mass matrix $M$ is, in this case, a complex self-adjoint $(3\times 3)$-matrix that transforms under $A\in\text{SU}(3)$ as:
\begin{gather*}
M^\prime = A\cdot M\cdot A^\dagger.
\end{gather*}
Note that, while the triplet $3$ is a complex representation acting on $\mathbb{C}^3$, the transformation of $M$ is a real representation of \text{SU}(3), as the space of complex self-adjoint $(3\times 3)$-matrices is a real nine-dimensional vector space.\par
First, let us consider the case where $M = m\mathbb{1}$ is a real factor times the identity. Then:
\begin{gather*}
M^\prime = A\cdot m\mathbb{1}\cdot A^\dagger = mA A^\dagger = m\mathbb{1} = M\quad\text{for } A\in\text{SU}(3).
\end{gather*}
As we can see, $M = m\mathbb{1}$ transforms trivially under \text{SU}(3). Therefore, the multiples of the identity are an invariant subspace under the transformation of $M$ and, since this subspace is one-dimensional, they even furnish an irreducible representation.\par
Next, consider $M$ to be a traceless self-adjoint matrix, i.e., $M^\dagger = M$ and\linebreak $\text{Tr}(M) = 0$. As stated earlier, $M^\prime$ is self-adjoint, if $M$ is self-adjoint. Furthermore, we find:
\begin{gather*}
\text{Tr}(M^\prime) = \text{Tr}(A\cdot M\cdot A^\dagger) = \text{Tr}(M\cdot A^\dagger\cdot A) = \text{Tr}(M) = 0\quad\text{for } A\in\text{SU}(3).
\end{gather*}
This shows that the real eight-dimensional vector space of traceless self-adjoint $(3\times 3)$-matrices is also an invariant subspace under the transformation of $M$. One can show that this subspace even furnishes an irreducible representation under the transformation of $M$. This irreducible representation is called octet or, equivalently, the adjoint representation of \text{SU}(3), denoted by $8$. A basis for the octet, i.e., a basis for the traceless self-adjoint $(3\times 3)$-matrices is given by the Gell-Mann matrices (cf. \cite{Langacker2017}):
\begin{align*}
\lambda_1 &= \begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix},\ \lambda_2 = \begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix},\ \lambda_3 = \begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}\\
\lambda_4 &= \begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix},\ \lambda_5 = \begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix},\ \lambda_6 = \begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}\\
\lambda_7 &= \begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix},\ \lambda_8 = \frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}.
\end{align*}
With these considerations in mind, we can now decompose a general self-adjoint $(3\times 3)$-mass matrix $M$ as:
\begin{gather*}
M = \frac{\text{Tr}(M)}{3}\mathbb{1} + \tilde{M}
\end{gather*}
with $\tilde{M}\coloneqq M - \frac{\text{Tr}(M)}{3}\mathbb{1}$. As we can see, the first term in the decomposition is just a real factor times the identity, while the second term is a traceless self-adjoint matrix. Therefore, we have found a decomposition of $M$ into terms that transform as irreducible representations under the transformation of $M$.\par
There is also a more general, but abstract way that leads to this decomposition. If we take a look at the transformation of $M$ in components (\autoref{eq:M_compon_field} and \autoref{eq:M_compon_state}), we can see that the components of the matrix $D^{(M)}(A)$ that transforms $M$ is a product of the components of the (complex) conjugate representation $\bar{\rho}$ of $\rho$ and of the components of the representation $\rho$ itself:
\begin{gather*}
\left(D^{(M)}(A)\right)_{(ab)(cd)} = \left(D^{(\rho)}(A)\right)_{ac}\cdot \left(D^{(\rho)}(A)\right)^\ast_{bd}\eqqcolon \left(D^{(\rho)}(A)\right)_{ac}\cdot \left(D^{(\bar{\rho})}(A)\right)_{bd}.
\end{gather*}
This equation implies that the representation $D^{(M)}$ is (equivalent to) the tensor product $\rho\otimes\bar{\rho}$. Since every finite-dimensional representation of a compact Lie group decomposes into a (direct) sum of irreducible representations (cf. \cite{Hamilton2017}), $\rho\otimes \bar{\rho}$ as a representation of \text{SU}(3) decomposes into irreducible representations. This decomposition of tensor products of representations into irreducible representations is called \textit{Clebsch-Gordan series} (cf. \cite{DeSwart1963}).\par
In the case of the triplet, this means that the transformation of $M$ is given by $3\otimes \bar{3}$. The Clebsch-Gordan series of $3\otimes \bar{3}$ can be calculated in multiple ways, for instance by using Young tableaux (cf. \cite{Lichtenberg}). One obtains:
\begin{gather*}
3\otimes \bar{3} = 1 \oplus 8.\label{eq:3ast3}
\end{gather*}
The Clebsch-Gordan series of $3\otimes \bar{3}$ implies that the transformation of $M$ contains one singlet and one octet. This result coincides with our prior analysis of the mass matrix $M$, where we identified the singlet with the multiples of the identity and the octet with the traceless self-adjoint $(3\times 3)$-matrices.\par
So far, we have only rewritten a nine-dimensional vector space as a sum of an one-dimensional and an eight-dimensional vector space with interesting transformation properties, but not gained any information about hadron masses or the entries of $M$. In order to do this, we have to make additional assumptions about the transformation behavior of $M$. For instance, we may assume that \text{SU}(3) is an exact symmetry, i.e., that $M$ is invariant under any \text{SU}(3)-transformation: $M^\prime = M\ \text{for } A\in\text{SU}(3)$. Then, we can directly deduce that $M$ has to be an element of the singlet and, therefore, a multiple of the identity, because the octet as an irreducible representation cannot contain an element that transforms trivially under \text{SU}(3). If the octet contained such an element, the span of this element would be a singlet and, hence, an invariant subspace of the octet which contradicts the fact that the octet is irreducible. In this regard, exact \text{SU}(3)-symmetry implies that all hadrons in the triplet have to have the same mass.\par
However, we often find that symmetries are broken in Nature. Therefore, let us consider the case where \text{SU}(3) is not taken to be an exact, but only an approximate symmetry. This means that the term which explicitly breaks the \text{SU}(3)-symmetry, i.e., the octet is small in comparison to the \text{SU}(3)-invariant term, i.e., the singlet:
\begin{gather*}
M = m_0\mathbb{1} + \sum\limits^8_{i=1}m_i\frac{\lambda_i}{2}
\end{gather*}
with $m_i\ll m_0$. Nonetheless, we have to have some residual symmetry, since without it we cannot make any further statement about the masses of the hadrons in the triplet. Hence, we require the subgroup of \text{SU}(3)-transformations that at most mix two specific hadrons, say the hadrons described by $(1,0,0)^\text{T}$ and $(0,1,0)^\text{T}$, to leave the mass matrix $M$ invariant. This subgroup contains \text{SU}(2)-transformations mixing $(1,0,0)^\text{T}$ and $(0,1,0)^\text{T}$:
\begin{gather*}
A = \begin{pmatrix}\tilde{A}&0\\0&1\end{pmatrix}\quad\text{for }\tilde{A}\in\text{SU}(2)
\end{gather*}
and phase modulations not mixing any hadrons:
\begin{gather*}
A = \begin{pmatrix}e^{i\alpha}&0&0\\0&e^{i\alpha}&0\\0&0&e^{-2i\alpha}\end{pmatrix}\quad\text{for }\alpha\in\mathbb{R}.
\end{gather*}
Therefore, all transformations that shall leave $M$ invariant are given by the subgroup
\begin{gather*}
E\coloneqq \left\{\begin{pmatrix}e^{i\alpha}\tilde{A}&0\\0&e^{-2i\alpha}\end{pmatrix}\mid\tilde{A}\in\text{SU}(2),\ \alpha\in\mathbb{R}\right\}.
\end{gather*}
Considering the map
\begin{gather*}
f:\text{SU}(2)\times\text{U}(1)\rightarrow E,\ (\tilde{A},\, e^{i\alpha})\mapsto\begin{pmatrix}e^{i\alpha}\tilde{A}&0\\0&e^{-2i\alpha}\end{pmatrix},
\end{gather*}
we can see that $\text{SU}(2)\times\text{U}(1)$ is the covering group of $E$ covering $E$ twice, as $f(\tilde{A},\, e^{i\alpha})$ is equal to $f(-\tilde{A},\, -e^{i\alpha})$. However, we will ignore this in the future and simply identify $\text{SU}(2)\times\text{U}(1)$ with $E$, because they have the same or similar properties regarding our purposes\footnote{We are mostly interested in the irreducible representations of $E$. Since the map $f$ connecting $\text{SU}(2)\times\text{U}(1)$ and $E$ is a surjective Lie group homomorphism, we can promote every irreducible representation of $E$ via $f$ to an irreducible representation of $\text{SU}(2)\times\text{U}(1)$. This allows us to identify the irreducible representations of $E$ with irreducible representations of $\text{SU}(2)\times\text{U}(1)$. The identification of the irreducible representations is unique, i.e., there are no two distinct irreducible representations of $E$ which are identified with the same irreducible representation of {${\text{SU}(2)\times\text{U}(1)}$}. Note, however, that not every irreducible representation of $\text{SU}(2)\times\text{U}(1)$ corresponds to a representation of $E$. Also, note that $E\cong \text{U}(2)$.}.\par
In this sense, we consider a \text{SU}(3)-symmetry breaking to its conserved subgroup $\text{SU}(2)\times\text{U}(1)$ here. To see the impact of the symmetry breaking on the hadron masses, we have to parametrize all mass matrices $M$ that are invariant under $\text{SU}(2)\times\text{U}(1)$. Clearly, the singlet is invariant under $\text{SU}(2)\times\text{U}(1)$, as it is already invariant under \text{SU}(3). Furthermore, the Gell-Mann matrix $\lambda_8$ is invariant under $\text{SU}(2)\times\text{U}(1)$, since it leaves the first two components unchanged, aside from a factor. Indeed, no other Gell-Mann matrix satisfies this condition and, hence, can be invariant under $\text{SU}(2)\times\text{U}(1)$. This means that $M$ is only parametrized by $\mathbb{1}$ and $\lambda_8$:
\begin{gather*}
M = m_0\mathbb{1} + m_8\frac{\lambda_8}{2} = \begin{pmatrix}m_0 + m_8/2\sqrt{3}&0&0\\0&m_0 + m_8/2\sqrt{3}&0\\0&0&m_0 - m_8/\sqrt{3}\end{pmatrix}.
\end{gather*}
As $M$ is already diagonal, we can directly read off the hadron masses: The masses of $(1,0,0)^\text{T}$ and $(0,1,0)^\text{T}$ -- the two hadrons that mix under $\text{SU}(2)\times\text{U}(1)$ -- are degenerate, while the third hadron has a different mass. However, the difference $\sqrt{3}m_8/2$ between the two non-degenerate masses is small in comparison to the average mass $m_0$, as we assume that the breaking is small.\par
The triplet as a model for hadron masses is realized in both the mesonic and baryonic sector. The $B$- and $D$-mesons show a mass structure very similar to the one described above and likewise the $\Lambda_c/\Xi_c$- and $\Lambda_b/\Xi_b$-baryons. The only difference for both mesons and baryons is that the hadrons corresponding to $(1,0,0)^\text{T}$ and $(0,1,0)^\text{T}$ are not exactly degenerate and show a very small deviation. This difference is related to isospin symmetry breaking.
\subsection*{Sextet}
Although the triplet exhibits a non-trivial mass structure, the triplet as well as the singlet do not lead to any mass relation. The lowest dimensional multiplet $D^{(\rho)}$ which does lead to mass relations is the sextet, denoted by $\rho=6$. It operates on the six-dimensional vector space $\mathbb{C}^6$ and, therefore, describes six hadrons. Like all representations of \text{SU}(3), the sextet can be chosen to be unitary, however, its explicit dependence on $A\in\text{SU}(3)$ is not as simple as in the case of the singlet or triplet. The transformation of the mass matrix $M$, a self-adjoint $(6\times 6)$-matrix now, is obviously given by:
\begin{gather*}
M^\prime = D^{(6)}(A)\cdot M\cdot D^{(6)}(A)^\dagger.
\end{gather*}
Similar to the triplet, the multiples of the identity and the traceless self-adjoint matrices are invariant subspaces of the transformation of $M$. Again, the multiples of the identity furnish an irreducible representation equivalent to the singlet. However, the traceless self-adjoint matrices only furnish a reducible representation, i.e., they contain non-trivial invariant subspaces. We can see this by calculating the Clebsch-Gordan series of $6\otimes \bar{6}$, as explained for the triplet:
\begin{gather*}
6\otimes \bar{6} = 1\oplus 8\oplus 27.
\end{gather*}
The 35 dimensional space of traceless self-adjoint $(6\times 6)$-matrices decomposes into an octet and a 27-dimensional irreducible representation of \text{SU}(3), denoted by $27$. By choosing a basis of the singlet, the octet, and $27$, we can parametrize the mass matrix $M$ in terms of irreducible representations.\par
As for the triplet, we can now consider exact and approximate \text{SU}(3)-symmetry. If the \text{SU}(3)-symmetry is exact, the mass matrix $M$ has to transform trivially and, hence, be an element of the singlet. This means that $M$ is a multiple of the identity and all hadrons in the sextet have the same mass.\par
If we take \text{SU}(3) to be an approximate symmetry, the \text{SU}(3)-breaking contributions of the octet and $27$ to the mass matrix are small in comparison to the contribution of the \text{SU}(3)-invariant singlet. Even though \text{SU}(3)-symmetry is broken, we want to have some residual symmetry. Again, we consider the case where the subgroup $\text{SU}(2)\times \text{U}(1)$ of \text{SU}(3) is an exact symmetry. Group theoretical considerations\footnote{Every representation of a group is also a representation of any subgroup of that group. We can use this here for \text{SU}(3) and $\text{SU}(2)\times\text{U}(1)\subset\text{SU}(3)$: 1, 8, and 27 are representations of \text{SU}(3) and, thus, furnish representations of $\text{SU}(2)\times\text{U}(1)$. These representations decompose into irreducible representations of $\text{SU}(2)\times\text{U}(1)$. Each decomposition of 1, 8, or 27 contains exactly one trivial representation of $\text{SU}(2)\times\text{U}(1)$, each corresponding to one $\text{SU}(2)\times\text{U}(1)$-invariant matrix. One can show this by noting that the sextet 6 decomposes into three different irreducible representations of $\text{SU}(2)\times\text{U}(1)$ and by using a lemma we prove in \autoref{sec:GMO_formula}.} then imply that the mass matrix $M$ is parametrized by three linearly independent matrices, one from each representation 1, 8, and 27.\par
At this point, there is no hierarchy between the contributions of 8 and 27, both could be of the same order of magnitude. However, it was observed in the 1960s that, for hadrons, the contribution from octets is dominant in comparison to contributions from other non-trivial representations. This phenomenological observation, known as \textit{octet enhancement} (cf. \cite{Lichtenberg}), was not understood at that time (cf. \cite{Langacker2017}), but could be explained years later in the framework of QCD. Since we want to apply our considerations to hadron masses, we assume octet enhancement from now on, meaning that we take all contributions to the mass matrix aside from singlets and octets to be negligible. The origin of octet enhancement will be the concern of the last two sections (\autoref{sec:Trafo_QCD} and \autoref{sec:EFT+H_Pert}) in this chapter. Note that the singlet and triplet are trivially subject to octet enhancement, as there are no other contributions aside from singlets and octets in these cases.\par
The mass matrix $M$ can now be parametrized by only two matrices, one corresponding to the singlet and one corresponding to the octet. If we denote the octet matrix with $F^{(6\otimes\bar{6})}_8$, we can write:
\begin{gather*}
M = m_0\mathbb{1} + m_8F^{(6\otimes\bar{6})}_8
\end{gather*}
with $m_8\ll m_0$. There are a lot of unitary representations acting on $\mathbb{C}^6$ that are equivalent to the sextet, all linked by similarity transformations. Every one of these representations is a possible candidate for $D^{(6)}$. In general, $F^{(6\otimes\bar{6})}_8$ is not diagonal for an arbitrary choice of $D^{(6)}$, but we will see in \autoref{sec:GMO_formula} (and by using the weight diagrams in \autoref{chap:mass_relations}) that one can choose $D^{(6)}$ such that $F^{(6\otimes\bar{6})}_8$ is given by:
\begin{gather*}
F^{(6\otimes\bar{6})}_8 = \text{diag}\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-1}{2\sqrt{3}},\frac{-1}{2\sqrt{3}},\frac{-2}{\sqrt{3}}\right).
\end{gather*}
For this choice, $M$ is diagonal and we can read off the hadron masses. There are three different masses: $m_1 = m_0 + \frac{m_8}{\sqrt{3}}$ is the mass of three hadrons in the sextet, $m_2 = m_0 - \frac{m_8}{2\sqrt{3}}$ is the mass of two hadrons, and $m_3 = m_0 - \frac{2m_8}{\sqrt{3}}$ is the mass of the remaining hadron. As for the triplet, the difference between two hadron masses is small in comparison to the average mass $m_0$ of the hadrons in the sextet.\par
On top of that, we see that there are three distinct hadron masses which are given by only two parameters. This means that we can find a relation between the hadron masses. The existence of a mass relation follows from assuming $\text{SU}(3)\rightarrow \text{SU}(2)\times \text{U}(1)$ symmetry breaking and octet enhancement. In this work, we call mass relations that follow from these two assumptions \textit{Gell-Mann--Okubo mass relations}. The relation for the sextet is given by:
\begin{gather*}
m_1 - m_2 = \frac{\sqrt{3}}{2}m_8 = m_2 - m_3.
\end{gather*}
This relation states that the difference between two neighboring hadron masses is the same for the entire sextet. Mass relations of this kind are known as \textit{equal spacing rules} and common for totally symmetric multiplets like sextets, decuplets etc.\par
The sextet as a model for hadrons is, to our knowledge, only realized in the baryonic sector. Baryons like $\Sigma_c,\ \Xi^\prime_c,\ \Omega_c$ and $\Sigma_b,\ \Xi^\prime_b,\ \Omega_b$, for instance, exhibit a structure very similar to the sextet structure described above. However, the equal spacing rule is not exactly satisfied, but only holds true to good approximation. As for triplets, the mass degeneracy is lifted in Nature due to isospin symmetry breaking.
\subsection*{Octet}
The octet $D^{(8)}$ operates on an eight-dimensional vector space. As shown in the discussion of the triplet, we can take this vector space to be the space of traceless self-adjoint $(3\times 3)$-matrices. In this case, $D^{(8)}$ is a real representation of \text{SU}(3) as this vector space is a real eight-dimensional space. If we let \text{SU}(3) act on the complexification of this space, i.e., the space of traceless complex $(3\times 3)$-matrices, in the same way, we obtain the octet $D^{(8)}$ as a complex representation of \text{SU}(3). We now identify the space of traceless complex $(3\times 3)$-matrices with the space $\mathbb{C}^8$ to get the octet $D^{(8)}$ as a unitary representation acting on $\mathbb{C}^8$. With this identification, the mass matrix $M$ is a self-adjoint $(8\times 8)$-matrix with the transformation:
\begin{gather*}
M^\prime = D^{(8)}(A)\cdot M\cdot D^{(8)}(A)^\dagger.
\end{gather*}
As for the other multiplets, the decomposition of $M$ into irreducible representations is given by the Clebsch-Gordan series of $8\otimes \bar{8}$. Note, however, that the octet is equivalent to a purely real representation, like explained above. This means that $\bar{8}$ and $8$ are equivalent and it is sufficient to consider $8\otimes 8$ instead of $8\otimes \bar{8}$:
\begin{gather*}
8\otimes 8 = 1\oplus 8\oplus 8\oplus 10\oplus \overline{10}\oplus 27,
\end{gather*}
where $10$ is a ten-dimensional irreducible representation of \text{SU}(3) called decuplet and $\overline{10}$ is its conjugate representation. Again, the singlet $1$ is given by the multiples of the identity which transform trivially and choosing a basis of each irreducible representation gives a parametrization of $M$ in terms of irreducible representations.\par
Let us consider exact \text{SU}(3)-symmetry now. In this case, $M$ has to be an element of the singlet and, hence, a multiple of the identity which means that all hadrons in the octet have the same mass.\par
If we only have approximate \text{SU}(3)-symmetry where the subgroup $\text{SU}(2)\times \text{U}(1)$ is an exact symmetry, group theory dictates that the mass matrix $M$ is parametrized by four linearly independent matrices, each from one of the representations 1, 8, 8, and 27. Imposing octet enhancement, we are left with only three contributions, one from the singlet and two from the two octets. In contrast to the triplet and the sextet, $M$ is given by three terms now instead of two after imposing octet enhancement. Actually, the mass matrix $M$ is only given by either two or three terms for arbitrary multiplets (except for the singlet), if octet enhancement is assumed. Depending on this, the multiplet satisfies different mass relations. We will show this in \autoref{sec:GMO_formula}.\par
If we denote the contribution from the first octet with $F^{(8\otimes\bar{8})}_8$ and the contribution from the second with $D^{(8\otimes\bar{8})}_8$, we can write:
\begin{gather*}
M = m_0\mathbb{1} + m^F_8F^{(8\otimes\bar{8})}_8 + m^D_8D^{(8\otimes\bar{8})}_8,
\end{gather*}
where $m^{F/D}_8\ll m_0$. One can choose $D^{(8)}$ such that $F^{(8\otimes\bar{8})}_8$ and $D^{(8\otimes\bar{8})}_8$ are diagonal, as we will see in \autoref{sec:GMO_formula}. If we make such a choice for $D^{(8)}$, we find:
\begin{align*}
F^{(8\otimes\bar{8})}_8 &= \text{diag}\left(\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2},0,0,0,0,\frac{-\sqrt{3}}{2},\frac{-\sqrt{3}}{2}\right),\\
D^{(8\otimes\bar{8})}_8 &= \text{diag}\left(\frac{-1}{2\sqrt{3}},\frac{-1}{2\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{-1}{2\sqrt{3}},\frac{-1}{2\sqrt{3}}\right).
\end{align*}
For this choice, $M$ is diagonal and we can read off the hadron masses:
\begin{align*}
m_1 &= m_0 + \frac{\sqrt{3}}{2}m^F_8 - \frac{1}{2\sqrt{3}}m^D_8,\\
m_2 &= m_0 + \frac{1}{\sqrt{3}}m^D_8,\\
m_3 &= m_0 - \frac{1}{\sqrt{3}}m^D_8,\\
m_4 &= m_0 - \frac{\sqrt{3}}{2}m^F_8 - \frac{1}{2\sqrt{3}}m^D_8,
\end{align*}
where $m_1$ is the mass of the first two hadrons, $m_2$ is the mass of the following three hadrons, $m_3$ is the mass of the only non-degenerate hadron, and $m_4$ is the mass of the last two hadrons. Again, the average mass $m_0$ of the hadrons is way bigger than the difference of any two hadron masses in the octet and, like for the sextet, there is one relation between the masses, as four masses are parametrized by only three quantities. The relation can be expressed as:
\begin{gather*}
2m_1 + 2m_4 = 4m_0 - \frac{2}{\sqrt{3}}m^D_8 = m_2 + 3m_3.
\end{gather*}
This equation is the mass relation Gell-Mann wrote down when examining \text{SU}(3)-symmetries in the strong sector (cf. \cite{Gell-Mann1961}) and the original GMO mass relation.\par
The octet as a model for hadrons is very much present in both the mesonic and baryonic sector. Examples include the baryon octet with $J^P = 1/2^+$ containing proton and neutron and the pseudoscalar meson octet containing the pions. However, both of these octets deviate from the described model. While, for the baryon octet, the deviations are small and the GMO mass relation applies with a precision of few percents (cf. \autoref{sec:mass_testing}), the pseudoscalar meson octet shows a large discrepancy from the octet model described above: The average mass of the mesons in the pseudoscalar meson octet -- roughly \SI{400}{MeV} -- is in the same order of magnitude as the difference between meson masses and the GMO mass relation is rather strongly violated (broken by about 15\%; cf. \autoref{sec:mass_testing}). It is also noteworthy that the quadratic GMO relation, i.e., the mass relation where the masses in the GMO relation above are replaced by squared masses, is much better satisfied by the pseudoscalar meson octet than the linear GMO relation, even though it is still slightly broken (cf. \autoref{sec:mass_testing}). Both relations are broken due to $\eta$-$\eta^\prime$-mixing. The difference between linear and quadratic relations regarding the pseudoscalar meson octet will be the concern of discussions in \autoref{sec:EFT+H_Pert} and \autoref{sec:mass_testing}.
\subsection*{Decuplet}
The decuplet $D^{(10)}$ operates, as already stated, on a ten-dimensional vector space. The mass matrix $M$, a self-adjoint $(10\times 10)$-matrix now, is subject to the transformation
\begin{gather*}
M^\prime = D^{(10)}(A)\cdot M\cdot D^{(10)}(A)^\dagger
\end{gather*}
under $A\in \text{SU}(3)$. Its decomposition into irreducible representations is given by the Clebsch-Gordan series of $10\otimes \overline{10}$:
\begin{gather*}
10\otimes \overline{10} = 1\oplus 8\oplus 27\oplus 64,
\end{gather*}
where $64$ is a 64-dimensional irreducible representation of \text{SU}(3). Once again, the singlet $1$ is the space spanned by the identity. If \text{SU}(3) is an exact symmetry, $M$ can only be an element of this singlet which means that, yet again, all hadrons in the decuplet have the same mass.\par
If we consider approximate \text{SU}(3)-symmetry with exact $\text{SU}(2)\times\text{U}(1)$-symmetry, group theoretical considerations show that $M$ is a linear combination of four linearly independent matrices, one from each representation in the Clebsch-Gordan series. Octet enhancement limits the number of free parameters to two, since only the singlet and octet contribution remain. As for the triplet and sextet, the mass matrix is given by two contributions. Denoting the octet contribution with $F^{(10\otimes\overline{10})}_8$, we find:
\begin{gather*}
M = m_0\mathbb{1} + m_8F^{(10\otimes\overline{10})}_8
\end{gather*}
with $m_8\ll m_0$. If $D^{(10)}$ is chosen such that $F^{(10\otimes\overline{10})}_8$ is diagonal, one obtains:
\begin{gather*}
F^{(10\otimes\overline{10})}_8 = \text{diag}\left(\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2},0,0,0,\frac{-\sqrt{3}}{2},\frac{-\sqrt{3}}{2},-\sqrt{3}\right).
\end{gather*}
We can directly read off the hadron masses:
\begin{align*}
m_1 &= m_0 + \frac{\sqrt{3}}{2}m_8,\\
m_2 &= m_0,\\
m_3 &= m_0 - \frac{\sqrt{3}}{2}m_8,\\
m_4 &= m_0 - \sqrt{3}m_8,
\end{align*}
where $m_1$ is the mass of four hadrons, $m_2$ is the mass of three hadrons, $m_3$ is the mass of two hadrons, and $m_4$ is the mass of the only non-degenerate hadron. As two parameters describe four masses, there are two mass relations this time:
\begin{align*}
m_1 - m_2 &= \frac{\sqrt{3}}{2}m_8 = m_2 - m_3,\\
m_2 - m_3 &= \frac{\sqrt{3}}{2}m_8 = m_3 - m_4.
\end{align*}
Like in the case of the sextet, these two relations are equivalent to an equal spacing rule:
\begin{gather*}
m_1 - m_2 = m_2 - m_3 = m_3 - m_4.
\end{gather*}
To our knowledge, decuplets as a model for hadrons are only realized in the baryonic sector. The baryon decuplet with $J^P = 3/2^+$ made up out of the $\Delta$-, $\Sigma^\ast$-, $\Xi^\ast$-, and $\Omega$-baryons is the most prominent example for this. In this decuplet, the equal spacing rule holds true to good approximation. Like for the other multiplets, the degeneracy is lifted due to isospin symmetry breaking.\\\par
The discussion of the hadronic multiplets and the GMO mass relations revealed some reoccurring features. In particular, the previous segments indicate that:
\begin{itemize}
\item The mass matrix of every multiplet contains exactly one singlet $1$. This singlet is given by the multiples of the identity. For exact \text{SU}(3)-symmetry, this implies that every hadron in the multiplet has to have the same mass.
\item The mass matrix of every non-trivial multiplet contains at least one, but at most two octets.
\item The multiplets whose mass matrix contains only one octet are exactly the totally symmetric multiplets like triplets, sextets, decuplets, and so on.
\item Aside from the triplet, all totally symmetric multiplets satisfy equal spacing rules.
\end{itemize}
We will prove these properties in \autoref{sec:GMO_formula}.
\newpage
\section{Transformation Behavior of a 3-Flavors Lagrangian}\label{sec:Trafo_QCD}
As we have seen in the previous section, assuming $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking and octet enhancement is crucial for the derivation of the GMO mass relations. The origin of these two assumptions lies in the Lagrangian governing the dynamics of the hadrons and their constituents. In order to understand how \text{SU}(3)-symmetry breaking and octet enhancement arise from a Lagrangian, we have to specify the Lagrangian first. In principle, the Lagrangian we need to investigate would be the Lagrangian of the Standard Model (SM), if we assume that the SM Lagrangian describes hadrons and quarks as their constituents to good approximation. For the sake of simplicity, let us drop everything in the SM Lagrangian which is not of immediate interest for the consideration at hand, meaning that we only keep the three light quarks u, d, and s and the strong interaction. By doing this, we are left with a QCD Lagrangian describing three flavors:\footnote{In the following calculations, all indices like spinor and color indices that are unchanged under flavor transformations are suppressed for the sake of clarity. Only the flavor is given explicitly.}
\begin{gather*}
\mathcal{L}_{\text{QCD}}(\bar{q},q) = \sum\limits_{q\in\{\text{u,d,s}\}}\bar{q}\left(i\slashed{D} - m_q\right)q + \mathcal{L}_\text{YM},
\end{gather*}
where $q\in\{\text{u, d, s}\}$ ($q\coloneqq q_\text{L} + q_\text{R}$) is the field of the u, d, or s quark, $D_\mu$ is the covariant derivative of QCD containing $\text{SU}(3)_\text{c}$-gauge fields\footnote{$\text{SU}(3)_\text{c}$ is the gauge group of QCD.}, $m_q$ is -- at this stage, i.e., without formulating a renormalization scheme -- the bare quark mass of u, d, or s generated by spontaneous symmetry breaking of the Higgs field, and $\mathcal{L}_\text{YM}$ is the Yang-Mills Lagrangian of QCD containing the kinetic terms and self-interaction of the gauge fields. For physical applications of the theories we present, we need finite values of quark masses, thus, we need to choose a renormalization scheme. The quark mass values we use and reference throughout this work are $\overline{\text{MS}}$ masses at some scale $\mu$ ($\mu = \SI{2}{GeV}$ for u, d, and s quark and $\mu = \overline{m}_Q$ for heavy quarks $Q$) and taken from review \textit{66. Quark Masses} in \cite{PDG}.\par
We immediately see that all terms in the Lagrangian besides the mass term of the quarks are flavor symmetric\footnote{We will see later on in this section how we have to understand this statement.}. For our investigations in \autoref{chap:hadron_masses} and \autoref{chap:GMO_formula}, we only use that the interaction governing the dynamics of the quarks is flavor symmetric. Indeed, the considerations we make in the course of this work (mostly) apply to all theories where the interaction is flavor symmetric. This is the reason why we suppressed all indices but the flavor indices. If we want to restore the color indices and write the gluon fields explicitly, we have to make the following replacements:
\begin{gather*}
\bar{q}\rightarrow\bar{q}_i,\quad q\rightarrow q_j,\\
\slashed{D}\rightarrow \slashed{D}_{ij} = \gamma^\mu\partial_\mu\delta_{ij} - ig_s\gamma^\mu A^a_\mu T^a_{ij},\\
m_q\rightarrow m_q\delta_{ij},\\
\mathcal{L}_\text{YM} = -\frac{1}{4}G^{\mu\nu;\, a}G^a_{\mu\nu}\quad\text{with}\\
G^{a}_{\mu\nu} \coloneqq \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g_sf^{abc}A^b_\mu A^c_\nu,
\end{gather*}
where we sum over doubly occurring indices, $i$ and $j$ are color indices of the quarks, i.e., ($i$) $j$ is an index transforming under the (anti)fundamental representation of $\text{SU}(3)_\text{c}$, $a$, $b$, and $c$ are color indices of the gluon, i.e., $a$, $b$, and $c$ are indices transforming under the adjoint representation of $\text{SU}(3)_\text{c}$, $g_s$ is the coupling constant of QCD, the fields $A^a_\mu$ are the gauge fields of QCD, the matrices $T^a_{ij}$ are the generators of the adjoint representation of $\text{SU}(3)_\text{c}$, and the constants $f^{abc}$ are the structure constants of $\text{SU}(3)_\text{c}$. Even though the particular kind of interaction is not important for the following considerations as long as the interaction is flavor symmetric, we still label the Lagrangian $\mathcal{L}_\text{QCD}$ with ``QCD'' to indicate that, for our purposes, this Lagrangian already takes the strong interaction into account.\par
Let us now turn to the transformation behavior of $\mathcal{L}_\text{QCD}$ under flavor transformations. By rewriting the Lagrangian as
\begin{gather*}
\mathcal{L}_{\text{QCD}} = \sum\limits_{p,q\in\{\text{u,d,s}\}}\bar{p}\left(i\slashed{D}\delta_{pq} - \mathscr{M}_{pq}\right)q + \mathcal{L}_\text{YM},
\end{gather*}
we see that the quark mass matrix $\mathscr{M}$ is given by:
\begin{gather*}
\mathscr{M} = \begin{pmatrix} m_\text{u} & 0 & 0 \\ 0 & m_\text{d} & 0 \\ 0 & 0 & m_\text{s} \end{pmatrix}.
\end{gather*}
As $\mathscr{M}$ is a diagonal, self-adjoint $(3\times 3)$-matrix, we can express it via $\mathbb{1},\ \lambda_3$ and $\lambda_8$:
\begin{gather}\label{eq:quark_mass_matrix}
\mathscr{M} = \frac{m_\text{u} + m_\text{d} + m_\text{s}}{3}\cdot\mathbb{1} + (m_\text{u} - m_\text{d})\cdot\frac{\lambda_3}{2} + \frac{m_\text{u} + m_\text{d} - 2m_\text{s}}{\sqrt{3}}\cdot\frac{\lambda_8}{2}.
\end{gather}
With this in mind, we can define a flavor transformation of the fields $q$ and the matrix $\mathscr{M}$ under $A\in\text{SU}(3)$:
\begin{align*}
q^\prime &\coloneqq \sum\limits_{\tilde{q}\in\{\text{u,d,s}\}} A_{q\tilde{q}}\, \tilde{q},\\
\sum\limits_{p,q\in\{\text{u,d,s}\}}\bar{p}^{\, \prime}\,\mathscr{M}^\prime_{pq}\, q^\prime &\coloneqq \sum\limits_{p,q\in\{\text{u,d,s}\}}\bar{p}\,\mathscr{M}_{pq}\, q.
\end{align*}
The transformation of $\mathscr{M}$ is then given by:
\begin{gather*}
\mathscr{M}^\prime = A\cdot\mathscr{M}\cdot A^\dagger.
\end{gather*}
The quark mass matrix transformation coincides with the transformation of the triplet mass matrix from \autoref{sec:mass_matrix}. This means that the quark mass matrix decomposes under \text{SU}(3)-flavor transformations into a singlet and an octet, similar to what we have seen in the previous section. The other terms in $\mathcal{L}_{\text{QCD}}$, i.e., $\sum_{q}\bar{q}i\slashed{D}q$ and $\mathcal{L}_\text{YM}$, do not change at all under flavor transformations of the fields $q$ and, hence, transform as a singlet of $\text{SU}(3)$. In this sense, we can say that the Lagrangian decomposes under global flavor transformations, given by
\begin{gather*}
\mathcal{L}^\prime_{\text{QCD}}(\bar{q}^{\, \prime}, q^\prime) \coloneqq \mathcal{L}_{\text{QCD}}(\bar{q}, q),
\end{gather*}
into a singlet and an octet. Under appropriate assumptions, this flavor transformation behavior of the Lagrangian (or of any Lagrangian describing hadrons that decomposes similarly) can be linked to octet enhancement. We will explore this in greater detail in \autoref{sec:EFT+H_Pert}.\par
But before we move on, we take a moment to consider different quark mass configurations and the symmetries/symmetry breakings that arise from these configurations. Let us start by considering the case where all quark masses are equal, i.e., $m_\text{u} = m_\text{d} = m_\text{s}$. In this case, the matrix $\mathscr{M}$ is just the quark mass times the identity and, therefore, a singlet under \text{SU}(3)-flavor transformations. This means that the \text{SU}(3)-flavor transformations are an exact, global symmetry of the Lagrangian.\par
Next, let us consider the case where only two quarks have the same mass. Typically, these quarks are chosen to be u and d meaning $m_\text{u} = m_\text{d}$. Then, according to \autoref{eq:quark_mass_matrix}, $\mathscr{M}$ is a linear combination of $\mathbb{1}$ and $\lambda_8$. Similar to the case of the triplet, the \text{SU}(3)-flavor symmetry is broken now. However, there is still a residual flavor symmetry, namely $\text{SU}(2)\times\text{U}(1)$. Hence, the case where two quark masses are equal is described by $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking and leads to the GMO mass relations, as we will see in \autoref{sec:EFT+H_Pert}.\par
Lastly, let us consider the case where all quark masses are taken to be different. Even though $\mathscr{M}$ is a linear combination of $\mathbb{1}$, $\lambda_3$ and $\lambda_8$ and the \text{SU}(3)-flavor symmetry is clearly broken now, a residual symmetry remains. The transformations of $\mathscr{M}$ where $A\in\text{SU}(3)$ is chosen to describe a change of phases along the diagonal,
\begin{gather*}
A = \begin{pmatrix} e^{i\alpha} & 0 & 0 \\ 0 & e^{i\beta} & 0 \\ 0 & 0 & e^{-i(\alpha + \beta)} \end{pmatrix},
\end{gather*}
leave the quark mass matrix invariant. This group is equivalent to $\text{U}(1)\times \text{U}(1)$. In this sense, the case of the most general parametrization of $\mathscr{M}$ deals with\linebreak $\text{SU}(3)\rightarrow \text{U}(1)\times \text{U}(1)$ symmetry breaking. This case will be important in \autoref{sec:add_con}.
\newpage
\section{\text{SU}(3)-Flavor Symmetry Breaking of Hadron Masses}
\label{sec:EFT+H_Pert}
In this section, we want to obtain the hadron mass patterns and relations discussed in \autoref{sec:mass_matrix} from the Lagrangian
of \autoref{sec:Trafo_QCD}. To do so, we have to link the Lagrangian and its transformation behavior under \text{SU}(3)-flavor transformations to hadron masses. We will investigate two approaches to this problem. For the first approach, one assumes that the hadrons we want to consider are described by fields of an EFT and that the transformation behavior of $\mathcal{L}_\text{QCD}$ ``carries over'' to the EFT Lagrangian. This EFT approach is especially interesting as Feynman's distinction between baryons and mesons arises naturally in this approach. The second approach, which we will call \textit{state formalism} for the remainder of this thesis, is based on the assumption that there are states describing the hadrons. We will see that the state formalism predicts GMO relations without distinguishing between baryons and mesons. We conclude this section with a discussion of Feynman's distinction and problems each approach faces.
\subsection*{EFT Approach}
In the EFT approach, the hadrons are described by fields in an EFT Lagrangian $\mathcal{L}$. For the sake of simplicity, we restrict ourselves to a finite number of scalar mesons represented by scalar fields $\phi_i$ and spin-$\frac{1}{2}$ baryons represented by Dirac fields $\psi_i$. $\mathcal{L}$ contains kinetic terms for the fields and an interaction part $\mathcal{L}_\text{Int}$:
\begin{gather*}
\mathcal{L} = \sum^{n_\text{M}}_{i,j=1}\left(\delta_{ij}\left(\partial_\mu\phi_i\right)^\dagger\left(\partial^\mu\phi_j\right) - \left(M^2_\text{M}\right)_{ij}\phi_i^\dagger\phi_j\right) + \sum^{n_\text{B}}_{i,j=1}\xbar\psi_i\left(\delta_{ij}i\slashed\partial - \left(M_\text{B}\right)_{ij}\right)\psi_j + \mathcal{L}_\text{Int},
\end{gather*}
where $M^2_{\text{M}}$ and $M_{\text{B}}$ are the mass matrices of the mesons and baryons and $n_M$ and $n_B$ are the numbers of mesons and baryons described by the EFT, respectively. We assume now that the fields $\phi_i$ and $\psi_i$ transform via unitary representations\linebreak $D^{(\rho_\text{M})}:\text{SU}(3)\rightarrow \text{GL}(\mathbb{C}^{n_\text{M}})$ and $D^{(\rho_\text{B})}:\text{SU}(3)\rightarrow \text{GL}(\mathbb{C}^{n_\text{B}})$ of \text{SU}(3):
\begin{gather*}
\phi_i \xrightarrow{A\,\in\,\text{SU}(3)} \phi^\prime_i = \sum^{n_\text{M}}_{j=1} \left(D^{(\rho_\text{M})}(A)\right)_{ij}\phi_j\quad\text{and}\quad\psi_i \xrightarrow{A\,\in\,\text{SU}(3)} \psi^\prime_i = \sum^{n_\text{B}}_{j=1} \left(D^{(\rho_\text{B})}(A)\right)_{ij}\psi_j.
\end{gather*}
This allows us to define a transformation of $\mathcal{L}$:
\begin{gather*}
\mathcal{L}^\prime(\phi_i^\prime, \psi_i^\prime) \coloneqq \mathcal{L}(\phi_i, \psi_i).
\end{gather*}
Let us now assume that the transformation behavior of $\mathcal{L}_\text{QCD}$ under $A\in\text{SU}(3)$ ``carries over'' to $\mathcal{L}$. As explained in \autoref{sec:Trafo_QCD}, $\mathcal{L}_\text{QCD}$ decomposes under \text{SU}(3)-flavor transformations into a singlet and an octet:
\begin{gather}\label{eq:L_QCD_decomp}
\mathcal{L}_\text{QCD} = \mathcal{L}^{0}_\text{QCD} + \varepsilon_3\cdot\mathcal{L}^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot\mathcal{L}^{8}_{\text{QCD};\, 8},
\end{gather}
where $\varepsilon_3 \coloneqq m_\text{u} - m_\text{d}$ and $\varepsilon_8 \coloneqq \frac{m_\text{u} + m_\text{d} - 2m_\text{s}}{\sqrt{3}}$ and
\begin{align*}
\mathcal{L}^{0}_{\text{QCD}} &\coloneqq \sum\limits_{q\in\{\text{u,d,s}\}}\bar{q}\left(i\slashed{D} - \frac{m_\text{u} + m_\text{d} + m_\text{s}}{3}\right)q + \mathcal{L}_\text{YM},\\
\mathcal{L}^{8}_{\text{QCD};\, k} &\coloneqq -\sum\limits_{p,q\in\{\text{u,d,s}\}}\frac{\bar{p}\left(\lambda_k\right)_{pq}q}{2}\quad\forall k\in\{1,\ldots, 8\}.
\end{align*}
The singlet term of $\mathcal{L}_{\text{QCD}}$ is given by $\mathcal{L}^{0}_{\text{QCD}}$, while the octet term is a linear combination of $\mathcal{L}^{8}_{\text{QCD};\, 3}$ and $\mathcal{L}^{8}_{\text{QCD};\, 8}$. We assume now that $\mathcal{L}$ transforms to first order in $\text{SU}(3)$-symmetry breaking in a similar way under \text{SU}(3):
\begin{gather}\label{eq:Sing+Oct}
\mathcal{L} = \mathcal{L}^{0} + \tilde{\varepsilon}_3\cdot\mathcal{L}^{8}_{3} + \tilde{\varepsilon}_8\cdot\mathcal{L}^{8}_{8} + \mathcal{O}(\tilde{\varepsilon}_i\tilde{\varepsilon}_j),
\end{gather}
where $\mathcal{L}^{0}$ is a singlet under \text{SU}(3), $\mathcal{L}^{8}_{3}$ and $\mathcal{L}^{8}_{8}$ are the 3rd and 8th component of an octet, and $\tilde{\varepsilon}_3\propto \varepsilon_3$ and $\tilde{\varepsilon}_8\propto \varepsilon_8$ are parameters governing how strongly \text{SU}(3) is broken. This equation with $\tilde{\varepsilon}_3 = 0$ corresponds to $\text{SU}(3)\rightarrow\text{SU}(2)\times \text{U}(1)$ symmetry breaking to first order in $\tilde{\varepsilon}_8$. Since the kinetic terms in $\mathcal{L}$ for both $\phi_i$ and $\psi_i$ and the interaction term $\mathcal{L}_\text{Int}$ do not mix into each other under \text{SU}(3)-transformations of the fields, both kinetic terms and $\mathcal{L}_\text{Int}$ have to transform analogously to \autoref{eq:Sing+Oct} under \text{SU}(3)-transformations. The part containing derivatives of fields in each kinetic term is a singlet under \text{SU}(3), hence, the mass terms have the same decomposition as in \autoref{eq:Sing+Oct}:
\begin{align*}
\sum^{n_\text{M}}_{i,j=1} \left(M^2_\text{M}\right)_{ij}\phi_i^\dagger\phi_j &= \mathcal{L}^{0}_{\text{M}} + \tilde{\varepsilon}_3\cdot\mathcal{L}^{8}_{\text{M};\, 3} + \tilde{\varepsilon}_8\cdot\mathcal{L}^{8}_{\text{M};\, 8} + \mathcal{O}(\tilde{\varepsilon}_i\tilde{\varepsilon}_j),\\
\sum^{n_\text{B}}_{i,j=1}\bar{\psi}_i \left(M_\text{B}\right)_{ij}\psi_j &= \mathcal{L}^{0}_{\text{B}} + \tilde{\varepsilon}_3\cdot\mathcal{L}^{8}_{\text{B};\, 3} + \tilde{\varepsilon}_8\cdot\mathcal{L}^{8}_{\text{B};\, 8} + \mathcal{O}(\tilde{\varepsilon}_i\tilde{\varepsilon}_j),
\end{align*}
where $\mathcal{L}^{0}_{\text{M}/\text{B}}$, $\mathcal{L}^{8}_{\text{M}/\text{B};\, 3}$, and $\mathcal{L}^{8}_{\text{M}/\text{B};\, 8}$ behave similar to \autoref{eq:Sing+Oct}. The transformation of the mass terms under $A\in \text{SU}(3)$ can be expressed as a transformation of the mass matrices:
\begin{align*}
\sum^{n_\text{M}}_{i,j=1} \left(M^{2\,\prime}_\text{M}\right)_{ij}\phi_i^{\prime\, \dagger}\phi^\prime_j = \sum^{n_\text{M}}_{i,j=1} \left(M^{2}_\text{M}\right)_{ij}\phi_i^{\dagger}\phi_j &\Leftrightarrow M^{2\,\prime}_\text{M} = D^{(\rho_\text{M})}(A)\cdot M^2_\text{M}\cdot D^{(\rho_\text{M})}(A)^\dagger,\\
\sum^{n_\text{B}}_{i,j=1}\bar{\psi}^\prime_i \left(M^\prime_\text{B}\right)_{ij}\psi^\prime_j = \sum^{n_\text{B}}_{i,j=1}\bar{\psi}_i \left(M_\text{B}\right)_{ij}\psi_j &\Leftrightarrow M^{\prime}_\text{B} = D^{(\rho_\text{B})}(A)\cdot M_\text{B}\cdot D^{(\rho_\text{B})}(A)^\dagger.
\end{align*}
It is now easy to see that the mass matrices have to satisfy the following structure:
\begin{align*}
M^2_\text{M} &= M^{2;\, 0}_\text{M} + \tilde{\varepsilon}_3\cdot M^{2;\, 8}_{\text{M};\, 3} + \tilde{\varepsilon}_8\cdot M^{2;\, 8}_{\text{M};\, 8} + \mathcal{O}(\tilde{\varepsilon}_i\tilde{\varepsilon}_j),\\
M_\text{B} &= M^{0}_\text{B} + \tilde{\varepsilon}_3\cdot M^{8}_{\text{B};\, 3} + \tilde{\varepsilon}_8\cdot M^{8}_{\text{B};\, 8} + \mathcal{O}(\tilde{\varepsilon}_i\tilde{\varepsilon}_j),\\
\end{align*}
with $M^{2;\, 0}_\text{M}$/$M^{0}_\text{B}$ being singlets of \text{SU}(3) and $M^{2;\, 8}_{\text{M};\, 3}$/$M^{8}_{\text{B};\, 3}$ and $M^{2;\, 8}_{\text{M};\, 8}$/$M^{8}_{\text{B};\, 8}$ being the 3rd and 8th component of an octet of \text{SU}(3), respectively.\par
To proceed, we need to require that \text{SU}(3) is an approximate symmetry, i.e., the \text{SU}(3)-invariant singlet contribution is much larger than the \text{SU}(3)-breaking contribution, in particular the octet contribution. For most hadrons like baryons or heavy mesons, this holds true. Usually, the octet contribution in \text{SU}(3)-multiplets, roughly given by their mass splitting, is of the order of \SI{100}{MeV}, while the singlet contribution given by the average of the masses in the multiplet is about \SI{1}{GeV} or higher. Furthermore, we set $\tilde{\varepsilon}_3$ equal to 0, as the 3rd component of the octet leads to isospin symmetry breaking which we want to discuss in \autoref{sec:add_con}.\par
With this, we can calculate the eigenvalues of the mass matrices, i.e., the hadron masses, by treating the \text{SU}(3)-breaking term as a small perturbation. In order to do so, we need to know the eigenstates and eigenvalues of the unperturbed matrices first. The unperturbed terms are just the singlets. Since $D^{(\rho_{\text{M}/\text{B}})}$ is unitary, we find:
\begin{align*}
D^{(\rho_\text{M})}(A)\cdot M^{2;\, 0}_\text{M}\cdot D^{(\rho_\text{M})}(A)^{-1} &= M^{2;\, 0}_\text{M}\quad\forall A\in \text{SU}(3),\\
D^{(\rho_\text{B})}(A)\cdot M^0_\text{B}\cdot D^{(\rho_\text{B})}(A)^{-1} &= M^0_\text{B}\quad\forall A\in \text{SU}(3)
\end{align*}
or equivalently:
\begin{gather*}
\left[M^{2;\, 0}_\text{M}, D^{(\rho_\text{M})}(A)\right] = 0\quad\text{and}\quad\left[M^0_\text{B}, D^{(\rho_\text{B})}(A)\right] = 0\quad\forall A\in \text{SU}(3).
\end{gather*}
As $D^{(\rho_\text{M})}$ and $D^{(\rho_\text{B})}$ are finite-dimensional, unitary representations, they decompose completely into a direct sum of irreducible representations or multiplets of \text{SU}(3). We will show in \autoref{sec:GMO_formula} that the commutation relations imply the existence of complete orthonormal eigenbases {${\{v^0_{\text{M};\, i}\mid i = 1,\ldots, n_M\}}$} of $M^{2;\, 0}_\text{M}$ and {${\{v^0_{\text{B};\, i}\mid i = 1,\ldots, n_B\}}$} of $M^0_\text{B}$ such that each eigenbasis consists of bases for the multiplets from the decomposition of $D^{(\rho_\text{M})}$ and $D^{(\rho_\text{B})}$ and that the eigenvalues of eigenvectors in the same multiplet are equal. Hence, $M^{2;\, 0}_\text{M}$ and $M^0_\text{B}$ are constant on those multiplets. This implies that the hadron masses are degenerate in each \text{SU}(3)-flavor multiplet, if there is no perturbation, i.e., if \text{SU}(3) is an exact symmetry.\par
With this knowledge, we can calculate the hadron masses to first order in perturbation theory:
\begin{align*}
m^2_{\text{M};\, i} &= v^{0\, \dagger}_{\text{M};\, i}M^{2;\, 0}_\text{M} v^{0}_{\text{M};\, i} + \tilde{\varepsilon}_8\cdot v^{0\, \dagger}_{\text{M};\, i} M^{2;\, 8}_{\text{M};\, 8} v^{0}_{\text{M};\, i} + \mathcal{O}(\tilde{\varepsilon}^{\, 2}_8),\\
m_{\text{B};\, i} &= v^{0\, \dagger}_{\text{B};\, i}M^{0}_\text{B} v^{0}_{\text{B};\, i} + \tilde{\varepsilon}_8\cdot v^{0\, \dagger}_{\text{B};\, i} M^{8}_{\text{B};\, 8} v^{0}_{\text{B};\, i} + \mathcal{O}(\tilde{\varepsilon}^{\, 2}_8),
\end{align*}
where $m^2_{\text{M};\, i}$ is the mass squared of the meson $i$ and $m_{\text{B};\, i}$ is the mass of the baryon $i$. To obtain these formulae, we assumed that it is sufficient to consider the \text{SU}(3)-multiplets separately. We will see in \autoref{sec:GMO_formula} how we can understand this statement. At this point, it should be mentioned that these two formulae are only correct if the eigenbases $\{v^0_{\text{M};\, i}\mid i = 1,\ldots, n_M\}$ and $\{v^0_{\text{B};\, i}\mid i = 1,\ldots, n_B\}$ are chosen such that
\begin{align*}
v^{0\, \dagger}_{\text{M};\, i} M^{2;\, 8}_{\text{M};\, 8} v^{0}_{\text{M};\, j} &= \delta_{ij}\cdot v^{0\, \dagger}_{\text{M};\, i} M^{2;\, 8}_{\text{M};\, 8} v^{0}_{\text{M};\, i}\quad\text{and}\\
v^{0\, \dagger}_{\text{B};\, i} M^{8}_{\text{B};\, 8} v^{0}_{\text{B};\, j} &= \delta_{ij}\cdot v^{0\, \dagger}_{\text{B};\, i} M^{8}_{\text{B};\, 8} v^{0}_{\text{B};\, i},
\end{align*}
if $v^{0}_{\text{M/B};\, i}$ and $v^{0}_{\text{M/B};\, j}$ are elements of the same multiplet. These two conditions are a consequent of degenerate perturbation theory (cf. \autoref{sec:GMO_formula}).\pa
Now we need to convince ourselves that the given expressions for the masses actually reflect the patterns explained in \autoref{sec:mass_matrix}. We only consider the baryonic case for this, the mesonic case works analogously. To do so, we consider a multiplet $D^{(\sigma)}$ from the decomposition of $D^{(\rho_\text{B})}$. Without loss of generality, we can take the first $d = \text{dim}(V^{(\sigma)})$ vectors of $\{v^0_{\text{B};\, i}\mid i = 1,\ldots, n_B\}$ to be a basis of the vector space $V^{(\sigma)}$ on which $D^{(\sigma)}$ acts. Let us define the matrix $m^{(\sigma)}_\text{B}$ by:
\begin{gather*}
m^{(\sigma)}_{\text{B};\, kl} \coloneqq v^{0\, \dagger}_{\text{B};\, k}M^{0}_\text{B} v^{0}_{\text{B};\, l} + \tilde{\varepsilon}_8\cdot v^{0\, \dagger}_{\text{B};\, k} M^{8}_{\text{B};\, 8} v^{0}_{\text{B};\, l}
\end{gather*}
with $k,l\in\{1,\ldots,d\}$. The diagonal elements of the already diagonalized matrix $m^{(\sigma)}_\text{B}$ coincide with the baryon masses of the multiplet $D^{(\sigma)}$ to first order in $\tilde{\varepsilon}_8$. We can define a transformation of $m^{(\sigma)}_\text{B}$ under $A\in\text{SU}(3)$:
\begin{gather*}
m^{(\sigma)\,\prime}_\text{B} \coloneqq D^{(\sigma)}(A)\cdot m^{(\sigma)}_\text{B}\cdot D^{(\sigma)}(A)^\dagger\\
\text{with } \left(D^{(\sigma)}(A)\right)_{kl} \coloneqq v^{0\,\dagger}_{\text{B};\, k}D^{(\rho_\text{B})}(A)v^{0}_{\text{B};\, l}.
\end{gather*}
This transformation corresponds to the transformation of mass matrices defined in \autoref{sec:mass_matrix}. A quick calculation now shows that this transformation is equivalent to a transformation of $M^0_\text{B} + \tilde{\varepsilon}_8\cdot M^{8}_{\text{B};\, 8}$ under $A\in\text{SU}(3)$:
\begin{align*}
m^{(\sigma)\,\prime}_{\text{B};\, kl} &= \sum\limits^d_{r,s = 1} \left(D^{(\sigma)}(A)\right)_{kr}\cdot m^{(\sigma)}_{\text{B};\, rs}\cdot \left(D^{(\sigma)}(A)\right)^\ast_{ls}\\
&= \left[\sum\limits^d_{r=1}\left(D^{(\sigma)}(A)\right)^\ast_{kr} v^{0}_{\text{B};\, r}\right]^\dagger M^0_\text{B} + \tilde{\varepsilon}_8\cdot M^{8}_{\text{B};\, 8}\left[\sum\limits^d_{s=1}\left(D^{(\sigma)}(A)\right)^\ast_{ls} v^{0}_{\text{B};\, s}\right]\\
&= v^{0\,\dagger}_{\text{B};\, k}D^{(\rho_\text{B})}(A)\left(M^0_\text{B} + \tilde{\varepsilon}_8\cdot M^{8}_{\text{B};\, 8}\right)D^{(\rho_\text{B})}(A)^\dagger v^{0}_{\text{B};\, l}\\
&= v^{0\,\dagger}_{\text{B};\, k}\left(M^{0\,\prime}_\text{B} + \tilde{\varepsilon}_8\cdot M^{8\,\prime}_{\text{B};\, 8}\right)v^{0}_{\text{B};\, l},
\end{align*}
where we used
\begin{align*}
\sum\limits^d_{r=1}\left(D^{(\sigma)}(A)\right)^\ast_{kr} v^{0}_{\text{B};\, r} &= \sum\limits^d_{r=1}\left(v^{0\,\dagger}_{\text{B};\, k}D^{(\rho_\text{B})}(A)v^{0}_{\text{B};\, r}\right)^\ast v^{0}_{\text{B};\, r}\\
&= \sum\limits^{n_\text{B}}_{r=1}\left(v^{0\,\dagger}_{\text{B};\, k}D^{(\rho_\text{B})}(A)v^{0}_{\text{B};\, r}\right)^\dagger v^{0}_{\text{B};\, r}\\
&= \sum\limits^{n_\text{B}}_{r=1}\left(v^{0\,\dagger}_{\text{B};\, r}D^{(\rho_\text{B})}(A)^\dagger v^{0}_{\text{B};\, k}\right) v^{0}_{\text{B};\, r}\\
&= \left(\sum\limits^{n_\text{B}}_{r=1} v^{0}_{\text{B};\, r}\cdot v^{0\,\dagger}_{\text{B};\, r}\right)D^{(\rho_\text{B})}(A)^\dagger v^{0}_{\text{B};\, k}\\
&= D^{(\rho_\text{B})}(A)^\dagger v^{0}_{\text{B};\, k}.
\end{align*}
In the second line of the second calculation, we used the fact that the first $d$ eigenvectors $v^0_{\text{B};\, k}$ live in the same invariant subspace and that the eigenbasis is orthogonal, meaning $v^{0\,\dagger}_{\text{B};\, k}D^{(\rho_\text{B})}(A)v^{0}_{\text{B};\, r} = 0$ for $k\leq d$ and $r> d$. In the last line, we used the completeness of the eigenbasis.\par
As $M^0_\text{B}$ transforms as a singlet and $M^8_{\text{B};\, 8}$ transforms as the 8th component of an octet, $m^{(\sigma)}_\text{B}$ transforms as a singlet plus an octet under \text{SU}(3). This means that octet enhancement applies to the transformation of $m^{(\sigma)}_\text{B}$. Furthermore, the 8th component of an octet is invariant under $\text{SU}(2)\times \text{U}(1)$, as we have seen in \autoref{sec:mass_matrix}. Therefore, $m^{(\sigma)}_\text{B}$ is invariant under $\text{SU}(2)\times \text{U}(1)$ as well, since $M^0_\text{B}$ is invariant under any \text{SU}(3)-transformation. Lastly, the \text{SU}(3)-breaking contribution to $m^{(\sigma)}_\text{B}$ is small in comparison to the \text{SU}(3)-invariant term, hence, \text{SU}(3) is an approximate symmetry. In total, the mass matrix $m^{(\sigma)}_\text{B}$ is subject to $\text{SU}(3)\rightarrow \text{SU}(2)\times \text{U}(1)$ symmetry breaking with octet enhancement and approximate \text{SU}(3)-symmetry. As seen in \autoref{sec:mass_matrix}, these properties lead to the GMO mass relations for sextets, octets, and decuplets.\par
Let us summarize what we have found: By describing the hadrons in an EFT and requiring that the transformation behavior of the EFT Lagrangian coincides with the transformation behavior of the fundamental Lagrangian $\mathcal{L}_\text{QCD}$ to first order in flavor symmetry breaking, we found that the baryon masses obey linear GMO relations, whilst meson masses obey quadratic GMO relations to first order in \text{SU}(3)-symmetry breaking. The difference between baryons and mesons can be traced back to their treatment in the EFT Lagrangian: Since baryons are fermions, their mass enters linearly into the Lagrangian. In contrast to that, mesons are bosons and, hence, their mass enters quadratically into the Lagrangian. This means that Feynman's distinction arises naturally in the EFT approach. However, this does not imply that, in Nature, baryons are only subject to linear relations, while mesons only follow quadratic relations. We will discuss this point in detail at the end of this section.
\subsection*{State Formalism}
Alternative to the EFT approach, we can try to link the masses of the hadrons to $\mathcal{L}_\text{QCD}$ via the corresponding Hamilton operator $H_\text{QCD}$. In this approach, the state formalism, we describe the hadrons by eigenstates of $H_\text{QCD}$ where the eigenvalues of these states correspond to the masses of the hadrons. We then calculate these eigenvalues in a perturbative treatment of $H_\text{QCD}$. $H_\text{QCD}$ is given by:
\begin{gather*}
H_\text{QCD} = \int d^3x\, \mathcal{H}_\text{QCD};\quad \mathcal{H}_\text{QCD} = \sum_{q\in\{\text{u,d,s}\}} \frac{\partial \mathcal{L}_\text{QCD}}{\partial \dot{q}}\dot{q} - (\mathcal{L}_\text{QCD} - \mathcal{L}_\text{YM}) + \mathcal{H}_\text{YM},
\end{gather*}
where $\mathcal{H}_\text{YM}$ is the Hamiltonian of $\mathcal{L}_\text{YM}$. With $\frac{\partial \mathcal{L}_\text{QCD}}{\partial \dot{q}} = iq^\dagger$, we obtain:
\begin{gather*}
\mathcal{H}_\text{QCD} = \sum\limits_{q\in\{\text{u,d,s}\}} iq^\dagger\dot{q} + \mathcal{L}_\text{YM} + \mathcal{H}_\text{YM} - \mathcal{L}_\text{QCD}.
\end{gather*}
Since $\sum\limits_{q\in\{\text{u,d,s}\}} iq^\dagger\dot{q} + \mathcal{L}_\text{YM} + \mathcal{H}_\text{YM}$ transforms as a singlet under \text{SU}(3)-transformations and $\mathcal{L}_\text{QCD}$ transforms as a singlet plus an octet, $\mathcal{H}_\text{QCD}$ transforms as a singlet plus an octet as well. Flavor transformations are global transformations, therefore, $H_\text{QCD}$ exhibits the same transformation behavior as $\mathcal{H}_\text{QCD}$ and, in turn, $\mathcal{L}_\text{QCD}$:
\begin{gather*}
H_\text{QCD} = H^{0}_\text{QCD} + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8},
\end{gather*}
where $H^{0}_\text{QCD}$, $H^{8}_{\text{QCD};\, 3}$, and $H^{8}_{\text{QCD};\, 8}$ are defined analogously to \autoref{eq:L_QCD_decomp}.\par
Now, in order to make statements about hadron masses by applying a perturbative treatment to the Hamilton operator, we need to make three assumptions:\footnote{The assumptions listed here do not apply to QFTs in a rigorous mathematical sense. However, they may hold true to good approximation. One can find a discussion of this point among others in \autoref{app:stateform}.}
\begin{itemize}
\item[1)] For every hadron $a$, there exists an eigenstate $\Ket{a}$ with $\Braket{a|a} = 1$ of the Hamilton operator $H_\text{QCD}$ from which the vacuum energy is already subtracted such that the mass $m_a$ of the hadron $a$ is given by
\begin{gather*}
m_a = \Bra{a}H_\text{QCD} \Ket{a}.
\end{gather*}
\item[2)] The subspace $V$ of the physical states which is spanned by the states $\Ket{a}$ from 1), i.e., {${V:= \overline{\text{Span}\left\{\Ket{a}\mid a\text{ hadron}\right\}}}$}, is a Hilbert space.
\item[3)] There is a unitary representation $D^{(\rho)}:V\rightarrow V$ of \text{SU}(3) on $V$ such that the following equation holds for every $A\in\text{SU}(3)$:
\begin{gather*}
\Bra{a} D^{(\rho)}(A)^\dagger\circ H_\text{QCD}\left(\bar{q},\, q\right)\circ D^{(\rho)}(A) \Ket{b} = \Bra{a}H_\text{QCD}\left(\bar{q}^{\, \prime},\, q^\prime\right) \Ket{b}\ \ \forall\Ket{a},\Ket{b}\in V
\end{gather*}
where $q^\prime\coloneqq \sum\limits_{\tilde{q}\in\{\text{u,d,s}\}}A_{q \tilde{q}}\cdot \tilde{q}$.
\end{itemize}
I cannot prove these assumptions, but they are explored and motivated in \autoref{app:stateform}. It should be noted at this point that we have yet to give a proper definition of the hadron mass. We postpone this discussion to \autoref{sec:polemass}.\par
Given these assumptions and assuming that the subtraction of vacuum energy does not spoil the transformation behavior of $H_\text{QCD}$, we can calculate the hadron masses. The following steps are very similar to the calculation of hadron masses in the EFT approach: For the computation of the masses, we make use of perturbation theory again. Like for the EFT approach, we take the \text{SU}(3)-symmetry breaking to be small such that \text{SU}(3) is an approximate symmetry. This means that the eigenvalues of $H^0_\text{QCD}$ are much larger than the eigenvalues of $\varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}$. Furthermore, we take $\varepsilon_3$ to be zero, as, for now, we do not consider isospin symmetry breaking.\par
For a perturbative treatment, we first have to know the eigenvalues and -vectors of the unperturbed operator, i.e., $H^0_\text{QCD}$. However, we are not interested in the entire spectrum of $H_\text{QCD}$, but only in its eigenvalues on $V$. Therefore, we can restrict $H_\text{QCD}$ and likewise $H^0_\text{QCD}$ and $H^8_{\text{QCD};8}$ to $V$. Hereby, we understand the restriction\footnote{The steps performed here may not be rigorous in a mathematical sense. Their purpose is rather to guide our intuition.} $H_\text{QCD}\vert_V$ of $H_\text{QCD}$ to $V$ to be
\begin{gather*}
H_\text{QCD}\vert_V\ket{b} \coloneqq\sum_a \Braket{a|H_\text{QCD}|b}\cdot\ket{a}\quad\forall\ket{b}\in V,
\end{gather*}
where $\{\ket{a}\}$ is any complete orthonormal basis of $V$. We define $H^0_\text{QCD}\vert_V$ and $H^8_{\text{QCD};8}\vert_V$ in a similar way. For the remainder of this section, we will only consider the restricted operators, therefore, we are going to drop ``$\vert_V$'' from now on. Note that we already had to understand $H_\text{QCD}$ in assumption 3) as $H_\text{QCD}\vert_V$, but suppressed ``$\vert_V$''.\par
Using assumption 3) and the fact that $H^0_\text{QCD}$ is a singlet under \text{SU}(3), we find for the unperturbed operator:
\begin{gather*}
D^{(\rho)}(A)\, H^0_\text{QCD}\, D^{(\rho)}(A)^\dagger = H^0_\text{QCD}\ \Leftrightarrow\ \left[D^{(\rho)}(A),\, H^0_\text{QCD}\right] = 0\quad\forall A\in\text{SU}(3).
\end{gather*}
As we will see in \autoref{sec:GMO_formula}, this implies that (the closure of) each eigenspace of $H^0_\text{QCD}$ is an invariant subspace of $D^{(\rho)}$. Similar to the EFT approach, $D^{(\rho)}$ decomposes into multiplets of \text{SU}(3) on this space. This means that $H^0_\text{QCD}$ is constant on multiplets of \text{SU}(3) which again implies that the hadron masses are degenerate in each \text{SU}(3)-flavor multiplet, if there is no perturbation, i.e., if \text{SU}(3) is an exact symmetry.\par
This allows us to compute the hadron masses to first order in perturbation theory. If we consider each multiplet separately (cf. \autoref{sec:GMO_formula}), we find for the hadron masses in a multiplet $D^{(\sigma)}$:
\begin{gather*}
m_{a} = m^{(0)} + \varepsilon_8\cdot\Braket{a^{(\sigma)}|H^8_{\text{QCD};8}|a^{(\sigma)}} + \mathcal{O}\left(\varepsilon_8^2\right),
\end{gather*}
where $m_a$ is the mass of the hadron $a$ in the multiplet $D^{(\sigma)}$, $m^{(0)}$ is the eigenvalue of $H^0_\text{QCD}$ corresponding to the multiplet $D^{(\sigma)}$ and {${\left\{\Ket{a^{(\sigma)}}\mid a\text{ is a hadron in }D^{(\sigma)}\right\}}$} is an orthonormal basis of the multiplet $D^{(\sigma)}$. As for the EFT approach, we made a special choice for the basis $\{\ket{a^{(\sigma)}}\}$ such that $H^8_{\text{QCD};8}$ is diagonal on the multiplet $D^{(\sigma)}$ in this basis.\par
Lastly, it remains to show that the given mass expressions exhibit the very same symmetry structures that led us to the GMO relations in \autoref{sec:mass_matrix}. We do this in the same way, as we did it in the EFT approach.
Let us define the mass matrix $m^{(\sigma)}$ by:
\begin{gather*}
m^{(\sigma)}_{ab} \coloneqq m^{(0)}\cdot\delta_{ab} + \varepsilon_8\cdot\Braket{a^{(\sigma)}|H^8_{\text{QCD};8}|b^{(\sigma)}}.
\end{gather*}
Earlier, we have chosen the basis $\{\ket{a^{(\sigma)}}\}$ such that $m^{(\sigma)}$ is diagonal. To first order in perturbation theory, the eigenvalues of $m^{(\sigma)}$, i.e., its diagonal entries, coincide with the hadron masses. We can define a transformation of $m^{(\sigma)}$ under $A\in\text{SU}(3)$:
\begin{gather*}
m^{(\sigma)\,\prime} \coloneqq D^{(\sigma)}(A)\cdot m^{(\sigma)}\cdot D^{(\sigma)}(A)^\dagger\\
\text{with } \left(D^{(\sigma)}(A)\right)_{ab} \coloneqq \Braket{a^{(\sigma)}|D^{(\rho)}(A)|b^{(\sigma)}}.
\end{gather*}
This transformation corresponds to the transformation of mass matrices defined in \autoref{sec:mass_matrix}. It is fairly easy to see that the transformation of $m^{(\sigma)}$ is given by the transformation of $m^{(0)}\mathbb{1} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}$ (cf. \autoref{sec:GMO_formula}):
\begin{gather*}
m^{(\sigma)\,\prime}_{ab} \coloneqq \Braket{a^{(\sigma)}|m^{(0)}\mathbb{1} + \varepsilon_8\cdot H^{8\,\prime}_{\text{QCD};\, 8}|b^{(\sigma)}}.
\end{gather*}
This means that the mass matrix $m^{(\sigma)}$ transforms as a singlet plus the 8th component of an octet. As the 8th component of an octet is $\text{SU}(2)\times\text{U}(1)$-invariant, $m^{(\sigma)}$ is subject to $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking with octet enhancement and approximate \text{SU}(3)-symmetry. In \autoref{sec:mass_matrix}, we saw that these properties are sufficient to obtain the GMO mass relations. Note that the state formalism makes no difference between fermions and bosons, meaning between baryons and mesons. All hadrons satisfy linear GMO relations in this approach.
\subsection*{Is Feynman's Distinction Physical?}
We have seen that Feynman's distinction arises naturally in the EFT approach. However, this alone is no compelling reason to distinguish the GMO relations for baryons and mesons, that is to say that baryons only satisfy linear relations, while mesons are only described by quadratic relations. In particular, such a statement faces three problems:\par
First of all, it is questionable whether the EFT approach represents a valid model for hadron masses and, if it does, how precise this model is. Hadrons are composite, often unstable particles which are mostly just experimentally accessible as resonances. It is not clear at all that these resonances are ``good'' degrees of freedom, i.e., that it is possible to describe the hadrons as fields. Additionally, it is unclear whether the mass parameters appearing in the EFT Lagrangian are connected to the mass-like quantities of these resonances that are determined in experiments and, if so, how they are connected. Lastly, it is unknown how to obtain the EFT Lagrangian from the fundamental SM Lagrangian or from $\mathcal{L}_\text{QCD}$. It is especially not obvious whether this process spoils the symmetry structure and transformation behavior of the fundamental Lagrangian. If it spoils the symmetry structure, the transformation behavior of the fundamental Lagrangian does not ``carry over'' to the EFT Lagrangian and the EFT approach is invalid. For some multiplets like the pseudoscalar meson octet, one can find in the framework of theories like chiral perturbation theory (cf. \cite{Scherer2011}) that the transformation behavior is actually preserved. Nonetheless, it is not clear whether this holds true for all multiplets. In total, the validity of the EFT approach is at least debatable.\par
Secondly, we have already seen that there is an alternative way of deriving the GMO mass relations which does not lead to Feynman's distinction, namely the state formalism. Truly, this model also faces several problems. There are technical difficulties like the subtraction of vacuum energy that may spoil the symmetry structure of the Hamilton operator and unproven assumptions like the assumptions 1)-3) which enter the state formalism. However, we can at least motivate these assumptions (cf. \autoref{app:stateform}).\par
Lastly, we have to note that both quadratic and linear GMO mass relations are equivalent approximations, if the \text{SU}(3)-symmetry breaking contribution is small enough. This is illustrated by the following consideration: In both the EFT approach and the state formalism, we were able to boil the calculation of the hadron masses down to the computation of eigenvalues of some matrix $N$ that transforms with
\begin{gather*}
N^\prime = D^{(\rho)}(A)\cdot N\cdot D^{(\rho)}(A)^\dagger\quad\text{for } A\in\text{SU}(3),
\end{gather*}
where $D^{(\rho)}$ is a multiplet of \text{SU}(3). This matrix $N$ was in both cases subject to $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking and octet enhancement to first order in some small parameter $\epsilon$ governing the symmetry breaking:
\begin{gather*}
N = N_0 + \epsilon\cdot N^8_8 + \mathcal{O}(\epsilon^2),
\end{gather*}
where $N_0$ is a singlet and $N^8_8$ is the 8th component of an octet. Depending on the approach and hadron multiplet, the matrix $N$ was either the hadron mass matrix or its square. If $N$ is the hadron mass matrix, its square is given by:
\begin{gather*}
N^2 = N^2_0 + \epsilon\left(N_0\cdot N^8_8 + N^8_8\cdot N_0\right) + \mathcal{O}(\epsilon^2).
\end{gather*}
Now, the transformation of $N$ implies a transformation of $N^2$:
\begin{gather*}
N^{2\,\prime} \coloneqq N^\prime\cdot N^\prime = D^{(\rho)}(A)\cdot N^2\cdot D^{(\rho)}(A)^\dagger\quad\text{for } A\in\text{SU}(3)
\end{gather*}
using the unitary of $D^{(\rho)}$. Under this transformation, $N^2_0$ transforms as a singlet and $N_0\cdot N^8_8 + N^8_8\cdot N_0$ transforms as the 8th component of an octet. This means that the eigenvalues of $N^2$, in our example the hadron masses squared, also satisfy GMO relations to first order in symmetry breaking. Hence, the hadrons in a multiplet satisfy quadratic GMO relations to first order in symmetry breaking, if they satisfy linear GMO relations to first order in symmetry breaking and the symmetry breaking in this multiplet is small. Likewise, if $N$ is the hadron mass matrix squared, we can take the square root of the eigenvalues of $N$ and perform a Taylor expansion of the square root about the \text{SU}(3)-invariant term. Note that the symmetry breaking term has to be small for the Taylor expansion to converge. We then find that the square roots of the eigenvalues of $N$, i.e., the hadron masses, coincide with the eigenvalues of the matrix $\sqrt{N}$:
\begin{gather*}
\sqrt{N}\coloneqq \sqrt{n_0}\mathbb{1} + \frac{\epsilon}{2\sqrt{n_0}}\cdot N^8_8 + \mathcal{O}(\epsilon^2),
\end{gather*}
where $N_0 \eqqcolon n_0\mathbb{1}$. Since $\sqrt{N}$ is a singlet plus the 8th component of an octet to first order in $\epsilon$, its eigenvalues and, therefore, the hadron masses satisfy GMO relations to first order in symmetry breaking. Again, this means that the hadrons in a multiplet satisfy linear GMO relations to first order in symmetry breaking, if they satisfy quadratic GMO relations to first order in symmetry breaking and the symmetry breaking in this multiplet is small. With this, we arrive at the statement that both quadratic and linear GMO relations are equivalent approximations in a hadronic multiplet, if the \text{SU}(3)-symmetry breaking contribution in this multiplet is small enough. We will see in \autoref{sec:mass_testing} that this is the case for most known hadronic multiplets.\par
With these three problems regarding Feynman's distinction in mind, we arrive at a rather surprising result: Most of the time, it does not appear to be sensible to ask which kind of mass relation, linear or quadratic, applies to which kind of particle, baryons or mesons, as most hadrons satisfy both relations within the same range of validity. For those hadrons, both kinds of GMO relations are satisfied to first order in flavor symmetry breaking. In this sense, we can say that the EFT approach and the state formalism are not contradicting considerations, but rather complementary and equivalent to some extent. Still, we have to note that one multiplet poses an exception to this statement: For the pseudoscalar meson octet, the quadratic GMO relation is clearly favored over the linear GMO relation (cf. \autoref{sec:mass_testing}). The linear and quadratic GMO mass relation are not equivalent in this case, as the symmetry breaking contribution in this multiplet is roughly of the same size as the \text{SU}(3)-invariant contribution (cf. \autoref{sec:mass_testing}). Nevertheless, the pseudoscalar meson octet cannot be seen as a confirmation of Feynman's distinction, since the linear and quadratic GMO mass relation are not inequivalent because of the spin, but because of the size of the flavor symmetry breaking: For heavier meson octets like the vector meson octet, the equivalence of the linear and quadratic GMO mass relation is restored (cf. \autoref{sec:mass_testing}). To this end, one might say that Feynman's distinction is artificial as a distinction of baryons and mesons into linear and quadratic GMO mass relations is not observable\footnote{We also have to exclude the mass relations following from heavy quark symmetry from this statement (cf. \autoref{sec:heavy_quark} and \autoref{sec:mass_testing}). However, these mass relations are not GMO mass relations as defined in \autoref{sec:mass_matrix}.} for the hadronic multiplets aside from the pseudoscalar meson octet.\par
Even though a global distinction of baryons and mesons into linear and quadratic relations, respectively, cannot be observed, the question remains whether there are reasons to prefer one relation over the other for certain multiplets or for a certain class of multiplets that is not only characterized by spin. We will investigate this question, among other questions, in the following chapters, but in particular in \autoref{sec:mass_testing}.
\newpage
\chapter{Mathematical Derivation of Hadronic Mass Formulae}
\label{chap:GMO_formula}
The main aspects of the derivation of the GMO mass relations were outlined in \autoref{chap:hadron_masses}, but a lot of mostly mathematical details were only glanced over in favor of clarity and simplicity. In this chapter, we want to take a closer look at these aspects on a deeper mathematical level. To do this, we repeat the derivation of the GMO mass relations from \autoref{chap:hadron_masses} in \autoref{sec:GMO_formula}. This time, we start with $\mathcal{L}_\text{QCD}$ and then proceed by linking this Lagrangian to the hadron masses and calculating them in a perturbative treatment. The perturbative description of hadron masses leads us to the notion of multiplets. We find that the hadron masses in a multiplet $\sigma$ transform as $\sigma\otimes\bar{\sigma}$ under $\text{SU}(3)$-flavor transformations and are subject to $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking and octet enhancement. These properties allow us to parametrize the hadron masses in a multiplet.
The mass parametrization gives rise to the GMO mass formula which implies the GMO mass relations. However, we only repeat the derivation of the GMO mass relations for the state formalism and not for the EFT approach. The mathematical statements we derive apply almost in the same manner to the EFT approach as they do to the state formalism, so an additional derivation for the EFT approach would be superficial. The differences between the EFT approach and the state formalism were already presented and discussed in \autoref{sec:EFT+H_Pert}.\par
In \autoref{sec:add_con}, we want to consider additional contributions to the GMO mass formula. In particular, we want to consider isospin symmetry breaking and electromagnetic contributions. In \autoref{sec:heavy_quark}, we will see how we can use heavy quark symmetry to link hadronic mass formulae of different multiplets that only differ by the exchange of a charm with a bottom quark.
\section{The Gell-Mann--Okubo Mass Formula}
\label{sec:GMO_formula}
As already stated, we want to repeat the derivation of the GMO mass relations by starting with $\mathcal{L}_\text{QCD}$:
\begin{gather*}
\mathcal{L}_{\text{QCD}}(\bar{q},q) = \sum\limits_{q\in\{\text{u,d,s}\}}\bar{q}\left(i\slashed{D} - m_q\right)q + \mathcal{L}_\text{YM},
\end{gather*}
where we used the same notations as in \autoref{sec:Trafo_QCD}. Defining \text{SU}(3)-flavor transformations of the fields $q$ implies \text{SU}(3)-transformations of $\mathcal{L}_\text{QCD}$:
\begin{align*}
q&\xrightarrow{A\in\text{SU}(3)}q^\prime \coloneqq \sum\limits_{p\in\{\text{u,d,s}\}} A_{qp}p,\\
\bar{q}&\xrightarrow{A\in\text{SU}(3)}\bar{q}^\prime \coloneqq \sum\limits_{p\in\{\text{u,d,s}\}} A^\ast_{qp}\bar{p}\\
\Rightarrow\mathcal{L}_\text{QCD}&\xrightarrow{A\in\text{SU}(3)}\mathcal{L}^\prime_\text{QCD}\text{ with } \mathcal{L}^\prime_\text{QCD}(\bar{q}^\prime, q^\prime) \coloneqq \mathcal{L}_\text{QCD}(\bar{q},q).
\end{align*}
We have seen in \autoref{sec:Trafo_QCD} and \autoref{sec:EFT+H_Pert} that $\mathcal{L}_\text{QCD}$ transforms under \text{SU}(3)-flavor transformations as a singlet plus an octet:
\begin{gather*}
\mathcal{L}_\text{QCD} = \mathcal{L}^{0}_\text{QCD} + \varepsilon_3\cdot\mathcal{L}^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot\mathcal{L}^{8}_{\text{QCD};\, 8},
\end{gather*}
where $\varepsilon_3 \coloneqq m_\text{u} - m_\text{d}$ and $\varepsilon_8 \coloneqq \frac{m_\text{u} + m_\text{d} - 2m_\text{s}}{\sqrt{3}}$ and
\begin{align*}
\mathcal{L}^{0}_{\text{QCD}} &\coloneqq \sum\limits_{q\in\{\text{u,d,s}\}}\bar{q}\left(i\slashed{D} - \frac{m_\text{u} + m_\text{d} + m_\text{s}}{3}\right)q + \mathcal{L}_\text{YM},\\
\mathcal{L}^{8}_{\text{QCD};\, k} &\coloneqq -\sum\limits_{p,q\in\{\text{u,d,s}\}}\frac{\bar{p}\left(\lambda_k\right)_{pq}q}{2}\quad\forall k\in\{1,\ldots, 8\}.
\end{align*}
In \autoref{sec:EFT+H_Pert}, we have seen that this transformation behavior of $\mathcal{L}_\text{QCD}$ also applies to the corresponding Hamilton operator $H_\text{QCD}$ of $\mathcal{L}_\text{QCD}$:
\begin{gather*}
H_\text{QCD} = H^{0}_\text{QCD} + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8},
\end{gather*}
where $H^{0}_\text{QCD}$ is a singlet of \text{SU}(3) and $H^{8}_{\text{QCD};\, 3}$ and $H^{8}_{\text{QCD};\, 8}$ are the 3rd and 8th component of an octet, respectively. For now, we set $m_\text{u} = m_\text{d}$. This implies that $\varepsilon_3 = 0$. In this case, $\text{SU}(2)\times\text{U}(1)\subset\text{SU}(3)$ is an exact global symmetry of $H_\text{QCD}$. We are going to investigate the case of $m_\text{u}\neq m_\text{d}$, i.e., the case of isospin symmetry breaking, in \autoref{sec:add_con}.\par
Like explained, we mainly want to consider the state formalism in this section. The state formalism makes the following three assumptions:
\begin{itemize}
\item[1)] For every hadron $a$, there exists an eigenstate $\Ket{a}$ with $\Braket{a|a} = 1$ of the Hamilton operator $H_\text{QCD}$ from which the vacuum energy is already subtracted such that the mass $m_a$ of the hadron $a$ is given by
\begin{gather*}
m_a = \Bra{a}H_\text{QCD} \Ket{a}.
\end{gather*}
\item[2)] The subspace $V$ of the physical states which is spanned by the states $\Ket{a}$ from 1), i.e., {${V:= \overline{\text{Span}\left\{\Ket{a}\mid a\text{ hadron}\right\}}}$}, is a Hilbert space.
\item[3)] There is a unitary representation $D^{(\rho)}:V\rightarrow V$ of \text{SU}(3) on $V$ such that the following equation holds for every $A\in\text{SU}(3)$:
\begin{gather*}
\Bra{a} D^{(\rho)}(A)^\dagger\circ H_\text{QCD}\left(\bar{q},\, q\right)\circ D^{(\rho)}(A) \Ket{b} = \Bra{a}H_\text{QCD}\left(\bar{q}^{\, \prime},\, q^\prime\right) \Ket{b}\ \ \forall\Ket{a},\Ket{b}\in V
\end{gather*}
where $q^\prime\coloneqq \sum\limits_{\tilde{q}\in\{\text{u,d,s}\}}A_{q \tilde{q}}\cdot \tilde{q}$.
\end{itemize}
Again, we understand $H_\text{QCD}$, $H^0_\text{QCD}$, $H^8_{\text{QCD};3}$, and $H^8_{\text{QCD};8}$ in assumption 3) and from now on as operators restricted and projected down to $V$, like explained in \autoref{sec:EFT+H_Pert}. A motivation of these assumptions is given in \autoref{app:stateform}. Before we can apply a perturbative treatment to $H_\text{QCD}$ to calculate the hadron masses, we need two more statements. The first one is a statement about the theory at hand and depends on the free parameters in $\mathcal{L}_\text{QCD}$. It is the assumption that the octet term $\varepsilon_8\cdot H^8_{\text{QCD};8}$ is actually just a small correction to the Hamilton operator $H_\text{QCD}$ such that we can treat it as a small perturbation. In Nature, we find this to be a sensible assumption for most hadron masses. The second statement is the assumption that the subtraction of vacuum energy does not spoil the transformation behavior of $H_\text{QCD}$. Like the assumptions 1)-3), I cannot prove this, as, in general, the subtraction of vacuum energy for an arbitrary QFT is not known. However, the subtraction of vacuum energy for a free QFT is accomplished by taking the normal ordered product of the Hamilton operator. The normal ordered product does not spoil the transformation behavior of the Hamilton operator, hence, the second statement is at least true for free QFTs. This makes it seem likely that the second statement is also true for other QFTs.\par
Now, we want to calculate the hadron masses in a perturbative treatment of $H_\text{QCD}$ with $\varepsilon_8\cdot H^8_{\text{QCD};8}$ as a small perturbation. Obviously, we need to know (degenerate) perturbation theory for this. At this point, it is instructive to quickly recapitulate perturbation theory.
\subsection*{Degenerate Perturbation Theory}
Let us consider linear operators $H(\varepsilon):V\rightarrow V$ on a Hilbert space $V$. Suppose that $H(\varepsilon)$ is given by:
\begin{gather*}
H(\varepsilon) = H_0 + \varepsilon\cdot\Delta H,
\end{gather*}
where $H_0$ is a self-adjoint linear operator on $V$, $\Delta H$ is a linear operator on $V$, and $\varepsilon$ is a ``small'' parameter, i.e., the contribution of $\varepsilon\cdot\Delta H$ to the eigenvectors and -values of $H(\varepsilon)$ is small and can be written as a Taylor series in $\varepsilon$. Now suppose that $\Ket{a(\varepsilon)}$ is such an eigenvector of $H(\varepsilon)$ with eigenvalue $m(\varepsilon)$ which can be expanded in a Taylor series:
\begin{align*}
\Ket{a(\varepsilon)} &= \ket{a^{(0)}} + \varepsilon\cdot \ket{a^{(1)}} + \mathcal{O}(\varepsilon^2),\\
m(\varepsilon) &= m^{(0)} + \varepsilon\cdot m^{(1)} + \mathcal{O}(\varepsilon^2).
\end{align*}
The eigenvector equation for $\Ket{a(\varepsilon)}$ then reads:
\begin{align*}
&H(\varepsilon)\Ket{a(\varepsilon)} = m(\varepsilon) \Ket{a(\varepsilon)}\\
\Rightarrow\ &H_0\ket{a^{(0)}} + \varepsilon\cdot (\Delta H\ket{a^{(0)}} + H_0\ket{a^{(1)}}) + \mathcal{O}(\varepsilon^2)\\
&= m^{(0)}\ket{a^{(0)}} + \varepsilon\cdot (m^{(1)}\ket{a^{(0)}} + m^{(0)}\ket{a^{(1)}}) + \mathcal{O}(\varepsilon^2).
\end{align*}
Note that the last equation holds true if we can commute $H_0$ and $\Delta H$ with the infinite sum of the Taylor expansion. This is, for instance, possible, if $H_0$ and $\Delta H$ are continuous or, equivalently, bounded which is always the case, if $V$ is finite-dimensional. As the coefficients of a Taylor series are unique, the equation above has to be satisfied to every order in $\varepsilon$. The equation for the zeroth order reads:
\begin{gather*}
H_0\ket{a^{(0)}} = m^{(0)}\ket{a^{(0)}}.
\end{gather*}
This equation states that $\ket{a^{(0)}}$ is an eigenvector of $H_0$ with eigenvalue $m^{(0)}$. The first order equation is given by:
\begin{gather}\label{eq:pert_e1}
\Delta H\ket{a^{(0)}} + H_0\ket{a^{(1)}} = m^{(1)}\ket{a^{(0)}} + m^{(0)}\ket{a^{(1)}}.
\end{gather}
We want to solve \autoref{eq:pert_e1} for $m^{(1)}$ now. This task is rather easy, if $\ket{a^{(0)}}$ is non-degenerate, i.e., if the eigenspace $V_{m^{(0)}}$ of $H_0$ with eigenvalue $m^{(0)}$ is just given by $\text{Span}\{\ket{a^{(0)}}\}$. In this case, we simply act with $\bra{a^{(0)}}$ on \autoref{eq:pert_e1}:
\begin{align*}
\braket{a^{(0)}|\Delta H|a^{(0)}} + \braket{a^{(0)}|H_0|a^{(1)}} &= \braket{a^{(0)}|\Delta H|a^{(0)}} + m^{(0)}\braket{a^{(0)}|a^{(1)}}\\
&= m^{(1)}\braket{a^{(0)}|a^{(0)}} + m^{(0)}\braket{a^{(0)}|a^{(1)}}\\
\Rightarrow m^{(1)} &= \frac{\braket{a^{(0)}|\Delta H|a^{(0)}}}{\braket{a^{(0)}|a^{(0)}}},
\end{align*}
where we used the hermicity of $H_0$. If we insert $\ket{\tilde{a}^{(0)}} = c\cdot \ket{a^{(0)}}$ ($c\in\mathbb{C}\backslash\{0\}$) instead of $\ket{a^{(0)}}$ into the expression for $m^{(1)}$, the equation does not change. This means that we can use any non-zero vector from the eigenspace $V_{m^{(0)}} = \text{Span}\{\ket{a^{(0)}}\}$ to calculate $m^{(1)}$.\par
However, if $\ket{a^{(0)}}$ is degenerate, which, in general, it is, the eigenspace $V_{m^{(0)}}$ is not just one-dimensional and it is not immediately obvious how to choose $\ket{a^{(0)}}$. In this case, let us suppose that there is an orthonormal basis $\{\ket{\alpha}\mid \alpha\in I\}$ of $V_{m^{(0)}}$ for some index set $I$.
We then have:
\begin{gather*}
\ket{a^{(0)}} = \sum\limits_{\alpha\in I} \braket{\alpha|a^{(0)}}\cdot \ket{\alpha}.
\end{gather*}
Let us now act with $\bra{\alpha}$ on \autoref{eq:pert_e1}:
\begin{gather*}
\braket{\alpha|\Delta H|a^{(0)}} + \braket{\alpha|H_0|a^{(1)}} = \braket{\alpha|\Delta H|a^{(0)}} + m^{(0)}\braket{\alpha|a^{(1)}} = m^{(1)}\braket{\alpha|a^{(0)}} + m^{(0)}\braket{\alpha|a^{(1)}}\\
\Rightarrow \braket{\alpha|\Delta H|a^{(0)}} = m^{(1)}\braket{\alpha|a^{(0)}}\\
\Rightarrow \sum\limits_{\alpha\in I} \braket{\alpha|\Delta H|a^{(0)}}\cdot \ket{\alpha} = m^{(1)}\sum\limits_{\alpha\in I} \braket{\alpha|a^{(0)}}\cdot \ket{\alpha} = m^{(1)}\ket{a^{(0)}}.
\end{gather*}
If we define the linear operator\footnote{Of course, this only makes sense, if the operator $\Delta H\vert_{\overline{V_{m^{(0)}}}}$ exists, which is not clear at this stage.} $\Delta H\vert_{\overline{V_{m^{(0)}}}}:\overline{V_{m^{(0)}}}\rightarrow\overline{V_{m^{(0)}}}, \ket{b}\mapsto \sum\limits_{\alpha\in I} \braket{\alpha|\Delta H|b}\cdot \ket{\alpha}$ on the closure $\overline{V_{m^{(0)}}}$ of $V_{m^{(0)}}$, the equation above states that $\ket{a^{(0)}}$ is an eigenvector of $\Delta H\vert_{\overline{V_{m^{(0)}}}}$ with eigenvalue $m^{(1)}$.\par
These considerations give us a method on how to calculate the eigenvalue $m(\varepsilon)$ to first order in $\varepsilon$: First, we have to calculate the eigenvalues and -spaces of $H_0$ to obtain the possible values for the zeroth order contribution $m^{(0)}$ of $m(\varepsilon)$. Next, we have to diagonalize the restriction $\Delta H\vert_{\overline{V_{m^{(0)}}}}$ of $\Delta H$ to (the closure of) each eigenspace $\overline{V_{m^{(0)}}}$ to determine the possible values for the first order contribution $m^{(1)}$ of $m(\varepsilon)$. Again, we are going to drop the restriction ``$\vert_{\overline{V_{m^{(0)}}}}$'' in further discussions. We should note at this point that we have to be cautious, if there are eigenstates of $H_0$ outside of $V_{m^{(0)}}$ that are ``almost degenerate'' to $V_{m^{(0)}}$, i.e, if the difference between the eigenvalues of these eigenstates and the eigenvalue $m^{(0)}$ of $V_{m^{(0)}}$ is in the order of or small in comparison to $m^{(1)}$ originating from the perturbation $\varepsilon\cdot\Delta H$. In this case, we can split up $H_0$ into $H^\prime_0$ and $\Delta H^\prime$ such that all ``nearly degenerate'' states of $H_0$ are now exactly degenerate for $H^\prime_0$ and $\Delta H^\prime$ is a small perturbation in the order of the difference between the eigenvalues. We then treat $H^\prime_0$ as the large contribution and $\Delta H^\prime + \varepsilon\cdot\Delta H$ as the perturbation.\\\par
We want to apply this now to $H_\text{QCD} = H^0_\text{QCD} + \varepsilon_8
\cdot H^8_{\text{QCD};8}$, where we treat $\varepsilon_8\cdot H^8_{\text{QCD};8}$ as a small perturbation. For this, we have to investigate the eigenvalues and -spaces of $H^0_\text{QCD}$.
\subsection*{Eigenspaces of $H^0_\text{\normalfont{QCD}}$}
$H^0_\text{QCD}$ is a singlet under \text{SU}(3). Together with assumption 3), we can rewrite this property as:
\begin{gather*}
D^{(\rho)}(A)\, H^0_\text{QCD}\, D^{(\rho)}(A)^\dagger = H^0_\text{QCD}\ \Leftrightarrow\ \left[D^{(\rho)}(A),\, H^0_\text{QCD}\right] = 0\quad\forall A\in\text{SU}(3).
\end{gather*}
Now let $V_m$ be the eigenspace of $H^0_\text{QCD}$ to the eigenvalue $m$ and $\ket{\alpha}$ an element of $V_m$, then:
\begin{gather*}
H^0_\text{QCD}D^{(\rho)}(A)\ket{\alpha} = D^{(\rho)}(A)H^0_\text{QCD}\ket{\alpha} = D^{(\rho)}(A)m\ket{\alpha} = m D^{(\rho)}(A)\ket{\alpha}\quad\forall A\in\text{SU}(3).
\end{gather*}
This equation states that $D^{(\rho)}(A)\ket{\alpha}$ is also an eigenvector of $H^0_\text{QCD}$ with eigenvalue $m$, if $\ket{\alpha}$ is an eigenvector with eigenvalue $m$, hence, $D^{(\rho)}(A)(V_m) = V_m\quad\forall A\in\text{SU}(3)$. Now consider the closure $\overline{V_m}$ of $V_m$. Let $\ket{\alpha}$ be an element of $\overline{V_m}$, then we can write $\ket{\alpha} = \lim_{n\to\infty} \ket{\alpha_n}$ with $(\ket{\alpha_n})_{n\in\mathbb{N}}$ being a Cauchy sequence in $V_m$. For every $A\in\text{SU}(3)$, $D^{(\rho)}(A)$ is unitary, hence, $D^{(\rho)}(A)$ is bounded and, therefore, continuous. With this, we obtain:
\begin{gather*}
D^{(\rho)}(A)\ket{\alpha} = D^{(\rho)}(A)(\lim_{n\to\infty} \ket{\alpha_n}) = \lim_{n\to\infty} D^{(\rho)}(A)\ket{\alpha_n}\quad\forall A\in\text{SU}(3).
\end{gather*}
This means that $D^{(\rho)}(A)\ket{\alpha}$ is an element of $\overline{V_m}$ for every $A\in\text{SU}(3)$ which implies that $D^{(\rho)}(A)(\overline{V_m}) = \overline{V_m}\ \forall A\in\text{SU}(3)$. As a closed subspace of the Hilbert space $V$, $\overline{V_m}$ itself is a Hilbert space. Therefore, the restriction $D^{(\rho)}\vert_{\overline{V_m}}:\text{SU}(3)\rightarrow\text{GL}(\overline{V_m})$, $A\mapsto D^{(\rho)}(A)\vert_{\overline{V_m}}$ of $D^{(\rho)}$ is a unitary representation of the compact Lie group \text{SU}(3) on the Hilbert space $\overline{V_m}$. Applying the Peter-Weyl theorem (cf. \cite{Knapp2001}), $\overline{V_m}$ can be written as the closure of the direct sum of irreducible, finite-dimensional, and orthogonal spaces $W_i$:
\begin{gather*}
\overline{V_m} = \overline{\bigoplus_i W_i}.
\end{gather*}
This means that if we choose an orthonormal basis for every $W_i$ (which is possible as they all are finite-dimensional) and combine all these bases into one set, we obtain a complete orthonormal basis for $\overline{V_m}$ only consisting of complete multiplets of \text{SU}(3).\par
Note that all multiplets in the complete orthonormal basis are complex representations, as $D^{(\rho)}$ is a complex representation. Also note that either $W_i\cap V_m = \{0\}$ or $W_i\subset V_m$ for every i, because if there is an element $\ket{\alpha}$ of $W_i\backslash\{0\}$ that is an eigenvector with eigenvalue $m$, then $W\coloneqq \text{Span}\{D^{(\rho)}(A)\ket{\alpha}\mid A\in\text{SU}(3)\}$ is a subset of $V_m$ and a closed invariant subspace of $W_i$. However, the only closed invariant subspaces of the irreducible representation $W_i$ are $\{0\}$ and $W_i$, hence, $W_i = W$ because of $0\neq \ket{\alpha}\in W$. If $V_m$ is closed, all $W_i$ are trivially contained in the eigenspace $V_m$ as, in this case, $\overline{V_m} = V_m$. $V_m$ is closed, if $H^0_\text{QCD}\vert_{\overline{V_m}}:\overline{V_m}\rightarrow H^0_\text{QCD}(\overline{V_m})$ is bounded (or, equivalently, continuous) or if $V_m$ is finite-dimensional.\par
Now suppose that there is a complete orthonormal eigenbasis $\{\ket{a}\mid a\in I\}$ of $H^0_\text{QCD}$ for some index set $I$, then every $\ket{a}$ is contained in an eigenspace $V_{m_a}$ for some eigenvalue $m_a$. Now choose an orthonormal basis for every eigenspace $\overline{V_{m_a}}$ such that each basis only consists of complete multiplets of \text{SU}(3). Combine all these bases into one set and denote it by $\{\ket{\alpha}\mid \alpha\in\tilde{I}\}$ for some index set $\tilde{I}$. All vectors of {${\{\ket{\alpha}\mid \alpha\in\tilde{I}\}}$} are pairwise orthogonal, as the spaces $\overline{V_{m_a}}$ are pairwise orthogonal because of the hermicity of $H^0_\text{QCD}$. Furthermore, $\ket{a}\in\overline{\text{Span}\{\ket{\alpha}\mid \alpha\in\tilde{I}\}}$ for every {${a\in I}$}, as {${\{\ket{\alpha}\mid \alpha\in\tilde{I}\}}$} contains an orthonormal basis for every $\overline{V_{m_a}}$ and {${\ket{a}\in V_{m_a}\subset\overline{V_{m_a}}}$}. This implies:
\begin{gather*}
V = \overline{\text{Span}\{\ket{a}\mid a\in I\}} \subset \overline{\text{Span}\left(\overline{\text{Span}\{\ket{\alpha}\mid \alpha\in\tilde{I}\}}\right)} = \overline{\text{Span}\{\ket{\alpha}\mid \alpha\in\tilde{I}\}}\subset V\\
\Rightarrow V = \overline{\text{Span}\{\ket{\alpha}\mid \alpha\in\tilde{I}\}}.
\end{gather*}
This means that $\{\ket{\alpha}\mid \alpha\in\tilde{I}\}$ is a complete orthonormal basis of $V$ that only consists of complete multiplets of \text{SU}(3) where every such multiplet is completely contained in one eigenspace $\overline{V_{m_a}}$.\\\par
Let us summarize what we found: There is a complete orthonormal basis for every closure $\overline{V_m}$ of an eigenspace $V_m$ of $H^0_\text{QCD}$ where this basis only consists of complete finite-dimensional multiplets of \text{SU}(3). Furthermore, if $H^0_\text{QCD}$ is diagonalizable, i.e., if there is a complete orthonormal eigenbasis of $H^0_\text{QCD}$, there is a complete orthonormal basis of $V$ only consisting of complete finite-dimensional multiplets of \text{SU}(3) where every such multiplet is completely contained in the closure of some eigenspace of $H^0_\text{QCD}$. Before we move on, let us make some remarks:
\begin{enumerate}
\item We only need the hermicity and the \text{SU}(3)-invariance of $H^0_\text{QCD}$ for the eigenspace analysis given above. Therefore, the very same statements apply to $M^{2;0}_\text{M}$ and $M^0_\text{B}$ from the EFT approach in \autoref{sec:EFT+H_Pert}. For both matrices $M^{2;0}_\text{M}$ and $M^0_\text{B}$, the underlying Hilbert spaces were assumed to be finite-dimensional, so all eigenspaces are closed and $M^{2;0}_\text{M}$ and $M^0_\text{B}$ are diagonalizable. This means that there is a complete orthonormal eigenbasis for each matrix $M^{2;0}_\text{M}$ and $M^0_\text{B}$ such that each basis consists of complete multiplets of \text{SU}(3) which each are completely contained in some eigenspace.
\item If there are multiple ``nearly degenerate'' eigenspaces of $H^0_\text{QCD}$ with respect to the perturbation $\varepsilon_8\cdot H^8_{\text{QCD};8}$, we need to diagonalize the perturbation on the closure of the direct sum of the ``nearly degenerate'' eigenspaces. One can show in similar fashion to the considerations above that there is a complete orthonormal basis for the closure of any direct sum of eigenspaces only consisting of complete finite-dimensional multiplets of \text{SU}(3).
\end{enumerate}
Now, we want to calculate the contribution of the perturbation $\varepsilon_8\cdot H^8_{\text{QCD};8}$ to the hadron masses in first order. In order to do so, we have to diagonalize $H^8_{\text{QCD};8}$ on (the closure of the direct sum of) some (``nearly degenerate'') eigenspace(s) of $H^0_\text{QCD}$. Let us call this space $W$. As we have just seen, there exists a complete orthonormal basis of $W$ only consisting of complete finite-dimensional multiplets of \text{SU}(3). In general, however, $H^8_{\text{QCD};8}$ is not diagonal in such a basis. It is possible that multiple multiplets in $W$ contribute to the same hadron mass. Nevertheless, we only want to consider the case where the diagonalization of $H^8_{\text{QCD};8}$ on $W$ is compatible with the multiplet structure of $W$, i.e, $H^8_{\text{QCD};8}$ on $W$ is diagonal in multiplets of $\text{SU}(3)$. In this case, there exists a complete orthonormal basis of $W$ only consisting of complete finite-dimensional multiplets of \text{SU}(3) such that this basis is also an eigenbasis of $H^8_{\text{QCD};8}$ on $W$. This assumption is a good approximation in multiple scenarios: Firstly, it is possible that the space $W$ at hand only consists of one multiplet. Then, $H^8_{\text{QCD};8}$ on $W$ is trivially diagonal in multiplets of \text{SU}(3). In Nature, this scenario applies when the mass difference between the average mass of the hadron multiplet at hand and any other hadron multiplet is larger than the mass contribution of the perturbation $\varepsilon_8\cdot H^8_{\text{QCD};8}$. Secondly, even if the space $W$ consists of several multiplets, the mixing of these multiplets in $H^8_{\text{QCD};8}$ might be negligibly small or suppressed such that we can diagonalize $H^8_{\text{QCD};8}$ on each multiplet separately. This suppression may originate from quantum numbers that are preserved under $H^8_{\text{QCD};8}$ like baryon number, total angular momentum, parity, and so on. For instance, the contribution of mesons to baryon masses should be zero, as mesons and baryons have different total angular momenta and, therefore, cannot mix.\par
The assumption that $H^8_{\text{QCD};8}$ on $W$ is diagonal in \text{SU}(3)-multiplets allows us to compute the hadron masses in each multiplet individually. Thus, we only have to parametrize the hadron masses in each of these multiplets in order to find the GMO mass formula. However, this task is still quite difficult and it is not clear yet how we can accomplish this. Therefore, we wish to rephrase the problem to clearly see what we have to do. For this, we consider the transformation behavior of the hadron masses in \text{SU}(3)-multiplets.
\subsection*{Flavor Transformation Behavior of Hadronic Mass Multiplets}
Applying the assumption that $H^8_{\text{QCD};8}$ on $W$ is diagonal in multiplets of \text{SU}(3), we can diagonalize $H^8_{\text{QCD};8}$ on each of these multiplets separately. Let us now pick out one of these finite-dimensional multiplets and call it $D^{(\sigma)}$. Then, the mass of the hadron $a$ in the multiplet $D^{(\sigma)}$ is given by:
\begin{gather*}
m_{a} = m^{(0)} + \varepsilon_8\cdot\Braket{a^{(\sigma)}|H^8_{\text{QCD};8}|a^{(\sigma)}} + \mathcal{O}\left(\varepsilon_8^2\right),
\end{gather*}
where $m^{(0)}$ is the eigenvalue of $H^0_\text{QCD}$ corresponding to the multiplet $D^{(\sigma)}$ and\linebreak $\left\{\Ket{a^{(\sigma)}}\mid a\text{ is a hadron in }D^{(\sigma)}\right\}$ is a basis of the multiplet $D^{(\sigma)}$ chosen such that it is an orthonormal eigenbasis of $H^8_{\text{QCD};8}$ restricted to the multiplet $D^{(\sigma)}$.\par
Although we have found this formula, we still need to parametrize\linebreak $\Braket{a^{(\sigma)}|H^8_{\text{QCD};8}|a^{(\sigma)}}$ to obtain the GMO mass formula. It is easier to solve this problem, if we rephrase it. Consider the matrix $m^{(\sigma)}$:
\begin{gather*}
m^{(\sigma)}_{ab} \coloneqq m^{(0)}\cdot\delta_{ab} + \varepsilon_8\cdot\Braket{a^{(\sigma)}|H^8_{\text{QCD};8}|b^{(\sigma)}}.
\end{gather*}
Since $\left\{\Ket{a^{(\sigma)}}\mid a\text{ is a hadron in }D^{(\sigma)}\right\}$ is an eigenbasis of $H^8_{\text{QCD};8}$ on $D^{(\sigma)}$, $m^{(\sigma)}$ is diagonal. This means that the eigenvalues of $m^{(\sigma)}$, i.e., its diagonal entries coincide with the hadron masses of the hadrons in $D^{(\sigma)}$ to first order. Furthermore, we can define the following transformation of $m^{(\sigma)}$ under $A\in\text{SU}(3)$:
\begin{gather*}
m^{(\sigma)\,\prime} \coloneqq D^{(\sigma)}(A)\cdot m^{(\sigma)}\cdot D^{(\sigma)}(A)^\dagger\\
\text{with } \left(D^{(\sigma)}(A)\right)_{ab} \coloneqq \Braket{a^{(\sigma)}|D^{(\rho)}(A)|b^{(\sigma)}}.
\end{gather*}
This makes $m^{(\sigma)}$ a hadronic mass matrix transforming under $\sigma\otimes\bar{\sigma}$. We investigated such mass matrices in \autoref{sec:mass_matrix}. $\left\{\Ket{a^{(\sigma)}}\mid a\text{ is a hadron in }D^{(\sigma)}\right\}$ is a complete orthonormal basis of the multiplet $D^{(\sigma)}$. We can extend such a basis into a complete orthonormal basis $\{\Ket{\alpha}\}$ of the entire Hilbert space $V$ $D^{(\rho)}$ is acting on. With this, we see that $m^{(\sigma)}$ transforms as a singlet plus the 8th component of an octet under $A\in\text{SU}(3)$:
\begin{align*}
m^{(\sigma)\,\prime}_{ab} &= \sum\limits_{c,d} \left(D^{(\sigma)}(A)\right)_{ac}\cdot m^{(\sigma)}_{cd}\cdot \left(D^{(\sigma)}(A)\right)^\ast_{bd}\\
&= \left[\sum\limits_{c}\left(D^{(\sigma)}(A)\right)^\ast_{ac} \Ket{c^{(\sigma)}}\right]^\dagger m^{(0)}\mathbb{1} + \varepsilon_8\cdot H^8_{\text{QCD};8}\left[\sum\limits_{d}\left(D^{(\sigma)}(A)\right)^\ast_{bd} \Ket{d^{(\sigma)}}\right]\\
&= \Braket{a^{(\sigma)}|D^{(\rho)}(A)\left(m^{(0)}\mathbb{1} + \varepsilon_8\cdot H^8_{\text{QCD};8}\right)D^{(\rho)}(A)^\dagger|b^{(\sigma)}}\\
&= \Braket{a^{(\sigma)}|m^{(0)}\mathbb{1} + \varepsilon_8\cdot H^{8\,\prime}_{\text{QCD};8}|b^{(\sigma)}},
\end{align*}
where we used
\begin{align*}
\sum\limits_{c}\left(D^{(\sigma)}(A)\right)^\ast_{ac} \Ket{c^{(\sigma)}} &= \sum\limits_{c}\Braket{a^{(\sigma)}|D^{(\rho)}(A)|c^{(\sigma)}}^\ast \Ket{c^{(\sigma)}}\\
&= \sum\limits_{c}\Braket{a^{(\sigma)}|D^{(\rho)}(A)|c^{(\sigma)}}^\dagger \Ket{c^{(\sigma)}}\\
&= \sum\limits_{c}\Ket{c^{(\sigma)}}\Braket{c^{(\sigma)}|D^{(\rho)}(A)^\dagger|a^{(\sigma)}}\\
&= \sum\limits_{\alpha}\left(\Ket{\alpha^{\color{white}(}}\Bra{\alpha^{\color{white}(}}\right)D^{(\rho)}(A)^\dagger\Ket{a^{(\sigma)}}\\
&= D^{(\rho)}(A)^\dagger\Ket{a^{(\sigma)}}.
\end{align*}
In the fourth line of the calculation, we used the unitarity of $D^{(\rho)}$ and the fact that $D^{(\sigma)}$ as a multiplet in $D^{(\rho)}$ is an invariant subspace of $D^{(\rho)}$. In the last line, we used the completeness of $\{\Ket{\alpha}\}$.\par
With this consideration, we find that the masses of the hadrons in the multiplet $D^{(\sigma)}$ are given to first order in flavor symmetry breaking by the eigenvalues of a mass matrix transforming under $\sigma\otimes\bar{\sigma}$ and decomposing into a singlet plus the 8th component of an octet under this transformation. As we have seen in \autoref{chap:hadron_masses}, this implies octet enhancement and $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking. Therefore, we now want to parametrize and diagonalize mass matrices transforming under $\sigma\otimes\bar{\sigma}$ for arbitrary complex finite-dimensional multiplets $\sigma$ which are subject to octet enhancement and $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking.
\subsection*{Multiplet Classification}
In \autoref{sec:mass_matrix}, we already studied mass matrices transforming under $\sigma\otimes\bar{\sigma}$ for a small selection of multiplets $\sigma$, namely for singlets, triplets, sextets, octets, and decuplets. In the course of this investigation, we observed some reoccurring features which we listed at the end of \autoref{sec:mass_matrix}. This list provides a guideline for the remainder of this section and tells us which aspects we need to investigate. We start by examining how many singlets and octets occur in the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ for arbitrary complex finite-dimensional multiplets $\sigma$. To answer this question, we derive the following lemma:
\begin{Lem}
Let $G$ be a compact Lie group, $\sigma$ a complex, finite-dimensional, and irreducible representation of $G$, and $\rho$ a complex finite-dimensional representation of $G$. Then $n_\sigma(\rho) = n_1(\bar{\sigma}\otimes\rho),$ where $1$ is the trivial representation of $G$ and $n_\mu(\nu)$ denotes the multiplicity of an irreducible representation $\mu$ of $G$ in the decomposition of a finite-dimensional representation $\nu$ of $G$ into irreducible representations, i.e., $n_\mu(\nu)$ denotes how often $\mu$ occurs in a decomposition of $\nu$ into irreducible representations.
\end{Lem}
\begin{proof}
Let $G$, $\sigma$, and $\rho$ be like above. As $\rho$ is a finite-dimensional representation of a compact Lie group, it decomposes completely into a direct sum of (finite-dimensional) irreducible representations of $G$ (cf. \cite{Knapp2001}):
\begin{gather*}
\rho = \bigoplus^n_{i=1}\rho_i,
\end{gather*}
where $\rho_i$ is an irreducible representation of $G$ for every $i\in\{1,\ldots,n\}$ and $n\in\mathbb{N}$. Then, the decomposition of $\bar{\sigma}\otimes\rho$ into irreducible representations can be obtained by decomposing $\bar{\sigma}\otimes\rho_i$ into irreducible representations for every $i$:
\begin{gather*}
\bar{\sigma}\otimes\rho = \bigoplus^n_{i=1}\bar{\sigma}\otimes\rho_i.
\end{gather*}
Denote the vector spaces $\sigma$ and $\rho_i$ are acting on with $V^{(\sigma)}$ and $V^{(\rho_i)}$, respectively. Then, $\bar{\sigma}\otimes\rho_i$ can be understood as:
\begin{gather*}
D^{(\bar{\sigma}\otimes\rho_i)}:G\rightarrow\text{GL}\left(\text{Hom}\left(V^{(\sigma)},V^{(\rho_i)}\right)\right),\\ g\mapsto\left(L\mapsto D^{(\bar{\sigma}\otimes\rho_i)}(g)(L) = D^{(\rho_i)}(g)\circ L\circ D^{(\sigma)}(g)^\dagger\right)
\end{gather*}
with Hom$\left(V^{(\sigma)}, V^{(\rho_i)}\right)\coloneqq\left\{L\mid L:V^{(\sigma)}\rightarrow V^{(\rho_i)}\text{ linear}\right\}$ being the vector space of linear maps from $V^{(\sigma)}$ to $V^{(\rho_i)}$. Suppose there is a $L\in\text{Hom}\left(V^{(\sigma)}, V^{(\rho_i)}\right)$ such that $D^{(\bar{\sigma}\otimes\rho_i)}(g)(L) = L\ \forall g\in G$, then:
\begin{gather*}
D^{(\rho_i)}(g)\circ L = L\circ D^{(\sigma)}(g)\ \forall g\in G,
\end{gather*}
since $\sigma$ can be chosen to be unitary, as it is finite-dimensional (cf. \cite{Knapp2001}). By Schur's lemma (cf. \cite{Knapp2001}), $\sigma$ and $\rho_i$ are either equivalent or $L = 0$. If $\sigma$ and $\rho_i$ are equivalent, we can assume $\sigma = \rho_i$ without loss of generality. Then:
\begin{gather*}
D^{(\sigma)}(g)\circ L = L\circ D^{(\sigma)}(g)\ \forall g\in G
\end{gather*}
Again, by Schur's lemma (cf. \cite{Knapp2001}), all maps $L$ that satisfy this equation are the multiples of the identity. This means that $\bar{\sigma}\otimes\rho_i$ contains exactly one trivial representation, if $\sigma$ and $\rho_i$ are equivalent, and no trivial representation, if they are not. Since $\bar{\sigma}\otimes\rho$ is the direct sum of all $\bar{\sigma}\otimes\rho_i$, the trivial representation $1$ occurs as often in the decomposition of $\bar{\sigma}\otimes\rho$ as $\sigma$ in the decomposition of $\rho$:
\begin{gather*}
n_\sigma(\rho) = n_1(\bar{\sigma}\otimes\rho).
\end{gather*}
\end{proof}
Using this lemma, we can easily calculate the number of singlets in the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ for arbitrary finite-dimensional multiplets $\sigma$ of \text{SU}(3):
\begin{gather*}
n_1(\sigma\otimes\bar{\sigma}) = n_1(\bar{\sigma}\otimes\sigma) = n_\sigma(\sigma) = 1.
\end{gather*}
This means that there is exactly one singlet in $\sigma\otimes\bar{\sigma}$. If we choose $\sigma$ to be unitary, which we always can and will do, the multiples of the identity form trivially a singlet and, therefore, are the only singlet in $\sigma\otimes\bar{\sigma}$.\par
Now, we want to determine the number of octets in $\sigma\otimes\bar{\sigma}$. Although this task is more laborious than calculating the number of singlets, we can again use the previous lemma:
\begin{gather*}
n_8(\sigma\otimes\bar{\sigma}) = n_1(\bar{8}\otimes(\sigma\otimes\bar{\sigma})) = n_1(\bar{\sigma}\otimes(\sigma\otimes\bar{8})) = n_\sigma(\sigma\otimes\bar{8}) = n_\sigma(\sigma\otimes 8),
\end{gather*}
where we used the fact that $8$ and $\bar{8}$ are equivalent in the last step. This equation allows us to calculate the number of multiplets $\sigma$ in $\sigma\otimes 8$ to obtain the number of octets in $\sigma\otimes\bar{\sigma}$. We will accomplish this by determining the Clebsch-Gordan series of $\sigma\otimes 8$ for arbitrary complex finite-dimensional multiplets $\sigma$. However, in order to do this, we need to classify the complex finite-dimensional multiplets of \text{SU}(3) first.\par
All complex finite-dimensional multiplets of \text{SU}(3) can be uniquely characterized by two non-negative integers $p$ and $q$ (cf. \cite{DeSwart1963}, \cite{Lichtenberg}, and review \textit{46. $\text{SU}(n)$ Multiplets and Young Diagrams} in \cite{PDG}). Given two numbers $p$ and $q$, we denote the corresponding multiplet by $D(p,q)$ following \cite{DeSwart1963}. In this notation, the complex conjugate representation of $D(p,q)$ is equivalent to $D(q,p)$, $\overline{D(p,q)} = D(q,p)$, and the dimension of $D(p,q)$ is given by $(p+1)(q+1)\frac{p+q+2}{2}$ (cf. {\cite{DeSwart1963}}). There is also a graphical representation of this classification. We can assign each multiplet $D(p,q)$ a Young tableau (cf. \cite{Lichtenberg}, \cite{sternberg1995}, and review \textit{46. $\text{SU}(n)$ Multiplets and Young Diagrams} in \cite{PDG}). A Young tableau is a diagram consisting of $n\in\mathbb{N}_0$ boxes which are ordered in rows and columns such that all rows are left-aligned. The number of boxes in one row cannot increase from top to bottom. For \text{SU}(3), a Young tableau has no more than three rows, but is not restricted in the number of columns. However, any column containing three boxes can be dropped (cf. \cite{Lichtenberg}). The Young tableau corresponding to $D(p,q)$ consists of $p+2q$ boxes arranged in the following way:
\begin{gather*}
D(p,q) = \ytableausetup{boxsize = 2.5em}\begin{ytableau} \, & \none[\dots] & \, & \, & \none[\dots] & \, \\ \, & \none[\dots] & \,\\ \none[1] & \none[\dots] & \none[q] & \none[q+1] & \none[\dots] & \none[q+p]\end{ytableau}
\end{gather*}
Of course, we can express the multiplets of \text{SU}(3) that already appeared in the course of this thesis, mainly the singlet, triplet, sextet, octet, and decuplet, in terms of $p$ and $q$ and Young tableaux:
\begin{gather*}
1 = D(0,0) = \emptyset\\
3 = D(1,0) = \ytableausetup{boxsize = 1em, centertableaux}\ydiagram{1},\quad \bar{3} = D(0,1) = \ydiagram{1,1}\\
6 = D(2,0) = \ydiagram{2},\quad \bar{6} = D(0,2) = \ydiagram{2,2}\\
8 = D(1,1) = \ydiagram{2,1}\\
10 = D(3,0) = \ydiagram{3},\quad \overline{10} = D(0,3) = \ydiagram{3,3}\\
27 = D(2,2) = \ydiagram{4,2}\\
64 = D(3,3) = \ydiagram{6,3},
\end{gather*}
where we denoted the Young tableau consisting of no boxes with $\emptyset$. Before we move on, we want to single out one important class of multiplets. We will see that the mass matrix decomposition for non-trivial multiplets with $p = 0$ or $q = 0$ is simpler than for the other non-trivial multiplets and all follow the same structure. Therefore, we want to denote them separately. We call a multiplet of the type $D(p,0)$ with $p\in\mathbb{N}$ or $D(0,q)$ with $q\in\mathbb{N}$ \textit{totally symmetric}.\par
One interesting aspect of Young tableaux is that they are very well suited for the calculation of the Clebsch-Gordan series of tensor product representations. One can derive a collection of rules for the Young tableaux -- readily available in literature (cf. \cite{Lichtenberg} and review \textit{46. $\text{SU}(n)$ Multiplets and Young Diagrams} in \cite{PDG}, for instance) -- on how to perform this calculation. We use these rules for the Young tableaux to determine the Clebsch-Gordan series of $\sigma\otimes 8$ for arbitrary complex finite-dimensional multiplets $\sigma$. For this, we represent $8$ by $D(1,1)$ and $\sigma$ by $D(p,q)$ with $p$ and $q$ being non-negative integers. As the actual calculations are rather lengthy, we only present the results here and display the full computation in \autoref{app:Young}. We find:\\\\
\textbf{Singlet}
\begin{flalign*}
D(0,0)\otimes D(1,1) &= D(1,1)&&
\end{flalign*}
\textbf{Totally symmetric multiplets}
\begin{flalign*}
D(1,0)\otimes D(1,1) &= D(1,0)\oplus D(0,2)\oplus D(2,1)&&\\
D(p,0)\otimes D(1,1) &= D(p-2,1)\oplus D(p,0)\oplus D(p-1,2)\oplus D(p+1,1),\quad p\geq 2&&\\
D(0,1)\otimes D(1,1) &= D(0,1)\oplus D(2,0)\oplus D(1,2)&&\\
D(0,q)\otimes D(1,1) &= D(1,q-2)\oplus D(0,q)\oplus D(2,q-1)\oplus D(1,q+1),\quad q\geq 2&&
\end{flalign*}
\textbf{Remaining multiplets}
\begin{flalign*}
D(1,1)\otimes D(1,1) &= D(0,0)\oplus D(1,1)\oplus D(1,1)\oplus D(3,0)\oplus D(0,3)\oplus D(2,2)&&\\
D(p,1)\otimes D(1,1) &= D(p-1,0)\oplus D(p-2,2)\oplus D(p,1)\oplus D(p,1)\oplus D(p+2,0)&&\\
&\ \ \ \oplus D(p-1,3)\oplus D(p+1,2),\quad p\geq 2&&\\
D(1,q)\otimes D(1,1) &= D(0,q-1)\oplus D(2,q-2)\oplus D(1,q)\oplus D(1,q)\oplus D(0,q+2)&&\\
&\ \ \ \oplus D(3,q-1)\oplus D(2,q+1),\quad q\geq 2&&\\
D(p,q)\otimes D(1,1) &= D(p-1,q-1)\oplus D(p+1,q-2)\oplus D(p-2,q+1)&&\\
&\ \ \ \oplus D(p,q)\oplus D(p,q)\oplus D(p+2,q-1)\oplus D(p-1,q+2)&&\\
&\ \ \ \oplus D(p+1,q+1),\quad p,q\geq 2&&
\end{flalign*}
Looking at the Clebsch-Gordan series, we see that the singlet $1$ does not occur in the decomposition of $1\otimes 8$, that every totally symmetric multiplet $\sigma$ appears exactly once in the decomposition of $\sigma\otimes 8$, and that every remaining multiplet $\sigma$ appears exactly twice in the decomposition of $\sigma\otimes 8$. Therefore, the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ contains no octet for $\sigma$ being the singlet, exactly one octet for totally symmetric multiplets $\sigma$, and exactly two octets for the remaining multiplets $\sigma$.\par
Although we have found the number of octets in every hadronic mass matrix, we still need to identify the $\text{SU}(2)\times\text{U}(1)$-symmetric elements of these octets and parametrize them. When discussing the structure of the octet in \autoref{sec:rel_within_multiplets}, we will observe that the octet contains exactly one $\text{SU}(2)\times\text{U}(1)$-singlet. This means that all $\text{SU}(2)\times\text{U}(1)$-symmetric elements in the octet are given by one non-zero element, aside from multiplication with scalars. Hence, every octet in the decomposition of the mass matrix gives rise to one contribution to the hadron masses. We now describe these contributions by parametrizing every octet appearing in the hadronic mass matrices transforming under $\sigma\otimes\bar{\sigma}$.
\subsection*{Mass matrix Parametrization \Romannum{1}: Constructing Octets}
For the parametrization of the octets in $\sigma\otimes\bar{\sigma}$, we need some mathematical background knowledge of Lie group theory. Consider a Lie group $G$, for instance \text{SU}(3). As $G$ is a differentiable manifold, we can assign every point $g\in G$ a tangent space $T_gG$, a real finite-dimensional vector space, and define the tangent bundle {${TG\coloneqq \bigcup_{g\in G}T_gG}$}. With this, we can define vector fields on $G$ as smooth sections of the tangent bundle $TG$. There exists a Lie bracket on the space of vector fields on $G$, called the commutator. Since $G$ is also a Lie group, one can define an important class of vector fields on $G$, the left-invariant vector fields (cf. \cite{Hamilton2017}). They span a real finite-dimensional subspace of the vector fields on $G$ and are closed under the commutator, i.e., the commutator of two left-invariant vector fields is also left-invariant. This turns the restriction of the commutator to the left-invariant vector fields into a Lie bracket on the left-invariant vector fields. Furthermore, the subspace of left-invariant vector fields is isomorphic to the tangent space $T_eG$ of $G$ at the neutral element $e\in G$. With this, we can define a Lie bracket on $T_eG$ by identifying the Lie bracket/commutator on $T_eG$ with the commutator on the left-invariant vector fields via the canonical isomorphism. We call the real finite-dimensional vector space $T_eG$ equipped with this Lie bracket/commutator the Lie algebra of $G$ (cf. \cite{Hamilton2017}). Usually, we denote the Lie algebra of a Lie group with lower case letters, often in Fraktur, for short. For instance, we denote the Lie algebra of $G$ with $\mathfrak{g}$ and the Lie algebra of \text{SU}(3) with $\mathfrak{su}(3)$.\par
If $G$ is a matrix group, i.e., if $G$ is a(n) (embedded) Lie subgroup of the Lie group $\text{GL}(V)$ for a finite-dimensional, real or complex vector space $V$, we can make an additional identification of the Lie algebra $\mathfrak{g}$: For matrix groups $G\subset\text{GL}(V)$, the tangent space $T_eG$ is isomorphic to a real subspace of {${\text{End}(V)\coloneqq\{A:V\rightarrow V\mid \text{A linear}\}}$}. Using the canonical isomorphism between this subspace and $T_eG$, we can equip this subspace with a Lie bracket. This Lie bracket then coincides with the standard commutator for matrices (cf. \cite{Hamilton2017}):
\begin{gather*}
[A,B]\equiv A\circ B - B\circ A\quad\forall A,B\in\text{End}(V).
\end{gather*}
Therefore, we also call this real subspace of $\text{End}(V)$ equipped with the commutator the Lie algebra $\mathfrak{g}$ of the Lie group $G$, if $G\subset\text{GL}(V)$ is a matrix group. In this sense, the Lie algebra $\mathfrak{gl}(V)$ of the Lie group $\text{GL}(V)$ and $\mathfrak{su}(3)$ of \text{SU}(3) are given by (cf. \cite{Hamilton2017}):
\begin{align*}
\mathfrak{gl}(V) &= \text{End}(V),\\
\mathfrak{su}(3) &= \{A\in\text{Mat}(3\times 3,\mathbb{C})\mid A^\dagger = -A\text{ and }\text{Tr}(A) = 0\}.
\end{align*}
We will use both identifications of the Lie algebra -- the Lie algebra as the tangent space at the neutral element and as a subspace of $\text{End}(V)$ -- in this work.\par
Let us now consider a multiplet $D^{(\sigma)}:\text{SU}(3)\rightarrow\text{GL}(V)$ of \text{SU}(3) on a finite-dimensional, complex vector space $V$.
By the definition of a representation, the map $\Phi^{(\sigma)}:\text{SU}(3)\times V\rightarrow V, (A,v)\mapsto D^{(\sigma)}(A)(v)$ is continuous. As $V$ is a finite-dimensional vector space, the continuity of $\Phi^{(\sigma)}$ implies the continuity of $D^{(\sigma)}$. $D^{(\sigma)}$ is a continuous group homomorphism between Lie groups, hence, $D^{(\sigma)}$ is also smooth and a Lie group homomorphism (cf. \cite{Hamilton2017}). \text{SU}(3) and $\text{GL}(V)$ are differentiable manifolds and $D^{(\sigma)}$ is smooth, therefore, we can define a differential $DD^{(\sigma)}\vert_A$ of $D^{(\sigma)}$ for every $A\in\text{SU}(3)$ that maps vectors from the tangent space $T_A\text{SU}(3)$ linearly into the tangent space $T_{D^{(\sigma)}(A)}\text{GL}(V)$. Hence, the differential $DD^{(\sigma)}\vert_\mathbb{1}$ at $\mathbb{1}\in\text{SU}(3)$ is a linear map between Lie algebras:
\begin{gather*}
DD^{(\sigma)}\vert_\mathbb{1}:\mathfrak{su}(3)\rightarrow\mathfrak{gl}(V).
\end{gather*}
As $D^{(\sigma)}$ is a Lie group homomorphism, $DD^{(\sigma)}\vert_\mathbb{1}$ is a Lie algebra homomorphism (cf. \cite{Hamilton2017}), i.e.:
\begin{gather*}
\left[DD^{(\sigma)}\vert_\mathbb{1}(X),DD^{(\sigma)}\vert_\mathbb{1}(Y)\right] = DD^{(\sigma)}\vert_\mathbb{1}\left(\left[X,Y\right]\right)\quad\forall X,Y\in\mathfrak{su}(3).
\end{gather*}
Furthermore, the image {${\text{Im}(D^{(\sigma)})\coloneqq\{D^{(\sigma)}(A)\mid A\in\text{SU}(3)\}}$} of $D^{(\sigma)}$ is a compact subgroup of $\text{GL}(V)$, since \text{SU}(3) is compact and $D^{(\sigma)}$ is a continuous group homomorphism. By Cartan's theorem (cf. \cite{Hamilton2017}), this makes $\text{Im}(D^{(\sigma)})$ an (embedded) Lie subgroup of $\text{GL}(V)$. The Lie algebra $\mathfrak{im}(D^{(\sigma)})$ of $\text{Im}(D^{(\sigma)})$ is, therefore, a real subspace of the Lie algebra $\mathfrak{gl}(V)$ of $\text{GL}(V)$. The differential $DD^{(\sigma)}\vert_\mathbb{1}$ only maps into $\mathfrak{im}(D^{(\sigma)})$, hence, $DD^{(\sigma)}\vert_\mathbb{1}$ is a Lie algebra homomorphism between $\mathfrak{su}(3)$ and $\mathfrak{im}(D^{(\sigma)})$:
\begin{gather*}
DD^{(\sigma)}\vert_\mathbb{1}:\mathfrak{su}(3)\rightarrow\mathfrak{im}(D^{(\sigma)}).
\end{gather*}
One can further show that the map $DD^{(\sigma)}\vert_\mathbb{1}$ is surjective on $\mathfrak{im}(D^{(\sigma)})$\footnote{One can see this in the following way: If $\sigma$ is the trivial representation, the statement is obvious. If $\sigma$ is not trivial, one can apply Sard's theorem to show that $DD^{(\sigma)}\vert_A$ has to be surjective on $T_{D^{(\sigma)}(A)}\text{Im}(D^{(\sigma)})$ for at least one $A\in\text{SU}(3)$. As $D^{(\sigma)}$ is a Lie group homomorphism, this has to be true for every $A\in\text{SU}(3)$, so in particular for $A = \mathbb{1}$.}\linebreak (cf. Ch. \Romannum{3}, § 3, no. 8, Proposition 28 in \cite{Bourbaki}). Now consider the kernel\linebreak {${K\coloneqq\left\{X\in\mathfrak{su}(3)\mid DD^{(\sigma)}\vert_\mathbb{1}(X)=0\right\}}$} of $DD^{(\sigma)}\vert_\mathbb{1}$. The kernel $K$ is a subspace of $\mathfrak{su}(3)$ satisfying:
\begin{gather*}
DD^{(\sigma)}\vert_\mathbb{1}\left(\left[X,Y\right]\right) = \left[DD^{(\sigma)}\vert_\mathbb{1}(X),0\right] = 0\quad\forall X\in\mathfrak{su}(3)\,\forall Y\in K\\
\Rightarrow \left[X,Y\right]\in K\quad\forall X\in\mathfrak{su}(3)\,\forall Y\in K.
\end{gather*}
This implies that $K$ is an ideal of $\mathfrak{su}(3)$. However, \text{SU}(3) is a simple Lie group, hence, its Lie algebra $\mathfrak{su}(3)$ is also simple. A simple Lie algebra has no proper ideals, i.e., every ideal is either $\{0\}$ or the entire Lie algebra. Therefore, we find $K=\{0\}$ or $K=\mathfrak{su}(3)$. If $K$ equates to $\mathfrak{su}(3)$, $DD^{(\sigma)}\vert_\mathbb{1}(X)$ is zero for every $X\in\mathfrak{su}(3)$. It is easy to see that this implies $DD^{(\sigma)}\vert_A = 0\forall A\in\text{SU}(3)$. Using the connectedness of \text{SU}(3) and the fact that $D^{(\sigma)}$ is a Lie group homomorphism, we thus obtain $D^{(\sigma)}(A)=\mathbb{1}\forall A\in\text{SU}(3)$. Therefore, every finite-dimensional complex multiplet $D^{(\sigma)}$ of \text{SU}(3) for which $K$ is equal to $\mathfrak{su}(3)$ is equivalent to the trivial representation of \text{SU}(3).\par
If $D^{(\sigma)}$ is not trivial or, equivalently, if the kernel $K$ is $\{0\}$, the Lie algebra homomorphism $DD^{(\sigma)}\vert_\mathbb{1}$ is bijective, thus, the Lie algebras $\mathfrak{su}(3)$ and $\mathfrak{im}(D^{(\sigma)})$ are isomorphic. This implies that $D^{(\sigma)}$ is a local diffeo- and isomorphism on $\text{Im}(D^{(\sigma)})$ and that the Lie groups \text{SU}(3) and $\text{Im}(D^{(\sigma)})$ are locally diffeo- and isomorphic.\par
From now on, we want to choose $D^{(\sigma)}$ such that it is unitary. As \text{SU}(3) is compact, we can always achieve this by a similarity transformation of $D^{(\sigma)}$ (cf. \cite{Knapp2001}). If $D^{(\sigma)}$ is not unitary, one has to transform the octets in $\sigma\otimes\bar{\sigma}$ we are going to identify with the help of such a similarity transformation. Define the representation $D^{(\sigma\otimes\bar{\sigma})}$:
\begin{gather*}
D^{(\sigma\otimes\bar{\sigma})}:\text{SU}(3)\rightarrow\text{GL}\left(\text{End}(V)\right),\\
D^{(\sigma\otimes\bar{\sigma})}(A)(X)\coloneqq D^{(\sigma)}(A)\circ X\circ D^{(\sigma)}(A)^\dagger = D^{(\sigma)}(A)\circ X\circ D^{(\sigma)}(A)^{-1}.
\end{gather*}
As $\mathfrak{gl}(V)=\text{End}(V)$ is the Lie algebra of $\text{GL}(V)$, $\mathfrak{im}(D^{(\sigma)})$ is a real subspace of $\text{End}(V)$. Let $\tilde{X}$ be an element of $\mathfrak{im}(D^{(\sigma)})$, then we can express it as $\tilde{X} = DD^{(\sigma)}\vert_\mathbb{1}(X)$ with $X\in\mathfrak{su}(3)$. If we insert $\tilde{X}$ in $D^{(\sigma\otimes\bar{\sigma})}$, we find for $A\in\text{SU}(3)$:
\begin{align*}
D^{(\sigma\otimes\bar{\sigma})}(A)(\tilde{X}) &= \left(D^{(\sigma\otimes\bar{\sigma})}(A)\circ DD^{(\sigma)}\vert_\mathbb{1}\right)(X)\\
&= D^{(\sigma)}(A)\circ DD^{(\sigma)}\vert_\mathbb{1}(X)\circ D^{(\sigma)}(A)^{-1}\\
&= D^{(\sigma)}(A)\circ \left(\left.\frac{d}{dt}\right\vert_{t=0}D^{(\sigma)}\left(\exp(tX)\right)\right)\circ D^{(\sigma)}(A^{-1})\\
&= \left.\frac{d}{dt}\right\vert_{t=0} D^{(\sigma)}(A)\circ D^{(\sigma)}\left(\exp(tX)\right)\circ D^{(\sigma)}(A^{-1})\\
&= \left.\frac{d}{dt}\right\vert_{t=0}D^{(\sigma)}\left(A\exp(tX)A^{-1}\right)\\
&= DD^{(\sigma)}\vert_\mathbb{1}\left(AXA^{-1}\right)\\
&= DD^{(\sigma)}\vert_\mathbb{1}\left(D^{(3\otimes\bar{3})}(A)(X)\right)\\
&= DD^{(\sigma)}\vert_\mathbb{1}\left(D^{(8)}(A)(X)\right)\\
&= \left(DD^{(\sigma)}\vert_\mathbb{1}\circ D^{(8)}(A)\right)(X),
\end{align*}
where $D^{(3\otimes\bar{3})}(A)(X)\coloneqq AXA^\dagger = AXA^{-1}$ is the tensor product representation of the fundamental representation $3$ of \text{SU}(3) with its conjugate representation $\bar{3}$. Before we discuss the last two lines of this calculation, we have to pay attention to the difference between real and complex vector spaces: The calculation shows that every element of $\mathfrak{im}(D^{(\sigma)})$ is mapped into $\mathfrak{im}(D^{(\sigma)})$ under $D^{(\sigma\otimes\bar{\sigma})}$. However, $\mathfrak{im}(D^{(\sigma)})$ is not an invariant subspace of $\sigma\otimes\bar{\sigma}$, as $\mathfrak{im}(D^{(\sigma)})$ is a real vector space and $\sigma\otimes\bar{\sigma}$ is a complex representation, i.e., operates on a complex vector space. Nevertheless, we can extend $\mathfrak{im}(D^{(\sigma)})$ to an invariant subspace of $\sigma\otimes\bar{\sigma}$ via complexification. If we take the complexification of $\mathfrak{im}(D^{(\sigma)})$ to be {${\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})}$}, the calculation above and the linearity of $D^{(\sigma\otimes\bar{\sigma})}(A)$ are sufficient to show that {${\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})}$} is an invariant subspace of $\sigma\otimes\bar{\sigma}$.\par
Similar remarks are important for the understanding of the last two lines: We have seen in \autoref{sec:mass_matrix} that $3\otimes\bar{3}$ as a real representation, i.e., $3\otimes\bar{3}$ only acting on Hermitian $(3\times 3)$-matrices, admits a non-trivial invariant subspace, the space of traceless, Hermitian $(3\times 3)$-matrices. $3\otimes\bar{3}$ restricted to this subspace is a real irreducible representation which we identified as the (real) octet $8$. If we complexify all the spaces, similar statements apply. $3\otimes\bar{3}$ as a complex representation, i.e., $3\otimes\bar{3}$ acting on all complex $(3\times 3)$-matrices, admits the space of traceless, complex $(3\times 3)$-matrices as an invariant subspace and $3\otimes\bar{3}$ restricted to this subspace is a complex multiplet, the (complex) octet $8$. We can easily see that the complexification of $\mathfrak{su}(3)$ is just the space of traceless, complex $(3\times 3)$-matrices. Thus, $3\otimes\bar{3}$ on $\mathfrak{su}(3)$ is just (equivalent to) the real octet and $3\otimes\bar{3}$ on the complexification of $\mathfrak{su}(3)$ is just the complex octet. This fact was used in the last two lines.\par
If we complexify the map $DD^{(\sigma)}\vert_\mathbb{1}$:
\begin{gather*}
DD^{(\sigma)}\vert_\mathbb{1}:\mathfrak{su}(3)+i\cdot\mathfrak{su}(3)\rightarrow\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)}),\\
DD^{(\sigma)}\vert_\mathbb{1}(X+iY)\coloneqq DD^{(\sigma)}\vert_\mathbb{1}(X) + iDD^{(\sigma)}\vert_\mathbb{1}(Y)\quad\forall X,Y\in\mathfrak{su}(3),
\end{gather*}
we can now see that the prior calculation also holds true, if we take $\tilde{X}$ and $X$ to be elements of the complexified Lie algebras. With this, we find an interesting result: If $\sigma$ is trivial, the invariant subspace $\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})$ of $\sigma\otimes\bar{\sigma}$ is trivial as well. However, if $\sigma$ is non-trivial, the complexified map $DD^{(\sigma)}\vert_\mathbb{1}$ is an isomorphism, as the real map $DD^{(\sigma)}\vert_\mathbb{1}$ is an isomorphism. If we now compare the first and last line of the prior calculation, we see that $\sigma\otimes\bar{\sigma}$ restricted to $\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})$ is equivalent to the octet $8$, i.e, $D^{(8)}(A) = DD^{(\sigma)}\vert^{-1}_\mathbb{1}\circ D^{(\sigma\otimes\bar{\sigma})}(A)\circ DD^{(\sigma)}\vert_\mathbb{1}$ for every $A\in\text{SU}(3)$. This means that we can parametrize one octet in $\sigma\otimes\bar{\sigma}$ with a (complex) basis of $\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})$ for every non-trivial, complex, and finite-dimensional multiplet $\sigma$.\par
Let us now choose a (complex) basis for $\mathfrak{su}(3) + i\cdot\mathfrak{su}(3)$. As explained in \autoref{sec:mass_matrix}, the Gell-Mann matrices $\lambda_k$, $k\in\{1,\ldots, 8\}$, constitute a (real) basis of the space of traceless, Hermitian $(3\times 3)$-matrices, thus, they constitute a (complex) basis of its complexification which coincides with $\mathfrak{su}(3) + i\cdot\mathfrak{su}(3)$. We modify the Gell-Mann matrices to obtain the basis $\{F_k\mid k\in\{1,\ldots, 8\}\}$ of $\mathfrak{su}(3) + i\cdot\mathfrak{su}(3)$ following \cite{Lichtenberg}:
\begin{gather*}
F_k\coloneqq \frac{\lambda_k}{2}\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
This gives us a (complex) basis $\{F^{(\sigma\otimes\bar{\sigma})}_k\mid k\in\{1,\ldots,8\}\}$ of $\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})$ for non-trivial multiplets $\sigma$:
\begin{gather*}
F^{(\sigma\otimes\bar{\sigma})}_k\coloneqq DD^{(\sigma)}\vert_\mathbb{1}\left(F_k\right)\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
Note that $F^{(3\otimes\bar{3})}_k \equiv F_k$. The matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ parametrize one octet in $\sigma\otimes\bar{\sigma}$ for non-trivial multiplets $\sigma$.\par
However, as we have seen prior in this section, $\sigma\otimes\bar{\sigma}$ contains, in general, two octets. The remaining octet can be constructed out of products of $F^{(\sigma\otimes\bar{\sigma})}_k$. Nevertheless, not any product of $F^{(\sigma\otimes\bar{\sigma})}_k$ is an element of an octet. In this regard, it is instructive to first consider products of $F_k$ to guide our intuition on how to form products of $F^{(\sigma\otimes\bar{\sigma})}_k$. Consider the product of two matrices $F_k$.
We can write any product of operators in terms of commutators and anticommutators:
\begin{gather*}
F_kF_l = \frac{1}{2}[F_k,F_l] + \frac{1}{2}\{F_k,F_l\}\quad\forall k,l\in\{1,\ldots,8\}
\end{gather*}
with $\{A,B\}\coloneqq AB + BA$ being the anticommutator. The commutator is closed in $\mathfrak{su}(3)$, therefore, we can write the commutator $[F_k,F_l]$ as a linear combination of $F_m$ with purely imaginary coefficients:
\begin{gather*}
[F_k,F_l] = \sum^{8}_{m=1} if_{klm}F_m\quad\forall k,l\in\{1,\ldots,8\},
\end{gather*}
where $f_{klm}$ are real constants defining the commutation relations of the $F_k$-matrices. $f_{klm}$ are called structure constants. Using the orthogonality of the $F_k$-matrices under the Hermitian form $\text{Tr}\left(A^\dagger B\right)$ (cf. \cite{Lichtenberg}) and the cyclicity of the trace, one can easily see that the structure constants $f_{klm}$ are totally antisymmetric in $k,\ l,\text{ and }m$:
\begin{gather*}
\text{Tr}(F_kF_l) = \frac{\delta_{kl}}{2}\quad\Rightarrow f_{klm} = -2i\text{Tr}\left([F_k,F_l]F_m\right)\quad\forall k,l,m\in\{1,\ldots,8\}.
\end{gather*}
The anticommutator $\{F_k,F_l\}$ is just an Hermitian $(3\times 3)$-matrix, hence, can be written as a real linear combination of $\mathbb{1}$ and $F_m$:
\begin{gather*}
\{F_k,F_l\} = a_{kl}\mathbb{1} + \sum^{8}_{m=1} d_{klm}F_m\quad\forall k,l\in\{1,\ldots, 8\},
\end{gather*}
where $a_{kl}$ and $d_{klm}$ are real constants. Again, using the orthogonality of the $F_k$-matrices and the fact that they are traceless, we can determine $a_{kl}$ and $d_{klm}$:
\begin{gather*}
3a_{kl} = \text{Tr}\left(\{F_k,F_l\}\right) = \delta_{kl}\Rightarrow a_{kl} = \frac{\delta_{kl}}{3}\quad\forall k,l\in\{1,\ldots, 8\},\\
d_{klm} = 2\text{Tr}\left(\{F_k,F_l\}F_m\right)\quad\forall k,l\in\{1,\ldots, 8\}.
\end{gather*}
Similar to $f_{klm}$, we can directly follow from this that $d_{klm}$ is totally symmetric in $k,\ l,\text{ and }m$ by using the cyclicity of the trace.\par
The analysis of products of $F_k$ has provided us with three tensors: $\delta_{kl} = 3a_{kl}$, $f_{klm}$, and $d_{klm}$. We can now try to contract these tensors with products of $F^{(\sigma\otimes\bar{\sigma})}_k$ to see what kind of objects we can form. First, consider the contraction with $\delta_{kl}$:
\begin{gather*}
C^{(\sigma\otimes\bar{\sigma})}\coloneqq \sum^{8}_{k,l = 1}\delta_{kl}F^{(\sigma\otimes\bar{\sigma})}_kF^{(\sigma\otimes\bar{\sigma})}_l.
\end{gather*}
$C^{(\sigma\otimes\bar{\sigma})}$ commutes with $F^{(\sigma\otimes\bar{\sigma})}_k$ for every $k\in\{1,\ldots,8\}$ (cf. \autoref{app:F-and_D-symbols}). Therefore, $C^{(\sigma\otimes\bar{\sigma})}$ is a Casimir operator. It is known as the quadratic Casimir operator. Casimir operators are constant on multiplets, i.e., $C^{(\sigma\otimes\bar{\sigma})} = c^{(\sigma\otimes\bar{\sigma})}\cdot\mathbb{1}$ for multiplets $\sigma$ (cf. Schur's lemma for Lie algebra representations in \cite{Hall2015}). This means that $C^{(\sigma\otimes\bar{\sigma})}$ spans the singlet in $\sigma\otimes\bar{\sigma}$ for non-trivial multiplets $\sigma$ (cf. \autoref{app:F-and_D-symbols}).\par
Now consider the contraction with $f_{klm}$:
\begin{gather*}
\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k\coloneqq \sum^{8}_{l,m=1}f_{klm}F^{(\sigma\otimes\bar{\sigma})}_lF^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
As the tensor $f_{klm}$ is totally antisymmetric, we have:
\begin{gather*}
\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k = \sum^{8}_{l,m=1}\frac{f_{klm}}{2}\left[F^{(\sigma\otimes\bar{\sigma})}_l, F^{(\sigma\otimes\bar{\sigma})}_m\right]\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
The map $DD^{(\sigma)}\vert_\mathbb{1}$ that connects $F_k$ and $F^{(\sigma\otimes\bar{\sigma})}_k$ is (the complexification of) a Lie algebra homomorphism, hence:
\begin{gather*}
\left[F^{(\sigma\otimes\bar{\sigma})}_l, F^{(\sigma\otimes\bar{\sigma})}_m\right] = \sum^8_{n=1}if_{lmn}F^{(\sigma\otimes\bar{\sigma})}_n\quad\forall l,m\in\{1,\ldots,8\}.
\end{gather*}
We obtain:
\begin{gather*}
\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k = \sum^{8}_{l,m,n=1}\frac{if_{klm}f_{lmn}}{2}F^{(\sigma\otimes\bar{\sigma})}_n\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
We can see that $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ is just a linear combination of $F^{(\sigma\otimes\bar{\sigma})}_k$. It is easy to show that the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ are all linearly independent, if $\sigma$ is non-trivial (cf. \autoref{app:F-and_D-symbols}). As the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ span an octet in $\sigma\otimes\bar{\sigma}$ for non-trivial multiplets $\sigma$, the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ form the very same octet.\par
This leaves us with the contraction with the last tensor $d_{klm}$:
\begin{gather*}
D^{(\sigma\otimes\bar{\sigma})}_k\coloneqq \frac{2}{3}\sum^{8}_{l,m=1}d_{klm}F^{(\sigma\otimes\bar{\sigma})}_lF^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
As the products of $F^{(\sigma\otimes\bar{\sigma})}_k$ contracted with the tensors $\delta_{kl}$ and $f_{klm}$ give rise to multiplets in $\sigma\otimes\bar{\sigma}$, it seems reasonable to conjecture that the same applies to $d_{klm}$, i.e, the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ form a multiplet in $\sigma\otimes\bar{\sigma}$. As we will see, this is the case for non-trivial multiplets $\sigma$. However, the full proof of this requires some formulae whose derivations involve lengthy calculations. We will skip a part of these calculations in this section, but display them in \autoref{app:F-and_D-symbols}. Partially, these computations can be found in \cite{Lichtenberg}.\par
First, we want to consider the commutator of $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_l$. To compute the commutator of these matrices, we need the following formula (cf. \autoref{app:F-and_D-symbols} and \cite{Lichtenberg}):
\begin{gather*}
\sum^8_{m=1}f_{klm}d_{nrm} = -\sum^8_{m=1}\left(f_{nlm}d_{krm} +f_{rlm}d_{nkm}\right)\quad\forall k,l,n,r\in\{1,\ldots,8\}.
\end{gather*}
Using this formula, we can calculate (cf. \autoref{app:F-and_D-symbols} and \cite{Lichtenberg}):
\begin{gather}
\left[F^{(\sigma\otimes\bar{\sigma})}_k,D^{(\sigma\otimes\bar{\sigma})}_l\right] = \sum^8_{m=1}if_{klm}D^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k,l\in\{1,\ldots,8\}.\label{eq:D-F_com}
\end{gather}
Now define the real vector space $V_D\coloneqq\left\{\sum^8_{k=1}ia_kD^{(\sigma\otimes\bar{\sigma})}_k\mid a_i\in\mathbb{R}\, \forall i\in\{1,\ldots,8\}\right\}$ and consider the (unique) $\mathbb{R}$-linear map $f$ defined by:
\begin{gather*}
f:\mathfrak{su}(3)\rightarrow V_D,\\
f(-iF_k)\coloneqq -iD^{(\sigma\otimes\bar{\sigma})}_k\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
Consider the kernel $K_D\coloneqq\text{ker}(f)$ of $f$ and let $X\in\mathfrak{su}(3)$ and $Y\in K_D$. We then find:
\begin{align*}
f\left(\left[-iF_k, -iF_l\right]\right) &= f\left(\sum^8_{m=1}f_{klm}(-iF_m)\right) = -\sum^8_{m=1}if_{klm}D^{(\sigma\otimes\bar{\sigma})}_m\\
&= -\left[F^{(\sigma\otimes\bar{\sigma})}_k,D^{(\sigma\otimes\bar{\sigma})}_l\right]\\
&= \left[DD^{(\sigma)}\vert_\mathbb{1}\left(-iF_k\right),f\left(-iF_l\right)\right]\quad\forall k,l\in\{1,\ldots,8\}\\
\Rightarrow f([X,Y]) &= [DD^{(\sigma)}\vert_\mathbb{1}(X),f(Y)] = [DD^{(\sigma)}\vert_\mathbb{1}(X),0] = 0\\
&\Rightarrow [X,Y]\in K_D.
\end{align*}
This means that the kernel $K_D$ is an ideal of $\mathfrak{su}(3)$. However, $\mathfrak{su}(3)$ is simple, therefore, $K_D$ is either $\{0\}$ or $\mathfrak{su}(3)$. Hence, $V_D$ is either isomorphic to $\mathfrak{su}(3)$ or $\{0\}$. Thus, all matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are either linearly independent
in the sense of a real vector space or zero. Moreover, we can show for unitary multiplets $\sigma$ that all matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are also linearly independent in the sense of a complex vector space, if they all are linearly independent in the sense of a real vector space: First, note that the Lie group $\text{Im}(D^{(\sigma)})$ is a(n) (embedded) Lie subgroup of the Lie group $\text{U}(V)\coloneqq{\{A:V\rightarrow V\mid A\text{ linear and unitary}\}}$, if $D^{(\sigma)}$ is a unitary representation. Therefore:
\begin{gather*}
\mathfrak{im}(D^{(\sigma)})\subset\mathfrak{u}(V) = \{A:V\rightarrow V\mid A^\dagger = -A\}.
\end{gather*}
Hence, the matrices $iF^{(\sigma\otimes\bar{\sigma})}_k$ are antihermitian:
\begin{gather*}
iF_k\in\mathfrak{su}(3)\Rightarrow iF^{(\sigma\otimes\bar{\sigma})}_k = DD^{(\sigma)}\vert_\mathbb{1}(iF_k)\in\mathfrak{im}(D^{(\sigma)})\subset\mathfrak{u}(V)\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
Thus, the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ are Hermitian. Now let $a_k$ and $b_k$ be real numbers and define $c_k\coloneqq a_k +ib_k$ for every $k\in\{1,\ldots,8\}$. Choose $a_k$ and $b_k$ such that:
\begin{gather*}
\sum^8_{k=1}c_kD^{(\sigma\otimes\bar{\sigma})}_k = \sum^8_{k=1}a_kD^{(\sigma\otimes\bar{\sigma})}_k + i\sum^8_{k=1}b_kD^{(\sigma\otimes\bar{\sigma})}_k = 0\\
\Rightarrow\left(\sum^8_{k=1}c_kD^{(\sigma\otimes\bar{\sigma})}_k\right)^\dagger = \sum^8_{k=1}a_kD^{(\sigma\otimes\bar{\sigma})}_k - i\sum^8_{k=1}b_kD^{(\sigma\otimes\bar{\sigma})}_k = 0\\
\Rightarrow \sum^8_{k=1}a_kD^{(\sigma\otimes\bar{\sigma})}_k = 0\ \wedge\ \sum^8_{k=1}b_kD^{(\sigma\otimes\bar{\sigma})}_k = 0.
\end{gather*}
Since the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are linearly independent in the sense of a real vector space, the coefficients $a_k$ and $b_k$ have to be all zero. This makes the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ linearly independent in the sense of a complex vector space.\par
Prior in this section, we have seen that the matrices $F_k$ transform as an octet under $A\in\text{SU}(3)$:
\begin{gather*}
\sum^{8}_{l=1}\left(D^{(8)}(A)\right)_{kl}F_l \coloneqq D^{(8)}(A)(F_k) = AF_kA^\dagger\quad\forall k\in\{1,\ldots,8\},
\end{gather*}
where $\left(D^{(8)}(A)\right)_{kl}$ are the coefficients mediating the octet transformation of $F_k$. We show in \autoref{app:F-and_D-symbols} that the same coefficients mediate the transformation of $D^{(\sigma\otimes\bar{\sigma})}_k$ under $A\in\text{SU}(3)$:
\begin{gather*}
D^{(\sigma\otimes\bar{\sigma})}(A)\left(D^{(\sigma\otimes\bar{\sigma})}_k\right) = \sum^{8}_{l=1}\left(D^{(8)}(A)\right)_{kl}D^{(\sigma\otimes\bar{\sigma})}_l\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
Thus, the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ form an invariant subspace in $\sigma\otimes\bar{\sigma}$. Furthermore, we can see that they either span an octet or are all zero, as the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are either linearly independent or zero and as their transformation coefficients correspond to the coefficients of an octet.
\subsection*{Mass matrix Parametrization \Romannum{2}: Multiplet Classification using Octets}
To summarize our results so far, we have found that both the $F^{(\sigma\otimes\bar{\sigma})}_k$- and $D^{(\sigma\otimes\bar{\sigma})}_k$-matrices span an invariant subspace of $\sigma\otimes\bar{\sigma}$ for \text{SU}(3)-multiplets $\sigma$ that either is $\{0\}$ or an octet. If $\sigma$ is a non-trivial multiplet, the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ span an octet. What we do not know so far is for which cases the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ span an octet and, if they do, when this octet is different from the octet of the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$. Therefore, in the next step, we want to proof the following classification for \text{SU}(3)-multiplets $\sigma$:
\begin{align*}
\text{1. }&F^{(\sigma\otimes\bar{\sigma})}_k=0\,\forall k\in\{1,\ldots,8\}\Leftrightarrow D^{(\sigma\otimes\bar{\sigma})}_k=0\,\forall k\in\{1,\ldots,8\}\Leftrightarrow \sigma\text{ trivial}\\
\text{2. }&F^{(\sigma\otimes\bar{\sigma})}_k = c\cdot D^{(\sigma\otimes\bar{\sigma})}_k\neq0\,\forall k\in\{1,\ldots,8\}\text{ for }c\in\mathbb{R}\backslash\{0\}\Leftrightarrow \sigma\text{ totally symmetric}\\
\text{3. }&F^{(\sigma\otimes\bar{\sigma})}_k\text{ and }D^{(\sigma\otimes\bar{\sigma})}_k\text{ are all linearly independent}\Leftrightarrow \sigma\text{ neither trivial nor tot. sym.}
\end{align*}
To show this classification, we need to know how the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ act on $V$ which is the vector space the multiplet $D^{(\sigma)}$ acts on. For this, it is helpful to choose an orthonormal basis of $V$ such that the greatest possible number of matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ is diagonal in this basis. To find this basis, we need to introduce the mathematical concept of weights. We will only provide a short overview over the most important results for weights here. These results are mostly taken from \cite{Lichtenberg} where a more in-depth discussion of weights can be found. Let us start by determining the greatest possible number of simultaneously diagonalizable matrices $F^{(\sigma\otimes\bar{\sigma})}_k$. Clearly, the condition that two Hermitian matrices can be diagonalized in the same basis is equivalent to requirement that their commutator has to be zero. Thus, we are looking for the maximal number of mutually commuting matrices $F^{(\sigma\otimes\bar{\sigma})}_k$. As the map $DD^{(\sigma)}\vert_\mathbb{1}$ connecting the matrices $F_k$ and $F^{(\sigma\otimes\bar{\sigma})}_k$ is a Lie algebra isomorphism for non-trivial multiplets $\sigma$, two matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $F^{(\sigma\otimes\bar{\sigma})}_l$ commute for non-trivial multiplets $\sigma$ if and only if the matrices $F_k$ and $F_l$ commute. Therefore, the maximal number of commuting matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ is the same as the maximal number of commuting matrices $F_k$. The maximal number of mutually commuting basis elements in a Lie algebra of a Lie group is called the \textit{rank} of that group. The rank of \text{SU}(3) is 2 meaning that at most two matrices $F_k$ and, in return, $F^{(\sigma\otimes\bar{\sigma})}_k$ can mutually commute. As we already saw in \autoref{sec:mass_matrix}, the Gell-Mann matrices $\lambda_3$ and $\lambda_8$ are diagonal and, therefore, commute. This implies that also the matrices $F_3$ and $F_8$ and likewise the matrices $F^{(\sigma\otimes\bar{\sigma})}_3$ and $F^{(\sigma\otimes\bar{\sigma})}_8$ commute. With this, we have found the set of matrices we need to diagonalize simultaneously. Now consider a non-zero vector $v$ in $V$ such that $v$ is both an eigenvector of $F^{(\sigma\otimes\bar{\sigma})}_3$ and of $F^{(\sigma\otimes\bar{\sigma})}_8$, i.e.:
\begin{gather*}
F^{(\sigma\otimes\bar{\sigma})}_3(v) = w^{(\sigma\otimes\bar{\sigma})}_3\cdot v\quad\text{and}\quad F^{(\sigma\otimes\bar{\sigma})}_8(v) = w^{(\sigma\otimes\bar{\sigma})}_8\cdot v,
\end{gather*}
where $w^{(\sigma\otimes\bar{\sigma})}_3$ and $w^{(\sigma\otimes\bar{\sigma})}_8$ are the corresponding eigenvalues of $F^{(\sigma\otimes\bar{\sigma})}_3$ and $F^{(\sigma\otimes\bar{\sigma})}_8$, respectively. The two-dimensional vector $w^{(\sigma\otimes\bar{\sigma})}\coloneqq (w^{(\sigma\otimes\bar{\sigma})}_3,w^{(\sigma\otimes\bar{\sigma})}_8)^\text{T}$ and, sometimes, its components are called \textit{weights} (cf. \cite{Lichtenberg}). The set of all weights for a \text{SU}(3)-multiplet $\sigma$ completely determines the multiplet $\sigma$. Therefore, multiplets are often just characterized by a graphical representation of their weights. This graphical representation, called \textit{weight diagram}, is usually just a coordinate system where each weight is indicated by a dot. Often, the multiplicity of each weight $w^{(\sigma\otimes\bar{\sigma})}$, i.e., the dimension of the set of vectors with weight $w^{(\sigma\otimes\bar{\sigma})}$, is encoded in a weight diagram, too. We will consider the weight diagrams for the triplet, sextet, octet, and decuplet in \autoref{chap:mass_relations}.\par
Physically, weights correspond to quantum numbers of a state. To explore this statement, consider the matrices $F_1,\ F_2\text{, and }F_3$. Under the commutator, they form a subalgebra of $\mathfrak{su}(3) + i\cdot\mathfrak{su}(3)$. This subalgebra is isomorphic to the (complexified) Lie algebra $\mathfrak{su}(2)+i\cdot\mathfrak{su}(2)$. As the Lie algebra $\mathfrak{su}(2)$ of the group \text{SU}(2) represents the spin algebra, we say that $F_1,\ F_2\text{, and }F_3$ also form a spin algebra. This spin algebra follows from flavor transformations and not from a spacetime symmetry, therefore, the matrices $F_1,\ F_2\text{, and }F_3$ are said to constitute the \textit{isospin} instead of the spin. Isospin related properties are usually denoted by a capital ``$I$'', for instance, the isospin generators $I_k$ and the total isospin $I^2$:
\begin{gather*}
I_k \coloneqq F_k\quad\forall k\in\{1,2,3\}\quad\text{and}\quad I^2 \coloneqq \sum^3_{k=1} I^2_k.
\end{gather*}
As the map $DD^{(\sigma)}\vert_\mathbb{1}$ mediating the transition from $F_k$ to $F^{(\sigma\otimes\bar{\sigma})}_k$ is a Lie algebra isomorphism for non-trivial multiplets $\sigma$, the isospin algebra for a non-trivial multiplet $\sigma$ is described by:
\begin{gather*}
I^{(\sigma\otimes\bar{\sigma})}_k \coloneqq F^{(\sigma\otimes\bar{\sigma})}_k\quad\forall k\in\{1,2,3\}\quad\text{and}\quad I^{2;\,(\sigma\otimes\bar{\sigma})} \coloneqq \sum^3_{k=1} \left(I^{(\sigma\otimes\bar{\sigma})}_k\right)^2.
\end{gather*}
In this sense, the weight $w^{(\sigma\otimes\bar{\sigma})}_3$ is the third component of the isospin and, hence, a spin-like quantum number.\par
To interpret the weight $w^{(\sigma\otimes\bar{\sigma})}_8$ in a similar way, we need to consider what kind of algebra $F_1,\ F_2,\ F_3\text{, and }F_8$ form. By looking at the Gell-Mann matrices from \autoref{sec:mass_matrix}, we can easily see that the matrix $F_8$ commutes with every matrix $F_k$ for $k\in\{1,2,3,\}$. Of course, this also applies to the matrices $F^{(\sigma\otimes\bar{\sigma})}_1,$ $F^{(\sigma\otimes\bar{\sigma})}_2,$ $F^{(\sigma\otimes\bar{\sigma})}_3\text{, and }F^{(\sigma\otimes\bar{\sigma})}_8$.
This means that $F^{(\sigma\otimes\bar{\sigma})}_8$ has to be constant on the isospin multiplets making up the multiplet $\sigma$. The operator $F^{(\sigma\otimes\bar{\sigma})}_8$ behaves like a \text{U}(1)-charge in the sense that its eigenvalues are an integer times a constant that is independent of the multiplet $\sigma$. Therefore, we can enforce the eigenvalues to be 1/3 times an integer by changing the normalization of $F^{(\sigma\otimes\bar{\sigma})}_8$ (cf. \cite{Lichtenberg}):
\begin{gather*}
Y\coloneqq \frac{2}{\sqrt{3}}F_8\quad\text{and}\quad Y^{(\sigma\otimes\bar{\sigma})}\coloneqq \frac{2}{\sqrt{3}}F^{(\sigma\otimes\bar{\sigma})}_8.
\end{gather*}
The third-integer-valued quantum number corresponding to $Y$ and $Y^{(\sigma\otimes\bar{\sigma})}$ is called \textit{hypercharge}. Note that the hypercharge follows from flavor transformations and not from a gauge symmetry like a lot of other charges. Now we can see that the weight $w^{(\sigma\otimes\bar{\sigma})}_8$ is related to the hypercharge, a charge-like quantum number.\par
In total, this means that a non-zero vector $v$ in $V$ with weight $w^{(\sigma\otimes\bar{\sigma})}$ has the following isospin and hypercharge quantum numbers:
\begin{gather*}
I_3 = w^{(\sigma\otimes\bar{\sigma})}_3\quad\text{and}\quad Y = \frac{2}{\sqrt{3}}w^{(\sigma\otimes\bar{\sigma})}_8.
\end{gather*}
Now consider an orthonormal basis of $V$ such that $F^{(\sigma\otimes\bar{\sigma})}_3$ and $F^{(\sigma\otimes\bar{\sigma})}_8$ are diagonal in this basis. Then, every vector in this basis is an eigenvector of both $F^{(\sigma\otimes\bar{\sigma})}_3$ and $F^{(\sigma\otimes\bar{\sigma})}_8$ and, thus, has a weight $w^{(\sigma\otimes\bar{\sigma})}$. For the trivial multiplet and for totally symmetric multiplets, such a basis is unique up to phases. For the remaining multiplets, this is not the case. To obtain a unique basis (up to phases), one additionally requires the basis of $V$ to be an eigenbasis of $I^{2;\,(\sigma\otimes\bar{\sigma})}$. This is always possible as the matrices $F^{(\sigma\otimes\bar{\sigma})}_3$ and $F^{(\sigma\otimes\bar{\sigma})}_8$ commute with $I^{2;\,(\sigma\otimes\bar{\sigma})}$. In the following, we are going to label the elements $v$ of such a basis of $V$ by three quantum numbers:
\begin{gather*}
v\equiv\Ket{Y,I,I_3}
\end{gather*}
where $Y$ is the hypercharge of $v$, $I$ the isospin quantum number of $v$, and $I_3$ the third component of the isospin of $v$. In summary, the three quantum numbers $Y$, $I$, and $I_3$ give rise to an orthonormal basis of $V$ that is unique up to phases.\par
One can define a special order on the weights of the multiplet $\sigma$. The first weight in this order is called the highest weight (cf. \cite{Lichtenberg}). The highest weight completely characterizes a multiplet in the sense that two multiplets are equivalent, if they have the same highest weight. If the multiplet $\sigma$ is equivalent to the multiplet $D(p,q)$ for two non-negative integers $p$ and $q$, its highest weight $w^{(\sigma\otimes\bar{\sigma})}_h$ is given by (cf. \cite{Lichtenberg}):
\begin{gather*}
w^{(\sigma\otimes\bar{\sigma})}_h = p\cdot\begin{pmatrix}\frac{1}{2}\\\frac{1}{2\sqrt{3}}\end{pmatrix} + q\cdot\begin{pmatrix}0\\\frac{1}{\sqrt{3}}\end{pmatrix}.
\end{gather*}
For every multiplet, a normalized vector $v_1\in V$ whose weight is the highest weight is unique up to a phase. For the multiplet $D(p,q)$, it is given by:
\begin{gather*}
v_1 = \Ket{Y = \frac{p+2q}{3}, I=\frac{p}{2}, I_3=\frac{p}{2}}.
\end{gather*}
As every weight diagram of \text{SU}(3) is invariant under rotations by $\frac{2\pi}{3}$ (cf. \cite{Lichtenberg}), the highest weight in a multiplet generates two more weights in the multiplet by rotation. For the multiplet $D(p,q)$, the corresponding vectors $v_2$ and $v_3$ of these rotated weights are:
\begin{align*}
v_2 &= \Ket{Y = \frac{p-q}{3}, I=\frac{p+q}{2}, I_3=-\frac{p+q}{2}},\\
v_3 &= \Ket{Y = -\frac{2p+q}{3}, I=\frac{q}{2}, I_3=\frac{q}{2}}.
\end{align*}
Let us now consider the matrix $D^{(\sigma\otimes\bar{\sigma})}_8$. By using the definition of the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ and a list of the constants $d_{klm}$ (cf. \cite{Lichtenberg}), we find:
\begin{align*}
D^{(\sigma\otimes\bar{\sigma})}_8 &= \frac{2}{3}\left(\frac{1}{\sqrt{3}}\sum^3_{k=1}\left(F^{(\sigma\otimes\bar{\sigma})}_k\right)^2 - \frac{1}{2\sqrt{3}}\sum^7_{k=4}\left(F^{(\sigma\otimes\bar{\sigma})}_k\right)^2 - \frac{1}{\sqrt{3}}\left(F^{(\sigma\otimes\bar{\sigma})}_8\right)^2\right)\\
&= \frac{2}{3}\left(\frac{\sqrt{3}}{2}\sum^3_{k=1}\left(F^{(\sigma\otimes\bar{\sigma})}_k\right)^2 - \frac{1}{2\sqrt{3}}\left(F^{(\sigma\otimes\bar{\sigma})}_8\right)^2 - \frac{1}{2\sqrt{3}}\sum^8_{k=1}\left(F^{(\sigma\otimes\bar{\sigma})}_k\right)^2\right)\\
&= \frac{1}{\sqrt{3}}\left(I^{2;\, (\sigma\otimes\bar{\sigma})} - \frac{1}{4}\left(Y^{(\sigma\otimes\bar{\sigma})}\right)^2 - \frac{1}{3}C^{(\sigma\otimes\bar{\sigma})}\right),
\end{align*}
where $C^{(\sigma\otimes\bar{\sigma})}$ is the quadratic Casimir operator. As already mentioned, the quadratic Casimir operator is constant on multiplets. For $\sigma$ being the multiplet $D(p,q)$, one finds (cf. \cite{Pais1966})\footnote{In this article, $p$ is used differently. The integer $p$ in \cite{Pais1966} has to be understood as $p+q$ in our notation.}:
\begin{gather*}
C^{(\sigma\otimes\bar{\sigma})} = \frac{1}{3}\left(p^2 + q^2 + pq + 3p + 3q\right)\mathbb{1}.
\end{gather*}
With this, we find for multiplets $\sigma = D(p,q)$:
\begin{align}
v^\dagger_1D^{(\sigma\otimes\bar{\sigma})}_8v_1 &= \frac{1}{\sqrt{3}}\left(\frac{p}{2}\left(\frac{p}{2}+1\right) - \frac{1}{4}\cdot\frac{(p+2q)^2}{9} - \frac{1}{3}\cdot\frac{p^2 + q^2 + pq + 3p + 3q}{3}\right)\nonumber\\
\Rightarrow v^\dagger_1D^{(\sigma\otimes\bar{\sigma})}_8v_1 &= \frac{1}{3\sqrt{3}}\left(\frac{p^2 -2pq -2q^2}{3} + \frac{p}{2} - q\right),\label{eq:v_1}\\
v^\dagger_2D^{(\sigma\otimes\bar{\sigma})}_8v_2 &= \frac{1}{3\sqrt{3}}\left(\frac{p^2 -4pq +q^2}{3} + \frac{p+q}{2}\right),\label{eq:v_2}\\
v^\dagger_3D^{(\sigma\otimes\bar{\sigma})}_8v_3 &= \frac{1}{3\sqrt{3}}\left(\frac{q^2 -2pq -2p^2}{3} + \frac{q}{2} - p\right)\label{eq:v_3}.
\end{align}
\autoref{eq:v_1}, \autoref{eq:v_2}, and \autoref{eq:v_3} are three equations that have to be satisfied for every multiplet $\sigma = D(p,q)$. These equations allow us now to prove the classification for \text{SU}(3)-multiplets mentioned above. Let us start with the first statement of the classification:
\begin{gather*}
\text{1. }F^{(\sigma\otimes\bar{\sigma})}_k=0\,\forall k\in\{1,\ldots,8\}\Leftrightarrow D^{(\sigma\otimes\bar{\sigma})}_k=0\,\forall k\in\{1,\ldots,8\}\Leftrightarrow \sigma\text{ trivial.}
\end{gather*}
First, suppose $\sigma$ is trivial. Then, all matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and, thus, $D^{(\sigma\otimes\bar{\sigma})}_k$ are trivially zero. Next, suppose all matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ are zero. Then, $DD^{(\sigma)}\vert_\mathbb{1}$ has to be zero. We have already seen that $\sigma$ needs to be trivial in this case. To conclude the proof of the first statement, we need to show that the multiplet $\sigma$ is trivial, if all matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are zero. Suppose that $\sigma$ is a multiplet such that all matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are zero. As $\sigma$ is a multiplet of \text{SU}(3), there are non-negative integers $p$ and $q$ such that $\sigma$ is equivalent to $D(p,q)$. All matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ are zero, so, in particular, $D^{(\sigma\otimes\bar{\sigma})}_8$ is zero. Thus, the left-hand sides of \autoref{eq:v_1}, \autoref{eq:v_2}, and \autoref{eq:v_3} are equal to zero:
\begin{align*}
0 &= \frac{1}{3\sqrt{3}}\left(\frac{p^2 -2pq -2q^2}{3} + \frac{p}{2} - q\right),\\
0 &= \frac{1}{3\sqrt{3}}\left(\frac{p^2 -4pq +q^2}{3} + \frac{p+q}{2}\right),\\
0 &= \frac{1}{3\sqrt{3}}\left(\frac{q^2 -2pq -2p^2}{3} + \frac{q}{2} - p\right).
\end{align*}
Subtracting the third from the first equation and multiplying the result with $3\sqrt{3}$ yields:
\begin{gather*}
0 = p^2 - q^2 + \frac{3}{2}\left(p-q\right) = (p-q)\left(p+q+\frac{3}{2}\right).
\end{gather*}
As $p$ and $q$ are non-negative numbers, we can divide by $p+q+3/2$ to find that $p$ is equal to $q$. Inserting this into the second equation gives:
\begin{gather*}
0 = \frac{1}{3\sqrt{3}}\left(\frac{-2}{3}p^2 + p\right).
\end{gather*}
The only integer-valued solution of this equation for $p$ is zero. Hence, $p$ and $q$ are zero implying that $\sigma$ is trivial. This concludes the proof for the first statement. Let us now turn to the proof of the second statement:
\begin{gather*}
\text{2. }F^{(\sigma\otimes\bar{\sigma})}_k = c\cdot D^{(\sigma\otimes\bar{\sigma})}_k\neq0\,\forall k\in\{1,\ldots,8\}\text{ for }c\in\mathbb{R}\backslash\{0\}\Leftrightarrow \sigma\text{ totally symmetric.}
\end{gather*}
First, suppose that $\sigma$ is a totally symmetric multiplet. We have seen prior in this section that the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ contains exactly one octet in this case. Furthermore, we know that the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ are either all linearly independent or zero. Likewise, the same statement applies to the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$. However, no matrix $F^{(\sigma\otimes\bar{\sigma})}_k$ or $D^{(\sigma\otimes\bar{\sigma})}_k$ can be zero, as then all matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ or $D^{(\sigma\otimes\bar{\sigma})}_k$ would have to be zero. By the first statement of the classification, this would imply that $\sigma$ is trivial which would be a contradiction. This means that the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ (and, likewise, $D^{(\sigma\otimes\bar{\sigma})}_k$) are linearly independent and span an octet in $\sigma\otimes\bar{\sigma}$. As $\sigma\otimes\bar{\sigma}$ only contains one octet, $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ have to span the same octet. We identified this octet with $\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})$ prior in this section. Now consider the linear map $f$ that is uniquely defined by:
\begin{gather*}
f:\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})\rightarrow\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)}),\\
f(D^{(\sigma\otimes\bar{\sigma})}_k)\coloneqq F^{(\sigma\otimes\bar{\sigma})}_k\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
As both matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ transform with the same transformation coefficients under $A\in\text{SU}(3)$ (cf. \autoref{app:F-and_D-symbols}):
\begin{align*}
&D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_k\right) = \sum^8_{l=1} \left(D^{(8)}(A)\right)_{kl} F^{(\sigma\otimes\bar{\sigma})}_l\quad\forall k\in\{1,\ldots,8\}\quad\text{and}\\
&D^{(\sigma\otimes\bar{\sigma})}(A)\left(D^{(\sigma\otimes\bar{\sigma})}_k\right) = \sum^8_{l=1} \left(D^{(8)}(A)\right)_{kl} D^{(\sigma\otimes\bar{\sigma})}_l\quad\forall k\in\{1,\ldots,8\},
\end{align*}
the map $f$ commutes with the octet transformation $D^{(8)}(A)$:
\begin{align*}
f\left(D^{(\sigma\otimes\bar{\sigma})}(A)\left(D^{(\sigma\otimes\bar{\sigma})}_k\right)\right) &= \sum^8_{l=1} \left(D^{(8)}(A)\right)_{kl} f\left(D^{(\sigma\otimes\bar{\sigma})}_l\right) = \sum^8_{l=1} \left(D^{(8)}(A)\right)_{kl} F^{(\sigma\otimes\bar{\sigma})}_l\\
&= D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_k\right)\\
&= D^{(\sigma\otimes\bar{\sigma})}(A)\left(f\left(D^{(\sigma\otimes\bar{\sigma})}_k\right)\right)\quad\forall k\in\{1,\ldots,8\}\\
\Rightarrow f\circ D^{(8)}(A) &= D^{(8)}(A)\circ f,
\end{align*}
where $D^{(8)}(A)\coloneqq D^{(\sigma\otimes\bar{\sigma})}(A)\vert_{\mathfrak{im}(D^{(\sigma)}) + i\cdot\mathfrak{im}(D^{(\sigma)})}$. By Schur's lemma (cf. \cite{Knapp2001}), every map that commutes with all transformations of a finite-dimensional, unitary, and irreducible representation of a compact Lie group is a multiple of the identity. Thus, $f=c\cdot\mathbb{1}$ for some constant $c\in\mathbb{C}$. However, $c$ has to be real, as all matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ are Hermitian. Moreover, $c$ cannot be zero, as the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ cannot be zero. With this, we find:
\begin{gather*}
F^{(\sigma\otimes\bar{\sigma})}_k = f\left(D^{(\sigma\otimes\bar{\sigma})}_k\right) = c\cdot D^{(\sigma\otimes\bar{\sigma})}_k\neq 0\ \forall k\in\{1,\ldots,8\}\text{ for some constant }c\in\mathbb{R}\backslash\{0\}.
\end{gather*}
Let us now suppose that $\sigma$ is a \text{SU}(3)-multiplet such that $F^{(\sigma\otimes\bar{\sigma})}_k = c\cdot D^{(\sigma\otimes\bar{\sigma})}_k\neq 0$ for every $k\in\{1,\ldots,8\}$ and some constant $c\in\mathbb{R}\backslash\{0\}$. As $\sigma$ is a multiplet of \text{SU}(3), there are non-negative integers $p$ and $q$ such that $\sigma$ is equivalent to $D(p,q)$. Consider \autoref{eq:v_1}, \autoref{eq:v_2}, and \autoref{eq:v_3}. We can calculate the left-hand sides by using $D^{(\sigma\otimes\bar{\sigma})}_8 = c^{-1}F^{(\sigma\otimes\bar{\sigma})}_8 =2/\sqrt{3}c\cdot Y^{(\sigma\otimes\bar{\sigma})}$:
\begin{align*}
\frac{2}{3\sqrt{3}c}(p+2q) &= \frac{1}{3\sqrt{3}}\left(\frac{p^2 -2pq -2q^2}{3} + \frac{p}{2} - q\right),\\
\frac{2}{3\sqrt{3}c}(p-q) &= \frac{1}{3\sqrt{3}}\left(\frac{p^2 -4pq +q^2}{3} + \frac{p+q}{2}\right),\\
\frac{-2}{3\sqrt{3}c}(2p+q) &= \frac{1}{3\sqrt{3}}\left(\frac{q^2 -2pq -2p^2}{3} + \frac{q}{2} - p\right).
\end{align*}
Subtracting the third from the first equation and multiplying the result with $3\sqrt{3}$ yields:
\begin{gather*}
\frac{6}{c}(p+q) = (p-q)\left(p+q+\frac{3}{2}\right).
\end{gather*}
Likewise, subtracting the third from the second equation and multiplying the result with $3\sqrt{3}$ yields:
\begin{gather*}
\frac{6}{c}p = p\left(p-\frac{2}{3}q+\frac{3}{2}\right).
\end{gather*}
Using the first statement of the classification, we know that $\sigma$ cannot be trivial, as the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ are not zero. Thus, $p$ or $q$ has to be greater than zero. First, suppose that $p$ is greater than zero. Then, $p>0$ and $p+q>0$. This allows us to solve the last two equations for $6/c$ and set the results equal to each other:
\begin{align*}
\frac{p-q}{p+q}\left(p+q+\frac{3}{2}\right) &= p -\frac{2}{3}q + \frac{3}{2}\\
\Rightarrow (p-q)\left(p+q+\frac{3}{2}\right) &= (p+q)\left(p -\frac{2}{3}q + \frac{3}{2}\right)\\
\Rightarrow q\left(\frac{p+q}{3} + 3\right) = 0.
\end{align*}
As $p$ and $q$ are non-negative, $(p+q)/3 + 3$ is greater than zero. This implies that $q$ has to be zero to satisfy the last equation. In this case, $\sigma$ is just $D(p,0)$ with $p>0$, i.e., $\sigma$ is totally symmetric. Now, consider the remaining case of $p=0$. Then, $q$ has to be greater than zero, as $\sigma$ is not trivial. Therefore, $\sigma$ is just $D(0,q)$ with $q>0$, i.e., $\sigma$ is totally symmetric. This concludes the proof of the second statement of the classification. Let us now turn to the proof of the last statement:
\begin{gather*}
\text{3. }F^{(\sigma\otimes\bar{\sigma})}_k\text{ and }D^{(\sigma\otimes\bar{\sigma})}_k\text{ are all linearly independent}\Leftrightarrow \sigma\text{ neither trivial nor tot. sym.}
\end{gather*}
First, suppose that $\sigma$ is a \text{SU}(3)-multiplet that is neither trivial nor totally symmetric. Then, by the first statement of the classification, none of the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ or $D^{(\sigma\otimes\bar{\sigma})}_k$ can be zero. Therefore, both the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ have to span an octet in $\sigma\otimes\bar{\sigma}$. Let us call the octet spanned by $F^{(\sigma\otimes\bar{\sigma})}_k$ $V_F$ and the octet spanned by $D^{(\sigma\otimes\bar{\sigma})}_k$ $V_D$. Then, we either have $V_F = V_D$ or $V_F\cap V_D = \{0\}$, since the vector space $V_F\cap V_D$ is an invariant subspace of the irreducible representation $V_{F/D}$, so it has to be $V_{F/D}$ or $\{0\}$. However, $V_F$ cannot be equal to $V_D$, as, in this case, we would be able to repeat the first part of the proof of the second statement to show that $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ are proportional to each other. But if $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ were proportional to each other, we would find with the second statement that $\sigma$ is totally symmetric contradicting our assumption. Hence, $V_F\cap V_D$ has to be $\{0\}$. This implies that combining a basis of $V_F$ and a basis of $V_D$ gives a set of linearly independent vectors. With this, we obtain that the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ are all linearly independent. To conclude this proof, we need to show the converse. Therefore, suppose that $\sigma$ is a \text{SU}(3)-multiplet such that the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ are all linearly independent. This implies that both the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and the matrices $D^{(\sigma\otimes\bar{\sigma})}_k$ span an octet in $\sigma\otimes\bar{\sigma}$. These octets cannot be equal, as the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ are all linearly independent. Hence, the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ contains at least two different octets. However, neither the trivial multiplet nor any totally symmetric multiplet contains more than one octet. Therefore, $\sigma$ is neither trivial nor totally symmetric. This concludes the proof of the classification.\par
Our initial motivation for classifying and parametrizing \text{SU}(3)-multiplets was to describe mass matrices transforming under $\sigma\otimes\bar{\sigma}$ for arbitrary complex finite-dimensional \text{SU}(3)-multiplets $\sigma$, where these mass matrices are, among others, subject to octet enhancement, i.e., are merely a sum of singlets and octets. In the course of our investigations, we have found that $\sigma\otimes\bar{\sigma}$ contains exactly one singlet for every \text{SU}(3)-multiplet $\sigma$, namely the multiples of the identity. Furthermore, we have seen that the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ contains no octet for $\sigma$ being the trivial multiplet, exactly one octet for $\sigma$ being totally symmetric, and exactly two octets for the remaining multiplets $\sigma$. After this classification, we aimed to parametrize the octets in $\sigma\otimes\bar{\sigma}$. We accomplished this by introducing matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$. We have found that each span of the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ is a candidate for an octet in $\sigma\otimes\bar{\sigma}$, i.e., is an octet or $\{0\}$. Lastly, we have proven a classification of \text{SU}(3)-multiplets $\sigma$ in terms of the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$:
\begin{align*}
\text{1. }&F^{(\sigma\otimes\bar{\sigma})}_k=0\,\forall k\in\{1,\ldots,8\}\Leftrightarrow D^{(\sigma\otimes\bar{\sigma})}_k=0\,\forall k\in\{1,\ldots,8\}\Leftrightarrow \sigma\text{ trivial}\\
\text{2. }&F^{(\sigma\otimes\bar{\sigma})}_k = c\cdot D^{(\sigma\otimes\bar{\sigma})}_k\neq0\,\forall k\in\{1,\ldots,8\}\text{ for }c\in\mathbb{R}\backslash\{0\}\Leftrightarrow \sigma\text{ totally symmetric}\\
\text{3. }&F^{(\sigma\otimes\bar{\sigma})}_k\text{ and }D^{(\sigma\otimes\bar{\sigma})}_k\text{ are all linearly independent}\Leftrightarrow \sigma\text{ neither trivial nor tot. sym.}
\end{align*}
We can now sum up this classification by saying that each set of matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ spans no octet for $\sigma$ being trivial, spans one (and the same) octet for $\sigma$ being totally symmetric, and each spans a different octet for the remaining multiplets $\sigma$. Together with the number of octets in $\sigma\otimes\bar{\sigma}$, we arrive at the conclusion that every matrix in $\sigma\otimes\bar{\sigma}$ only consisting of a sum of octets is a linear combination of the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ for complex finite-dimensional \text{SU}(3)-multiplets $\sigma$. In total, this means that every mass matrix transforming under $\sigma\otimes\bar{\sigma}$ for arbitrary complex finite-dimensional \text{SU}(3)-multiplets $\sigma$ and subject to octet enhancement is a linear combination of the 17 matrices $\mathbb{1}$, $F^{(\sigma\otimes\bar{\sigma})}_k$ ($k\in\{1,\ldots,8\}$), and $D^{(\sigma\otimes\bar{\sigma})}_l$ ({${l\in\{1,\ldots,8\}}$}).
\subsection*{GMO Mass Formula}
Now, we have all tools at hand to write down the Gell-Mann--Okubo mass formula. Prior in \autoref{sec:GMO_formula}, we have seen that we can group hadrons into complex finite-dimensional \text{SU}(3)-multiplets $\sigma$ such that the masses of these hadrons are given to first order in perturbation theory (neglecting isospin symmetry breaking) by the eigenvalues of a mass matrix transforming under $\sigma\otimes\bar{\sigma}$ which is subject to octet enhancement and $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$ symmetry breaking. We have just shown that every matrix transforming under $\sigma\otimes\bar{\sigma}$ and subject to octet enhancement is a linear combination of $\mathbb{1}$, $F^{(\sigma\otimes\bar{\sigma})}_k$, and $D^{(\sigma\otimes\bar{\sigma})}_l$. However, we are only interested in the $\text{SU}(2)\times\text{U}(1)$-invariant part of this parametrization. When discussing the weight diagram of the octet in \autoref{sec:rel_within_multiplets}, we will see that the octet only contains one $\text{SU}(2)\times\text{U}(1)$-invariant element, aside from multiplication with scalars. Therefore, only one matrix from each set of matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ contributes to the hadronic mass matrix. The matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ transform in the same way as $F_k$ under \text{SU}(3). As $F_8$ is invariant under $\text{SU}(2)\times\text{U}(1)$ (cf. \autoref{sec:mass_matrix}), $F^{(\sigma\otimes\bar{\sigma})}_8$ and $D^{(\sigma\otimes\bar{\sigma})}_8$ are invariant under $\text{SU}(2)\times\text{U}(1)$ as well. This means that the masses of hadrons in a \text{SU}(3)-multiplet $\sigma$ are given to first order in perturbation theory (neglecting isospin symmetry breaking) by the eigenvalues of the matrix $m^{(\sigma\otimes\bar{\sigma})}$:
\begin{gather*}
m^{(\sigma\otimes\bar{\sigma})} = m_0\mathbb{1} + m^F_8F^{(\sigma\otimes\bar{\sigma})}_8 + m^D_8D^{(\sigma\otimes\bar{\sigma})}_8,
\end{gather*}
where $m_0$, $m^F_8$, and $m^D_8$ are parameters which can, in general, be different for different multiplets $\sigma$. We can rewrite this in terms of isospin and hypercharge operators:
\begin{align*}
m^{(\sigma\otimes\bar{\sigma})} &= \left(m_0 - \frac{c^{(\sigma\otimes\bar{\sigma})}}{3\sqrt{3}}m^D_8\right)\mathbb{1} + \frac{\sqrt{3}m^F_8}{2}Y^{(\sigma\otimes\bar{\sigma})} + \frac{m^D_8}{\sqrt{3}}\left(I^{2;\, (\sigma\otimes\bar{\sigma})} - \frac{1}{4}\left(Y^{(\sigma\otimes\bar{\sigma})}\right)^2\right)\\
&\eqqcolon \tilde{m}_0\mathbb{1} + \tilde{m}^F_8Y^{(\sigma\otimes\bar{\sigma})} + \tilde{m}^D_8\left(I^{2;\, (\sigma\otimes\bar{\sigma})} - \frac{1}{4}\left(Y^{(\sigma\otimes\bar{\sigma})}\right)^2\right),
\end{align*}
where $c^{(\sigma\otimes\bar{\sigma})}$ is the constant describing the Casimir operator $C^{(\sigma\otimes\bar{\sigma})}\eqqcolon c^{(\sigma\otimes\bar{\sigma})}\mathbb{1}$. All that is left to do is to diagonalize the matrix $m^{(\sigma\otimes\bar{\sigma})}$ to obtain its eigenvalues. However, this is rather easy as we have already introduced the orthonormal basis $\{\Ket{Y,I,I_3}\}$ of $\sigma$. In this basis, $Y^{(\sigma\otimes\bar{\sigma})}$ and $I^{2;\, (\sigma\otimes\bar{\sigma})}$ are diagonal, hence, $m^{(\sigma\otimes\bar{\sigma})}$ is also diagonal in this basis. Let us denote the mass of the hadron in the multiplet $\sigma$ with hypercharge $Y$, total isospin $I$, third component $I_3$ of the isospin by $m(Y,I,I_3)$. Neglecting isospin symmetry breaking, $m(Y,I,I_3)$ is given by:
\begin{gather}
m(Y,I,I_3) = \tilde{m}_0 + \tilde{m}^F_8\cdot Y + \tilde{m}^D_8\left(I(I+1) - \frac{Y^2}{4}\right) + \mathcal{O}(\varepsilon^2_8).\label{eq:GMO_mass_formula}
\end{gather}
This is the famous \textit{Gell-Mann--Okubo mass formula} (cf. \cite{Okubo}). It implies all hadronic mass formulae and relations from \autoref{sec:mass_matrix}. In particular, the GMO mass formula predicts equal spacing rules for totally symmetric multiplets $\sigma$. As the Clebsch-Gordan series of $\sigma\otimes\bar{\sigma}$ for totally symmetric multiplets $\sigma$ contains exactly one octet, the parametrization of $m^{(\sigma\otimes\bar{\sigma})}$ with both $F^{(\sigma\otimes\bar{\sigma})}_8$ and $D^{(\sigma\otimes\bar{\sigma})}_8$ is redundant and we can freely choose one of the parameters $m^F_8$ and $m^D_8$. For instance, we can choose $m^D_8 = 0$ for totally symmetric multiplets $\sigma$. We then find for totally symmetric multiplets $\sigma$:
\begin{gather*}
m(Y,I,I_3) = \tilde{m}_0 + \tilde{m}^F_8\cdot Y + \mathcal{O}(\varepsilon^2_8).
\end{gather*}
In this case, the mass of a hadron in $\sigma$ only depends its hypercharge. If $Y_\text{min}$ and $Y_\text{max}$ are the minimal and maximal hypercharge in a totally symmetric multiplet $\sigma$, the other hypercharges occurring in $\sigma$ are element of $\{Y_\text{min}, Y_\text{min}+1,\ldots, Y_\text{max}-1, Y_\text{max}\}$ (cf. \cite{Lichtenberg}). Hence, all hadron masses in a totally symmetric multiplet $\sigma$ are equidistant and, thus, subject to an equal spacing rule.
\newpage
\section{Additional Mass Contributions}
\label{sec:add_con}
In \autoref{sec:GMO_formula}, we made a lot of assumptions and used a lot of approximations to arrive at the GMO mass formula. In particular, we considered a Lagrangian only consisting of kinetic and mass terms of three quark flavors (up, down, and strange) and of flavor symmetric interaction terms as a starting point for our investigations. However, the only (quark) flavor symmetric interaction in the SM is the strong interaction. Following such an approach, we clearly neglected the non-flavor symmetric electroweak interaction and the remaining particle content of the SM. Furthermore, we took the masses of the up and down quark to be equal, but we observe in Nature that the masses of the up and down quark are most likely to be different. Obviously, these effects give rise to corrections to the GMO mass formula. The goal of this section is to describe additional contributions to the GMO mass formula originating from these effects. In particular, we aim to derive expressions for the isospin symmetry breaking induced by the difference of the up and down quark mass and for electromagnetic corrections in first order perturbation theory.
\subsection*{Isospin Symmetry Breaking induced by the Up-Down-Mass Difference}
In \autoref{chap:hadron_masses}, we considered $\mathcal{L}_\text{QCD}$, a Lagrangian describing three quark flavors and a flavor symmetric interaction between them, and showed that its corresponding Hamilton operator $H_\text{QCD}$ decomposes as follows under \text{SU}(3)-flavor transformations:
\begin{gather*}
H_\text{QCD} = H^{0}_\text{QCD} + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8},
\end{gather*}
where $H^{0}_\text{QCD}$ is a singlet of \text{SU}(3), $H^{8}_{\text{QCD};\, 3}$ and $H^{8}_{\text{QCD};\, 8}$ are the 3rd and 8th component of an octet, respectively, $\varepsilon_3\coloneqq m_u-m_d$, and $\varepsilon_8\coloneqq (m_u+m_d-2m_s)/\sqrt{3}$. In \autoref{sec:GMO_formula}, we only investigated the case of $\varepsilon_3 = 0\Leftrightarrow m_u=m_d$ to ensure that the Hamilton operator $H_\text{QCD}$ is invariant under $\text{SU}(2)\times\text{U}(1)$-flavor transformations. The generators of this $\text{SU}(2)\times\text{U}(1)$-symmetry are the three isospin operators $I_1$, $I_2$, $I_3$, and the hypercharge operator $Y$. Therefore, the $\text{SU}(2)\times\text{U}(1)$-flavor symmetry of $H_\text{QCD}$ (or its subsymmetry $\text{SU}(2)$ to be precise) is often referred to as \textit{isospin symmetry}. In the most general case of all quark masses being different, the isospin symmetry is broken by $H^{8}_{\text{QCD};\, 3}$, as the third component of an octet is not isospin symmetric. Nevertheless, there is still a residual symmetry left in this case. As we have seen in \autoref{sec:Trafo_QCD}, the quark mass matrix $\mathcal{M}\coloneqq\text{diag}(m_u,m_d,m_s)$ is still invariant under diagonal \text{SU}(3)-phase modulations, i.e.:
\begin{gather*}
A\mathcal{M}A^\dagger = \mathcal{M}\text{ with }A=\text{diag}(e^{i\alpha},e^{i\beta},e^{-i(\alpha+\beta)})\in\text{SU}(3)\ \forall\alpha,\beta\in\mathbb{R}.
\end{gather*}
We can identify this residual symmetry with the Lie group $\text{U}(1)\times\text{U}(1)$. The generators of this group are the third component $I_3$ of the isospin and the hypercharge operator $Y$.\par
Now, we want to examine how the introduction of isospin symmetry breaking in terms of $H^{8}_{\text{QCD};\, 3}$ affects the derivation of the GMO mass formula from \autoref{sec:GMO_formula}. Even though one might think that this is rather laborious task given the length of \autoref{sec:GMO_formula}, it is actually surprisingly simple. All statements from \autoref{sec:GMO_formula} apply and can be derived in a very similar fashion for a Hamilton operator $H_\text{QCD}$ including a non-vanishing $\varepsilon_3\cdot H^{8}_{\text{QCD};\, 3}$ term. We just have to be cautious to include the third component of an octet when needed and be aware that the flavor symmetry of $H_\text{QCD}$ is now just $\text{U}(1)\times\text{U}(1)$ instead of $\text{SU}(2)\times\text{U}(1)$. In sloppy terms, one might say we just have to replace the phrases ``singlet plus the 8th component of an octet'', ``$\text{SU}(2)\times\text{U}(1)$'', and ``$\mathcal{O}\left(\varepsilon^2_8\right)$'' in \autoref{sec:GMO_formula} by ``singlet plus the 3rd and 8th component of an octet'', ``$\text{U}(1)\times\text{U}(1)$'', and ``$\mathcal{O}\left(\varepsilon^2_8\right)+\mathcal{O}\left(\varepsilon^2_3\right)+\mathcal{O}\left(\varepsilon_3\varepsilon_8\right)$'', respectively. Repeating \autoref{sec:GMO_formula} with these comments in mind, we find that we can group hadrons into complex finite-dimensional \text{SU}(3)-multiplets $\sigma$ such that the masses of the hadrons in $\sigma$ are given to first order in perturbation theory (meaning to first order in both $\varepsilon_3$ and $\varepsilon_8$ this time) by the eigenvalues of a mass matrix transforming under $\sigma\otimes\bar{\sigma}$ which is subject to octet enhancement and $\text{SU}(3)\rightarrow\text{U}(1)\times\text{U}(1)$ symmetry breaking. We already know that every mass matrix transforming under $\sigma\otimes\bar{\sigma}$ which is subject to octet enhancement is a linear combination of the matrices $\mathbb{1}$, $F^{(\sigma\otimes\bar{\sigma})}_k$, and $D^{(\sigma\otimes\bar{\sigma})}_k$. Hence, all that is left to do is to find the $\text{U}(1)\times\text{U}(1)$-invariant elements of this decomposition. When discussing the weight diagram of the octet in \autoref{sec:rel_within_multiplets}, we will see that every octet contains exactly two linearly independent elements that are $\text{U}(1)\times\text{U}(1)$-invariant. These elements correspond to the Gell-Mann matrices $\lambda_3$ and $\lambda_8$, i.e., $F_3$ and $F_8$. This means the only $F^{(\sigma\otimes\bar{\sigma})}_k$- and $D^{(\sigma\otimes\bar{\sigma})}_k$-matrices that are $\text{U}(1)\times\text{U}(1)$-invariant are $F^{(\sigma\otimes\bar{\sigma})}_3$, $F^{(\sigma\otimes\bar{\sigma})}_8$, $D^{(\sigma\otimes\bar{\sigma})}_3$, and $D^{(\sigma\otimes\bar{\sigma})}_8$. Thus, the masses of hadrons in a \text{SU}(3)-multiplet $\sigma$ are given to first order in perturbation theory by the eigenvalues of the matrix $m^{(\sigma\otimes\bar{\sigma})}$:
\begin{gather*}
m^{(\sigma\otimes\bar{\sigma})} = m_0\mathbb{1} + m^F_3F^{(\sigma\otimes\bar{\sigma})}_3 + m^F_8F^{(\sigma\otimes\bar{\sigma})}_8 + m^D_3D^{(\sigma\otimes\bar{\sigma})}_3 + m^D_8D^{(\sigma\otimes\bar{\sigma})}_8,
\end{gather*}
where $m_0$, $m^F_3$, $m^F_8$, $m^D_3$, and $m^D_8$ are parameters which can, in general, be different for different multiplets $\sigma$. Again, we only need the $F^{(\sigma\otimes\bar{\sigma})}_k$- or the $D^{(\sigma\otimes\bar{\sigma})}_k$-matrices for the description of totally symmetric multiplets $\sigma$. For instance, we can set $m^D_3 = m^D_8 = 0$:
\begin{align*}
m^{(\sigma\otimes\bar{\sigma})} &= m_0\mathbb{1} + m^F_3F^{(\sigma\otimes\bar{\sigma})}_3 + m^F_8F^{(\sigma\otimes\bar{\sigma})}_8\\
&= m_0\mathbb{1} + m^F_3I^{(\sigma\otimes\bar{\sigma})}_3 + \tilde{m}^F_8Y^{(\sigma\otimes\bar{\sigma})}
\end{align*}
with $\tilde{m}^F_8\coloneqq\sqrt{3}m^F_8/2$.\par
To find the eigenvalues of $m^{(\sigma\otimes\bar{\sigma})}$, we need to diagonalize it. For totally symmetric multiplets $\sigma$, this poses no problem, as $I^{(\sigma\otimes\bar{\sigma})}_3$ and $Y^{(\sigma\otimes\bar{\sigma})}$ are diagonal in the basis $\{\Ket{Y,I,I_3}\}$. If we denote by $m(Y,I,I_3)$ the mass of the hadron in the multiplet $\sigma$ whose hypercharge is $Y$, whose total isospin is $I$, and whose third component of the isospin is $I_3$, we obtain for totally symmetric multiplets $\sigma$:
\begin{gather}
m(Y,I,I_3) = m_0 + m^F_3\cdot I_3 + \tilde{m}^F_8\cdot Y + \mathcal{O}\left(\varepsilon^2_3\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right) + \mathcal{O}\left(\varepsilon^2_8\right).\label{eq:iso_tot_sym}
\end{gather}
If $\sigma$ is not totally symmetric, we need to take $D^{(\sigma\otimes\bar{\sigma})}_3$ and $D^{(\sigma\otimes\bar{\sigma})}_8$ into account as well. We have already seen that $D^{(\sigma\otimes\bar{\sigma})}_8$ is diagonal in the basis $\{\Ket{Y,I,I_3}\}$. However, $D^{(\sigma\otimes\bar{\sigma})}_3$ does not need to be diagonal in this basis, as $I^{2;\,(\sigma\otimes\bar{\sigma})}$ and $D^{(\sigma\otimes\bar{\sigma})}_3$ do not, in general, commute. Nevertheless, $D^{(\sigma\otimes\bar{\sigma})}_3$ does commute with $I^{(\sigma\otimes\bar{\sigma})}_3$ and $Y^{(\sigma\otimes\bar{\sigma})}$, since $F^{(\sigma\otimes\bar{\sigma})}_3$ commutes with $I^{(\sigma\otimes\bar{\sigma})}_3$ and $Y^{(\sigma\otimes\bar{\sigma})}$ (cf. \autoref{eq:D-F_com}). This means that $D^{(\sigma\otimes\bar{\sigma})}_3$ links hadrons with same hypercharge $Y$ and third isospin component $I_3$, but different total isospin $I$. In this regard, $D^{(\sigma\otimes\bar{\sigma})}_3$ is a source for $\Lambda^0$-$\Sigma^0$-mixing in the spin-$\frac{1}{2}$ baryon octet\footnote{In fact, one can compute the mixing angle $\alpha$ of the $\Lambda^0$-$\Sigma^0$-mixing by calculating the entries of $D^{(8\otimes \bar{8})}_3$. One obtains:
\[\frac{1}{2}\tan (2\alpha) = \frac{m_{\Sigma} - m_{\Sigma^+} + m_p - m_n}{\sqrt{3}(m_{\Sigma} - m_{\Lambda})},\]
where $m_{\Sigma^+}$, $m_p$, and $m_n$ are the masses of the baryons given in the index and $m_{\Sigma}$ and $m_{\Lambda}$ are the diagonal entries of $m^{(8\otimes \bar{8})}$ corresponding to $\ket{0,1,0}$ (roughly $\Sigma^0$) and $\ket{0,0,0}$ (roughly $\Lambda^0$) respectively. As the mixing and, thus, $\alpha$ is small, we have $m_{\Sigma} \approx m_{\Sigma^0}$ and $m_{\Lambda} \approx m_{\Lambda^0}$ to good approximation. Furthermore, one can use $\tan (2\alpha)/2 \approx \tan (\alpha)$ for small $\alpha$ to recover the formula of R. H. Dalitz and F. von Hippel for the mixing angle $\alpha$ (cf. \cite{Dalitz1964}; note the different sign convention for $\alpha$).}.\par
To diagonalize $m^{(\sigma\otimes\bar{\sigma})}$ for multiplets $\sigma$ that are not totally symmetric, we make use of the following trick: In physical application, we are only interested in the eigenvalues of $m^{(\sigma\otimes\bar{\sigma})}$ for isospin breaking terms that are way smaller than their $\text{SU}(2)\times\text{U}(1)$-invariant counterparts, i.e., for $\varepsilon_3\ll\varepsilon_8$. This allows us to treat the $D^{(\sigma\otimes\bar{\sigma})}_3$-term as a perturbation of $m^{(\sigma\otimes\bar{\sigma})}$. However, the eigenvectors of the unperturbed part of $m^{(\sigma\otimes\bar{\sigma})}$ are just $\Ket{Y,I,I_3}$. Hence, we obtain for the eigenvalue $m^{(\sigma\otimes\bar{\sigma})}_{Y,I,I_3}$ of $m^{(\sigma\otimes\bar{\sigma})}$ corresponding to $\Ket{Y,I,I_3}$:
\begin{align*}
m^{(\sigma\otimes\bar{\sigma})}_{Y,I,I_3} &= \tilde{m}_0 + m^F_3\cdot I_3 + \tilde{m}^F_8\cdot Y + \tilde{m}^D_8\left(I(I+1) - \frac{Y^2}{4}\right)\\
&\ \ \ + m^D_3\Braket{Y,I,I_3|D^{(\sigma\otimes\bar{\sigma})}_3|Y,I,I_3} + \mathcal{O}\left(\varepsilon_8\cdot\left(\frac{\varepsilon_3}{\varepsilon_8}\right)^2\right),
\end{align*}
where $\tilde{m}_0$, $\tilde{m}^F_8$, and $\tilde{m}^D_8$ are defined like in \autoref{sec:GMO_formula}. If $\sigma$ is an octet $8$, one can calculate:
\begin{gather*}
\Braket{Y,I,I_3|D^{(8\otimes\bar{8})}_3|Y,I,I_3} = I_3Y.
\end{gather*}
Denoting the mass of a hadron in an octet whose hypercharge is $Y$, whose total isospin is $I$, and whose third component of the isospin is $I_3$ by $m(Y,I,I_3)$, we find for the octet:
\begin{align}
m(Y,I,I_3) &= \tilde{m}_0 + m^F_3\cdot I_3 + \tilde{m}^F_8\cdot Y + m^D_3\cdot I_3Y + \tilde{m}^D_8\left(I(I+1) - \frac{Y^2}{4}\right)\nonumber\\
&\ \ \ + \mathcal{O}\left(\varepsilon^2_3\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right) + \mathcal{O}\left(\varepsilon^2_8\right) + \mathcal{O}\left(\varepsilon_8\cdot\left(\frac{\varepsilon_3}{\varepsilon_8}\right)^2\right).\label{eq:iso_octet}
\end{align}
\indent The mass difference between up and down quark is not the only source of isospin symmetry breaking. The electroweak interaction also breaks the isospin symmetry, as, for instance, the up and down quark carry a different electric charge. However, the electroweak interaction also breaks the residual $\text{U}(1)\times\text{U}(1)$-symmetry originating from the quark mass matrix $\mathcal{M}$, since the weak interaction couples quarks of different flavor to each other. Nevertheless, the contribution of the weak interaction to the hadron masses is parametrically suppressed by at least $(M/M_W)^2$, where $M\sim\tilde{m}_0$ is the mass scale of the hadronic multiplet at hand and $M_W$ is the mass of the $W$-bosons. Thus, we can safely assume that for most hadronic multiplets the contribution of the weak interaction to the hadron masses is very small in comparison to the other isospin symmetry breaking effects like up-down-mass difference and electromagnetism. This allows us to neglect all flavor changing currents in the SM and restore the residual $\text{U}(1)\times\text{U}(1)$-flavor symmetry. In principal, the restoration of this symmetry lets us describe electromagnetic contributions to the GMO mass formula in the same way as the contribution arising from the mass difference of the up and down quark. For this, we simply need to incorporate the expansion parameter of the electromagnetic interaction into $\varepsilon_3$. As the electromagnetic coupling constant is directly related to $\sqrt{\alpha}$, we just need to incorporate $\sqrt{\alpha}$ into $\varepsilon_3$. However, the GMO mass formula including isospin symmetry breaking only describes contributions to first order in $\varepsilon_3$. This way, we can only describe electromagnetic contributions at order $\sqrt{\alpha}$. Since we do not expect any electromagnetic contribution at order $\sqrt{\alpha}$, but first at order $\alpha$, we have to think of something else to describe electromagnetic contributions at order $\alpha$.
\subsection*{Electromagnetic Contributions}
Up to this point, we have aimed to be very precise and rigorous with every statement and tried to motivate every assumption we had to make. However, we now want to employ a more heuristic and phenomenological model to describe the corrections to the GMO mass formula arising from the electromagnetic interaction. We will see in \autoref{chap:mass_relations} that the model we present in this section leads to mass relations that are well known in the literature like the Coleman-Glashow mass relation (cf. \cite{coleman-glashow}). For our model, we make the assumption that the electromagnetic interaction inside hadrons can be described in an effective approach: We imagine that we are able to ``integrate out'' the photon field inside a hadron\footnote{We refrain from defining the meaning of the phrase ``integrate out'', as it is not crucial for our reasoning.}. This leaves us with an effective hadronic Hamilton operator $H$. Let us now suppose that $H$ takes the following form:
\begin{align*}
H &= H_\text{QCD} + \alpha\Delta H + \mathcal{O}\left(\alpha^2\right),
\end{align*}
where $H_\text{QCD}$ is the Hamilton operator from the previous section that led us to the GMO mass formula (including isospin symmetry breaking originating from the mass difference between up and down quark) and $\Delta H$ describes the electromagnetic interaction inside hadrons at order $\alpha$.
Again, we identify the mass of a hadron with an eigenvalue of $H$ and calculate the eigenvalues of $H$ in a perturbative treatment. The perturbative treatment of the electromagnetic interaction for the computation of hadron masses is justified, since we expect the electromagnetic interaction to play a secondary role in the formation of hadrons. This time, we take $\alpha$ to be the expansion parameter of the perturbation series. For a perturbative treatment of $H$, we need to know the eigenvalues and -spaces of the unperturbed part, $H_\text{QCD}$, first. The determination of these properties was the concern of the previous sections in this chapter. We have found that we can group hadrons, i.e., eigenstates of $H_\text{QCD}$, into \text{SU}(3)-multiplets $\sigma$ such that every hadron in $\sigma$ is fully determined by three quantum numbers $Y$, $I$, and $I_3$. Usually, we have to mind the degeneracy of the eigenvalues of the unperturbed operator when applying perturbation theory. If $H_\text{QCD}$ was completely flavor symmetric, i.e., invariant under \text{SU}(3)-flavor transformations, all hadrons in a multiplet $\sigma$ would be degenerate. However, the symmetry breaking induced by $\varepsilon_8$ lifts the degeneracy between different isospin multiplets\footnote{Isospin multiplets are the irreducible representations of the isospin symmetry group $\text{SU}(2)\times\text{U}(1)$. Isospin multiplets within a \text{SU}(3)-multiplet $\sigma$ are formed by hadrons which have the same hypercharge $Y$ and total isospin $I$ (cf. \autoref{sec:rel_within_multiplets}).}, i.e., only hadrons with the same $Y$ and $I$ would be degenerate, if the term proportional to $\varepsilon_8$ was the only term breaking the flavor symmetry (cf. \autoref{sec:GMO_formula}). Additionally, the symmetry breaking induced by $\varepsilon_3$ lifts the remaining degeneracy within those isospin multiplets (cf. first part of \autoref{sec:add_con}). Thus, we can take the degeneracy of hadrons in $\sigma$ to be already lifted by the symmetry breaking terms that are included in $H_\text{QCD}$\footnote{On a deeper level, this is not true. We can deduce from experimental data that the contributions arising from the electromagnetic interaction and from $\varepsilon_3$ are in the same order of magnitude, thus, it is not sensible to consider the electromagnetic interaction to be a perturbation of the $\varepsilon_3$-term included in $H_\text{QCD}$. However, one can use the residual $\text{U}(1)\times\text{U}(1)$-flavor symmetry of the electromagnetic interaction to show that the electromagnetic interaction as a perturbation of {${H^0_\text{QCD} + \varepsilon_8\cdot H^8_\text{QCD;8}}$} is diagonal in the basis $\Ket{Y,I,I_3}$ when restricted to the eigenspaces of {${H^0_\text{QCD} + \varepsilon_8\cdot H^8_\text{QCD;8}}$}. This way, we obtain the same results as if we considered the electromagnetic interaction to be a perturbation of $H_\text{QCD}$ including the $\varepsilon_3$-term.}. If we denote the mass (including electromagnetic corrections) of a hadron in $\sigma$ with hypercharge $Y$, total isospin $I$, and third component $I_3$ of the isospin by $m_\alpha(Y,I,I_3)$, we find to first order in perturbation theory:
\begin{align*}
m_\alpha(Y,I,I_3) = m(Y,I,I_3) + \alpha\cdot\Braket{Y,I,I_3|\Delta H|Y,I,I_3} + \mathcal{O}\left(\alpha^2\right) + \mathcal{O}\left(\alpha\varepsilon_3\right) + \mathcal{O}\left(\alpha\varepsilon_8\right),
\end{align*}
where $m(Y,I,I_3)$ is the mass of the hadron following from $H_\text{QCD}$, i.e., without electromagnetic corrections (cf. \autoref{eq:iso_tot_sym} and \autoref{eq:iso_octet}) and $\Ket{Y,I,I_3}$ is the eigenstate of the hadron in $\sigma$.\par
In order to calculate the first order correction of the electromagnetic interaction, we need some information about $\Delta H$. For the following arguments, we suppose that $\Delta H$ is given by:
\begin{align*}
\Delta H &=\sum_{f_1,f_2\in\{\text{u,d,s}\}}\frac{q_{f_1}q_{f_2}}{e^2}C_{f_1f_2},
\end{align*}
where $q_{f_1}$ and $q_{f_2}$ are the electric charges associated with the flavors $f_1$ and $f_2$ and the operators $C_{f_1f_2}$ mediate the electromagnetic interaction inside the hadrons at order $\alpha$. The choice of the form of $\Delta H$ is motivated by the following consideration: If one wants to introduce the electromagnetic interaction of the quarks to the Lagrangian $\mathcal{L}_\text{QCD}$, one has to add the Yang-Mills term $\mathcal{L}_\text{QED}^\text{YM}$ of the photon field and the coupling of the electromagnetic current $J_\text{QED}^\mu$ to the photon field $A_\mu$:
\begin{align*}
\mathcal{L}(x) &= \mathcal{L}_\text{QCD}(x) + \mathcal{L}_\text{QED}^\text{YM}(x) + J_\text{QED}^\mu(x) A_\mu(x)\quad\text{with}\\
J_\text{QED}^\mu(x) &\coloneqq \sum_{f\in\{\text{u,d,s}\}} q_f\bar{\psi}_f(x)\gamma^\mu\psi_f(x),
\end{align*}
where $\psi_q(x) \coloneqq q(x)$ for $q\in\{\text{u, d, s}\}$ with the notation of \autoref{sec:Trafo_QCD} and similar for $\bar{\psi}_q$. Commonly, perturbative QFT calculations involve the computation of matrix elements containing the time-ordered exponential of the interaction terms in the Lagrangian. By expanding the time-ordered exponential, we find an expansion of the matrix elements in the coupling constants, i.e., in $\alpha$ in the case of quantum electrodynamics (QED). Therefore, the terms at order $\alpha$ always involve a product of two electromagnetic currents $J^\mu_\text{QED}$:
\begin{gather*}
J^\mu_\text{QED}(x) J^\nu_\text{QED}(y) = \sum_{f_1, f_2\in\{\text{u,d,s}\}} q_{f_1}q_{f_2}\left(\bar{\psi}_{f_1}(x)\gamma^\mu\psi_{f_1}(x)\right)\left(\bar{\psi}_{f_2}(y)\gamma^\nu\psi_{f_2}(y)\right)
\end{gather*}
If we assume that the product of two electromagnetic currents also occurs in the operator $\Delta H$ describing the electromagnetic interaction inside hadrons at order $\alpha$, it seems plausible that $\Delta H$ takes the given form. Of course, the integration over the space-time variables $x$ and $y$, terms related to or arising from photon fields, and Wilson loops and lines guaranteeing the gauge invariance of $\Delta H$ also need to be considered for a complete description of $\Delta H$.\par
To proceed, we need to make use of the quark model. In the quark model, we think of baryons and mesons as composite particles and imagine the hadrons to be made out of valence and sea quarks (cf. \cite{Povh2014}). The valence quarks of a hadron dictate its properties and quantum numbers. The valence quark content of a baryon in the SM is given by three quarks, while the valence quark content of a meson consists of a quark and an antiquark. With the quark model in mind, we want to argue now that the following formula applies:
\begin{align*}
\alpha\cdot\Braket{Y,I,I_3|\Delta H|Y,I,I_3} = \Delta_\alpha\sum_{(i,j)}\frac{q_iq_j}{e^2} + \mathcal{O}(\alpha\varepsilon_3) + \mathcal{O}(\alpha\varepsilon_8),
\end{align*}
where $\Delta_\alpha$ is a quantity that is constant on $\sigma$, i.e., independent of $Y$, $I$, and $I_3$, $i$ and $j$ denote valence (anti)quarks of the hadron described by $\Ket{Y,I,I_3}$, $(i,j)$ is a pair of different valence (anti)quarks, the sum runs over every pair once, and $q_i$ and $q_j$ are the charges of the valence (anti)quarks $i$ and $j$, respectively.\par
To convince ourselves that this formula is reasonable, we have to compute:
\begin{gather*}
\alpha\cdot\Braket{Y,I,I_3|\Delta H|Y,I,I_3} = \sum_{f_1,f_2\in\{\text{u,d,s}\}}\Braket{Y,I,I_3|\alpha C_{f_1f_2}|Y,I,I_3}\frac{q_{f_1}q_{f_2}}{e^2}
\end{gather*}
For this, it is helpful to consider the electromagnetic current $J^\mu_\text{QED}$ again. $J^\mu_\text{QED}$ is a conserved Noether current. Its conserved Noether charge $Q_\text{QED}$ is just the electric charge operator, i.e, the operator counting the electric charges in a state when applied to it. In that regard, it seems plausible that the expectation value of the operator $\Delta H$ from which we expect to involve the product of two currents $J^\mu_\text{QED}$ is just the sum of some constants $\Delta_{\alpha;\, ij}(Y,I,I_3)$ times charge products $q_iq_j$ of pairs $(i,j)$ of valence and/or sea (anti)quarks in the hadron state:
\begin{align*}
\alpha\cdot\Braket{Y,I,I_3|\Delta H|Y,I,I_3} = \sum_{(i,j)}\Delta_{\alpha;\, ij}(Y,I,I_3)\frac{q_iq_j}{e^2}.
\end{align*}
We assume for the following considerations that the contribution of sea quarks to this sum is negligible. Whether this assumption is satisfied and, if not, how large the contribution of the sea quarks is, poses an interesting question for further investigations. Moreover, it is sensible to assume that the constants $\Delta_{\alpha;\, ij}(Y,I,I_3)$ are independent of $i$, $j$, $Y$, $I$, and $I_3$:
The \text{SU}(3)-flavor transformations form an approximate symmetry of the hadrons which is only broken by $\varepsilon_3$ and $\varepsilon_8$ neglecting the electromagnetic interaction. The electromagnetic interaction only breaks this approximate symmetry, since the quarks carry different electric charge. However, the constants $\Delta_{\alpha;\, ij}(Y,I,I_3)$ do not involve the electric charge of the quarks. To this end, it seems plausible that the constants $\Delta_{\alpha;\, ij}(Y,I,I_3)$ are independent of the flavor-dependent quantities $i$, $j$, $Y$, $I$, and $I_3$ aside from corrections proportional to products of $\alpha$ and $\varepsilon_{3/8}$:
\begin{gather*}
\Delta_{\alpha;\, ij}(Y,I,I_3) = \Delta_\alpha + \mathcal{O}(\alpha\varepsilon_3) + \mathcal{O}(\alpha\varepsilon_8).
\end{gather*}
This reproduces the initially given formula.
In total, the mass of a hadron including electromagnetic corrections of order $\alpha$ should be given by:
\begin{align}
m_\alpha(Y,I,I_3) = m(Y,I,I_3) + \Delta_\alpha\sum_{(i,j)}\frac{q_iq_j}{e^2} + \mathcal{O}\left(\alpha^2\right) + \mathcal{O}\left(\alpha\varepsilon_3\right) + \mathcal{O}\left(\alpha\varepsilon_8\right).\label{eq:mass_ele}
\end{align}
\section{Heavy Quark Symmetry}
\label{sec:heavy_quark}
So far, we have only considered light hadrons, i.e., hadrons that are formed out of the three light quarks up, down, and strange as valence quarks. The description of these hadrons led us to the classification of light hadrons into \text{SU}(3)-multiplets $\sigma$ and gave us a mass formula for the light hadrons (cf. \autoref{sec:GMO_formula} and \autoref{sec:add_con}). However, we can only use this mass formula to derive mass relations within a \text{SU}(3)-multiplet $\sigma$ (cf. \autoref{sec:mass_matrix} and \autoref{sec:rel_within_multiplets}): The mass formula is just a parametrization of the hadron masses in a \text{SU}(3)-multiplet $\sigma$ in terms of undetermined parameters like $m_0$, $m^F_3$, $m^F_8$, $m^D_3$, $m^D_8$, or $\Delta_\alpha$. Since for some \text{SU}(3)-multiplets $\sigma$ the number of undetermined parameters is smaller than the number of (degenerate) hadron masses in $\sigma$, the hadron masses in $\sigma$ have to satisfy mass relations to be in agreement with the mass formula. But since we have not found any relation between the undetermined parameters of different hadronic \text{SU}(3)-multiplets $\sigma$ yet, we are unable to formulate mass relations between different multiplets.\par
In this section, we want to incorporate the description of hadrons with heavy quarks like charm and bottom quark as part of their valence quark content into our model for the hadron masses. Furthermore, we will see that we can use a heuristic approach to hadrons containing a heavy quark to formulate mass relations between different $\text{SU}(3)$-multiplets. Let us start by considering a Lagrangian $\mathcal{L}^5_\text{QCD}$ that contains five quark flavors:
\begin{gather*}
\mathcal{L}^5_{\text{QCD}}(\bar{q},q) = \sum\limits_{q\in\{\text{u,d,s,c,b}\}}\bar{q}\left(i\slashed{D} - m_q\right)q + \mathcal{L}_\text{YM},
\end{gather*}
where we employ the same notation as in \autoref{sec:Trafo_QCD}. In the same way as for $\mathcal{L}_\text{QCD}$ (cf. \autoref{sec:Trafo_QCD}), we can define flavor transformations of the fields $q$. The only difference is that now the group of flavor transformations is given by \text{SU}(5) instead of \text{SU}(3). The flavor transformations of the fields $q$ furnish transformations of the quark mass matrix $\mathcal{M}$ that form a representation of \text{SU}(5). Likewise, the field flavor transformations give rise to transformations of the Lagrangian $\mathcal{L}^5_\text{QCD}$ and the corresponding Hamilton operator $H^5_\text{QCD}$. Similar to the case of three flavors, we can decompose the Hamilton operator $H^5_\text{QCD}$ into a singlet of \text{SU}(5) plus terms that transform like the adjoint representation of \text{SU}(5) under flavor transformations. If all quark masses in the theory are roughly the same, i.e., if the \text{SU}(5)-flavor transformations form an approximate global symmetry of the Lagrangian and if we can treat the non-singlet terms in the Hamilton operator $H^5_\text{QCD}$, which are proportional to the quark mass differences, as a perturbation of the Hamilton operator, one can proceed similarly to \autoref{sec:GMO_formula} to find that one can group the hadrons of this 5-flavors theory into \text{SU}(5)-multiplets and derive a mass formula for the hadrons in a \text{SU}(5)-multiplet.\par
In Nature, however, we observe that the charm and bottom quark are much heavier than up, down, and strange quark: $m_\text{b},m_\text{c}\gg m_\text{u},m_\text{d},m_\text{s}$. Even though this means that the \text{SU}(5)-flavor symmetry is severely broken in Nature and we cannot apply a perturbative treatment to the Hamilton operator $H^5_\text{QCD}$, the \text{SU}(5)-flavor group and its multiplets still provide a classification for hadrons (cf. $\text{SU}(4)$-flavor group in review \textit{105. Charmed Baryons} in \cite{PDG}). To describe states in a \text{SU}(5)-multiplet, we need quantum numbers additional to $Y$, $I$, and $I_3$. We can link two of these additional quantum numbers to the charm $C$ and to the bottomness $B$. Essentially, the charm $C$ and the bottomness $B$ of a hadron specify -- together with other quantum numbers -- how many charm and bottom (anti)quarks are part of the valence quark content of the hadron.\par
The flavor transformations of only the first three quarks u, d, and s form a Lie subgroup of the \text{SU}(5)-flavor transformations which is equivalent to the Lie group \text{SU}(3). Hence, the hadronic \text{SU}(5)-multiplets decompose into a direct sum of multiplets of this \text{SU}(3)-flavor subgroup. The decomposition of a \text{SU}(5)-multiplet can be done in such a way that all hadrons contained in a given \text{SU}(3)-multiplet of this decomposition have the same number of charm and bottom valence (anti)quarks. This allows us to group hadrons into \text{SU}(3)-multiplets where each hadron in a given \text{SU}(3)-multiplet has the same number of charm and bottom valence (anti)quarks. The \text{SU}(3)-flavor group of the quarks u, d, and s can be treated as an approximate global symmetry of the Lagrangian. This allows us to apply the state formalism we introduced in the previous chapters and sections to recover the GMO mass formula including isospin symmetry breaking and electromagnetic corrections for the hadronic \text{SU}(3)-multiplets with fixed number of charm and bottom valence (anti)quarks.\par
So far, we have found that we can apply the mass formula from \autoref{sec:add_con} to hadronic \text{SU}(3)-multiplets with fixed number of charm and bottom valence (anti)quarks. The mass formula from \autoref{sec:add_con} contains undetermined parameters that can, in general, be different for different hadronic \text{SU}(3)-multiplets. In the next step, we want to link the undetermined parameters of a hadronic \text{SU}(3)-multiplet with exactly one charm valence (anti)quark and no bottom valence (anti)quark to the undetermined parameters of the hadronic \text{SU}(3)-multiplet that is obtained from the first multiplet via the exchange of charm and bottom quarks. Before we consider these hadronic \text{SU}(3)-multiplets, it is instructive and helpful to investigate an easily accessible example of a composite particle, namely the hydrogen atom. We think of the hydrogen atom as a composite particle that is formed by a proton and an electron. The mass $m_\text{H}$ of the hydrogen atom is, in a naive picture, just the sum of the proton mass $m_p$, the electron mass $m_e$, and the binding energy $E$:
\begin{gather*}
m_\text{H} = m_p + m_e + E.
\end{gather*}
In non-relativistic quantum mechanics, one can determine the binding energy $E$ of the hydrogen atom by solving the Schr\"odinger equation of the proton-electron-system. Neglecting all additional terms arising from the finestructure and the hyperfinestructure, one obtains:
\begin{gather*}
E = -\mu\frac{\alpha^2}{2n}\quad\text{with}\quad\mu\coloneqq\frac{m_em_p}{m_e+m_p},
\end{gather*}
where $\mu$ is the reduced mass and $n\in\mathbb{N}$ is the principal quantum number. As the mass of the electron is much smaller than the mass of the proton, we can expand the reduced mass $\mu$ in a Taylor series:
\begin{gather*}
\mu = m_e\sum^\infty_{k=0}\left(-\frac{m_e}{m_p}\right)^k.
\end{gather*}
Thus, we can write the mass $m_\text{H}$ of the hydrogen atom as the proton mass plus a power series in $m_e/m_p$:
\begin{gather*}
m_\text{H}(n) = m_p + m_e\left(1 - \frac{\alpha^2}{2n} + \mathcal{O}\left(\frac{m_e}{m_p}\right)\right).
\end{gather*}
Now consider an atom where we have exchanged the proton of the hydrogen atom by a particle with same electric charge, but different mass $m_d>m_e$ like, for instance, the deuteron. The mass $m_\text{D}$ of this atom is given by an expression similar to the one for the mass $m_\text{H}$ of the hydrogen atom; We simply have to replace the proton mass $m_p$ in the expression for $m_\text{H}$ with the new mass $m_d$:
\begin{gather*}
m_\text{D}(n) = m_d + m_e\left(1 - \frac{\alpha^2}{2n} + \mathcal{O}\left(\frac{m_e}{m_d}\right)\right).
\end{gather*}
We can easily see that the mass formulae for $m_\text{H}$ and $m_\text{D}$ give rise to a mass relation that is satisfied to lowest order in $m_e/m_p$ and $m_e/m_d$:
\begin{gather*}
m_\text{H}(n^\prime) - m_\text{H}(n) = m_\text{D}(n^\prime) - m_\text{D}(n) + \mathcal{O}\left(\frac{m_e}{m_p}\right) + \mathcal{O}\left(\frac{m_e}{m_d}\right),
\end{gather*}
where $n$ and $n^\prime$ are natural numbers. We can even say that the corrections to this mass relation have to be of order $m_e(1/m_p - 1/m_d)$, since this relation is trivially true, if $m_p = m_d$:
\begin{gather*}
m_\text{H}(n^\prime) - m_\text{H}(n) = m_\text{D}(n^\prime) - m_\text{D}(n) + \mathcal{O}\left(m_e\left(\frac{1}{m_p} - \frac{1}{m_d}\right)\right).
\end{gather*}
We are able to write down this mass relation, since the binding energy of a hydrogen-like atom exhibits a ``heavy nucleus symmetry'': If the nucleus of a hydrogen-like atom is much heavier than the surrounding electron, the binding energy of this hydrogen-like atom depends very little on the exact mass of the nucleus. This dependence on the mass of the nucleus is suppressed by factors of $m_e/m_N$, where $m_N$ is the mass of the nucleus. Therefore, the mass difference between two excitations of a hydrogen-like atom with a heavy nucleus is approximately symmetric under the exchange of the heavy nucleus, i.e., picks up corrections in the order of $m_e(1/m_N - 1/m_{N^\prime})$ under the exchange of a heavy nucleus $N$ with a heavy nucleus $N^\prime$.\par
The example of the hydrogen-like atoms demonstrated that a $1/m$-expansion allows us to find mass relations. To this end, we should be able to find mass relations between hadrons containing heavy valence quarks like charm or bottom quarks (and a heavy quark symmetry\footnote{For a deeper discussion of heavy quark symmetry, confer \cite{Neubert1994}.}, in general), if the hadron masses can be expanded in $1/m_Q$ where $m_Q$ is the mass of a heavy quark. Indeed, one often finds in heavy quark effective theory (HQET; cf. \cite{Neubert1994} and \cite{Jenkins1996}) that objects like the Lagrangian, fields, operators, and hadron masses are described by a $1/m_Q$-expansion, implying the existence of mass relations between hadrons containing heavy valence quarks in HQET (cf. \cite{Neubert1994}). As the parameter $1/m_Q$ is dimensionful, we need some mass scale $\Lambda$ to form a dimensionless expansion parameter $\Lambda/m_Q$. In HQET, this mass scale is typically $\Lambda_\text{QCD}$ (cf. \cite{Jenkins2008}), the mass scale of QCD obtained from dimensional transmutation.\par
Motivated by HQET, we want to incorporate a heuristic description of hadrons containing a heavy quark into our model of hadron masses. For this, consider the following matrix $S$:
\begin{gather*}
S \coloneqq \begin{pmatrix}1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 0\end{pmatrix}.
\end{gather*}
If we label the columns from left to right and the rows from top to bottom with u, d, s, c, and b, the matrix $S$ defines a flavor transformation of the fields:
\begin{align*}
q&\xrightarrow{S\in\text{U}(5)}q^\prime \coloneqq \sum\limits_{p\in\{\text{u,d,s,c,b}\}} S_{qp}p,\\
\bar{q}&\xrightarrow{S\in\text{U}(5)}\bar{q}^\prime \coloneqq \sum\limits_{p\in\{\text{u,d,s,c,b}\}} S^\ast_{qp}\bar{p}.
\end{align*}
Note that $S$ is not a \text{SU}(5)-flavor transformation, but a \text{U}(5)-flavor transformation. The transformation given by $S$ only exchanges the charm with the bottom field, so it only exchanges charm and bottom quarks. Suppose that there exists an operator $\hat S$ such that:
\begin{gather*}
H^5_\text{QCD}(q^\prime,\bar{q}^\prime) = \hat{S}^\dagger H^5_\text{QCD}(q,\bar{q})\hat S.
\end{gather*}
We can interpret $\hat S$ as the operator that exchanges charm and bottom quarks when applied to hadronic states in the framework of the state formalism. Now consider a hadronic \text{SU}(3)-multiplet with exactly one charm and no bottom valence (anti)quarks and its bottom counterpart. Denote the hadronic states in the charm \text{SU}(3)-multiplet by $\Ket{Y,I,I_3}_\text{c}$ and the hadronic states in the bottom counterpart by $\Ket{Y,I,I_3}_\text{b}$. Motivated by HQET, we postulate that $\hat S$ links $\Ket{Y,I,I_3}_\text{c}$ and $\Ket{Y,I,I_3}_\text{b}$ in the following way:
\begin{gather*}
\hat S\Ket{Y,I,I_3}_\text{c} = \Ket{Y,I,I_3}_\text{b} + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right).
\end{gather*}
We chose to use $\mathcal{O}\left(\Lambda_\text{QCD}\left(1/m_\text{c} - 1/m_\text{b}\right)\right)$ instead of $\mathcal{O}\left(\Lambda_\text{QCD}/m_\text{c}\right) + \mathcal{O}\left(\Lambda_\text{QCD}/m_\text{b}\right)$ here, since we want to reflect the fact that there would be an exact global \text{U}(2)-flavor symmetry between charm and bottom quark, if the charm and bottom quark masses were equal (neglecting the electroweak interaction). If this symmetry was exact, the masses of the hadrons corresponding to $\Ket{Y,I,I_3}_\text{c}$ and $\Ket{Y,I,I_3}_\text{b}$ would have to be equal. In this case, we would expect the equation above to have no corrections. However, the \text{U}(2)-flavor symmetry between charm and bottom is broken by the quark mass difference $m_\text{b} - m_\text{c}$, so corrections to the equation above have to scale with $m_\text{b} - m_\text{c}$. This is reflected in $\Lambda_\text{QCD}(1/m_\text{c} - 1/m_\text{b})$, as $\Lambda_\text{QCD}(1/m_\text{c} - 1/m_\text{b}) = \Lambda_\text{QCD}(m_\text{b}-m_\text{c})/m_\text{c}m_\text{b}$. Note that we only would expect the $\text{U}(2)$-flavor symmetry breaking terms to be proportional to $m_\text{b}-m_\text{c}$, if the $\text{U}(2)$-flavor symmetry was still an approximate symmetry, i.e., if the $\text{U}(2)$-flavor symmetry breaking was small enough such that we can treat the symmetry breaking term as a perturbation. In Nature, however, we observe that the symmetry breaking is quite large, as $m_\text{b}\gg m_\text{c}$. To this end, we have to understand the last arguments with a grain of salt. Further notice that we have completely neglected any logarithmic correction or any logarithmic scale dependence originating from quantum loops in the discussion of the corrections, even though they might be quite large.\par
Let us now decompose the Hamilton operator $H^5_\text{QCD}$ into a \text{SU}(3)-singlet plus the 3rd and 8th component of a \text{SU}(3)-octet:
\begin{gather*}
H^5_\text{QCD} = H^{5;\,0}_\text{QCD} + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8},
\end{gather*}
where $\varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}$ coincides with the identically named term from \autoref{sec:EFT+H_Pert} and \autoref{sec:GMO_formula} and $H^{5;\,0}_\text{QCD}$ is the collection of the remaining terms. $H^{5;\,0}_\text{QCD}$ is a \text{SU}(3)-singlet under uds-flavor transformations, while $H^8_{\text{QCD};\, 3}$ and $H^8_{\text{QCD};\, 8}$ are the 3rd and 8th component of a \text{SU}(3)-octet under uds-flavor transformations, respectively. Let us denote the mass of the hadron corresponding to $\Ket{Y,I,I_3}_\text{c}$ by $m^\text{c}(Y,I,I_3)$ and the mass of the hadron corresponding to $\Ket{Y,I,I_3}_\text{b}$ by $m^\text{b}(Y,I,I_3)$. Following the state formalism, we find:
\begin{align*}
m^\text{b}(Y,I,I_3) &= \leftidx{_\text{b}}{\Braket{Y,I,I_3|H^{5;\,0}_\text{QCD}(q,\bar{q}) + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3}(q,\bar{q}) + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}(q,\bar{q})|Y,I,I_3}}{_\text{b}}\\
&\ \ \ + \mathcal{O}(\varepsilon_i\varepsilon_j)\\
&= \leftidx{_\text{b}}{\Braket{Y,I,I_3|H^5_\text{QCD}(q,\bar{q})|Y,I,I_3}}{_\text{b}} + \mathcal{O}(\varepsilon_i\varepsilon_j)\\
&= \leftidx{_\text{c}}{\Braket{Y,I,I_3|\hat{S}^\dagger H^5_\text{QCD}(q,\bar{q})\hat S|Y,I,I_3}}{_\text{c}} + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right) + \mathcal{O}(\varepsilon_i\varepsilon_j)\\
&= \leftidx{_\text{c}}{\Braket{Y,I,I_3|H^5_\text{QCD}(q^\prime,\bar{q}^\prime)|Y,I,I_3}}{_\text{c}} + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right) + \mathcal{O}(\varepsilon_i\varepsilon_j)\\
&= \leftidx{_\text{c}}{\Braket{Y,I,I_3|H^{5;\,0}_\text{QCD}(q^\prime,\bar{q}^\prime) + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3}(q,\bar{q}) + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}(q,\bar{q})|Y,I,I_3}}{_\text{c}}\\
&\ \ \ + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right) + \mathcal{O}(\varepsilon_i\varepsilon_j),
\end{align*}
where $\mathcal{O}(\varepsilon_i\varepsilon_j)$ combines all higher order corrections from \autoref{eq:iso_tot_sym} and \autoref{eq:iso_octet}. The last line follows, as the flavor transformation given by $S$ only affects the charm and bottom fields, but only the \text{SU}(3)-singlet $H^{5;\, 0}_\text{QCD}$ contains charm and bottom fields. The mass $m^\text{c}(Y,I,I_3)$ of the charmed hadrons is similarly given by:
\begin{align*}
m^\text{c}(Y,I,I_3) &= \leftidx{_\text{c}}{\Braket{Y,I,I_3|H^{5;\,0}_\text{QCD}(q,\bar{q}) + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3}(q,\bar{q}) + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}(q,\bar{q})|Y,I,I_3}}{_\text{c}}\\
&\ \ \ + \mathcal{O}(\varepsilon_i\varepsilon_j).
\end{align*}
We can see that $m^\text{c}(Y,I,I_3)$ and $m^\text{b}(Y,I,I_3)$ only differ in the \text{SU}(3)-singlet except for higher order corrections. All other terms that appear in the mass formulae are equal. This allows us to parametrize $m^\text{c}(Y,I,I_3)$ and $m^\text{b}(Y,I,I_3)$ for totally symmetric \text{SU}(3)-multiplets in the following way (cf. \autoref{eq:iso_tot_sym}):
\begin{align}
m^\text{c}(Y,I,I_3) &= m^\text{c}_0 + m^F_3\cdot I_3 + \tilde{m}^F_8\cdot Y + \mathcal{O}\left(\varepsilon_i\varepsilon_j\right) + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right),\label{eq:mass_tot_sym_c}\\
m^\text{b}(Y,I,I_3) &= m^\text{b}_0 + m^F_3\cdot I_3 + \tilde{m}^F_8\cdot Y + \mathcal{O}\left(\varepsilon_i\varepsilon_j\right) + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right),\label{eq:mass_tot_sym_b}
\end{align}
where $m^\text{c}_0$ and $m^\text{b}_0$ are unrelated parameters. One can write down similar parametrizations for the other \text{SU}(3)-multiplets.\par
If we want to include electromagnetic corrections in the mass parametrization, we have to consider two important points: Firstly, the charm and bottom quark carry different electric charge. Secondly, we were able to parametrize electromagnetic corrections within a hadronic \text{SU}(3)-multiplet in \autoref{sec:add_con}, as the flavors of all valence quarks within the multiplet were part of the approximate global \text{SU}(3)-flavor symmetry. This does not apply to hadronic \text{SU}(3)-multiplets containing heavy quarks. Therefore, we have to introduce two contributions to the mass formula of hadronic \text{SU}(3)-multiplets containing exactly one heavy quark: One term $\Delta^{LL}_\alpha$ that describes the electromagnetic interaction between two light quarks and another term $\Delta^{LH}_\alpha$ that describes the electromagnetic interaction between a light and a heavy quark. Note that $\Delta^{LL}_\alpha$ does not arise for mesons. With these remarks in mind, we can repeat the parametrization of electromagnetic contributions from \autoref{sec:add_con} to find for baryons:
\begin{align}
m^\text{c}_\alpha(Y,I,I_3) &= m^\text{c}(Y,I,I_3) + \Delta^{LH}_\alpha\sum_{j\in\{k,l\}}\frac{q_jq_\text{c}}{e^2} + \Delta^{LL}_\alpha\frac{q_kq_l}{e^2}\label{eq:mass_ele_c}\\
&\ \ \ + \mathcal{O}\left(\alpha\varepsilon_i\right) + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right),\nonumber\\
m^\text{b}_\alpha(Y,I,I_3) &= m^\text{b}(Y,I,I_3) + \Delta^{LH}_\alpha\sum_{j\in\{k,l\}}\frac{q_jq_\text{b}}{e^2} + \Delta^{LL}_\alpha\frac{q_kq_l}{e^2}\label{eq:mass_ele_b}\\
&\ \ \ + \mathcal{O}\left(\alpha\varepsilon_i\right) + \mathcal{O}\left(\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)\right),\nonumber
\end{align}
where we employ a notation similar to \autoref{eq:mass_ele}. $\mathcal{O}\left(\alpha\varepsilon_i\right)$ combines all higher order corrections involving $\alpha$ (cf. \autoref{eq:mass_ele}). $k$ and $l$ denote the light valence quarks of the hadron, $q_k$ and $q_l$ denote their charge. In the case of mesons, we have to slightly modify the formula: There is only one light valence quark for mesons and we have to set $\Delta^{LL}_\alpha = 0$.
\newpage
\chapter{Overview and Discussion of Mass Relations}\label{chap:mass_relations}
In \autoref{chap:hadron_masses} and \autoref{chap:GMO_formula}, we explored how $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)\rightarrow\text{U}(1)\times\text{U}(1)$ flavor symmetry breaking together with electromagnetic corrections and heavy quark symmetry enters the mass parametrization of hadrons in multiplets. We want to investigate the mass relations arising from the hadronic mass parametrizations in this chapter. In particular, we examine the mass relations of hadrons within sextets, octets, and decuplets in \autoref{sec:rel_within_multiplets} and the mass relations between charm and bottom antitriplets and sextets in \autoref{sec:rel_between_multiplets}. For each multiplet or pair of charm and bottom multiplet, we present the weight diagram(s) of the multiplet(s), the mass parametrization of the hadrons within the multiplet(s), and the mass relations following from this parametrization. To make the weight diagram(s) and mass relations more accessible, we label each weight in the weight diagram(s) with the corresponding hadron of a prominent example for that multiplet/pair of multiplets. After the presentation of the mass relations, we discuss the order of magnitude of the dominant correction for every mass relation within/between that multiplet/pair of multiplets.
\section{Mass Relations within Multiplets}\label{sec:rel_within_multiplets}
Before we dive into the discussion of the multiplets, let us first consider which multiplets we expect to be realized in Nature and make some remarks about weight diagrams and notations. As already explained, we think of hadrons as composite particles in the quark model. In this picture, we imagine the hadrons to be made out of valence and sea quarks. The valence quarks dictate the flavor structure of the hadron in this model, i.e., tell us how the hadron transforms under flavor transformations. In regard to $\text{SU}(3)$-flavor transformations, the light quarks and antiquarks up, down, and strange transform under the fundamental representations $3$ and $\bar{3}$, respectively. Baryons are considered to be made out of three valence quarks. If the baryon at hand is a light baryon, i.e., if the valence quark content of the baryon only consists of light quarks (u, d, or s), the baryon transforms under the tensor product representation of three fundamental representations $3$. Using Young tableaux, one then finds:
\begin{gather*}
3\otimes 3\otimes 3 = 1\oplus 8\oplus 8\oplus 10.
\end{gather*}
If the baryon consists of one charm or bottom and two light valence quarks, it transforms under:
\begin{gather*}
3\otimes 3 = \bar{3}\oplus 6.
\end{gather*}
From this consideration, we expect the light baryons to form singlets, octets, and decuplets and the baryons containing exactly one charm or bottom valence quark to form antitriplets and sextets. However, light baryon singlets are not yet discovered in Nature\footnote{And we do not expect to find one, as the symmetry and statistics of baryonic states forbid the existence of baryon singlets.}.\par
We can determine the multiplets formed by mesons in a similar way: The valence quark content of mesons consists of a quark and an antiquark. Therefore, light mesons, i.e., mesons only containing light valence quarks transform under:
\begin{gather*}
3\otimes \bar{3} = 1\oplus 8,
\end{gather*}
while mesons containing exactly one charm/bottom valence quark or antiquark simply transform under $\bar{3}$ or $3$, respectively. This means that light mesons form singlets and octets, while mesons containing exactly one charm/bottom valence quark or antiquark form antitriplets or triplets, respectively. Note that the hadronic (anti)triplets do not contain enough hadrons to form mass relations within the (anti)triplet.\par
When we introduced weights in \autoref{sec:GMO_formula}, we stated that the collection of all weights of one \text{SU}(3)-multiplet uniquely characterizes that multiplet. Hence, it is sufficient to consider just the weights if one wishes to describe the multiplets of $\text{SU}(3)$. Commonly, the weights of a \text{SU}(3)-multiplet are presented in graphical form, i.e., by a weight diagram. A weight diagram of a \text{SU}(3)-multiplet is a two-dimensional coordinate system where the axes correspond to the components of the weights and each weight is indicated by a dot at the appropriate position. In the following sections, we rescale the axes of the weight diagrams we display such that the axes of the weight diagram coincide with the hypercharge $Y$ and the third isospin component $I_3$. Note that the rescaling of the axes breaks the rotational symmetry of the weight diagrams (the weight diagrams of \text{SU}(3)-multiplets are invariant under rotations by $\frac{2\pi}{3}$; cf. \autoref{sec:GMO_formula} and \cite{Lichtenberg}).\par
Hadrons correspond to vectors in the multiplet, thus, each hadron in a multiplet has a weight. However, multiple hadrons in a multiplet might have the same weight. If the multiplicity of a weight in a multiplet is larger than one, i.e, if there are multiple linearly independent vectors and, thus, hadrons in the multiplet which correspond to that weight, we add a dot near that weight for every additional linearly independent vector/hadron to indicate the multiplicity in the weight diagram.\par
As {${\text{SU}(2)\times\text{U}(1)}$} is a Lie subgroup of $\text{SU}(3)$, we can group basis vectors of a $\text{SU}(3)$-multiplet and, thus, the corresponding weights in a weight diagram into {${\text{SU}(2)\times\text{U}(1)}$}-multiplets. All vectors with same hypercharge $Y$ and total isospin $I$ form a {${\text{SU}(2)\times\text{U}(1)}$-}multiplet. The trivial representation of {${\text{SU}(2)\times\text{U}(1)}$} has a hypercharge and total isospin of $0$. In the following weight diagrams, we connect weights corresponding to the same {${\text{SU}(2)\times\text{U}(1)}$-}multiplet with red lines, if the {${\text{SU}(2)\times\text{U}(1)}$-}multiplet consists of more than one weight.\par
Now consider the group of flavor transformations between the up and strange quark, denoted by {${\text{SU}(2)_\text{us}\times\text{U}(1)}$}, and the group of flavor transformations between the down and strange quark, denoted by {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$}:
\begin{align*}
\text{SU}(2)_\text{us}\times\text{U}(1) &\coloneqq \left\{\begin{pmatrix}e^{i\alpha}A_{11} & 0 & e^{i\alpha}A_{12}\\ 0 & e^{-2i\alpha} & 0\\ e^{i\alpha}A_{21} & 0 & e^{i\alpha}A_{22}\end{pmatrix}\middle| \alpha\in\mathbb{R};\, \begin{pmatrix}A_{11} & A_{12}\\ A_{21} & A_{22}\end{pmatrix}\in\text{SU}(2)\right\},\\
\text{SU}(2)_\text{ds}\times\text{U}(1) &\coloneqq \left\{\begin{pmatrix}e^{-2i\alpha} & 0\\ 0 & e^{i\alpha}A\end{pmatrix}\middle| \alpha\in\mathbb{R};\, A\in\text{SU}(2)\right\}.
\end{align*}
As the groups {${\text{SU}(2)_\text{us}\times\text{U}(1)}$} and {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$} are also Lie subgroups of $\text{SU}(3)$, we can likewise group the weights into {${\text{SU}(2)_\text{us}\times\text{U}(1)}$}- and {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$}-multiplets, respectively. If needed, we connect weights corresponding to the same {${\text{SU}(2)_\text{us}\times\text{U}(1)}$}-multiplet with green lines and weights corresponding to the same {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$}-multiplet with blue lines. A recipe for the construction of weight diagrams of $\text{SU}(3)$ and for grouping the weights of such a weight diagram into {${\text{SU}(2)\times\text{U}(1)}$-},\linebreak {${\text{SU}(2)_\text{us}\times\text{U}(1)}$-}, or {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$-}multiplets can be found in \cite{Lichtenberg}.\par
$\text{U}(1)\times\text{U}(1)$ is also a Lie subgroup of $\text{SU}(3)$, so $\text{SU}(3)$-multiplets decompose into $\text{U}(1)\times\text{U}(1)$-multiplets. However, $\text{U}(1)\times\text{U}(1)$ is an Abelian Lie group, hence, all multiplets are one-dimensional. All vectors in a $\text{U}(1)\times\text{U}(1)$-multiplet are eigenvectors of $I^{(\sigma\otimes\bar{\sigma})}_3$ and $Y^{(\sigma\otimes\bar{\sigma})}$, therefore, every weight in a weight diagram corresponds to a $\text{U}(1)\times\text{U}(1)$-multiplet. The trivial representation of $\text{U}(1)\times\text{U}(1)$ has a hypercharge and third isospin component of $0$.\par
Lastly, we need to clarify some details regarding the (expansion) parameters $\varepsilon_3$, $\varepsilon_8$, $\alpha$, and $\varepsilon_\text{cb}\coloneqq\Lambda_\text{QCD}\left(1/m_\text{c} - 1/m_\text{b}\right)$ we used throughout this work before we can begin the discussion of the mass relations. $\varepsilon_3$ and $\varepsilon_8$ have mass dimension 1, while $\alpha$ and $\varepsilon_\text{cb}$ are dimensionless. A dimensionful quantity as an expansion parameter has the advantage that it allows us to at least estimate how large the first order contribution is, but it does not immediately tell us by which factor the higher order contributions are suppressed. Conversely, a dimensionless quantity as an expansion parameter does not allow us to estimate the first order contribution, but can be used as a suppression factor for the higher order contributions. For instance, we expect the first order contribution of $\varepsilon_8$ to a hadron mass to be of the order of \SI{100}{MeV}, as the current quark mass difference between strange and up/down quark is of that order (cf. comments on quark masses in \autoref{sec:Trafo_QCD} and review \textit{66. Quark Masses} in \cite{PDG}). Likewise, the first order contribution of $\varepsilon_3$ to a hadron mass should be in the order of up-down current quark mass difference, i.e., in the order of \SI{1}{MeV} (cf. comments on quark masses in \autoref{sec:Trafo_QCD} and review \textit{66. Quark Masses} in \cite{PDG}). However, $\varepsilon_3$ and $\varepsilon_8$ per se do not indicate by which factor higher order contributions are suppressed. To obtain the suppression factors of higher order terms of $\varepsilon_3$ and $\varepsilon_8$, we need to know the (unperturbed) singlet contribution $m_0$ to the mass in a hadron multiplet. As we expanded the hadron masses at $m_0$, higher order contributions are suppressed by powers of $\varepsilon_{3/8}/m_0$. In the following sections, we will use $\varepsilon_{3/8}$ both in the sense of a dimensionful and dimensionless parameter, i.e., we will use $\varepsilon_{3/8}$ to denote both $\varepsilon_{3/8}$ and $\varepsilon_{3/8}/m_0$. For $m_0\sim\SI{1}{GeV}$, the order of magnitude of the dimensionless parameters $\varepsilon_3$, $\varepsilon_8$, $\alpha$, and $\varepsilon_\text{cb}$ is given in \autoref{tab:exp_param}. The parameters are ordered from highest to lowest. The values used for this estimate can be found in \cite{PDG}.
\begin{table}[t!]
\centering
\caption{Order of magnitude of several expansion parameters for hadrons with a mass of roughly \SI{1}{GeV}.}
\begin{tabular}{|c|c|}
\hline
Expansion or suppression parameter & Order of magnitude\\\hline
$\varepsilon_\text{cb}\coloneqq\Lambda_\text{QCD}\left(\frac{1}{m_\text{c}} - \frac{1}{m_\text{b}}\right)$ & $\mathcal{O}\left(10\%\right)$\\
$\varepsilon_8$ & $\mathcal{O}\left(10\%\right)$\\
$\alpha$ & $\mathcal{O}\left(1\%\right)$\\
$\varepsilon_3$ & $\mathcal{O}\left(1\%\right)$ to $\mathcal{O}\left(0.1\%\right)$\\\hline
\end{tabular}
\label{tab:exp_param}
\end{table}
Note that the hadron masses only enter linearly in all following mass formulae and relations. Aside from a few exceptions, however, the following formulae and relations are also valid, if one replaces the hadron masses with their squares. For a deeper discussion, confer \autoref{chap:hadron_masses} and \autoref{chap:data}.
\subsection*{Sextet}
An example for a baryonic sextet is the lowest-energy charm sextet with\footnote{Note that the quantum numbers $J^P$ we present in this work are not measured directly for most hadrons, but assumed based on quark model predictions (cf. \cite{PDG}).} $J^P = 1/2^+$. Its weight diagram is shown in \autoref{fig:c_sextet}.
\begin{figure}[htpb]
\centering
\includegraphics[width=\textwidth]{c_sextet.png}
\caption{The weight diagram of a sextet. For every weight, the name of the corresponding baryon from the charm sextet with $J^P = 1/2^+$ is included. The red lines visualize isospin multiplets of $\text{SU}(2)\times\text{U}(1)$. All weights not connected by red lines are one-dimensional isospin multiplets.}
\label{fig:c_sextet}
\end{figure}
Using \autoref{eq:mass_tot_sym_c} and \autoref{eq:mass_ele_c}, we find for the masses in the sextet:
\begin{alignat*}{4}
&m^\text{c}_\alpha\left(\frac{2}{3},1,1\right)&&\equiv m_{\Sigma^{++}_\text{c}} &&= m^\text{c}_0 + m^F_3 &&+ \frac{2}{3}\tilde{m}^F_8 + \frac{8}{9}\Delta^{LH}_\alpha + \frac{4}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(\frac{2}{3},1,0\right)&&\equiv m_{\Sigma^{+}_\text{c}} &&= m^\text{c}_0 &&+ \frac{2}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha - \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(\frac{2}{3},1,-1\right)&&\equiv m_{\Sigma^{0}_\text{c}} &&= m^\text{c}_0 - m^F_3 &&+ \frac{2}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3},\frac{1}{2},\frac{1}{2}\right)&&\equiv m_{\Xi^{\prime\, +}_\text{c}} &&= m^\text{c}_0 + \frac{1}{2}m^F_3 &&- \frac{1}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha - \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3},\frac{1}{2},-\frac{1}{2}\right)&&\equiv m_{\Xi^{\prime\, 0}_\text{c}} &&= m^\text{c}_0 - \frac{1}{2}m^F_3 &&- \frac{1}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{4}{3},0,0\right)&&\equiv m_{\Omega^{0}_\text{c}} &&= m^\text{c}_0 &&- \frac{4}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,
\end{alignat*}
where we omitted all ``$\mathcal{O}$'' for the sake of clarity. This mass parametrization allows us to find two inequivalent mass relations:
\begin{align}
m_{\Sigma^+_\text{c}} - m_{\Sigma^0_\text{c}} &= m_{\Xi^{\prime\, +}_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:sextet_c_iso_bre}\\
m_{\Sigma^0_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} &= m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Omega^0_\text{c}} + \mathcal{O}\left(\varepsilon^2_8\right).\label{eq:sextet_c_GMO_equal_spacing}
\end{align}
This time, we have included the order of magnitude of the dominant correction(s) in the mass relations. Of course, analogous mass relations apply to all baryonic charm and bottom sextets. \autoref{eq:sextet_c_GMO_equal_spacing} is just the equal spacing rule for sextets following from the GMO mass formula (cf. \autoref{eq:GMO_mass_formula}). \autoref{eq:sextet_c_iso_bre} relates the mass splittings of different isospin multiplets in the sextet. The particular form of this mass relation is actually rather interesting. It allows us to derive \autoref{eq:sextet_c_iso_bre} and the order of magnitude of its dominant corrections without the knowledge of a mass parametrization. For this, suppose that the $\text{SU}(2)\times\text{U}(1)$-isospin symmetry was exact. Then, \autoref{eq:sextet_c_iso_bre} would have to be exact, as all baryons in one isospin multiplet would have to have the same mass. One can show this analogously to the case of $\text{SU}(3)$, for which we showed that all hadrons in one $\text{SU}(3)$-multiplet would have to have the same mass, if $\text{SU}(3)$ was an exact symmetry (cf. \autoref{sec:GMO_formula}). However, only the mass and charge difference between the up and down quark break the isospin symmetry (neglecting weak interaction), meaning that only $\varepsilon_3$ and $\alpha$ break the isospin symmetry. Hence, every correction to \autoref{eq:sextet_c_iso_bre} has to be proportional to $\varepsilon_3$ or $\alpha$. Furthermore, \autoref{eq:sextet_c_iso_bre} would also be exact, if the $\text{SU}(2)_\text{ds}\times\text{U}(1)$-flavor transformations of the down and strange quark were an exact symmetry. We can see this by rewriting \autoref{eq:sextet_c_iso_bre}:
\begin{gather*}
m_{\Sigma^+_\text{c}} - m_{\Xi^{\prime\, +}_\text{c}} = m_{\Sigma^0_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}}.
\end{gather*}
This equation is exactly satisfied for exact $\text{SU}(2)_\text{ds}\times\text{U}(1)$-symmetry, as $\Sigma^+_\text{c}$ and $\Xi^{\prime\, +}_\text{c}$ and likewise $\Sigma^0_\text{c}$ and $\Xi^{\prime\, 0}_\text{c}$ have the same mass for exact $\text{SU}(2)_\text{ds}\times\text{U}(1)$-symmetry, since they are contained in the same $\text{SU}(2)_\text{ds}\times\text{U}(1)$-multiplet (cf. \autoref{fig:sextet_ds}).
\begin{figure}[b!]
\centering
\includegraphics[width=0.7\textwidth]{sextet_240degree.png}
\caption{The weight diagram of a sextet. The blue lines visualize the multiplets of $\text{SU}(2)_\text{ds}\times\text{U}(1)$. All weights not connected by blue lines are one-dimensional multiplets.}
\label{fig:sextet_ds}
\end{figure}
However, the $\text{SU}(2)_\text{ds}\times\text{U}(1)$-symmetry is only broken by the mass difference between down and strange quark, so roughly by $\varepsilon_8$. Hence, the corrections to \autoref{eq:sextet_c_iso_bre} have to be proportional to $\varepsilon_8$. In total, this means that every correction to \autoref{eq:sextet_c_iso_bre} has to be proportional to $\alpha\varepsilon_8$ or $\varepsilon_3\varepsilon_8$.\par
It is easy to see that the dominant correction to \autoref{eq:sextet_c_GMO_equal_spacing} is of order $\varepsilon^2_8$: Neglecting the weak interaction, the only parameters that give rise to corrections to \autoref{eq:mass_tot_sym_c} and \autoref{eq:mass_ele_c} are $\varepsilon_3$, $\varepsilon_8$, $\alpha$, and $\varepsilon_\text{cb}$. $\varepsilon_\text{cb}$ and $\varepsilon_8$ induce the largest corrections (cf. \autoref{tab:exp_param}). However, \autoref{eq:sextet_c_GMO_equal_spacing} does not contain any bottom hadrons, so corrections proportional to $\varepsilon_\text{cb}$ do not occur. As terms in the order of $\varepsilon_8$, $\alpha$, and $\varepsilon_3$ are respected by \autoref{eq:mass_ele_c} and, thus, by \autoref{eq:sextet_c_GMO_equal_spacing}, the dominant correction is of order $\varepsilon^2_8$.
\subsection*{Octet}
The $J^P = 1/2^+$ baryon octet is the lightest and probably one of the most known baryon multiplets. Its weight diagram is displayed in \autoref{fig:baryon_octet}.
\begin{figure}[htpb]
\centering
\includegraphics[width=\textwidth]{baryon_octet.png}
\caption{The weight diagram of an octet. For every weight, the name of the corresponding baryon from the $J^P = 1/2^+$ baryon octet is included. The red lines visualize isospin multiplets of $\text{SU}(2)\times\text{U}(1)$. All weights not connected by red lines are isospin singlets.}
\label{fig:baryon_octet}
\end{figure}
Note that both $\Sigma^0$ as well as $\Lambda^0$ have a hypercharge and third isospin component of $0$. The difference between the two particles is that $\Lambda^0$ also has a total isospin of $0$ making it a singlet of $\text{SU}(2)\times\text{U}(1)$, while $\Sigma^0$ has a total isospin of $1$ forming together with $\Sigma^+$ and $\Sigma^-$ an isospin multiplet. We can sum this up in a very elegant way: The octet has two $\text{U}(1)\times\text{U}(1)$-singlets, $\Sigma^0$ and $\Lambda^0$, but only one $\text{SU}(2)\times\text{U}(1)$-isospin singlet, $\Lambda^0$. Using \autoref{eq:iso_octet} and \autoref{eq:mass_ele}, we find for the masses in the octet:
\begin{alignat*}{6}
&m_\alpha\left(1,\frac{1}{2},\frac{1}{2}\right)&&\equiv m_{p} &&= \tilde{m}_0 + \frac{1}{2}m^F_3 &&+ \tilde{m}^F_8 + \frac{1}{2}m^D_3 &&+ \frac{1}{2}\tilde{m}^D_8,&&\\
&m_\alpha\left(1,\frac{1}{2},-\frac{1}{2}\right)&&\equiv m_{n} &&= \tilde{m}_0 - \frac{1}{2}m^F_3 &&+ \tilde{m}^F_8 - \frac{1}{2}m^D_3 &&+ \frac{1}{2}\tilde{m}^D_8 &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(0,1,1\right)&&\equiv m_{\Sigma^{+}} &&= \tilde{m}_0 + m^F_3 && &&+ 2\tilde{m}^D_8,&&\\
&m_\alpha\left(0,1,0\right)&&\equiv m_{\Sigma^{0}} &&= \tilde{m}_0 && &&+ 2\tilde{m}^D_8 &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(0,1,-1\right)&&\equiv m_{\Sigma^{-}} &&= \tilde{m}_0 - m^F_3 && &&+ 2\tilde{m}^D_8 &&+ \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(0,0,0\right)&&\equiv m_{\Lambda^{0}} &&= \tilde{m}_0 && && &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(-1,\frac{1}{2},\frac{1}{2}\right)&&\equiv m_{\Xi^{0}} &&= \tilde{m}_0 + \frac{1}{2}m^F_3 &&- \tilde{m}^F_8 - \frac{1}{2}m^D_3 &&+ \frac{1}{2}\tilde{m}^D_8 &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(-1,\frac{1}{2},-\frac{1}{2}\right)&&\equiv m_{\Xi^{-}} &&= \tilde{m}_0 - \frac{1}{2}m^F_3 &&- \tilde{m}^F_8 + \frac{1}{2}m^D_3 &&+ \frac{1}{2}\tilde{m}^D_8 &&+ \frac{1}{3}\Delta_\alpha,
\end{alignat*}
where we omitted all ``$\mathcal{O}$'' for the sake of clarity. This mass parametrization allows us to find two inequivalent mass relations:
\begin{align}
m_{p} - m_{n} + m_{\Xi^0} - m_{\Xi^-} &= m_{\Sigma^+} - m_{\Sigma^-} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:Coleman-Glashow}\\
m_{p} + m_{n} + m_{\Xi^0} + m_{\Xi^-} &= 3m_{\Lambda^0} + m_{\Sigma^+} + m_{\Sigma^-} - m_{\Sigma^0} + \mathcal{O}\left(\varepsilon^2_8\right).\label{eq:Gell-Mann--Okubo}
\end{align}
This time, we have included the order of magnitude of the dominant correction(s) in the mass relations. Of course, analogous mass relations apply to all baryonic octets. For mesonic octets, especially for the pseudoscalar meson octet, one has to mind the octet-singlet-mixing ($\eta$-$\eta^\prime$-mixing for the pseudoscalar meson octet) and the power of the meson masses.
\autoref{eq:Gell-Mann--Okubo} corresponds to the original Gell-Mann--Okubo mass relation (cf. \cite{Gell-Mann1961}). Often, exact $\text{SU}(2)\times\text{U}(1)$-isospin symmetry is assumed such that the relation can be presented in the following form:
\begin{gather*}
2(m_N + m_\Xi) = 3m_{\Lambda^0} + m_{\Sigma},
\end{gather*}
where $m_N$ is the mass of the $p$-$n$-isospin multiplet, $m_\Xi$ is the mass of the $\Xi$-isospin multiplet, and $m_{\Sigma}$ is the mass of the $\Sigma$-multiplet. In the same way as for \autoref{eq:sextet_c_GMO_equal_spacing}, we find that the dominant correction to \autoref{eq:Gell-Mann--Okubo} is of the order $\varepsilon^2_8$.\par
\autoref{eq:Coleman-Glashow} is the famous and very precise Coleman-Glashow mass relation (cf. \cite{coleman-glashow}). Like for \autoref{eq:sextet_c_iso_bre}, we can derive the Coleman-Glashow mass relation and the order of magnitude of its dominant corrections from purely group theoretical considerations: If any of the flavor transformation groups {${\text{SU}(2)\times\text{U}(1)}$,} {${\text{SU}(2)_\text{us}\times\text{U}(1)}$,} or {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$} was an exact symmetry, all baryons in one {${\text{SU}(2)\times\text{U}(1)}$\text{-,}}\linebreak {${\text{SU}(2)_\text{us}\times\text{U}(1)}$\text{-,}} or $\text{SU}(2)_\text{ds}\times\text{U}(1)$-multiplet would have to have the same mass, respectively. Thus, considering the combination of masses that occur in \autoref{eq:Coleman-Glashow} and the different submultiplets in the octet (cf. \autoref{fig:baryon_octet} and \autoref{fig:baryon_octet_us_ds}), we easily find that the Coleman-Glashow mass relation would be exact, if any of the flavor transformation groups $\text{SU}(2)\times\text{U}(1)$, $\text{SU}(2)_\text{us}\times\text{U}(1)$, or $\text{SU}(2)_\text{ds}\times\text{U}(1)$ was an exact symmetry.
\begin{figure}[t!]
\hfill
\subfigure{\includegraphics[width=0.45\textwidth]{octet_240degree.png}}
\hfill
\subfigure{\includegraphics[width=0.45\textwidth]{octet_120degree.png}}
\hfill
\caption{Two weight diagrams of an octet. The blue lines visualize the multiplets of {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$}, while the green lines visualize the multiplets of {${\text{SU}(2)_\text{us}\times\text{U}(1)}$}. All weights not connected by blue lines or green lines are singlets. Note that neither $\Sigma^0$ nor $\Lambda^0$ are part of a {${\text{SU}(2)_\text{us}\times\text{U}(1)}$-} or {${\text{SU}(2)_\text{ds}\times\text{U}(1)}$-}multiplet, but only a mixture of $\Sigma^0$ and $\Lambda^0$.}
\label{fig:baryon_octet_us_ds}
\end{figure}
However, all these symmetries are broken, but the $\text{SU}(2)\times\text{U}(1)$-symmetry is only broken by $\varepsilon_3$ and $\alpha$, while the $\text{SU}(2)_\text{ds}\times\text{U}(1)$-symmetry is only broken by $\varepsilon_8$. As for \autoref{eq:sextet_c_iso_bre}, this means that every correction to the Coleman-Glashow mass relation has to be proportional to $\alpha\varepsilon_8$ or $\varepsilon_3\varepsilon_8$.
\subsection*{Decuplet}
The $J^P = 3/2^+$ baryon decuplet is a textbook example for the application of the GMO mass formula, as Gell-Mann was able to predict the $\Omega^-$-particle contained in the $J^P = 3/2^+$ baryon decuplet and its mass using the equal spacing rules within the decuplet, before the $\Omega^-$-particle was discovered (cf. \cite{Zee2016} and \cite{Langacker2017}). Its weight diagram is displayed in \autoref{fig:baryon_decuplet}.
\begin{figure}[htpb]
\centering
\includegraphics[width=\textwidth]{decuplet.png}
\caption{The weight diagram of a decuplet. For every weight, the name of the corresponding baryon from the $J^P = 3/2^+$ baryon decuplet is included. The red lines visualize isospin multiplets of $\text{SU}(2)\times\text{U}(1)$. All weights not connected by red lines are one-dimensional isospin multiplets.}
\label{fig:baryon_decuplet}
\end{figure}
Using \autoref{eq:iso_tot_sym} and \autoref{eq:mass_ele}, we find for the masses in the decuplet:
\begin{alignat*}{6}
&m_\alpha\left(1,\frac{3}{2},\frac{3}{2}\right)&&\equiv m_{\Delta^{++}} &&= m_0 &&+ \frac{3}{2}m^F_3 &&+ \tilde{m}^F_8 &&+ \frac{4}{3}\Delta_\alpha,\\
&m_\alpha\left(1,\frac{3}{2},\frac{1}{2}\right)&&\equiv m_{\Delta^{+}} &&= m_0 &&+ \frac{1}{2}m^F_3 &&+ \tilde{m}^F_8,&&\\
&m_\alpha\left(1,\frac{3}{2},-\frac{1}{2}\right)&&\equiv m_{\Delta^{0}} &&= m_0 &&- \frac{1}{2}m^F_3 &&+ \tilde{m}^F_8 &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(1,\frac{3}{2},-\frac{3}{2}\right)&&\equiv m_{\Delta^{-}} &&= m_0 &&- \frac{3}{2}m^F_3 &&+ \tilde{m}^F_8 &&+ \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(0,1,1\right)&&\equiv m_{\Sigma^{\ast\, +}} &&= m_0 &&+ m^F_3,&& &&\\
&m_\alpha\left(0,1,0\right)&&\equiv m_{\Sigma^{\ast\, 0}} &&= m_0 && && &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(0,1,-1\right)&&\equiv m_{\Sigma^{\ast\, -}} &&= m_0 &&- m^F_3 && &&+ \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(-1,\frac{1}{2},\frac{1}{2}\right)&&\equiv m_{\Xi^{\ast\, 0}} &&= m_0 &&+ \frac{1}{2}m^F_3 &&- \tilde{m}^F_8 &&- \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(-1,\frac{1}{2},-\frac{1}{2}\right)&&\equiv m_{\Xi^{\ast\, -}} &&= m_0 &&- \frac{1}{2}m^F_3 &&- \tilde{m}^F_8 &&+ \frac{1}{3}\Delta_\alpha,\\
&m_\alpha\left(-2,0,0\right)&&\equiv m_{\Omega^{-}} &&= m_0 && &&- 2\tilde{m}^F_8 &&+ \frac{1}{3}\Delta_\alpha,
\end{alignat*}
where we omitted all ``$\mathcal{O}$'' for the sake of clarity. This mass parametrization allows us to find seven mass relations:
\begin{gather}\small
m_{\Sigma^{\ast\, +}} - m_{\Sigma^{\ast\, -}} = m_{\Delta^{+}} - m_{\Delta^{0}} + m_{\Xi^{\ast\, 0}} - m_{\Xi^{\ast\, -}} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:Coleman-Glashow_decuplet}\\
m_{\Delta^{-}} = m_{\Delta^{++}} + 3\left( m_{\Delta^{0}} - m_{\Delta^{+}}\right) + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:Delta-}\\
m_{\Delta^{++}} + m_{\Delta^{0}} - 2m_{\Delta^{+}} = m_{\Sigma^{\ast\, +}} + m_{\Sigma^{\ast\, -}} - 2m_{\Sigma^{\ast\, 0}} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:iso1_decuplet}\\
m_{\Sigma^{\ast\, -}} - m_{\Sigma^{\ast\, +}} = m_{\Delta^{++}} + 3m_{\Delta^{0}} - 4m_{\Delta^{+}} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:iso2_decuplet}\\
m_{\Delta^{+}} - m_{\Sigma^{\ast\, +}} = m_{\Sigma^{\ast\, -}} - m_{\Xi^{\ast\, -}} + \mathcal{O}\left(\varepsilon^2_8\right),\label{eq:equal_spacing1_decuplet}\\
m_{\Sigma^{\ast\, -}} - m_{\Xi^{\ast\, -}} = m_{\Xi^{\ast\, -}} - m_{\Omega^{-}} + \mathcal{O}\left(\varepsilon^2_8\right),\label{eq:equal_spacing2_decuplet}\\
4m_{\Delta^{++}} - 6\left(m_{\Delta^{+}} - m_{\Delta^{0}}\right) - 4m_{\Omega^{-}} = 6\left(m_{\Sigma^{\ast\, +}} + m_{\Sigma^{\ast\, -}}\right) - 6\left(m_{\Xi^{\ast\, 0}} + m_{\Xi^{\ast\, -}}\right)\label{eq:better_GMO_decuplet}\\
+ \mathcal{O}\left(\varepsilon^3_8\right) + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right).\nonumber
\end{gather}
This time, we have included the order of magnitude of the dominant correction(s) in the mass relations. Of course, analogous mass relations apply to all baryonic decuplets. In contrast to the case of the sextet and octet, only the first six mass relations are inequivalent in the sense that none of the first six mass relations follows from each other. The seventh mass relation, however, can be formed out of the first six relations.\par
\autoref{eq:Coleman-Glashow_decuplet} is the companion piece to the Coleman-Glashow mass relation in the decuplet. It and its dominant corrections can be derived in the same way as for the Coleman-Glashow mass relation of the octet (cf. the previous section ``Octet'', \autoref{fig:baryon_decuplet}, and \autoref{fig:baryon_decuplet_us_ds}).\par
\begin{figure}[htbp]
\hfill
\subfigure{\includegraphics[width=0.45\textwidth]{decuplet_240degree.png}}
\hfill
\subfigure{\includegraphics[width=0.45\textwidth]{decuplet_120degree.png}}
\hfill
\caption{Two weight diagrams of a decuplet. The blue lines visualize the multiplets of $\text{SU}(2)_\text{ds}\times\text{U}(1)$, while the green lines visualize the multiplets of $\text{SU}(2)_\text{us}\times\text{U}(1)$. All weights not connected by blue lines or green lines are one-dimensional multiplets.}
\label{fig:baryon_decuplet_us_ds}
\end{figure}
The dominant corrections to \autoref{eq:Delta-}, \autoref{eq:iso1_decuplet}, and \autoref{eq:iso2_decuplet} are in the order of $\alpha\varepsilon_8$ and $\varepsilon_3\varepsilon_8$. As all of these relations are exact in the limit of exact isospin symmetry, every correction to these relations has to be proportional to $\alpha$ or $\varepsilon_3$. However, the mass parametrization we used to derive these relations already includes the first order contributions of $\alpha$ and $\varepsilon_3$, hence, the dominant corrections are in the order of $\alpha\varepsilon_8$ and $\varepsilon_3\varepsilon_8$. At this point, it should be noted that it is also possible to derive \autoref{eq:Coleman-Glashow_decuplet}, \autoref{eq:Delta-}, and \autoref{eq:iso1_decuplet} from quark model calculations (cf. \cite{Ishida1966} and \cite{Gal1967}) and baryonic chiral perturbation theory (cf. \cite{Lebed1994}).\par
\autoref{eq:equal_spacing1_decuplet} and \autoref{eq:equal_spacing2_decuplet} are the equal spacing rules for the decuplet following from the GMO mass formula (cf. \autoref{eq:GMO_mass_formula}). Their dominant correction is in the order of $\varepsilon^2_8$. We can show this in the same way as for the equal spacing rule of the sextet.\par
\autoref{eq:better_GMO_decuplet} seems to be redundant, as this mass relation follows from the other six decuplet mass relations. However, this relation is more precise than the equal spacing rules, since it holds true to second order in $\text{SU}(3)\rightarrow\text{SU}(2)\times\text{U}(1)$-flavor symmetry breaking (for the derivation of \autoref{eq:better_GMO_decuplet} and its dominant corrections, cf. \cite{Okubo1963} and \cite{Lebed1994}), thus, the dominant corrections to \autoref{eq:better_GMO_decuplet} are in the order of $\varepsilon^3_8$, $\alpha\varepsilon_8$, and $\varepsilon_3\varepsilon_8$.
\section{Mass Relations between Multiplets}\label{sec:rel_between_multiplets}
In this section, we want to take a closer look at pairs of charm and bottom multiplets. For each pair of charm and bottom multiplets, heavy quark symmetry allows us to formulate mass relations that involve hadrons from both the charm and bottom multiplet. This is interesting for multiple reasons: First off, all mass relations we have found in \autoref{sec:rel_within_multiplets} only relate hadrons in the same multiplet, so the mass relations we introduce in this section expand our description of hadron masses. Secondly, considering several multiplets to form mass relations gives us the possibility to use hadronic multiplets that do not contain enough hadrons to give rise to a mass relation within them. This applies, in particular, to hadronic charm and bottom (anti)triplets. Lastly, the mass relations following from heavy quark symmetry provide us with a way to investigate mesonic mass relations that are not spoiled by octet-singlet-mixing in contrast to the mass relations within mesonic octets.\par
The discussions of the charm and bottom multiplets and the mass relations between them are structured very similar to \autoref{sec:rel_within_multiplets}. All remarks at the beginning of \autoref{sec:rel_within_multiplets} apply in very similar fashion to the following discussions.
\subsection*{Baryonic Charm and Bottom Antitriplets}
The lightest charm and bottom baryons form antitriplets with $J^P = 1/2^+$. Their weight diagrams are displayed in \autoref{fig:c_b_baryon_triplets}.
\begin{figure}
\centering
\subfigure{\includegraphics[width=0.85\textwidth]{c_baryon_triplet.png}}
\vspace{0.1cm}
\subfigure{\includegraphics[width=0.85\textwidth]{b_baryon_triplet.png}}
\caption{Two weight diagrams of an antitriplet. For every weight, the name of the corresponding baryon from the charm or bottom antitriplet with $J^P = 1/2^+$ is included in the upper or lower weight diagram, respectively. The red lines visualize isospin multiplets of $\text{SU}(2)\times\text{U}(1)$. All weights not connected by red lines are one-dimensional isospin multiplets.}
\label{fig:c_b_baryon_triplets}
\end{figure}
Using \autoref{eq:mass_tot_sym_c}, \autoref{eq:mass_tot_sym_b}, \autoref{eq:mass_ele_c}, and \autoref{eq:mass_ele_b}, we find for the masses in the charm and bottom antitriplets:
\begin{alignat*}{7}
&m^\text{c}_\alpha\left(\frac{2}{3}, 0, 0\right) &&\equiv m_{\Lambda^{+}_\text{c}} &&= m^\text{c}_0 && &&+\frac{2}{3}\tilde{m}^F_8 &&+ \frac{2}{9}\Delta^{LH}_\alpha &&- \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, \frac{1}{2}\right) &&\equiv m_{\Xi^{+}_\text{c}} &&= m^\text{c}_0 &&+\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&+ \frac{2}{9}\Delta^{LH}_\alpha &&- \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, -\frac{1}{2}\right) &&\equiv m_{\Xi^{0}_\text{c}} &&= m^\text{c}_0 &&-\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&- \frac{4}{9}\Delta^{LH}_\alpha &&+ \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(\frac{2}{3}, 0, 0\right) &&\equiv m_{\Lambda^{0}_\text{b}} &&= m^\text{b}_0 && &&+\frac{2}{3}\tilde{m}^F_8 &&- \frac{1}{9}\Delta^{LH}_\alpha &&- \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, \frac{1}{2}\right) &&\equiv m_{\Xi^{0}_\text{b}} &&= m^\text{b}_0 &&+\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&- \frac{1}{9}\Delta^{LH}_\alpha &&- \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, -\frac{1}{2}\right) &&\equiv m_{\Xi^{-}_\text{b}} &&= m^\text{b}_0 &&-\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&+ \frac{2}{9}\Delta^{LH}_\alpha &&+ \frac{1}{9}\Delta^{LL}_\alpha,
\end{alignat*}
where we omitted all ``$\mathcal{O}$'' for the sake of clarity. This mass parametrization allows us to formulate one mass relation:
\begin{gather}
m_{\Lambda^{+}_\text{c}} - m_{\Xi^{+}_\text{c}} = m_{\Lambda^{0}_\text{b}} - m_{\Xi^{0}_\text{b}} + \mathcal{O}\left(\varepsilon_\text{cb}\varepsilon_8\right).\label{eq:bary_c_b_tri}
\end{gather}
This time, we have included the order of magnitude of the dominant correction in the mass relation. Of course, an analogous mass relation applies to all baryonic charm and bottom antitriplets. \autoref{eq:bary_c_b_tri} states that the mass splittings between the isospin multiplets in the charm and bottom antitriplets are equal. The dominant correction to \autoref{eq:bary_c_b_tri} is in the order of $\varepsilon_\text{cb}\varepsilon_8$. This can be shown by the following consideration: $\Lambda^+_\text{c}$ and $\Xi^+_\text{c}$ as well as $\Lambda^0_\text{b}$ and $\Xi^0_\text{b}$ form a multiplet of $\text{SU}(2)_\text{ds}\times\text{U}(1)$. Therefore, their masses would be equal and \autoref{eq:bary_c_b_tri} would be exact, if $\text{SU}(2)_\text{ds}\times\text{U}(1)$ was an exact symmetry. Neglecting the weak interaction, this symmetry is only broken by the mass difference of the down and strange quark, so roughly by $\varepsilon_8$. Thus, every correction to \autoref{eq:bary_c_b_tri} has to be proportional to $\varepsilon_8$. Following the line of reasoning from \autoref{sec:heavy_quark}, \autoref{eq:bary_c_b_tri} only picks up corrections proportional to $\varepsilon_\text{cb}$ or to a product of at least two of the parameters $\varepsilon_3$, $\varepsilon_8$, and $\alpha$. As every correction has to be proportional to $\varepsilon_8$, corrections proportional to $\varepsilon_\text{cb}$ have to be proportional to $\varepsilon_\text{cb}\varepsilon_8$ as well. This leaves us with $\varepsilon_\text{cb}\varepsilon_8$ and $\varepsilon^2_8$ as the largest corrections to \autoref{eq:bary_c_b_tri}. However, we have an exact $\text{SU}(2)_\text{cb}\times\text{U}(1)$-flavor symmetry between charm and bottom quark in the limit of $m_\text{c} = m_\text{b}$ and $\alpha = 0$. In this limit, the corresponding charm and bottom baryons of the charm and bottom antitriplet have to have the same mass. This implies that \autoref{eq:bary_c_b_tri} is exact in this limit. Nevertheless, corrections proportional to $\varepsilon^2_8$ do not vanish in this limit which means that they cannot be present in the first place. Therefore, the dominant correction to \autoref{eq:bary_c_b_tri} is in the order of $\varepsilon_\text{cb}\varepsilon_8$.
\subsection*{Mesonic Charm and Bottom Antitriplets}
Mesonic multiplets include both triplets and antitriplets. However, we can restrict ourselves to considering just triplets or antitriplets, since for every mesonic triplet there is a mesonic antitriplet that contains the antiparticles of the mesons in the triplet. As CPT-invariance dictates that particles and their corresponding antiparticles have to have the same mass, the mass relations of triplets also apply to antitriplets and vice versa. In reference to the baryonic case, we restrict ourselves to antitriplets.\par
The lightest charm and bottom mesons form (anti)triplets with $J^P = 0^-$. Their weight diagrams are displayed in \autoref{fig:c_b_meson_triplets}.
\begin{figure}
\centering
\subfigure{\includegraphics[width=0.85\textwidth]{c_meson_triplet.png}}
\vspace{0.1cm}
\subfigure{\includegraphics[width=0.85\textwidth]{b_meson_triplet.png}}
\caption{Two weight diagrams of an antitriplet. For every weight, the name of the corresponding meson from the charm or bottom antitriplet with $J^P = 0^-$ is included in the upper or lower weight diagram, respectively. The red lines visualize isospin multiplets of $\text{SU}(2)\times\text{U}(1)$. All weights not connected by red lines are one-dimensional isospin multiplets.}
\label{fig:c_b_meson_triplets}
\end{figure}
Using \autoref{eq:mass_tot_sym_c}, \autoref{eq:mass_tot_sym_b}, \autoref{eq:mass_ele_c}, and \autoref{eq:mass_ele_b}, we find for the masses in the charm and bottom antitriplets:
\begin{alignat*}{6}
&m^\text{c}_\alpha\left(\frac{2}{3}, 0, 0\right) &&\equiv m_{D^{+}_\text{s}} &&= m^\text{c}_0 && &&+\frac{2}{3}\tilde{m}^F_8 &&+ \frac{2}{9}\Delta^{LH}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, \frac{1}{2}\right) &&\equiv m_{D^{+}} &&= m^\text{c}_0 &&+\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&+ \frac{2}{9}\Delta^{LH}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, -\frac{1}{2}\right) &&\equiv m_{D^{0}} &&= m^\text{c}_0 &&-\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&- \frac{4}{9}\Delta^{LH}_\alpha,\\
&m^\text{b}_\alpha\left(\frac{2}{3}, 0, 0\right) &&\equiv m_{\bar{B}^{0}_\text{s}} &&= m^\text{b}_0 && &&+\frac{2}{3}\tilde{m}^F_8 &&- \frac{1}{9}\Delta^{LH}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, \frac{1}{2}\right) &&\equiv m_{\bar{B}^{0}} &&= m^\text{b}_0 &&+\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&- \frac{1}{9}\Delta^{LH}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{1}{3}, \frac{1}{2}, -\frac{1}{2}\right) &&\equiv m_{B^{-}} &&= m^\text{b}_0 &&-\frac{1}{2}m^F_3 &&-\frac{1}{3}\tilde{m}^F_8 &&+ \frac{2}{9}\Delta^{LH}_\alpha,
\end{alignat*}
where we omitted all ``$\mathcal{O}$'' for the sake of clarity. This mass parametrization allows us to formulate one mass relation:
\begin{gather}
m_{D^{+}_\text{s}} - m_{D^{+}} = m_{\bar{B}^{0}_\text{s}} - m_{\bar{B}^{0}} + \mathcal{O}\left(\varepsilon_\text{cb}\varepsilon_8\right).\label{eq:meso_c_b_tri}
\end{gather}
This time, we have included the order of magnitude of the dominant corrections in the mass relation. Of course, an analogous mass relation applies to all mesonic charm and bottom (anti)triplets. \autoref{eq:meso_c_b_tri} is the mesonic companion piece to \autoref{eq:bary_c_b_tri}. Indeed, its dominant corrections can be obtained in the same way. Relations like \autoref{eq:meso_c_b_tri} are a typical result of heavy quark symmetry (cf. \cite{Neubert1994}).
\subsection*{Charm and Bottom Sextets}
The lightest charm and bottom sextets are characterized by $J^P = 1/2^+$. Their weight diagrams are displayed in \autoref{fig:c_b_sextets}.
\begin{figure}
\centering
\subfigure{\includegraphics[width=0.85\textwidth]{c_sextet.png}}
\vspace{0.1cm}
\subfigure{\includegraphics[width=0.85\textwidth]{b_sextet.png}}
\caption{Two weight diagrams of a sextet. For every weight, the name of the corresponding baryon from the charm or bottom sextet with $J^P = 1/2^+$ is included in the upper or lower weight diagram, respectively. The red lines visualize isospin multiplets of $\text{SU}(2)\times\text{U}(1)$. All weights not connected by red lines are one-dimensional isospin multiplets.}
\label{fig:c_b_sextets}
\end{figure}
Using \autoref{eq:mass_tot_sym_c}, \autoref{eq:mass_tot_sym_b}, \autoref{eq:mass_ele_c}, and \autoref{eq:mass_ele_b}, we find for the masses in the charm and bottom sextets:
\begin{alignat*}{4}
&m^\text{c}_\alpha\left(\frac{2}{3},1,1\right)&&\equiv m_{\Sigma^{++}_\text{c}} &&= m^\text{c}_0 + m^F_3 &&+ \frac{2}{3}\tilde{m}^F_8 + \frac{8}{9}\Delta^{LH}_\alpha + \frac{4}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(\frac{2}{3},1,0\right)&&\equiv m_{\Sigma^{+}_\text{c}} &&= m^\text{c}_0 &&+ \frac{2}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha - \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(\frac{2}{3},1,-1\right)&&\equiv m_{\Sigma^{0}_\text{c}} &&= m^\text{c}_0 - m^F_3 &&+ \frac{2}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3},\frac{1}{2},\frac{1}{2}\right)&&\equiv m_{\Xi^{\prime\, +}_\text{c}} &&= m^\text{c}_0 + \frac{1}{2}m^F_3 &&- \frac{1}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha - \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{1}{3},\frac{1}{2},-\frac{1}{2}\right)&&\equiv m_{\Xi^{\prime\, 0}_\text{c}} &&= m^\text{c}_0 - \frac{1}{2}m^F_3 &&- \frac{1}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{c}_\alpha\left(-\frac{4}{3},0,0\right)&&\equiv m_{\Omega^{0}_\text{c}} &&= m^\text{c}_0 &&- \frac{4}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(\frac{2}{3},1,1\right)&&\equiv m_{\Sigma^{+}_\text{b}} &&= m^\text{b}_0 + m^F_3 &&+ \frac{2}{3}\tilde{m}^F_8 - \frac{4}{9}\Delta^{LH}_\alpha + \frac{4}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(\frac{2}{3},1,0\right)&&\equiv m_{\Sigma^{0}_\text{b}} &&= m^\text{b}_0 &&+ \frac{2}{3}\tilde{m}^F_8 - \frac{1}{9}\Delta^{LH}_\alpha - \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(\frac{2}{3},1,-1\right)&&\equiv m_{\Sigma^{-}_\text{b}} &&= m^\text{b}_0 - m^F_3 &&+ \frac{2}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{1}{3},\frac{1}{2},\frac{1}{2}\right)&&\equiv m_{\Xi^{\prime\, 0}_\text{b}} &&= m^\text{b}_0 + \frac{1}{2}m^F_3 &&- \frac{1}{3}\tilde{m}^F_8 - \frac{1}{9}\Delta^{LH}_\alpha - \frac{2}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{1}{3},\frac{1}{2},-\frac{1}{2}\right)&&\equiv m_{\Xi^{\prime\, -}_\text{b}} &&= m^\text{b}_0 - \frac{1}{2}m^F_3 &&- \frac{1}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,\\
&m^\text{b}_\alpha\left(-\frac{4}{3},0,0\right)&&\equiv m_{\Omega^{-}_\text{b}} &&= m^\text{b}_0 &&- \frac{4}{3}\tilde{m}^F_8 + \frac{2}{9}\Delta^{LH}_\alpha + \frac{1}{9}\Delta^{LL}_\alpha,
\end{alignat*}
where we omitted all ``$\mathcal{O}$'' for the sake of clarity. This mass parametrization allows us to find ten mass relations:
\begin{gather}
m_{\Sigma^+_\text{c}} - m_{\Sigma^0_\text{c}} = m_{\Xi^{\prime\, +}_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:sextet_c_iso_bre2}\\
m_{\Sigma^0_\text{b}} - m_{\Sigma^-_\text{b}} = m_{\Xi^{\prime\, 0}_\text{b}} - m_{\Xi^{\prime\, -}_\text{b}} + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\label{eq:sextet_b_iso_bre}\\
m_{\Sigma^0_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} = m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Omega^0_\text{c}} + \mathcal{O}\left(\varepsilon^2_8\right),\label{eq:sextet_c_GMO_equal_spacing2}\\
m_{\Sigma^-_\text{b}} - m_{\Xi^{\prime\, -}_\text{b}} = m_{\Xi^{\prime\, -}_\text{b}} - m_{\Omega^-_\text{b}} + \mathcal{O}\left(\varepsilon^2_8\right),\label{eq:sextet_b_GMO_equal_spacing}\\
m_{\Sigma^{++}_\text{c}} + m_{\Sigma^{0}_\text{c}} - 2m_{\Sigma^{+}_\text{c}} = m_{\Sigma^{+}_\text{b}} + m_{\Sigma^{-}_\text{b}} - 2m_{\Sigma^{0}_\text{b}} + \mathcal{O}\left(\alpha\varepsilon_\text{cb}\right) + \mathcal{O}\left(\alpha\varepsilon_8\right),\label{eq:sextet_c_b_sigma}\\
m_{\Sigma^0_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} = m_{\Sigma^-_\text{b}} - m_{\Xi^{\prime\, -}_\text{b}} + \mathcal{O}\left(\varepsilon_\text{cb}\varepsilon_8\right),\label{eq:sextet_c_b_spacing}\\
m_{\Sigma^+_\text{c}} - m_{\Sigma^0_\text{c}} + m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Xi^{\prime\, +}_\text{c}} = m_{\Sigma^0_\text{b}} - m_{\Sigma^-_\text{b}} + m_{\Xi^{\prime\, -}_\text{b}} - m_{\Xi^{\prime\, 0}_\text{b}} + \mathcal{O}\left(\alpha\varepsilon_8\right),\label{eq:sextet_c_b_very_precise}\\
2m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Sigma^0_\text{c}} - m_{\Omega^0_\text{c}} = 2m_{\Xi^{\prime\, -}_\text{b}} - m_{\Sigma^-_\text{b}} - m_{\Omega^-_\text{b}} + \mathcal{O}\left(\varepsilon_\text{cb}\varepsilon^2_8\right),\label{eq:sextet_c_b_precise}\\
m_{\Sigma^{0}_\text{b}} = m_{\Xi^{\prime\, +}_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} + \frac{1}{2}\left(m_{\Sigma^{+}_\text{b}} + m_{\Sigma^{-}_\text{b}} + m_{\Sigma^{0}_\text{c}} - m_{\Sigma^{++}_\text{c}}\right)\label{eq:sextet_c_b_sigma_b}\\
+ \mathcal{O}\left(\alpha\varepsilon_\text{cb}\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_\text{cb}\right) + \mathcal{O}\left(\alpha\varepsilon_8\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_8\right),\nonumber\\
m_{\Xi^{\prime\, 0}_\text{b}} = m_{\Xi^{\prime\, -}_\text{b}} + m_{\Xi^{\prime\, +}_\text{c}} - m_{\Xi^{\prime\, 0}_\text{c}} + \frac{1}{2}\left(m_{\Sigma^{+}_\text{b}} - m_{\Sigma^{-}_\text{b}} + m_{\Sigma^{0}_\text{c}} - m_{\Sigma^{++}_\text{c}}\right)\label{eq:sextet_c_b_xi}\\
+ \mathcal{O}\left(\alpha\varepsilon_\text{cb}\right) + \mathcal{O}\left(\varepsilon_3\varepsilon_\text{cb}\right) + \mathcal{O}\left(\alpha\varepsilon_8\right).\nonumber
\end{gather}
This time, we have included the order of magnitude of the dominant correction(s) in the mass relations. Of course, analogous mass relations apply to all pairs of baryonic charm and bottom sextets. Only the first six relations are inequivalent in the sense that none of the first six mass relations follows from each other. The seventh and eighth mass relation (\autoref{eq:sextet_c_b_very_precise} and \autoref{eq:sextet_c_b_precise}) are more precise variants of the first two mass relations (\autoref{eq:sextet_c_iso_bre2} and \autoref{eq:sextet_b_iso_bre}) and of the following two mass relations (\autoref{eq:sextet_c_GMO_equal_spacing2} and \autoref{eq:sextet_b_GMO_equal_spacing}), respectively. Experimentally, some baryon masses in some sextets are not measured yet, therefore, we will use the last two mass relations (\autoref{eq:sextet_c_b_sigma_b} and \autoref{eq:sextet_c_b_xi}) to calculate the masses of the missing baryons in \autoref{sec:mass_predictions}.\par
The first mass relation (\autoref{eq:sextet_c_iso_bre2}) is just \autoref{eq:sextet_c_iso_bre} from \autoref{sec:rel_within_multiplets} and listed here for completeness' sake. How to obtain the dominant corrections to this mass relation is explained in \autoref{sec:rel_within_multiplets}. The second mass relation (\autoref{eq:sextet_b_iso_bre}) is just the bottom companion piece to the first mass relation. Similar to the first two mass relations, the third mass relation (\autoref{eq:sextet_c_GMO_equal_spacing2}) is just \autoref{eq:sextet_c_GMO_equal_spacing} from \autoref{sec:rel_within_multiplets} and the fourth mass relation (\autoref{eq:sextet_b_GMO_equal_spacing}) is the bottom companion piece to the third mass relation.\par
The fifth and sixth mass relation (\autoref{eq:sextet_c_b_sigma} and \autoref{eq:sextet_c_b_spacing}) involve masses of both the charm and bottom sextet. To determine the dominant corrections to these mass relations, we note that both mass relations are exact in the limit of $m_\text{c} = m_\text{b}$ and $\alpha = 0$, since, in this limit, the mass of a baryon in the charm sextet is equal to the mass of the corresponding baryon in the bottom sextet. This means that every correction to \autoref{eq:sextet_c_b_sigma} and \autoref{eq:sextet_c_b_spacing} has to be proportional to $\varepsilon_\text{cb}$ or $\alpha$. Moreover, we find that \autoref{eq:sextet_c_b_sigma} would also be exact, if the $\text{SU}(2)\times\text{U}(1)$-isospin symmetry was exact. As this symmetry is only broken by $\varepsilon_3$ and $\alpha$, every correction to \autoref{eq:sextet_c_b_sigma} has to be proportional to $\varepsilon_3$ or $\alpha$ as well. Corrections in the order of $\alpha$ cannot occur, as the given mass parametrization already takes such correction into account. This implies that the dominant corrections to \autoref{eq:sextet_c_b_sigma} have to be in the order of $\alpha\varepsilon_\text{cb}$, $\alpha\varepsilon_8$, or $\varepsilon_3\varepsilon_\text{cb}$. However, a correction in the order of $\varepsilon_3\varepsilon_\text{cb}$ cannot occur, because neither the combination of all charm baryons nor the combination of all bottom baryons in \autoref{eq:sextet_c_b_sigma}, i.e., neither the left-hand side nor the right-hand side of \autoref{eq:sextet_c_b_sigma} contains any terms proportional to a single power of $\varepsilon_3$ (cf. \autoref{eq:iso_tot_sym}):
\begin{gather*}
m_{\Sigma^{++}_\text{c}} + m_{\Sigma^{0}_\text{c}} - 2m_{\Sigma^{+}_\text{c}} = 0 + \mathcal{O}\left(\varepsilon^2_3\right) + \mathcal{O}\left(\alpha\right),\\
m_{\Sigma^{+}_\text{b}} + m_{\Sigma^{-}_\text{b}} - 2m_{\Sigma^{0}_\text{b}} = 0 + \mathcal{O}\left(\varepsilon^2_3\right) + \mathcal{O}\left(\alpha\right).
\end{gather*}
Thus, the dominant corrections to \autoref{eq:sextet_c_b_sigma} are in the order of\linebreak {${\alpha\varepsilon_\text{cb}}$} and {${\alpha\varepsilon_8}$}.\par
The sixth mass relation (\autoref{eq:sextet_c_b_spacing}) is not only exact in the limit of $m_\text{c} = m_\text{b}$ and $\alpha = 0$, but also in the limit of exact $\text{SU}(2)_\text{ds}\times\text{U}(1)$-flavor symmetry (cf. \autoref{fig:sextet_ds}). This means that every correction to \autoref{eq:sextet_c_b_spacing} has to be proportional to $\varepsilon_8$ and proportional to $\varepsilon_\text{cb}$ or $\alpha$. Hence, the dominant correction to \autoref{eq:sextet_c_b_spacing} is in the order of $\varepsilon_\text{cb}\varepsilon_8$.\par
We obtain the seventh and eighth mass relation (\autoref{eq:sextet_c_b_very_precise} and \autoref{eq:sextet_c_b_precise}) by subtracting \autoref{eq:sextet_b_iso_bre} from \autoref{eq:sextet_c_iso_bre2} and by subtracting \autoref{eq:sextet_b_GMO_equal_spacing} from \autoref{eq:sextet_c_GMO_equal_spacing2}, respectively, and rewriting the results. This implies that both \autoref{eq:sextet_c_b_very_precise} and \autoref{eq:sextet_c_b_precise} are exact in the limit of exact heavy quark symmetry, i.e., in the limit of $m_\text{c}=m_\text{b}$ and $\alpha=0$. Thus, every correction to \autoref{eq:sextet_c_b_very_precise} and \autoref{eq:sextet_c_b_precise} has to be proportional to $\varepsilon_\text{cb}$ or $\alpha$. Furthermore, if the $\text{SU}(2)\times\text{U}(1)$-isospin symmetry or the $\text{SU}(2)_\text{ds}\times\text{U}(1)$-flavor symmetry was exact, \autoref{eq:sextet_c_b_very_precise} would also be exact. We can deduce from this that the dominant correction to \autoref{eq:sextet_c_b_very_precise} is in the order of $\alpha\varepsilon_8$. For \autoref{eq:sextet_c_b_precise}, we note that it is exact in the limit of exact $\text{SU}(2)_\text{ds}\times\text{U}(1)$-flavor symmetry and that both the left-hand and right-hand side of \autoref{eq:sextet_c_b_precise} only pick up corrections proportional to $\varepsilon^2_8$ (cf. \autoref{eq:sextet_c_GMO_equal_spacing2} and \autoref{eq:sextet_b_GMO_equal_spacing}):
\begin{gather*}
2m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Sigma^{0}_\text{c}} - m_{\Omega^{0}_\text{c}} = 0 + \mathcal{O}\left(\varepsilon^2_8\right),\\
2m_{\Xi^{\prime\, -}_\text{b}} - m_{\Sigma^{-}_\text{b}} - m_{\Omega^{-}_\text{b}} = 0 + \mathcal{O}\left(\varepsilon^2_8\right).
\end{gather*}
This implies that the dominant correction to \autoref{eq:sextet_c_b_precise} is in the order of $\varepsilon_\text{cb}\varepsilon^2_8$.\par
The last two mass relations (\autoref{eq:sextet_c_b_sigma_b} and \autoref{eq:sextet_c_b_xi}) follow from all previous sextet mass relations. Both relations are exact in the limit of exact $\text{SU}(2)\times\text{U}(1)$-isospin symmetry, so corrections to both relations have to be proportional to $\varepsilon_3$ or $\alpha$. The contributions in the order of $\varepsilon_3$ and $\alpha$ are already taken into account by the mass parametrization, so the dominant corrections are given by products of $\varepsilon_3$ or $\alpha$ with the remaining parameters\footnote{\autoref{eq:sextet_c_b_xi} is exact for $m_\text{c} = m_\text{b}$ and $\alpha=0$, thus the correction $\varepsilon_3\varepsilon_8$ cannot occur.}
\newpage
\chapter{Testing Mass Relations on Experimental Data}\label{chap:data}
In the previous chapters, we discussed the symmetry properties of hadron masses regarding global flavor transformations in different approaches to find mass parametrizations and relations. In the course of this process, especially in \autoref{sec:EFT+H_Pert}, the question arose whether these mass relations apply to the hadron masses or their squares. To answer this question, we referred to the experimentally determined values for the hadron masses. However, we have refrained from showing any numbers or results so far. Therefore, we want to test the hadronic mass relations from \autoref{chap:mass_relations} on experimental data in this chapter.\par
To compare the mass relations from \autoref{chap:mass_relations} with experimental values, we first have to say how hadron masses can be accessed experimentally. For this, we will motivate a general definition of particle masses in the framework of S-matrix theory, the pole mass. In the course of this process, we will see that we will be faced with several difficulties regarding the definition and determination of hadron masses. The discussion of mass determination and pole mass is the concern of \autoref{sec:polemass}.\par
In \autoref{sec:mass_testing}, we will discuss to which extent the mass relations from \autoref{chap:mass_relations} are satisfied for both linear and quadratic hadron masses. The aim of this section is two-fold: On the one hand, we want to convince ourselves that the mass relations from \autoref{chap:mass_relations} actually apply up to their dominant correction(s) and that the assumptions which guided us to these mass relations are at least somewhat justified, as they lead to reasonable results. On the other hand, we want to back up our claims concerning the relation between linear and quadratic mass relations we made throughout this work, in particular at the end of \autoref{sec:EFT+H_Pert}.\par
Lastly, we want to use the mass relations from \autoref{chap:mass_relations} and empirical observations to group yet unassigned hadrons into multiplets and to predict the mass of hadrons missing within almost complete multiplets in \autoref{sec:mass_predictions}.
\section{Pole Mass and the Experimental Accessibility of Particle Masses}\label{sec:polemass}
The mass of a point-like particle in a non-quantized, relativistic theory is an intrinsic property of the particle, i.e., independent of the choice of the inertial frame and invariant under all Poincar\'{e} transformations. Following the laws of special relativity, the mass $m$ of a point-like particle with four-momentum $p^\mu$ can be obtained by using the Minkowski metric: $p^2 = m^2$. Thus, it is sufficient to measure the four-momentum of a particle to determine its mass in the framework of such a theory. If we try to apply this method of mass determination to hadrons, we are faced with two problems: On the one hand, most hadrons we are interested in are very short-lived, so it is often not viable in experiments to create a hadron and measure its four-momentum. On the other hand, such an approach neglects the very nature of subatomic particles, namely that they are subject to the laws of the quantum regime.\par
Therefore, a more sophisticated ansatz is needed to describe the mass of particles. This ansatz should factor in both special relativity as well as the laws of the quantum regime. To extent special relativity to quantum theories, we have to incorporate Poincar\'{e} transformations into a quantized theory. We can do this by requiring that the Poincar\'{e} transformations act via a representation on a set of states. We then identify particle states with irreducible representations of the Poincar\'{e} group (cf. \cite{Weinberg1995}). Irreducible representations of the Poincar\'{e} group are characterized by eigenvalues of Casimir operators which are constant on irreducible representations. One Casimir operator of the Poincar\'{e} group is the quadratic Casimir operator $\hat P^\mu\hat P_\mu$ of the translation operators $\hat P^\mu$. Its eigenvalue, denoted by $m^2$, defines the squared mass of the particle.\par
Even though this approach is a more sophisticated, it neither gives us an experimental prescription to determine the mass of a particle nor is it clear whether unstable, composite particles are described by the same formalism. Therefore, we are interested in a more general definition of particle masses. To guide our intuition, consider the following Lagrangian of a free quantized scalar field $\Phi$:
\begin{gather*}
\mathcal{L} = (\partial_\mu\Phi)^\dagger(\partial^\mu\Phi) - m^2\Phi^\dagger\Phi,
\end{gather*}
where $m$ is a real positive parameter. If one requires the fields $\Phi$ and $\Phi^\dagger$ to satisfy the Euler-Lagrange equations following from $\mathcal{L}$ and canonical commutation relations, we can interpret the fields $\Phi$ and $\Phi^\dagger$ -- their Fourier modes to be precise -- as operators annihilating and creating states which we can identify as particle states with mass $m$. Again, we understand particle states as vectors ``living'' in irreducible representations of the Poincar\'{e} group and the mass as the eigenvalue of $\hat P^\mu\hat P_\mu$. Thus, the mass $m$ of a particle in a free theory is directly connected to the coefficients of quadratic field terms in the Lagrangian. Now, consider the propagator $G(p)$ of $\Phi$:
\begin{gather*}
G(p) = \frac{i}{p^2-m^2}.
\end{gather*}
We see that $G$ has a pole at $p^2 = m^2$. This illustrates that particle masses in a QFT are in some way related to poles.\par
Still, we have refrained from giving an experimental prescription for measuring hadron masses. To account for this, consider now a simplistic model for an interacting theory:
\begin{gather*}
\mathcal{L} = (\partial_\mu\Phi)^\dagger(\partial^\mu\Phi) - m^2\Phi^\dagger\Phi + \overline{\Psi}\left(i\slashed{\partial} - M\right)\Psi + g\Phi\overline{\Psi}\Psi,
\end{gather*}
where $\Phi$ is a scalar field, $\Psi$ is a fermionic field, and $m$, $M$, and $g$ are real parameters. Now imagine we make a scattering experiment $\bar{f}f\to\bar{f}f$ within this theory, where we denote the (anti)fermions of this theory by $(\bar{f})f$. Experimentally, we would be able to determine the cross section of this reaction. Theoretically, the cross section for the reaction $\bar{f}f\to\bar{f}f$ is given by a phase space integral of the modulus squared of the transition amplitude $\mathcal{M}(\bar{f}f\to\bar{f}f)$. At tree level, $\mathcal{M}(\bar{f}f\to\bar{f}f)$ is given by the following Feynman diagram:\\
\begin{minipage}{0.49\textwidth}
\centering
\begin{tikzpicture}
\begin{feynman}
\vertex (a);
\vertex [below left=of a] (b) {\(f\)};
\vertex [above left=of a] (c) {\(\bar{f}\)};
\vertex [right=of a] (d);
\vertex [above right=of d] (e) {\(\bar{f}\)};
\vertex [below right=of d] (f) {\(f\)};
\diagram* {
(b) -- [fermion, momentum' = \(p_1\)] (a) -- [fermion, rmomentum'=\(p_2\)] (c),
(a) -- [scalar] (d),
(d) -- [anti fermion] (e),
(d) -- [fermion] (f),
};
\end{feynman}
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\begin{flalign*}
\propto\frac{i}{(p_1+p_2)^2 - m^2}.
\end{flalign*}
\end{minipage}\\
In terms of Mandelstam variables, $(p_1+p_2)^2$ is the square $s$ of the center-of-mass energy. Thus, the cross section of the reaction $\bar{f}f\to\bar{f}f$ has -- at least at tree level -- a pole in $s$ at $s = m^2$. We would like to deduce from this result that cross sections have poles at particle masses -- as we will formulate later on --, since we have seen that the parameter $m$ corresponds to the mass of the particle associated with $\Phi$ in the case of a free field. However, we have to be cautious: The parameter $m$ is a bare parameter in the case of an interacting theory. This means that the parameter $m$ itself is not well defined and renormalized parameters need to be introduced to describe physical quantities. The definition of these parameters depends heavily on the chosen renormalization scheme. This means that the renormalized parameter $m$ might not bear much semblance, if any, to any pole of the cross section. In this regard, we can only understand the previous considerations as a motivation.\par
Nevertheless,
in the S-matrix formalism, particles are commonly related to s-poles of the scattering amplitude and particle masses are defined via the corresponding pole positions (cf. review \textit{48. Resonances} in \cite{PDG}). The notion of mass following this approach is known as \textit{pole mass}. There are two conventions for parametrizing a pole $s_R$ of the S-matrix:
\begin{align*}
s_R &= M_s^2 - iM_s\Gamma_s,\\
w_R :&= \sqrt{s_R} = M_w - i\frac{\Gamma_w}{2},
\end{align*}
whereby $M_{s/w}$ and $\Gamma_{s/w}$ define the pole mass and width of a particle, respectively, in the $s/w$-plane with $w\coloneqq\sqrt{s}$. Usually, the second parametrization is used and referred to as pole mass and decay width (cf. review \textit{48. Resonances} in \cite{PDG}). In both cases, the imaginary part is chosen to be negative which is always possible since $s_R^*$ is a pole, if $s_R$ is a pole (cf. review \textit{48. Resonances} in \cite{PDG}). The parametrizations can be converted into each other by:
\begin{align}
M_s &= \sqrt{M_w^2 - \left(\frac{\Gamma_w}{2}\right)^2},\label{eq:s-w-relation}\\
\Gamma_s &= \frac{M_w}{M_s}\Gamma_w.\nonumber
\end{align}
Stable particles do not decay, thus, $\Gamma_{s/w}$ equals zero and $s_R$ is real. In this case, both definitions of mass and width coincide. For a lot of hadrons, a similar statement applies: By expanding \autoref{eq:s-w-relation}, we see that the difference of $M_s$ and $M_w$ is proportional to $(\Gamma_w/M_w)^2$. This means that the difference between $M_s$ and $M_w$ is very small in comparison to $M_s$ and $M_w$ for $\Gamma_w\ll M_w$.
If the experimental uncertainties of $M_s$ and $M_w$ are larger than their difference, it does not really matter which convention for the pole mass one uses. This applies to most hadrons we consider in the following sections. One should note that under certain circumstances, usually for very broad resonances, the relation between the imaginary part of the pole and the decay width breaks down (cf. review \textit{48. Resonances} in \cite{PDG}). In these cases, the decay width is related to the residue of the S-matrix pole (cf. review \textit{48. Resonances} in \cite{PDG}).\par
Most hadrons are only experimentally accessible as resonances in formation experiments or as subchannel resonances together with spectator particles in scattering experiments (cf. review \textit{48. Resonances} in \cite{PDG}). Thus, most data on hadron masses come from those experiments. To extract information about hadron masses from the scattering cross section, one needs to describe the scattering amplitude $\mathcal{M}$ in the vicinity of the resonance. For simplicity, a decomposition of $\mathcal{M}$ into a part containing the pole and a background amplitude is often performed (cf. review \textit{48. Resonances} in \cite{PDG}):
\begin{gather*}
\mathcal{M} = \mathcal{M}_{\text{pole}} + \mathcal{M}_\text{B}.
\end{gather*}
Obviously, this decomposition is not unique without further restrictions. The common choice for the background amplitude $\mathcal{M}_\text{B}$ is either to take it to be constant near the pole or to omit $\mathcal{M}_\text{B}$ entirely. For instance, $\mathcal{M}$ for a single scalar resonance with small decay width neglecting the background $\mathcal{M}_\text{B}$ can be parametrized near the pole by (cf. review \textit{48. Resonances} in \cite{PDG}):
\begin{gather*}
\mathcal{M} = \frac{\mathcal{M}_0}{s - M^2_\text{BW} + i\sqrt{s}\Gamma_\text{BW}},
\end{gather*}
where $M_\text{BW}$ and $\Gamma_\text{BW}$ are the Breit-Wigner (BW) parameters of mass and width. Assuming this parametrization is exact, the Breit-Wigner parameters are directly linked to the pole position (for instance, in the $w$-plane) via:
\begin{align*}
M_w &= \sqrt{M_\text{BW}^2 - \left(\frac{\Gamma_\text{BW}}{2}\right)^2},\\
\Gamma_w &= \Gamma_\text{BW}.
\end{align*}
If $\Gamma_\text{BW}$ is much smaller than $M_\text{BW}$, $\sqrt{s}$ can be replaced by $M_\text{BW}$:
\begin{gather*}
\mathcal{M} = \frac{\mathcal{M}_0}{s - M^2_\text{BW} + iM_\text{BW}\Gamma_\text{BW}}.
\end{gather*}
In this case, the Breit-Wigner parameters coincide with the $M_s$-$\Gamma_s$-convention.\par
In the context of scattering experiments, the expression ``small decay width'' which was used laxly in the discussion of the Breit-Wigner parametrization refers to the width in relation to other resonances and thresholds nearby. A decay width is small if the width is much smaller than the difference between pole position and other resonances/thresholds, meaning e.g. ${M_s\Gamma_s\ll|s_\text{Res./Thres.} - M^2_s|}$ (cf. review \textit{48. Resonances} in \cite{PDG}). For some strongly decaying resonances, this relation is not satisfied. The $\Delta^+(1232)$-resonance with a width of about $\SI{130}{MeV}$ (taken from \cite{PDG}), for instance, in the photoreaction $p+\gamma$ is located closely to the $N+\pi$ threshold around approximately $\SI{1080}{MeV}$.\par
In general, the Breit-Wigner parametrization is a rather crude approximation and analysis method for resonances. The interaction of spins and angular momenta, the vicinity of other resonances and thresholds, the non-negligible size of decay widths and dominant background effects, among others, call for more sophisticated analysis procedures to obtain the physical pole positions (cf. review \textit{48. Resonances} in \cite{PDG}). This task which is partially the concern of current research is quite troublesome.
As stated in review \textit{98. $N$ and $\Delta$ Resonances} in \textit{The Review of Particle Physics} of the \textit{Particle Data Group} (cf. \cite{PDG}), ``the accurate determination of pole parameters from the analysis of data on the real energy axis is not necessarily simple, or even straightforward. It requires the implementation of the correct analytic structure of the relevant (often coupled) channels.'' Some methods for the experimental determination of pole parameters are faced with ``almost unsurmountable difficulties.'' (Quotes are taken from review \textit{98. $N$ and $\Delta$ Resonances} in \cite{PDG}.)\par
Next to these more technical problems, we are also faced with a theoretical problem on a deeper level: We still have to explain how the notion of pole mass is related to the notion of hadron mass we employed in the previous chapters. For the state formalism, this means to formulate a relation between the eigenvalues of the Hamilton operator and the poles of the scattering matrix. As it stands now, I am unable to give a complete and rigorous answer to this question. However, one might formulate the wrong question, if one asks how to relate the eigenvalues of the Hamilton operator to the poles of the scattering matrix: Historically, the \text{SU}(3)-structure present in the hadronic sector was first recognized for the positions of resonances related to hadrons, i.e., for the notion of pole mass. In this sense, we should start by assuming approximate \text{SU}(3)-symmetry for the pole masses instead of the Lagrangian, if we want to follow empirical observations very closely. We do not need to consider the relation between the eigenvalues of the Hamilton operator and the poles of the scattering matrix in this case. Obviously, we are faced with different complications in such an approach like, for instance, the description of electromagnetic contributions and heavy quark symmetry. Nevertheless, we operate on the assumption
that the mass formulae and relations from \autoref{chap:GMO_formula} and \autoref{chap:mass_relations} apply to pole masses for the time being.\par
The difficulties to define and determine the mass of a particle should be kept in mind when discussing the results of mass relations on experimental data and their predictions.
\section{Verification of Mass Relations}\label{sec:mass_testing}
We want to test the mass relations from \autoref{chap:mass_relations} for both linear and quadratic hadron masses in this section. Our goal is to verify that the mass relations given in \autoref{chap:mass_relations} apply within their range of validity, i.e., are correct up to the given corrections. Moreover, we want to back up our claims from \autoref{sec:EFT+H_Pert}: In this section, we stated that, in general, both linear and quadratic mass relations apply to both baryons and mesons. If they do not, we can argue that the distinction we have to make does not originate from the distinction in baryons and mesons.\par
There are multiple ways to check the mass formulae and relations from \autoref{chap:GMO_formula} and \autoref{chap:mass_relations} and to decide whether and in which cases linear mass formulae and relations are preferable to their quadratic counterparts or vice versa. One method is to determine the free parameters in the linear and quadratic mass formulae by fitting the formulae to the measured hadron masses using the $\chi^2$-method. One could then compare $\chi^2$ of the linear and quadratic mass formulae to find which mass formula is favored by the observed masses. Performing such a fit would also give an estimation for the free parameters in the mass formulae. However, there are a lot of drawbacks to such an approach. First off, the mass formulae from \autoref{chap:GMO_formula} pick up different corrections of different scale. However, a fit is not sensitive to every correction and dominated by the largest one. Moreover, the number of hadron masses in a multiplet or in a pair of multiplets exceeds the number of undetermined parameters in the corresponding mass formulae by at most 6. Considering that some multiplets are incomplete, i.e., that some experimental values for the hadron masses are missing, we lack a sufficient number of data points for the fit-method to be meaningful. For the same reasons, a statistical analysis is, in general, not advisable for testing the implications of the theory of global flavor symmetry breaking and, in particular, the state formalism.\par
Another more refined method
is to determine how ``good'' or ``bad'' the mass relations from \autoref{chap:mass_relations} are satisfied for linear and quadratic hadron masses. In contrast to the previous method, different mass relations are sensitive to different corrections. Additionally, the method of checking mass relations give us more control over the particles involved in the analysis: If the mass of a hadron in a multiplet is not measured, we might be able to find mass relations that do not involve this hadron. But in order to apply this method, we have to elaborate on what we mean by ``determine how `good' or `bad' the mass relations from \autoref{chap:mass_relations} are satisfied for linear and quadratic hadron masses''. Let us start this discussion by clarifying what we mean by ``linear'' and ``quadratic'' mass relations: We can write all mass relations from \autoref{chap:mass_relations} as:
\begin{gather*}
\sum_x k_x m_x = 0,
\end{gather*}
where $x$ denotes a hadron, the sum runs over all hadrons in a multiplet or in a pair of charm and bottom multiplets, $k_x$ is a rational number, and $m_x$ is the mass of the hadron $x$. This form can be achieved by, for instance, subtracting the right-hand side from the left-hand side of a mass relation from the previous chapter. Since such a mass relation is a linear combination of hadron masses, we call it linear mass relation. For every linear mass relation, there is a quadratic version of this mass relation which is obtained by replacing the hadron masses with the squared hadron masses:
\begin{gather*}
\sum_x k_x m^2_x = 0.
\end{gather*}
This means that we can describe linear and quadratic mass relations by:
\begin{gather*}
\sum_x k_x m^n_x = 0,
\end{gather*}
where the exponent $n\in\{1,2\}$ distinguishes between the linear and the quadratic version.\par
Next, we have to figure out how we can check the linear and quadratic mass relations. One of the easiest ways to do this is to simply insert the experimentally determined values for the hadron masses $m_x$ into the left-hand side of the linear and quadratic mass relations given in the form above and to see how much the calculated values deviate from zero. However, this procedure is not helpful in any way, as it suffers from two major problems: On the one hand, the values for the linear and quadratic mass relations have different units: The value obtained from the linear mass relation has a mass dimension of 1, while the value of the quadratic version has mass dimension 2. This makes the values incomparable. On the other hand, the values given by this method are not invariant under rescaling the mass relation, i.e., under multiplying the mass relation with a constant. This is especially troublesome, since the mass relations we found in \autoref{chap:mass_relations} are to some extent arbitrary. For instance, we could have used 1/4 times \autoref{eq:Gell-Mann--Okubo} instead of \autoref{eq:Gell-Mann--Okubo} which is done sometimes (cf. \cite{Gell-Mann1961}). Of course, this implies that the previously suggested procedure itself is arbitrary.\par
We can avoid these issues by employing a more sophisticated method. For this, consider a mass relation given in the following form:
\begin{gather*}
\sum_x k_x m^n_x = 0.
\end{gather*}
If this mass relation is not trivial, there is at least one hadron $y$ for which the coefficient $k_y$ does not vanish. Solving this mass relation for $m_y$ yields:
\begin{gather}
m_y = \sqrt[n]{\sum_{x\neq y}\frac{-k_x}{k_y}m^n_x}.\label{eq:prediction}
\end{gather}
Let us now denote the experimentally determined value for $m_y$ by $m^\text{exp}$ and the value for $m_y$ calculated from the remaining hadrons using \autoref{eq:prediction} by $m^\text{cal}$. We can then compute the following expression:
\begin{gather}
m^\text{cal} - m^\text{exp} = \sqrt[n]{\sum_{x\neq y}\frac{-k_x}{k_y}m^n_x} - m_y.\label{eq:cal-exp}
\end{gather}
If the mass relation was exact, the expression would be zero. Therefore, the deviation of this expression from zero tells us how strongly the mass relation is broken. Clearly, \autoref{eq:cal-exp} is invariant under rescaling the initial mass relation: If we were to multiply the entire mass relation with a constant, $k_x$ and $k_y$ would be multiplied with that constant as well. As only their ratio enters \autoref{eq:cal-exp}, $m^\text{cal}-m^\text{exp}$ remains unchanged. Furthermore, the value obtained from \autoref{eq:cal-exp} has mass dimension 1 for both linear and quadratic mass relations allowing us to compare both values. It becomes clear that the comparison of the linear ($n=1$) values with their quadratic ($n=2$) counterpart is meaningful, if we try to find a physical interpretation for the comparison. The physical relevance of \autoref{eq:cal-exp} is quite obvious: $m^\text{cal}$ is just the mass of the hadron $y$ predicted by the mass relation and the other hadron masses, thus, $m^\text{cal} - m^\text{exp}$ simply states how accurate the prediction for the mass of the hadron $y$ is. Comparing the linear and quadratic value for $m^\text{cal} - m^\text{exp}$ then tells us which mass relation is more accurate making $m^\text{cal} - m^\text{exp}$ a useful measure for comparing linear with quadratic mass relations.\par
To employ this method, we need to say which hadron $y$ we choose for computing $m^\text{cal}-m^\text{exp}$ for an arbitrary mass relation. In principle, there is no reason to favor one hadron over another (except for mesonic octets because of octet-singlet-mixing). We only have to be cautious not to employ systematics such that the selection of the hadron $y$ itself is biased and favors linear or quadratic mass relations. This would be the case, for instance, if we chose the hadron $y$ for every mass relation such that $m^\text{cal}-m^\text{exp}$ attains the smallest-possible value for the linear mass relation. For the following calculations of $m^\text{cal}-m^\text{exp}$ (except for the mesonic octets), we choose to solve the mass relations for the highest mass appearing in the relation, i.e, choose the hadron $y$ to be the hadron with the highest mass (and $k_y\neq 0$) in the multiplet or pair of multiplets.\par
Before we dive into the comparison of linear and quadratic mass relations, we should first tend to the problem the chosen comparison method faces: Even though $m^\text{cal}-m^\text{exp}$ allows us to compare a linear mass relation with its quadratic counterpart, one might struggle to use $m^\text{cal} - m^\text{exp}$ to compare the linear (or quadratic) versions of different mass relations with each other. The reason for this is illustrated by the following example: Consider the lowest-energy baryon octet and charm sextet with $J^P = 1/2^+$. Let us compute the values of $m^\text{cal}-m^\text{exp}$ corresponding to the linear versions of \autoref{eq:Gell-Mann--Okubo} and \autoref{eq:sextet_c_GMO_equal_spacing} for these multiplets, where we solve for the highest mass in each multiplet, i.e., for $y = \Xi^-$ and $y = \Omega^0_\text{c}$ in the case of the octet and sextet, respectively. We find for the values of $m^\text{cal}-m^\text{exp}$ (cf. \autoref{eq:cal-exp}):
\begin{gather*}
3m_{\Lambda^0} + m_{\Sigma^+} + m_{\Sigma^-} - m_{\Sigma^0} - m_{p} - m_{n} - m_{\Xi^0} - m_{\Xi^-} = (26.8\pm 0.2)~\si{MeV},\\
2m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Sigma^0_\text{c}} - m_{\Omega^0_\text{c}} = (9.4\pm 2.0)~\si{MeV}.
\end{gather*}
The given uncertainties follow from the uncertainties of the hadron masses via Gaussian error propagation. The values for the hadron masses and their uncertainties are taken from \cite{PDG}. We can see that in this case the value $m^\text{cal}-m^\text{exp}$ of the octet is three times larger than the corresponding value of the charm sextet. However, the observation we made depends heavily on the hadrons $y$ we solve for, since $m^\text{cal}-m^\text{exp}$ depends heavily on the hadron $y$. If we use $y = \Lambda^0$ and $y = \Omega^0_\text{c}$ for the determination of $m^\text{cal} - m^\text{exp}$ of the octet and sextet, respectively, we obtain:
\begin{gather*}
\frac{1}{3}\left(m_{\Sigma^0} + m_{p} + m_{n} + m_{\Xi^0} + m_{\Xi^-} - m_{\Sigma^+} - m_{\Sigma^-}\right) - m_{\Lambda^0} = (-8.9\pm 0.1)~\si{MeV},\\
2m_{\Xi^{\prime\, 0}_\text{c}} - m_{\Sigma^0_\text{c}} - m_{\Omega^0_\text{c}} = (9.4\pm 2.0)~\si{MeV}.
\end{gather*}
The absolute values of both $m^\text{cal} - m^\text{exp}$ now agree within their range of uncertainty and no significant difference can be observed.\par
As we have just seen, the dependence of $m^\text{cal} - m^\text{exp}$ on the hadron $y$ we solve for makes it difficult to compare $m^\text{cal} - m^\text{exp}$ for different mass relations. Indeed, other quantities were proposed to circumvent this problem. In \cite{Jenkins1995}, for instance, the \textit{experimental accuracy}\footnote{This expression is used in the paper \cite{Jenkins1995} itself.} $q$ is used as a quality measure for mass relations. The experimental accuracy $q$ of a mass relation
\begin{gather*}
\sum_x k_x m_x = 0
\end{gather*}
is in this paper defined to be:
\begin{gather*}
q\coloneqq \frac{\left|\sum_x k_x m_x\right|}{\frac{1}{2}\sum_x |k_x|m_x}.
\end{gather*}
Similar to $m^\text{cal} - m^\text{exp}$, $q$ is invariant under rescaling the mass relation, but in contrast to $m^\text{cal} - m^\text{exp}$ we do not have to single out a special hadron like $y$. One might also find appealing that $q$ is a dimensionless quantity, while $m^\text{cal} - m^\text{exp}$ has mass dimension 1. Nevertheless, we are able to relate $q$ and $m^\text{cal}-m^\text{exp}$. For $n=1$, we have:
\begin{gather*}
m^\text{cal} - m^\text{exp} = \sum_{x\neq y}\frac{-k_x}{k_y}m_x - m_y.
\end{gather*}
If the mass relation at hand only contains hadron masses from one $\text{SU}(3)$-multiplet which is not a mesonic octet, the hadron masses $m_x$ deviate very little from the average $M$ of the masses in the multiplet. Using this, we find:
\begin{gather*}
q = \frac{\left|\sum_x k_x m_x\right|}{\frac{1}{2}\sum_x |k_x|m_x} = \frac{\left|\sum_{x\neq y} \frac{-k_x}{k_y} m_x - m_y\right|}{\frac{1}{2}\left(\sum_{x\neq y} |\frac{k_x}{k_y}|m_x + m_y\right)}\approx \frac{|m^\text{cal}-m^\text{exp}|}{N\frac{M}{2}},
\end{gather*}
where $N$ is a normalization factor:
\begin{gather}
N\coloneqq \sum_{x\neq y}\left|\frac{k_x}{k_y}\right| + 1 = \sum_{x}\left|\frac{k_x}{k_y}\right|.\label{eq:normalization}
\end{gather}
The reason why we favor $m^\text{cal}-m^\text{exp}$ over $q$ in this work becomes clear when we try to extend $q$ to quadratic mass relations. The natural generalization of $q$ to both linear and quadratic mass relations is to replace the hadron masses in the original definition by the $n$-th power of the hadron masses:
\begin{gather*}
q\coloneqq \frac{\left|\sum_x k_x m^n_x\right|}{\frac{1}{2}\sum_x |k_x|m^n_x}.
\end{gather*}
This quantity $q$ is likewise invariant under rescaling of the mass relation and dimensionless. Let us try to relate $q$ to $m^\text{cal}-m^\text{exp}$ again. $m^\text{cal}-m^\text{exp}$ is for $n=2$ given by:
\begin{gather*}
m^\text{cal}-m^\text{exp} = \sqrt{\sum_{x\neq y}\frac{-k_x}{k_y}m^2_x} - m_y
\end{gather*}
Under the same assumptions as before, we find:
\begin{align*}
q &= \frac{\left|\sum_{x\neq y} \frac{-k_x}{k_y} m^2_x - m^2_y\right|}{\frac{1}{2}\left(\sum_{x\neq y} |\frac{k_x}{k_y}|m^2_x + m^2_y\right)} \approx \frac{\left|\sqrt{\sum_{x\neq y} \frac{-k_x}{k_y} m^2_x} - m_y\right|\cdot\left|\sqrt{\sum_{x\neq y} \frac{-k_x}{k_y} m^2_x} + m_y\right|}{N\frac{M^2}{2}}\\
&\approx \frac{|m^\text{cal} - m^\text{exp}|\cdot 2m_y}{N\frac{M^2}{2}}\approx 2\frac{|m^\text{cal} - m^\text{exp}|}{N\frac{M}{2}}.
\end{align*}
This allows us to sum up the relation between $q$ and $m^\text{cal}-m^\text{exp}$ for $n\in\{1,2\}$ in the following way:
\begin{gather*}
q\approx n\frac{|m^\text{cal} - m^\text{exp}|}{N\frac{M}{2}}.
\end{gather*}
Now one can see why we favor $m^\text{cal}-m^\text{exp}$ over $q$ for our purposes. If we chose $q$ as a quality measure for comparing linear with quadratic mass relations, we would find undesirable results: Since the relation between $q$ and $m^\text{cal}-m^\text{exp}$ is linear in $n$, it may occur that $q$ favors a linear mass relation over a quadratic one, even though the mass prediction of the quadratic relation is more accurate than the one of the linear relation. To prevent such unwanted results, we use $m^\text{cal}-m^\text{exp}$ instead of $q$ as a quality measure for the comparison of linear and quadratic mass relations.\par
To be able to compare the linear or quadratic versions of different mass relations with each other, we also provide the quantity $(m^\text{cal}-m^\text{exp})/N$ in the following sections. This quantity is also (mostly) independent of the choice of the hadron $y$, as $q$ is independent of that choice and $|m^\text{cal}-m^\text{exp}|/N$ and $q$ only differ by a factor of roughly $\frac{M}{2n}$. The normalization factors $N$ used for our analysis are given in \autoref{tab:normalization_factors}. If one wishes to retrieve $q$ from $(m^\text{cal}-m^\text{exp})/N$, one simply has to take the absolute value of $(m^\text{cal}-m^\text{exp})/N$ and divide by $\frac{M}{2n}$ (cf. \autoref{tab:mass_scales}).\par
\begin{table}[!htb]
\small
\centering
\caption{Normalization factors $N$ for hadronic mass relations. All mass relations aside from the ones of the mesonic octets are solved for the hadron with the highest mass, i.e., hadron $y$ is chosen to be the hadron with the highest mass. The value in brackets is only used for the mesonic octets.}
\begin{tabular}{|c|c|}
\hline
Mass relation & Normalization factor $N$\\
\hline\hline
\autoref{eq:Coleman-Glashow} & 6\\
\hline
\autoref{eq:Gell-Mann--Okubo} & 10 (10/3)\\
\hline
\autoref{eq:Coleman-Glashow_decuplet} & 6\\
\hline
\autoref{eq:Delta-} & 8\\
\hline
\autoref{eq:iso1_decuplet} & 8\\
\hline
\autoref{eq:iso2_decuplet} & 10\\
\hline
\autoref{eq:equal_spacing1_decuplet} & 4\\
\hline
\autoref{eq:equal_spacing2_decuplet} & 4\\
\hline
\autoref{eq:better_GMO_decuplet} & 11\\
\hline
\autoref{eq:bary_c_b_tri} & 4\\
\hline
\autoref{eq:meso_c_b_tri} & 4\\
\hline
\autoref{eq:sextet_c_iso_bre2} & 4\\
\hline
\autoref{eq:sextet_b_iso_bre} & 4\\
\hline
\autoref{eq:sextet_c_GMO_equal_spacing2} & 4\\
\hline
\autoref{eq:sextet_b_GMO_equal_spacing} & 4\\
\hline
\autoref{eq:sextet_c_b_sigma} & 8\\
\hline
\autoref{eq:sextet_c_b_spacing} & 4\\
\hline
\autoref{eq:sextet_c_b_very_precise} & 8\\
\hline
\autoref{eq:sextet_c_b_precise} & 8\\
\hline
\autoref{eq:sextet_c_b_sigma_b} & 5\\
\hline
\autoref{eq:sextet_c_b_xi} & 6\\
\hline
\end{tabular}
\label{tab:normalization_factors}
\end{table}
Let us now turn to the actual analysis of the mass relations. We begin with an overview over the hadronic multiplets we evaluate in this section. For our analysis, we choose to only use hadronic multiplets which satisfy three criteria:
\begin{itemize}
\item There has to be evidence for every multiplet that the assignment of hadrons to the multiplet is correct.
\item Enough hadron masses in each multiplet have to be measured to evaluate at least one mass relation.
\item The uncertainties of the hadron masses in each multiplet have to be small enough to make meaningful statements.
\end{itemize}
\newpage
\noindent These criteria leave us with 18 hadronic multiplets we use for our analysis:\par
The lowest-energy pseudoscalar and vector meson octet ($J^P = 0^-$ and $J^P = 1^-$), the lowest-energy baron octet ($J^P = 1/2^+$), the lowest-energy baryon decuplet ($J^P = 3/2^+$), the lowest-energy pair of charm and bottom baryon antitriplets ($J^P = 1/2^+$), four pairs of charm and bottom meson (anti)triplets ($J^P = 0^-$, $J^P = 1^-$, $J^P = 1^+$, and $J^P = 2^+$), and two pairs of charm and bottom baryon sextets ($J^P = 1/2^+$ and $J^P = 3/2^+$).\par
A tabular summary of this list together with the mass scales of the multiplets is given by \autoref{tab:mass_scales}.
\begin{table}[t!]
\small
\centering
\caption{Number of multiplets used for the analysis of mass relations together with a crude estimation for their mass average $M$. The last three entries in the table denote mesonic multiplets, the rest is baryonic. The charm and bottom multiplets come and are analyzed in pairs.}
\begin{tabular}{|c|c|c|}
\hline
\# & Hadronic multiplet & Mass average $M$ (roughly)\\
\hline\hline
1 & Baryon octet & \SI{1}{GeV}\\
\hline
1 & Baryon decuplet & \SI{1.4}{GeV}\\
\hline
1 & Charm baryon antitriplet & \SI{2.4}{GeV}\\
1 & Bottom baryon antitriplet & \SI{5.7}{GeV}\\
\hline
2 & Charm sextet & \SI{2.5}{GeV}\\
2 & Bottom sextet & \SI{6}{GeV}\\
\hline\hline
2 & Meson octet & \SI{0.4}{GeV} and \SI{0.8}{GeV}\\
\hline
4 & Charm meson (anti)triplet & \SI{2}{GeV} to \SI{2.5}{GeV}\\
4 & Bottom meson (anti)triplet & \SI{5.3}{GeV} to \SI{5.8}{GeV}\\
\hline
\end{tabular}
\label{tab:mass_scales}
\end{table}
The assignment of hadrons to these multiplets used for our analysis is based on the assignments and quantum numbers provided and favored by the \textit{Particle Data Group} (cf. \cite{PDG}), but is also based on the results from \cite{Faustov_HM} and \cite{Faustov_HB}. The experimentally determined values for the hadron masses and their uncertainties are taken from \cite{PDG} and from other references\footnote{\cite{Aaij2012}, \cite{Aaij2013}, \cite{Aaij2014}, \cite{Aaij2015}, \cite{Aaij2017}, \cite{Aaij2018}, \cite{Aaij2019}, \cite{Abe04}, \cite{Aubert2008}, \cite{Bernicha1995}, \cite{Ganenko1979}, \cite{Gridnev2006}, \cite{Hanstein1996}, \cite{Kato2016}, \cite{Li2018}, \cite{Lichtenberg1974}, \cite{Mohr2016}}\addtocounter{footnote}{-1}\addtocounter{Hfootnote}{-1}. If the experimental value for the mass of a hadron is given in \cite{PDG} and in one of the other references\footnotemark, the value given by \cite{PDG} is used. If \cite{PDG} gives two values for the mass of a hadron, a mass fit and a mass average (cf. \cite{PDG} for the meaning of ``mass fit'' and ``mass average''), the mass fit is used in instead of the mass average.\par
We present the results of our analysis in the form of tables (cf., for instance, \autoref{table:mass_relations_octet_decuplet} and \autoref{table:mass_relations_octet_decuplet_rescaled_2}). Each table has four columns: The first column specifies the mass relation by referencing the corresponding mass relation from \autoref{chap:mass_relations} and the hadronic multiplet or pair of charm and bottom multiplets the mass relation is applied to. The second column presents the value for $m^\text{cal}-m^\text{exp}$ or $(m^\text{cal}-m^\text{exp})/N$ together with its experimental uncertainty. This uncertainty is obtained from the experimental uncertainties\footnote{If the experimental uncertainty of the hadron mass is asymmetric, the larger uncertainty is used for the Gaussian error propagation. If the uncertainty is split up into a systematic and statistical uncertainty, the square root of the sum of the squared uncertainties is used.} of the hadron masses via Gaussian error propagation. The third column indicates whether the linear or quadratic version of the mass relation was used for the calculation of the second column by presenting the exponent $n$. In the last column, the order of the dominant correction(s) to the mass relation at hand is listed, where ``$\mathcal{O}$'' is dropped and $\varepsilon_\text{cb}\coloneqq \Lambda_\text{QCD}(1/m_\text{c} - 1/m_\text{b})$ is used for the sake of clarity.\par
First, consider the results for the baryon octet and decuplet ($J^P = 1/2^+$ and $J^P = 3/2^+$), shown in \autoref{table:mass_relations_octet_decuplet} and \autoref{table:mass_relations_octet_decuplet_rescaled_2}.
\input{mass_relations_octet_decuplet.tex}
We can directly see that both the linear and the quadratic version of the Coleman-Glashow mass relation (\autoref{eq:Coleman-Glashow}) in the baryon octet are outstandingly well satisfied. Both versions are violated by a value of $m^\text{cal}-m^\text{exp}$ below \SI{1}{MeV}. With a mass scale of roughly \SI{1}{GeV} for the baryon octet, the precision of their predictions is better than 0.1\%. Even though the value of $m^\text{cal}-m^\text{exp}$ for the quadratic version is about 8 times larger than the corresponding linear value, the difference between both values is not significant in consideration of their uncertainties, as the values agree within two $2\sigma$. Judging from \autoref{tab:exp_param}, the dominant corrections to the Coleman-Glashow relation should be in the order of \SI{1}{MeV}. This seems plausible, although it is difficult to make a definite statement, since the uncertainties are also of that order.\par
The GMO mass relation in the baryon octet (\autoref{eq:Gell-Mann--Okubo}) exhibits a different behavior. As expected, both the linear as well as the quadratic GMO mass relation are much stronger violated than the Coleman-Glashow relation. Using \autoref{tab:exp_param} again, we can estimate that the dominant correction to the GMO mass relation should be in the order of \SI{10}{MeV}. With values between \SI{25}{MeV} and \SI{30}{MeV} for $m^\text{cal}-m^\text{exp}$, this seems to be reasonable. This time, however, the difference of \SI{3}{MeV} to \SI{4}{MeV} between the linear and quadratic version of the mass relation is significant. One might be tempted to deduce from this that only the linear GMO mass relation applies to the baryon octet. Nonetheless, we only expect the GMO mass relation to be satisfied up to a correction that is in the order of \SI{10}{MeV}. In consideration of such a correction, a difference of \SI{3}{MeV} to \SI{4}{MeV} does not make the quadratic GMO relation inapplicable, even though the prediction of the linear mass relation is more precise.\par
To understand the results of the baryon decuplet, we have to note some important remarks. First off, the uncertainties of the baryon masses in the decuplet are much larger than the ones of the octet. This makes it more difficult to obtain meaningful results from the available data. Secondly, the mass of the $\Delta^-$-baryon is not measured yet which means that we cannot check \autoref{eq:Delta-}. Nevertheless, this mass relation can be used for the determination of the $\Delta^-$-mass (cf. \autoref{sec:mass_predictions}). Lastly, we have to note that the $\Delta$- and $\Sigma^\ast$-resonances are not only rather broad compared to the other baryons in the decuplet and to the octet baryons, but also in the sense of \autoref{sec:polemass} (cf. ``small decay width''). This is problematic for two reasons: On the one hand, there are two conventions $M_s$ and $M_w$ for defining the pole mass (cf. \autoref{sec:polemass}). We have seen that the mass conventions differ by terms in the order of $(\Gamma_{w}/M_{w})^2$. As we are not able to say to which definition of the pole mass the mass relations from \autoref{chap:mass_relations} apply, we have to expect to pick up corrections of the order of $(\Gamma_{w}/M_{w})^2$. This means that the relations involving $\Delta$-masses face additional corrections in the order of 1\% of the $\Delta$-masses or roughly \SI{10}{MeV}, while the additional corrections to $\Sigma^\ast$-mass relations are in the order of \SI{1}{MeV}. On the other hand, the accurate determination of the pole positions is more challenging for broader resonances (cf. \autoref{sec:polemass}). In particular, the parameters of a Breit-Wigner fit to a resonance do not represent the pole position accurately. However, there are still ways to retrieve the pole mass from the resonance (cf. \cite{Bernicha1995}, \cite{Hanstein1996}, and \cite{Lichtenberg1974}; the pole masses for the decuplet are taken from these papers as well). Nonetheless, the pole mass is not known for every baryon in the decuplet. The pole masses for $\Delta^-$, $\Sigma^{\ast\, 0}$, and $\Omega^-$ are missing, even though we can take the Breit-Wigner mass of the $\Omega^-$-baryon to be its pole mass, as the $\Omega^-$-resonance is rather narrow. In order to use the available data to full extent, we present $m^\text{cal}-m^\text{exp}$ and $(m^\text{cal}-m^\text{exp})/N$ for both Breit-Wigner and pole masses in the decuplet. To distinguish between relations using Breit-Wigner masses and relations using pole masses in \autoref{table:mass_relations_octet_decuplet} and \autoref{table:mass_relations_octet_decuplet_rescaled_2}, we indicate the pole mass relations with ``pole''.\par
Considering the values of $m^\text{cal}-m^\text{exp}$ for the baryon decuplet, we observe that the pole mass relations are -- aside from two exceptions, namely the linear versions of \autoref{eq:equal_spacing1_decuplet} and \autoref{eq:equal_spacing2_decuplet} -- better satisfied than their Breit-Wigner counterpart. Since we operate on the assumption that the mass relations apply to the pole masses (cf. \autoref{sec:polemass}), this matches our expectation. Moreover, we can see that every quadratic mass prediction -- aside from \autoref{eq:equal_spacing1_decuplet} involving Breit-Wigner masses -- is at least as precise as its linear counterpart within the range of uncertainty. We even find that the equal spacing rules (\autoref{eq:equal_spacing1_decuplet} and \autoref{eq:equal_spacing2_decuplet}) are -- aside from one exception\footnote{\autoref{eq:equal_spacing1_decuplet} involving Breit-Wigner masses. The observation that \autoref{eq:equal_spacing1_decuplet} seems to be an exception may be contributed to the fact that it is the only equal spacing rule involving the problematic $\Delta$-resonance.} -- significantly more precise than their linear counterparts. Furthermore, we observe that the linear and quadratic values of $m^\text{cal}-m^\text{exp}$ for the decuplet mass relations whose dominant correction is in the order of $\varepsilon^2_8$ fluctuate between roughly \SI{1}{MeV} and \SI{15}{MeV}. As corrections associated with $\varepsilon^2_8$ should be in the order of \SI{10}{MeV}, it seems plausible that the dominant correction to these mass relations is indeed\footnote{One might argue that the dominant correction to these mass relations appears to be smaller than $\varepsilon^2_8$, judging from the experimental data, but the experimental data lacks the precision to make a definite statement.} $\varepsilon^2_8$. However, we have to be cautious with our observations for the decuplet because of the problems explained above.\par
\input{mass_relations_octet_decuplet_rescaled_2.tex}
In principle, \autoref{table:mass_relations_octet_decuplet_rescaled_2} shows the same features as \autoref{table:mass_relations_octet_decuplet}, just scaled down. The only additional noteworthy aspect is that the values of $(m^\text{cal}-m^\text{exp})/N$ for mass relations whose dominant corrections are in the same order of magnitude seem to be more compatible with each other than the corresponding values of $m^\text{cal}-m^\text{exp}$, but, again, we have to be cautious with our observations for the decuplet.\par
Next, consider the charm and bottom sextets ($J^P = 1/2^+$ and $J^P = 3/2^+$). Let us start by evaluating the results for the mass relations that only involve baryons from one sextet (cf. \autoref{table:mass_relations_within_heavy_quarks} and \autoref{table:mass_relations_within_heavy_quarks_rescaled_2}). As we can see, every quadratic value of $m^\text{cal}-m^\text{exp}$ and $(m^\text{cal}-m^\text{exp})/N$ is smaller than its linear counterpart. In the case of the GMO equal spacing rules (\autoref{eq:sextet_c_GMO_equal_spacing2} and \autoref{eq:sextet_b_GMO_equal_spacing}), the difference is actually significant. This observation is particularly interesting in consideration of Feynman's distinction\footnote{With this expression, we denote the statement that baryons should be subject to linear mass relations, while mesons should satisfy quadratic mass relations.} of mass relations: Feynman's distinction predicts that sextet mass relations as baryonic mass relations have to be linear. As the experimental data clearly favor quadratic over linear mass relations, Feynman's distinction is in disagreement with the experimental data for sextets. Even though there is strong evidence that quadratic relations are more precise than linear ones for sextets, this does not necessarily mean that linear mass relations do not apply within their range of validity, i.e., do not apply up to their dominant correction. Indeed, the given results seem to indicate that both linear and quadratic mass relations apply within their range of validity, although we have to understand this statement with a grain of salt, since we can only estimate the size of the dominant corrections.\par
\input{mass_relations_within_heavy_quarks.tex}
\input{mass_relations_within_heavy_quarks_rescaled_2.tex}
It is also noteworthy that the mass relations whose dominant corrections are in the order of $\alpha\varepsilon_8$ and $\varepsilon_3\varepsilon_8$ are significantly more precise than the mass relations whose dominant correction is of the order of $\varepsilon^2_8$, matching our expectation.\par
As for the baryon octet and decuplet, \autoref{table:mass_relations_within_heavy_quarks_rescaled_2} shows very similar features to \autoref{table:mass_relations_within_heavy_quarks}. The only noteworthy difference is that the values of $(m^\text{cal}-m^\text{exp})/N$ for mass relations with the same dominant correction(s) (cf. \autoref{table:mass_relations_octet_decuplet_rescaled_2} and \autoref{table:mass_relations_within_heavy_quarks_rescaled_2}) seem to be more compatible with each other than the corresponding values of $m^\text{cal}-m^\text{exp}$ (cf. \autoref{table:mass_relations_octet_decuplet} and \autoref{table:mass_relations_within_heavy_quarks}). Before we move on, we should note that we can only check one mass relation for the bottom sextets ($J^P = 1/2^+$ and $J^P = 3/2^+$). The reason for this is that the bottom sextets are incomplete: In the bottom sextet with $J^P = 1/2^+$, the masses of $\Sigma^0_\text{b}$ and $\Xi^0_\text{b}$ are not measured yet, while the masses of $\Sigma^{\ast\, 0}_\text{b}$ and the counterpart to $\Omega^-_\text{b}$ (we will call this particle $\Omega^-_\text{b}(6070)$ in \autoref{sec:mass_predictions}) are missing for the bottom sextet with $J^P = 3/2^+$. We will use the mass relations from \autoref{chap:mass_relations} to predict the masses of these baryons in \autoref{sec:mass_predictions}.\par
So far, we have only considered mass relations that involve hadrons from only one multiplet. Thus, we now want to check mass relations that involve hadrons from a pair of charm and bottom multiplets. The results for these mass relations are displayed in \autoref{table:mass_relations_between_heavy_quarks} and \autoref{table:mass_relations_between_heavy_quarks_rescaled_2}.\par
\input{mass_relations_between_heavy_quarks.tex}
\input{mass_relations_between_heavy_quarks_rescaled_2.tex}
Immediately, we observe that the values of $m^\text{cal}-m^\text{exp}$ for the quadratic mass relations are all about 5 to 10 times larger than their linear counterparts. Naively, one might think that this behavior contradicts the predictions of the state formalism, however, it actually confirms the state formalism. To see this, we have to remind ourselves how we derived the mass relations connecting charm and bottom multiplets. In \autoref{sec:heavy_quark}, we identified the hadron masses with the eigenvalues of the Hamiltonian $H^5_\text{QCD}$ and determined them in a perturbative expansion:
\begin{gather*}
H^5_\text{QCD} = H^{5;\,0}_\text{QCD} + \varepsilon_3\cdot H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{8}_{\text{QCD};\, 8}.
\end{gather*}
The operators $H^{8}_{\text{QCD};\, 3}$ and $H^{8}_{\text{QCD};\, 8}$ which we treat as a perturbation are independent of the charm and bottom quark fields and masses. Therefore, only $H^{5;\, 0}_\text{QCD}$ changes under the exchange of charm and bottom quarks. We deduced from this that the first order contributions of the flavor symmetry breaking to the hadron masses are the same for pairs of charm and bottom multiplets, while the singlet mass term $m^0$ which directly corresponds to $H^{5;\, 0}_\text{QCD}$ changes. This allowed us to formulate mass relations involving hadrons from pairs of charm and bottom multiplets. However, that kind of reasoning only applies to linear mass relations. If we want to formulate quadratic mass relations, we have to consider the square of the Hamiltonian:
\begin{gather*}
\left(H^5_\text{QCD}\right)^2 = \left(H^{5;\,0}_\text{QCD}\right)^2 + \varepsilon_3\cdot H^{5;\,0}_\text{QCD} H^{8}_{\text{QCD};\, 3} + \varepsilon_8\cdot H^{5;\,0}_\text{QCD} H^{8}_{\text{QCD};\, 8} + \mathcal{O}\left(\varepsilon_i\varepsilon_j\right).
\end{gather*}
Now, the perturbation of the squared Hamilton operator varies under the exchange of charm and bottom quark. This implies that quadratic mass relations connecting charm and bottom multiplets do not apply with the same precision as linear mass relations: If the dominant corrections to a linear mass relation are suppressed by $\varepsilon_\text{cd}$ originating from heavy quark symmetry, the quadratic version of that mass relation does not have this suppression $\varepsilon_\text{cb}$, as heavy quark symmetry breaks down in the quadratic case. The dominant corrections to the quadratic mass relation are then given by the dominant corrections to the linear mass relation where we have to drop the factor $\varepsilon_\text{cb}$. This description matches the observed behavior: The values of $m^\text{cal}-m^\text{exp}$ for the quadratic mass relations are about 5 to 10 times higher than the corresponding linear values which coincides rather well with $\varepsilon^{-1}_\text{cb}$ (cf. \autoref{tab:exp_param}).\par
To this end, we can say that mass relations involving particles from different multiplets clearly favor linear over quadratic mass relations, but still agree with the state formalism. It is noteworthy that this preference of linear mass relations is reflected by baryons as well as mesons as one would expect following the state formalism.\par
Lastly, let us turn to the mesonic octets. We have pointed out multiple times throughout this thesis that the isospin singlet in a mesonic octet mixes with a meson which forms a \text{SU}(3)-singlet. While the GMO mass relation (cf. \autoref{eq:Gell-Mann--Okubo}) for mesonic octets is affected by this mixing, the Coleman-Glashow mass relation (cf. \autoref{eq:Coleman-Glashow}) is not, since it does not involve the isospin singlet. However, the Coleman-Glashow mass relation is trivially zero for mesonic octets. The reason for this is simple: For every meson in a mesonic octet, the corresponding antiparticle is contained in the same octet as well. In the weight diagram of a mesonic octet, a particle is diametrically opposed to its antiparticle. Thus, the Coleman-Glashow mass relation for mesonic octets only involves differences of particle and antiparticle masses. But since we have not observed CPT violation\footnote{The CPT invariance of most QFTs follows from the CPT theorem.} yet, we assume that particles and antiparticles have the same mass which is in agreement with all experiments so far. Hence, we find that the Coleman-Glashow mass relation is exactly zero for mesonic octets.\par
This leaves us with only the GMO mass relation for mesonic octets. As stated, this mass relation is affected by octet-singlet-mixing. Commonly, one takes the quadratic GMO mass relation for the mesonic octets to be exact in order to determine the mixing angle between the isospin singlet of the mesonic octet and the \text{SU}(3)-singlet meson. As we are aiming to compare linear and quadratic mass relations, this is not a viable approach. Nevertheless, there is still a way to analyze the mesonic octets: Prior in this section, we explained that we have to choose a hadron $y$ for whose mass we solve the mass relation at hand. If we take $y$ to be the isospin singlet for mesonic octets, the mass prediction $m^\text{cal}$ only involves particles which are not affected by mixing. Now, $m^\text{cal}$ does not predict the mass of an observable particle like $\eta$ or $\eta^\prime$, but the mass of a mixture of particles. Even though we do not know the mass of that mixture and, consequently, cannot determine the precision of the predictions $m^\text{cal}$, we still can compare the linear and quadratic versions of the mass prediction $m^\text{cal}$ with each other to determine how much the predictions $m^\text{cal}$ deviate: If the deviation of the predictions $m^\text{cal}$ is large in comparison to the meson masses, both mass relations cannot apply simultaneously, at least not with a high precision. If the predictions $m^\text{cal}$ are very similar, there is no reason to favor one mass relation over the other. In this regard, $m^\text{cal}$ is now the quantity of interest. Nevertheless, the values shown in \autoref{table:mass_relations_meson_octets} and \autoref{table:mass_relations_meson_octets_rescaled_2} are still $m^\text{cal}-m^\text{exp}$ and $(m^\text{cal}-m^\text{exp})/N$. For both linear and quadratic GMO mass relations, $m^\text{exp}$ is chosen to be the mass of $\eta$ in the case of the pseudoscalar meson octet and the mass of $\omega$ in the case of the vector meson octet. This choice corresponds to a naive picture of mesons in which we neglect mixing effects. Thus, $m^\text{cal}-m^\text{exp}$ tells us now how strongly the ``naive'' GMO mass relations are violated and, if the actual GMO mass relation is applicable, how large the mixing is. But since we have only shifted $m^\text{cal}$ by a constant, we can still take the difference of the linear and quadratic value for $m^\text{cal}-m^\text{exp}$ to obtain the corresponding difference of the values of $m^\text{cal}$.\par
\input{mass_relations_meson_octets.tex}
\input{mass_relations_meson_octets_rescaled_2.tex}
\autoref{table:mass_relations_meson_octets} depicts an interesting behavior of the meson masses: All values for $m^\text{cal}-m^\text{exp}$ -- aside from the quadratic case in the pseudoscalar meson octet -- show a violation of the ``naive'' GMO mass relation about 15\% or higher with respect to their mass scale $M$ (cf. \autoref{tab:mass_scales}). Only the violation of the ``naive'' quadratic GMO mass relation for the pseudoscalar meson octet is below 5\%. Furthermore, we see that the difference between the linear and quadratic value of $m^\text{cal}-m^\text{exp}$ is about 10\% for the pseudoscalar meson octet and below 1\% for the vector meson octet (in regard to their respective mass scales $M$). We can interpret this in the following way: The vector meson octet is subject to the state formalism, as the flavor symmetry breaking in this octet is rather small and can be treated as a perturbation. Therefore, the linear as well as the quadratic GMO mass relation apply and give similar results in the vector meson octet. In the pseudoscalar meson octet, however, the state formalism breaks down, since the flavor symmetry breaking is not small anymore. Nevertheless, the quadratic GMO mass relation for the pseudoscalar meson octet can still be derived in the framework of chiral perturbation theory (cf. \cite{Scherer2011}) and, thus, applies. The linear GMO mass relation cannot be obtained as demonstrated in \autoref{sec:EFT+H_Pert} from the quadratic GMO mass relation in the pseudoscalar meson octet, because the symmetry breaking is not small. The mixing in the pseudoscalar meson octet, however, is relatively small which explains why the ``naive'' quadratic GMO mass relation applies relatively well to the pseudoscalar meson octet.\par
This interpretation is supported by the mixing angles $\theta_\text{lin}$ and $\theta_\text{quad}$ one can calculate by taking the linear or quadratic GMO mass relations to be exact, respectively (cf. Ch. 6 of \cite{Oneda1985}). For the pseudoscalar meson octet, one finds $|\theta^\text{P}_\text{lin}|\approx 23\degree$ and $|\theta^\text{P}_\text{quad}|\approx 10\degree$, while we have $|\theta^\text{V}_\text{lin}|\approx 36\degree$ and $|\theta^\text{V}_\text{quad}|\approx 39\degree$ for the vector meson octet. The mixing angles show that the mixing obtained from the linear GMO mass relation deviates a lot from the quadratic mixing in the case of the pseudoscalar meson octet, while the difference is rather small in the case of the vector meson octet. To this end, the mixing angles match our interpretation. The mixing angles of other meson octets exhibit a similar behavior as the vector meson octet (cf. Ch. 6 of \cite{Oneda1985}).\par
In conclusion, we have seen that the mass relations from \autoref{chap:mass_relations} apply within their range of validity, i.e., are correct up to their dominant correction(s). Generally, this is true for both versions of mass relations -- linear as well as quadratic. Moreover, we were able to reject Feynman's distinction of baryons and mesons into linear and quadratic mass relations for three different reasons: Firstly, every time we had to make a distinction between the two versions of mass relations we could argue that this distinction does not need to arise from the distinction of baryons and mesons into different mass relations: We had to utilize the linear version of mass relations involving both charm and bottom multiplets, as heavy quark symmetry breaks down for quadratic relations, and we had to use the quadratic GMO mass relation for the pseudoscalar meson octet, since it is predicted by chiral perturbation theory and not equivalent to its linear version because of the large symmetry breaking. Secondly, linear as well as quadratic mass relations apply within the same range of validity for a lot of multiplets. In this sense, Feynman's distinction is not necessary and, thus, artificial. Lastly, we found that the mass relations which favor the linear or quadratic version do not always match Feynman's distinction: The mass relations within baryon sextets favor the quadratic version, even though they are baryons, and the mass relations involving both charm and bottom multiplets favor the linear version for both baryons and mesons.
\section{Multiplet Assignments and Mass Predictions}\label{sec:mass_predictions}
The goal of this section is to use the mass relations from \autoref{chap:mass_relations} to make predictions for hadrons. There are two kinds of predictions we want to make: We want to assign known and measured resonances to multiplets and/or to pairs of charm and bottom multiplets and we want to make predictions for yet undetermined hadron masses.\par
Let us begin with the assignments. The idea for this is to start with an ``educated guess'' for the assignment of yet ungrouped hadrons to multiplets. In the next step, we check the assignment by examining the mass relations this assignment implies: If the mass relations are satisfied within their range of validity, the assignment is more likely to be true. If not, this is evidence for the assignment being false. To find this ``educated guess'', we use the quantum numbers provided and favored by the \textit{Particle Data Group} (cf. \cite{PDG}) and results from \cite{Faustov_HM} and \cite{Faustov_HB}. On top of that, we make use of an empirical observation: Consider the mass splittings between the isospin multiplets of hadronic (anti)triplets which are listed in \autoref{table:mass_splitting_baryon_triplet} and \autoref{table:mass_splitting_meson_triplet}. The first column of each table shows the hadrons which we assign to the same (possible) (anti)triplet, the second column displays the mass difference of the hadrons in the first column, and the third column contains the favored value for $J^P$. The assignment of the hadrons to (anti)triplets is based on results provided by \cite{PDG}, if not marked by $(\ast)$, and on \cite{Faustov_HM} and \cite{Faustov_HB}, otherwise. The hadron masses and their uncertainties are taken from the same references as in \autoref{sec:mass_testing}. Again, all uncertainties in this section are obtained via Gaussian error propagation.\par
\input{mass_splitting_baryon_triplet.tex}
\input{mass_splitting_meson_triplet.tex}
Considering the mass splittings listed in these tables, it seems like the mass splittings depend little on $J^P$ and are mainly determined by the type of (anti)triplet: While the mass splittings of baryonic antitriplets range from roughly \SI{170}{MeV} to \SI{200}{MeV}, the mass splittings of mesonic (anti)triplets lie mostly\footnote{The most striking exceptions are the (anti)triplets with $J^P = 0^+$ and $J^P = 1^+$. These (anti)triplets are quite odd, as they also disagree with the predictions of \cite{Faustov_HM}.} between \SI{85}{MeV} and \SI{125}{MeV}. It is also noteworthy that the mass splittings of bottom (anti)triplets are always smaller than their charm counterparts. Usually, the difference between charm and bottom mass splitting is about \SI{10}{MeV}.\par
If this behavior generalizes to all or at least many hadronic multiplets, it gives us an additional tool for assigning hadrons to multiplets. Based on that assumption, we assign the hadrons $\Lambda^+_\text{c}(2940)$ and $\Xi_\text{c}(3123)$ to the same antitriplet with $J^P = 3/2^-$\footnote{The evidence for the existence of the $\Xi_\text{c}(3123)$ particle is rather weak (cf. \cite{PDG}) which is why the value for $J^P$ of the corresponding antitriplet is marked with a question mark in \autoref{table:mass_splitting_baryon_triplet}.}. The value of $J^P$ for that multiplet is based on $J^P$ of $\Lambda^+_\text{c}(2940)$ favored by \cite{PDG}. Unfortunately, we cannot check this assignment with the help of a mass relation, since we lack the corresponding bottom baryons.\par
The mass splittings of charm and bottom sextets behave very similar to what we have observed for the hadronic (anti)triplets (cf. \autoref{table:mass_splitting_sextet}).\par
\input{mass_splitting_sextet.tex}
Based on that and \cite{Faustov_HB}, we assign the hadrons $\Sigma_\text{c}(2800)$, $\Xi_\text{c}(2930)$, $\Omega^0_\text{c}(3050)$, $\Sigma_\text{b}(6097)$, and $\Xi_\text{b}(6227)$ to the same pair of charm and bottom sextets. The values\footnote{In \cite{Faustov_HB}, three pairs of charm and bottom sextets are predicted to be very close together. This is the reason why three values for $J^P$ are given for the newly assigned pair of sextets in \autoref{table:mass_splitting_sextet}.} of $J^P$ for this pair are based on \cite{Faustov_HB}. This time, there are mass relations we can use to test the new assignment. The mass relations we can check for this assignment are satisfied to the extent we expect them to be valid (cf. \autoref{table:mass_assignments} and \autoref{table:mass_assignments_rescaled_2}), although it is hard to tell because of the large uncertainties.\par
\input{mass_assignments.tex}
\input{mass_assignments_rescaled_2.tex}
\clearpage
Let us now turn to the mass predictions. How to use mass relations to make predictions for hadron masses is pretty straightforward: If we know the masses of at least a few hadrons in a multiplet, we can simply apply \autoref{eq:prediction}. This way, we can predict the masses of the hadrons which are missing to complete the multiplets from \autoref{sec:mass_testing}. Moreover, we can give an estimate for the mass of $\Omega^-_\text{b}(6350)$ which denotes the counterpart to the $\Omega^-_\text{b}$-baryon in the newly assigned pair of charm and bottom sextets. Furthermore, we can predict the masses of $\Xi^0_\text{b}(6112)$ and $\Xi^0_\text{b}(6109)$ which denote hadrons in the charm-bottom pairs $\Lambda^+_\text{c}(2595)$-$\Xi_\text{c}(2790)$-$\Lambda^0_\text{b}(5912)$ and $\Lambda^+_\text{c}(2625)$-$\Xi_\text{c}(2815)$-$\Lambda^0_\text{b}(5920)$ with $J^P = 1/2^-$ and $J^P = 3/2^-$, respectively. The predictions are displayed in \autoref{table:mass_predictions}. \autoref{table:mass_predictions} is organized similarly to the tables in \autoref{sec:mass_testing}. The uncertainties for the mass predictions $m^\text{cal}$ are obtained from the experimental uncertainties via Gaussian error propagation and do not include any theory or model error.\par
\input{mass_predictions.tex}
One might be confused that the mass of $\Xi^0_\text{b}(6112)$ is larger than the mass of $\Xi^0_\text{b}(6109)$, even though the order is reversed for the corresponding $\Lambda$-hadrons, i.e., the mass of $\Lambda^0_\text{b}(5912)$ is smaller than the mass of $\Lambda^0_\text{b}(5920)$. However, this behavior is also predicted by other authors (cf. \cite{Thakkar2017}).
\newpage
\chapter*{Summary}
\addcontentsline{toc}{chapter}{Summary}
\markboth{}{Summary}
The initial question of this work revolved around Feynman's distinction, i.e., the distinction of baryons and mesons into linear and quadratic GMO mass relations, respectively. As formulated in the introduction, we aimed to resolve the discrepancy between Feynman's distinction and the symmetry between fermion (baryon) and boson (meson) masses in a supersymmetrical world. To accomplish this task, we wanted to answer the question whether Feynman's distinction is a real physical distinction, merely artificial, or maybe even false. In the course of this thesis, we have found a clear answer to this question: While Feynman's distinction is not necessarily false, it is most certainly artificial. We have seen that this result holds true on a theoretical and experimental level.\par
From a theoretical perspective, we have considered two descriptions of hadron masses, the EFT approach and the state formalism. The EFT approach in which hadrons are identified with fields in an effective Lagrangian seemed to exhibit Feynman's distinction naturally, while Feynman's distinction did not arise in the state formalism that describes hadrons as eigenstates of the Hamilton operator. We were able to understand this difference between the EFT approach and the state formalism by considering the $\text{SU}(3)$-flavor symmetry breaking that led us to the formulation of the GMO mass relations in the first place: If the flavor symmetry breaking is small such that it can be treated as a perturbation and no heavy quark symmetry is involved, both linear and quadratic mass relations are valid to first order in flavor symmetry breaking. As the state formalism is a perturbative description of hadron masses and, thus, only applicable, if the symmetry breaking is small, every mass relation predicted by the state formalism -- omitting relations involving both charm and bottom hadrons -- is valid in both its linear and quadratic form. To this end, Feynman's distinction is artificial, because it is simply not necessary to introduce the distinction.\par
In spite of this result, we have seen that there are still multiplets where the results following from the different versions of mass relations -- linear and quadratic -- clearly differ and one version is most likely favored over the other. Examples for these multiplets include the pseudoscalar meson octet and the pairs of charm and bottom antitriplets and sextets. Nevertheless, we were able to explain the patterns exhibited by those multiplets:\par
The behavior of the pseudoscalar meson octet can be understood by considering the size of the symmetry breaking. The mesons in the pseudoscalar meson octet are not very heavy in comparison to the mass splitting induced by the flavor symmetry breaking. Hence, the symmetry breaking cannot be treated as a perturbation in this case and the different versions of the GMO mass relation in the pseudoscalar meson octet cannot be satisfied simultaneously. As the quadratic GMO mass relation of the pseudoscalar meson octet can be calculated in chiral perturbation theory (cf. \cite{Scherer2011}), the quadratic version is applicable, while the linear one is not.\par
The behavior of the pairs of charm and bottom multiplets is a direct consequence of the state formalism: Only the linear version is applicable for mass relations involving charm and bottom hadrons, because the heavy quark symmetry only holds true for linear mass relations and breaks down for quadratic ones. The interesting aspect of this result is that it applies to baryonic as well as mesonic pairs of charm and bottom multiplets.\par
We have argued in this thesis that these cases cannot be seen as a confirmation of Feynman's distinction, even though a distinction of mass relations is necessary in those cases. The found patterns simply do not match Feynman's distinction: The pseudoscalar meson octet favors the quadratic GMO mass relation over the linear one, but the heavier meson octets like the vector meson octet do not feature this preference. The baryonic pairs of charm and bottom multiplets clearly favor linear mass relations involving charm and bottom hadron masses, but so do the mesonic pairs.\par
Concerning the part of this work related to experimental data, we were able to support the presented discussion of Feynman's distinction with the most recent data on hadron masses. In the analysis of experimental data, we addressed several issues regarding the applicability of the mass relations to pole masses, the method for comparing mass relations, and the experimental data itself.\par
In addition to the discussion of Feynman's distinction, we have also incorporated the effects of isospin symmetry breaking, electromagnetic interaction, and heavy quark symmetry into the state formalism. This allowed us to obtain strongly satisfied and well known mass relations like the Coleman-Glashow relation (cf. \cite{coleman-glashow}). The validity of these mass relations could also be confirmed by the analysis of experimental data.\par
At the end of the thesis, we used the mass relations we derived to make predictions for the masses of yet undiscovered hadrons.
\newpage
\begin{appendix}
\chapter{Motivation for the State
Formalism}\label{app:stateform}
In \autoref{sec:polemass}, the mass of a particle or resonance is defined via poles of scattering amplitudes with respect to the Mandelstam variable $s$. However, it is not clear at all what the relation of these poles and the parameters in the Lagrangian is and how to calculate the position of the poles. Therefore, in order to make any statement about hadron masses, we have to make some assumptions on the relation of the Lagrangian and the hadron masses. In \autoref{sec:EFT+H_Pert}, two approaches to this problem are given. For the first one, one assumes that the hadrons can be described as fields in an EFT and that the symmetry structure of $\mathcal{L}_\text{QCD}$ ``carries over" to the EFT Lagrangian to first order in flavor symmetry breaking.\par
The second approach, the state formalism, which is used throughout my thesis is based on three assumptions:
\begin{itemize}
\item[1)] For every hadron $a$, there exists an eigenstate $\Ket{a}$ with $\Braket{a|a} = 1$ of the Hamilton operator $H$ from which the vacuum energy is already subtracted such that the mass $m_a$ of $a$ is given by
\begin{gather*}
m_a = \Bra{a}H \Ket{a}.
\end{gather*}
\item[2)] The subspace $V$ of the physical states which is spanned by the states $\Ket{a}$ from 1), i.e., {${V:= \overline{\text{Span}\left\{\Ket{a}\mid a\text{ hadron}\right\}}}$}, is a Hilbert space.
\item[3)] There is a unitary representation $D^{(\rho)}:V\rightarrow V$ of \text{SU}(3) on $V$ such that the following equation holds for every $A\in\text{SU}(3)$:
\begin{gather*}
\Bra{a} D^{(\rho)}(A)^\dagger\circ H\left(\bar{q}_{\text{L/R}},\, q_{\text{L/R}}\right)\circ D^{(\rho)}(A) \Ket{b} = \Bra{a}H\left(\bar{q}^{\, \prime}_{\text{L/R}},\, q^\prime_{\text{L/R}}\right) \Ket{b}\ \forall\Ket{a},\Ket{b}\in V
\end{gather*}
where $q_{\text{L/R}}$ are the left/right-handed fields of the light quarks $(q\in\{\text{u, d, s}\})$ and $q^\prime_{\text{L/R}}\coloneqq \sum\limits_{\tilde{q}\in\{\text{u,d,s}\}}A_{q \tilde{q}}\cdot \tilde{q}_{\text{L/R}}$.
\end{itemize}
I cannot prove these assumptions in the case of $\mathcal{L}_\text{QCD}$. However, we can consider the case of a non-interacting theory to see how we can understand and motivate the assumptions 1) to 3).\par
Let us consider a theory of three free spin-$\frac{1}{2}$ particles with fields $q$ and masses $m_q$, $q\in\{\text{u, d, s}\}$:
\begin{align*}
\mathcal{L} &= \sum\limits_{q\in\{\text{u,d,s}\}}\bar{q}(i\slashed{\partial} - m_q)q\\
&= \sum\limits_{q\in\{\text{u,d,s}\}}\bar{q}_Li\slashed{\partial}q_L + \bar{q}_Ri\slashed{\partial}q_R - \bar{q}_Lm_qq_R - \bar{q}_Rm_qq_L.
\end{align*}
Following \cite{Peskin}, we can express the field $q$ as
\begin{gather*}
q(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{p;q}}}\sum_{s=1,2}\left(a^s_{p;q}u_q^s(\vec p)e^{-ip\cdot x} + b^{s\dagger}_{p;q}v^s_q(\vec p)e^{ip\cdot x}\right)
\end{gather*}
with energy $p_0 \equiv E_{p;q}\coloneqq \sqrt{m_q^2 + {\vec p}^{\, 2}}$, annihilation and creation operators $a^s_{p,q}$, $a^{s\dagger}_{p;q}$, $b^s_{p,q}$, and $b^{s\dagger}_{p;q}$ of particles and antiparticles satisfying
\begin{gather*}
\left\{a^{r}_{p;q},a^{s\dagger}_{p^\prime;q^\prime}\right\} = \left\{b^{r}_{p;q},b^{s\dagger}_{p^\prime;q^\prime}\right\} = (2\pi)^3\delta^{(3)}(\vec p - {\vec p}^{\,\prime})\delta^{rs} \delta_{q q^\prime}\\
\text{+ all other anticommutators of $a$ and $b$ vanish,}
\end{gather*}
and spinors $u^s_q(\vec p)$ and $v^s_q(\vec p)$ normalized to $\bar{u}^r_q\left(\vec p\right)u^s_q\left(\vec p\right) = -\bar{v}^r_q\left(\vec p\right)v^s_q\left(\vec p\right) = 2m_q\delta^{rs}$ and satisfying the free Dirac equation.
The Hamilton operator $H$ is then given by (cf. \cite{Peskin}):
\begin{gather*}
H = \sum_{q\in\{\text{u,d,s}\}}\int d^3x\,\bar{q}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)q.
\end{gather*}
Subtracting vacuum energy, one obtains:
\begin{gather*}
H = \sum_{q\in\{\text{u,d,s}\}}\int\frac{d^3p}{(2\pi)^3}\sum_{s=1,2}E_{p;q}\left(a^{s\dagger}_{p;q}a^{s}_{p;q} + b^{s\dagger}_{p;q}b^{s}_{p;q}\right).
\end{gather*}
Let us now define the particle state $\Ket{\vec p;s,q}\coloneqq a^{s\dagger}_{p;q}\Ket{0}$ with $\Ket{0}$ being the vacuum state. The state $\Ket{\vec p;s,q}$ is an energy eigenstate with eigenvalue $E_{p;q}$. One maybe tempted to identify $\Ket{0;s,q}$ with a state from 1), however, it does not have the proper normalization as $\Braket{0;s,q|0;s,q} = (2\pi)^3\delta^{(3)}(0)$. In order to fix this, we consider wave packets now and define for $\varepsilon>0$:
\begin{gather*}
\Ket{\varepsilon;s,q}\coloneqq \frac{1}{\sqrt[4]{(2\pi)^9\varepsilon^3}}\int d^3p\ e^{-\frac{{\vec p}^{\, 2}}{4\varepsilon}}\Ket{\vec p;s,q}.
\end{gather*}
The state $\Ket{\varepsilon;s,q}$ is normalized to 1:
\begin{align*}
\Braket{\varepsilon;s,q|\varepsilon;s,q} &= \frac{1}{(2\pi)^3}(2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\int d^3p^\prime\ e^{-\frac{{\vec p}^{\, 2} + {\vec p}^{\,\prime 2}}{4\varepsilon}}\Braket{\vec p;s,q|{\vec p}^{\,\prime};s,q}\\
&= (2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\int d^3p^\prime\ e^{-\frac{{\vec p}^{\, 2} + {\vec p}^{\,\prime 2}}{4\varepsilon}}\delta^{(3)}\left(\vec p - {\vec p}^{\,\prime}\right)\\
&= (2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\ e^{-\frac{{\vec p}^{\, 2}}{2\varepsilon}}\\
&= (2\pi\varepsilon)^{-\frac{3}{2}}\left(\int\limits^{\infty}_{-\infty}dp\ e^{-\frac{p^2}{2\varepsilon}}\right)^3\\
&= (2\pi\varepsilon)^{-\frac{3}{2}}(2\pi\varepsilon)^{\frac{3}{2}} = 1.
\end{align*}
Let us calculate the energy of this state:
\begin{align*}
\Braket{\varepsilon;s,q|H|\varepsilon;s,q} &= \sum_{\tilde{q}\in\{\text{u,d,s}\}}\int\frac{d^3p}{(2\pi)^3}\sum_{r=1,2}E_{p;\tilde{q}}\Braket{\varepsilon;s,q|a^{r\dagger}_{p;\tilde{q}}a^{r}_{p;\tilde{q}}|\varepsilon;s,q}\\
&= \frac{1}{(2\pi)^6}(2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\int d^3k \int d^3k^\prime\sum_{\tilde{q}\in\{\text{u,d,s}\}}\sum_{r=1,2}E_{p;\tilde{q}}\, e^{-\frac{{\vec k}^{\, 2} + {\vec k}^{\,\prime 2}}{4\varepsilon}}\\
&\qquad\times\Braket{\vec k;s,q|a^{r\dagger}_{p;\tilde{q}}a^{r}_{p;\tilde{q}}|{\vec k}^{\,\prime};s,q}\\
&= \frac{1}{(2\pi)^6}(2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\int d^3k \int d^3k^\prime\sum_{\tilde{q}\in\{\text{u,d,s}\}}\sum_{r=1,2}E_{p;\tilde{q}}\, e^{-\frac{{\vec k}^{\, 2} + {\vec k}^{\,\prime 2}}{4\varepsilon}}\\
&\qquad\times\left\{a^{s}_{k;q},a^{r\dagger}_{p;\tilde{q}}\right\}\left\{a^{r}_{p;\tilde{q}},a^{s\dagger}_{k^\prime;q}\right\}\\
&= \frac{1}{(2\pi)^6}(2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\int d^3k \int d^3k^\prime\sum_{\tilde{q}\in\{\text{u,d,s}\}}\sum_{r=1,2}E_{p;\tilde{q}}\, e^{-\frac{{\vec k}^{\, 2} + {\vec k}^{\,\prime 2}}{4\varepsilon}}\\
&\qquad\times\left(2\pi\right)^6\,\delta^{rs}\,\delta_{q\tilde{q}}\,\delta^{(3)}\left(\vec k -\vec p\right)\delta^{(3)}\left(\vec p - {\vec k}^{\,\prime}\right)\\
&= (2\pi\varepsilon)^{-\frac{3}{2}}\int d^3p\, E_{p;q}\, e^{-\frac{{\vec p}^{\, 2}}{2\varepsilon}}\\
&= \frac{4\pi}{\left(2\pi\varepsilon\right)^{\frac{3}{2}}}\int\limits^{\infty}_0 dp\, p^2\sqrt{m^2_q + p^2}\, e^{-\frac{p^2}{2\varepsilon}}\\
&= \frac{4\pi}{\left(2\pi\varepsilon\right)^{\frac{3}{2}}}\int\limits^{\infty}_0 d \left(\sqrt{2\varepsilon}y\right)\, \left(\sqrt{2\varepsilon}y\right)^2\sqrt{m^2_q + \left(\sqrt{2\varepsilon}y\right)^2}\, e^{-y^2}\\
&= \frac{4}{\sqrt{\pi}}\int\limits^{\infty}_0 dy\, y^2\sqrt{m^2_q + 2\varepsilon y^2}\, e^{-y^2}\\
&= m_q\frac{4}{\sqrt{\pi}}\int\limits^{\infty}_0 dy\, e^{-y^2}\, y^2\, \sqrt{1 + \frac{2\varepsilon}{m^2_q}y^2}.
\end{align*}
With the help of mathematics and the definition $z\coloneqq \frac{2\varepsilon}{m^2_q}$, we can rewrite this expression in terms of the modified Bessel function $K_1$ of the second kind:
\begin{gather*}
\Braket{\varepsilon;s,q|H|\varepsilon;s,q} = m_q\frac{e^{\frac{1}{2z}}K_1\left(\frac{1}{2z}\right)}{\sqrt{\pi z}}.
\end{gather*}
If we consider the limit $\varepsilon\rightarrow 0^+$, $z$ also tends to $0$ and with $\lim\limits_{z\rightarrow 0^+}\frac{e^{\frac{1}{2z}}K_1\left(\frac{1}{2z}\right)}{\sqrt{\pi z}} = 1$ we obtain:
\begin{gather*}
\lim\limits_{\varepsilon\rightarrow 0^+}\Braket{\varepsilon;s,q|H|\varepsilon;s,q} = m_q.
\end{gather*}
Furthermore, a similar calculation shows:
\begin{align*}
\Braket{\varepsilon;s,q|H^2|\varepsilon;s,q} &= \frac{4}{\sqrt{\pi}}\int\limits^{\infty}_0 dy\, y^2\left(m^2_q + 2\varepsilon y^2\right)\, e^{-y^2}\\
&= m^2_q + 3\varepsilon.
\end{align*}
Therefore, we find:
\begin{gather*}
\lim\limits_{\varepsilon\rightarrow 0^+}\left(\Delta H^2(\varepsilon;s,q)\right) = \lim\limits_{\varepsilon\rightarrow 0^+}\left(\Braket{\varepsilon;s,q|H^2|\varepsilon;s,q} - \Braket{\varepsilon;s,q|H|\varepsilon;s,q}^2\right) = 0.
\end{gather*}
In this sense, the state $\Ket{\varepsilon;s,q}$ ``tends'' for $\varepsilon\rightarrow 0^+$ to a ``state'' whose energy expectation value is the mass $m_q$ and whose energy variance $\Delta H^2$ is zero. If the variance $\Delta A^2$ of any state for a given operator $A$ is zero, the state is an eigenstate of $A$. To this end, we can say that the normalized state $\Ket{\varepsilon;s,q}$ ``tends'' for $\varepsilon\rightarrow 0^+$ to an ``eigenstate'' of $H$ with ``eigenvalue'' $m_q$. These energy ``eigenstates'' are exactly the states we are looking for to fulfill 1). Note, however, that it is neither clear whether $\Ket{\varepsilon;s,q}$ converges in a mathematical sense for $\varepsilon\rightarrow 0^+$ nor obvious that the limit of $\Ket{\varepsilon;s,q}$ exhibits the described properties, if $\Ket{\varepsilon;s,q}$ converges. We are going to ignore this for now. Similarly, we can define anti-particle states $\Ket{\varepsilon; s, \bar{q}}$ by replacing $\Ket{\vec p; s, q}$ with $\Ket{\vec p; s, \bar{q}}\coloneqq b^{s\dagger}_{p;q}\Ket{0}$ in the definition of $\Ket{\varepsilon; s, q}$ and obtain the same results for anti-particles.\par
The subspace of these energy ``eigenstates'' which we need for 2) is then given by:
\begin{gather*}
V_\varepsilon \coloneqq \text{Span}\left\{\Ket{\varepsilon;s,q}\mid s\in\{1,2\};\ q\in\{\text{u,d,s,}\bar{\text{u}},\bar{\text{d}},\bar{\text{s}}\}\right\}.
\end{gather*}
$V_\varepsilon$ is a finite-dimensional complex vector space and, therefore, trivially a Hilbert space satisfying 2).\par
In order to satisfy 3), we have to find a representation $D^{(\rho)}:V_\varepsilon\rightarrow V_\varepsilon$ of \text{SU}(3) on $V_\varepsilon$. Let us define:
\begin{align*}
D^{(\rho)}(A)\Ket{\varepsilon;s,q}&\coloneqq \sum\limits_{p\in\{\text{u,d,s}\}} A_{pq}\Ket{\varepsilon;s,p}\quad\forall A\in\text{SU}(3)\,\forall s\in\{1,2\}\,\forall q\in\{\text{u,d,s}\},\\
D^{(\rho)}(A)\Ket{\varepsilon;s,\bar{q}}&\coloneqq \sum\limits_{p\in\{\text{u,d,s}\}} A^\ast_{pq}\Ket{\varepsilon;s,\bar{p}}\quad\forall A\in\text{SU}(3)\,\forall s\in\{1,2\}\,\forall q\in\{\text{u,d,s}\}.
\end{align*}
One can easily check that $D^{(\rho)}$ defined this way is a unitary representation. With this definition of $D^{(\rho)}$, we find:
\begin{align*}
&\Braket{\varepsilon;s,p|D^{(\rho)}(A)^\dagger\circ H\circ D^{(\rho)}(A)|\varepsilon;t,l}\\
= &\Braket{\varepsilon;s,p|D^{(\rho)}(A)^\dagger\left(\sum\limits_{q\in\{\text{u,d,s}\}}\int\frac{d^3k}{(2\pi)^3}\sum\limits_{r=1,2}E_{k;q}a^{r\dagger}_{k;q}a^r_{k;q}\right)D^{(\rho)}(A)|\varepsilon;t,l}\\
= &\left(\sum\limits_{\tilde{p}\in\{\text{u,d,s}\}} A_{\tilde{p}p} \Ket{\varepsilon;s,\tilde{p}}\right)^\dagger \left(\sum\limits_{q\in\{\text{u,d,s}\}}\int\frac{d^3k}{(2\pi)^3}\sum\limits_{r=1,2}E_{k;q}a^{r\dagger}_{k;q}a^r_{k;q}\right) \left(\sum\limits_{\tilde{l}\in\{\text{u,d,s}\}} A_{\tilde{l}l} \Ket{\varepsilon;t,\tilde{l}}\right)\\
= &\sum\limits_{\tilde{l},\tilde{p},q\in\{\text{u,d,s}\}} A^\ast_{\tilde{p}p} A_{\tilde{l}l} \int\frac{d^3k}{(2\pi)^3}\sum\limits_{r=1,2} E_{k;q}\Braket{\varepsilon;s,\tilde{p}|a^{r\dagger}_{k;q}a^r_{k;q}|\varepsilon;t,\tilde{l}}\\
= &\sum\limits_{\tilde{l},\tilde{p},q\in\{\text{u,d,s}\}} A^\ast_{\tilde{p}p} A_{\tilde{l}l} \int\frac{d^3k}{(2\pi)^3}\sum\limits_{r=1,2} E_{k;q}\ (2\pi)^3\ (2\pi\varepsilon)^{-\frac{3}{2}}\ \delta^{rs}\,\delta^{rt}\,\delta_{q\tilde{p}}\,\delta_{q\tilde{l}}\ e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}\\
= &\sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp}A_{ql} \int d^3k\ \frac{e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}}{(2\pi\varepsilon)^{\frac{3}{2}}}\ E_{k;q}\,\delta^{st},
\end{align*}
where we used the following relation:
\begin{align*}
&\Braket{\varepsilon;s,\tilde{p}|a^{r\dagger}_{k;q}a^r_{k;q}|\varepsilon;t,\tilde{l}}\\
= &\, (2\pi)^{-3}\, (2\pi\varepsilon)^{-\frac{3}{2}}\int d^3k^\prime\int d^3k^{\prime\prime} e^{-\frac{{\vec k}^{\, \prime 2} + {\vec k}^{\, \prime\prime 2}}{4\varepsilon}} \Braket{{\vec k}^{\, \prime}; s, \tilde{p}|a^{r\dagger}_{k;q}a^{r}_{k;q}|{\vec k}^{\, \prime\prime}; t, \tilde{l}}\\
= &\, (2\pi)^{-3}\, (2\pi\varepsilon)^{-\frac{3}{2}}\int d^3k^\prime\int d^3k^{\prime\prime} e^{-\frac{{\vec k}^{\, \prime 2} + {\vec k}^{\, \prime\prime 2}}{4\varepsilon}} \left\{a^{s}_{k^\prime; \tilde{p}}, a^{r\dagger}_{k;q}\right\} \left\{a^{r}_{k; q}, a^{t\dagger}_{k^{\prime\prime};\tilde{l}}\right\}\\
= &\, (2\pi)^{-3}\, (2\pi\varepsilon)^{-\frac{3}{2}}\int d^3k^\prime\int d^3k^{\prime\prime} e^{-\frac{{\vec k}^{\, \prime 2} + {\vec k}^{\, \prime\prime 2}}{4\varepsilon}} (2\pi)^6\delta^{sr}\delta^{rt}\delta_{\tilde{p}q}\delta_{q\tilde{l}}\delta^{(3)}\left({\vec k}^{\, \prime} - \vec k\right)\delta^{(3)}\left(\vec k - {\vec k}^{\, \prime\prime}\right)\\
= &\, (2\pi)^3\, (2\pi\varepsilon)^{-\frac{3}{2}}\ \delta^{rs}\,\delta^{rt}\,\delta_{q\tilde{p}}\,\delta_{q\tilde{l}}\ e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}.
\end{align*}
This result corresponds to the left-hand side of the equation form 3). In order to calculate the right-hand side, we have to calculate $H(\bar{q}^{\, \prime}, q^\prime)$ where\linebreak $q^\prime = \sum\limits_{\tilde{l}\in\{\text{u,d,s}\}} A_{q\tilde{l}}\, \tilde{l}$. $H(\bar{q},q)$ is the normal ordered Hamilton operator:
\begin{align*}
H(\bar{q},q) &= \mathopen{:} \sum_{q\in\{\text{u,d,s}\}}\int d^3x\,\bar{q}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)q \mathclose{:}\\
&= \sum_{q\in\{\text{u,d,s}\}}\int\frac{d^3p}{(2\pi)^3}\sum_{s=1,2}E_{p;q}\left(a^{s\dagger}_{p;q}a^{s}_{p;q} + b^{s\dagger}_{p;q}b^{s}_{p;q}\right).
\end{align*}
$H(\bar{q}^{\, \prime}, q^\prime)$ is, therefore, given by:
\begin{align*}
H(\bar{q}^{\, \prime}, q^\prime) &= \mathopen{:} \sum_{q\in\{\text{u,d,s}\}}\int d^3x\,\bar{q}^{\, \prime}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)q^\prime \mathclose{:}\\
&= \mathopen{:} \sum_{q\in\{\text{u,d,s}\}}\int d^3x\,\left(\sum\limits_{\tilde{p}\in\{\text{u,d,s}\}} A^\ast_{q\tilde{p}}\,\bar{\tilde{p}}\right)\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)\left(\sum\limits_{\tilde{l}\in\{\text{u,d,s}\}} A_{q\tilde{l}}\, \tilde{l}\right) \mathclose{:}\\
&= \sum\limits_{\tilde{l},\tilde{p},q\in\{\text{u,d,s}\}} A^\ast_{q\tilde{p}} A_{q\tilde{l}}\int d^3x\ \mathopen{:}\bar{\tilde{p}}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)\tilde{l}\mathclose{:}.
\end{align*}
The Fourier decomposition of $\tilde{l}(x)$ and $\bar{\tilde{p}}(x)$ then gives:
\begin{align*}
&\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)\tilde{l}(x)\\
=& \int\frac{d^3k^\prime}{(2\pi)^3}\, \frac{1}{\sqrt{2 E_{k^\prime;\tilde{l}}}}\\
&\times\sum\limits_{s=1,2}\left[a^s_{k^\prime;\tilde{l}}\left({\vec k}^{\, \prime}\cdot\vec\gamma + m_q\right)u^s_{\tilde{l}}({\vec k}^{\, \prime})e^{-ik^\prime\cdot x} + b^{s\dagger}_{k^\prime;\tilde{l}}\left(-{\vec k}^{\, \prime}\cdot\vec\gamma + m_q\right)v^s_{\tilde{l}}({\vec k}^{\, \prime})e^{ik^\prime\cdot x}\right],\\
\bar{\tilde{p}}(x) =& \int\frac{d^3k}{(2\pi)^3}\frac{1}{\sqrt{2E_{k;\tilde{p}}}}\sum_{r=1,2}\left(a^{r\dagger}_{k;\tilde{p}}\bar{u}_{\tilde{p}}^r(\vec k)e^{ik\cdot x} + b^{r}_{k;\tilde{p}}\bar{v}^r_{\tilde{p}}(\vec k)e^{-ik\cdot x}\right).
\end{align*}
Inserting this in $\mathopen{:}\bar{\tilde{p}}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)\tilde{l}\mathclose{:}$ from $H(\bar{q}^{\, \prime}, q^\prime)$ yields:
\begin{align*}
&\mathopen{:}\bar{\tilde{p}}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)\tilde{l}\mathclose{:}\\
=& \int\frac{d^3k}{(2\pi)^3}\int\frac{d^3k^\prime}{(2\pi)^3}\, \frac{1}{2\sqrt{E_{k;\tilde{p}} E_{k^\prime;\tilde{l}}}}\\
&\times\sum\limits_{r,s = 1,2}\left[a^{r\dagger}_{k;\tilde{p}}a^s_{k^\prime;\tilde{l}}\bar{u}^r_{\tilde{p}}(\vec k)\left({\vec k}^{\, \prime}\cdot\vec\gamma + m_q\right)u^s_{\tilde{l}}({\vec k}^{\, \prime}) e^{i(k-k^\prime)\cdot x}\ +\ \text{terms with }b\right].
\end{align*}
Using $\int d^3x\ e^{i(k-k^\prime)\cdot x} = (2\pi)^3 \delta^{(3)}({\vec k}^{\, \prime} - \vec k)e^{i(E_{k;\tilde{p}} - E_{k^\prime ;\tilde{l}})t}$, we obtain:
\begin{align*}
&\int d^3x\ \mathopen{:}\bar{\tilde{p}}\left(-i\vec\gamma\cdot\vec\nabla + m_q\right)\tilde{l}\mathclose{:}\\
=& \int\frac{d^3k}{(2\pi)^3}\, \frac{1}{2\sqrt{E_{k;\tilde{p}} E_{k;\tilde{l}}}}\\
&\times\sum\limits_{r,s = 1,2}\left[a^{r\dagger}_{k;\tilde{p}}a^s_{k;\tilde{l}}\bar{u}^r_{\tilde{p}}(\vec k)\left(\vec k\cdot\vec\gamma + m_q\right)u^s_{\tilde{l}}(\vec k) e^{i(E_{k;\tilde{p}} - E_{k;\tilde{l}})t}\ +\ \text{terms with }b\right].
\end{align*}
This implies that the operator $H(\bar{q}^{\, \prime}, q^\prime)$ has, in general, a non-trivial time dependence, while the left-hand side of the equation from 3) is time-independent. However, we only need assumption 3) to be true at one point in time. Therefore, we can choose $t=0$ and find for the right-hand side of the equation from 3):
\begin{align*}
& \Braket{\varepsilon;s,p|H(\bar{q}^{\, \prime}, q^\prime)|\varepsilon;t,l}\\
=& \sum\limits_{\tilde{l},\tilde{p},q\in\{\text{u,d,s}\}} A^\ast_{q\tilde{p}} A_{q\tilde{l}}\int\frac{d^3k}{(2\pi)^3}\, \frac{1}{2\sqrt{E_{k;\tilde{p}} E_{k;\tilde{l}}}}\\
&\times\sum\limits_{r,\tilde{r} = 1,2} \Braket{\varepsilon;s,p|a^{r\dagger}_{k;\tilde{p}}a^{\tilde{r}}_{k;\tilde{l}}|\varepsilon;t,l}\bar{u}^r_{\tilde{p}}(\vec k)\left(\vec k\cdot\vec\gamma + m_q\right)u^{\tilde{r}}_{\tilde{l}}(\vec k)\\
=& \sum\limits_{\tilde{l},\tilde{p},q\in\{\text{u,d,s}\}} A^\ast_{q\tilde{p}} A_{q\tilde{l}}\int\frac{d^3k}{(2\pi)^3}\, \frac{1}{2\sqrt{E_{k;\tilde{p}} E_{k;\tilde{l}}}}\\
&\times\sum\limits_{r,\tilde{r} = 1,2} (2\pi)^3\, (2\pi\varepsilon)^{-\frac{3}{2}}\ \delta^{rs}\,\delta^{\tilde{r}t}\,\delta_{p\tilde{p}}\,\delta_{l\tilde{l}}\ e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}\bar{u}^r_{\tilde{p}}(\vec k)\left(\vec k\cdot\vec\gamma + m_q\right)u^{\tilde{r}}_{\tilde{l}}(\vec k)\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp} A_{ql}\int d^3k\, \frac{e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}}{(2\pi\varepsilon)^{\frac{3}{2}}}\, \frac{1}{2\sqrt{E_{k;p} E_{k;l}}} \bar{u}^s_{p}(\vec k)\left(\vec k\cdot\vec\gamma + m_q\right)u^{t}_{l}(\vec k)\\
\eqqcolon& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp} A_{ql}\int d^3k\, \frac{e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}}{(2\pi\varepsilon)^{\frac{3}{2}}}\, \tilde{E}(\vec k;q,p,l,s,t).
\end{align*}
With this, we have written the left-hand and right-hand side of the equation from 3) in a very similar form. They only differ in the integrand of the 3-momentum integration. As we found that assumption 1) is fulfilled in the case of $\varepsilon\rightarrow 0^+$, we expect assumption 3) to be valid in the same limit. To show this, we note that the $\varepsilon$-dependence of both integrands is given by multiplication with:
\begin{gather*}
\frac{e^{-\frac{{\vec k}^{\, 2}}{2\varepsilon}}}{(2\pi\varepsilon)^{\frac{3}{2}}} = \prod\limits^3_{i=1} \frac{e^{-\frac{k^2_i}{2\varepsilon}}}{\sqrt{2\pi\varepsilon}}.
\end{gather*}
This is a nascent delta function known as the heat kernel. In the limit of $\varepsilon\rightarrow 0^+$, the heat kernel can be replaced by a delta distribution of $\vec k$. With this, one obtains:
\begin{align*}
&\lim\limits_{\varepsilon\rightarrow 0^+}\left(\Braket{\varepsilon;s,p|D^{(\rho)}(A)^\dagger\circ H\circ D^{(\rho)}(A)|\varepsilon;t,l}\right)\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp}A_{ql} \int d^3k\, \delta^{(3)}(\vec k)\, E_{k;q}\,\delta^{st}\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp}A_{ql}\, m_q\,\delta^{st},
\end{align*}
\begin{align*}
&\lim\limits_{\varepsilon\rightarrow 0^+}\left(\Braket{\varepsilon;s,p|H(\bar{q}^{\, \prime}, q^\prime)|\varepsilon;t,l}\right)\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp} A_{ql}\int d^3k\, \delta^{(3)}(\vec k)\, \tilde{E}(\vec k;q,p,l,s,t)\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp} A_{ql}\, \tilde{E}(0;q,p,l,s,t)\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp} A_{ql}\, \frac{1}{2\sqrt{m_p m_l}}m_q\, \bar{u}^s_{p}(0)u^{t}_{l}(0)\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp} A_{ql}\, \frac{1}{2\sqrt{m_p m_l}}m_q\, 2\sqrt{m_p m_l}\, \delta^{st}\\
=& \sum\limits_{q\in\{\text{u,d,s}\}} A^\ast_{qp}A_{ql}\, m_q\,\delta^{st}\\
=& \lim\limits_{\varepsilon\rightarrow 0^+}\left(\Braket{\varepsilon;s,p|D^{(\rho)}(A)^\dagger\circ H\circ D^{(\rho)}(A)|\varepsilon;t,l}\right),
\end{align*}
where used that the spinors are given by (cf. \cite{Peskin})
\begin{gather*}
\bar{u}^s_p(0) = \sqrt{m_p}(\xi^{s\dagger}\ \xi^{s\dagger})\text{ and }u^{t}_{l}(0) = \sqrt{m_l}\begin{pmatrix}\xi^t\\\xi^t\end{pmatrix} \text{ with } \xi^{s\dagger}\xi^t = \delta^{st}
\end{gather*}
in the Weyl representation. In the sense of the limit $\varepsilon\rightarrow 0^+$, we have shown assumption 3) for the particle states $\Ket{\varepsilon;s,q}$. For the other states in $V_\varepsilon$, we can proceed in similar fashion to show assumption 3).\par
At this point, it is helpful to recapitulate what we have done. For $\varepsilon>0$, we have defined normalized states $\Ket{\varepsilon;s,q}$ and $\Ket{\varepsilon;s,\bar{q}}$. As $\varepsilon$ tends to $0$, these states exhibit the very same properties as states satisfying assumption 1), 2), and 3). However, as stated earlier, it is neither clear that the states converge for $\varepsilon\rightarrow 0^+$ nor obvious that the limit, if it exists, has these desired properties. To this end, we have to understand the assumptions 1), 2), and 3) in a different way. For instance, we have to understand 1) as:
\begin{itemize}
\item[1)] For every hadron $a$, there exists a parameter $\varepsilon\geq0$ and a state $\Ket{\varepsilon;a}$ with $\Braket{\varepsilon;a|\varepsilon;a} = 1$ such that the mass $m_a$ of $a$ is given by
\begin{align*}
m_a + \mathcal{O}(\varepsilon) &= \Braket{\varepsilon;a|H|\varepsilon;a}\text{ with}\\
m^2_a + \mathcal{O}(\varepsilon) &= \Braket{\varepsilon;a|H^2|\varepsilon;a}.
\end{align*}
\end{itemize}
If $\varepsilon$ is 0, we recover the original formulation of 1). In this sense, we may view $\varepsilon$ as a parameter that states how ``strongly'' the assumptions 1), 2), and 3) are violated. In the case of the free fields, we have seen that we can choose $\varepsilon$ to be arbitrarily small. However, it is not clear that this is the case for all theories and, in particular, QCD to which we apply the assumptions 1), 2), and 3) in this work. Nevertheless, the assumptions 1), 2), and 3) are used throughout this thesis as they were presented in the beginning of this section, since these assumptions are only used to calculate relations of the hadron masses in a perturbative approach. If $\varepsilon$ is indeed greater than 0, we might pick up corrections of order $\varepsilon$ in the mass relations.\par
This is a rather interesting point, especially if we try to interpret $\varepsilon$. The $\varepsilon$ we introduced for free fields can be considered to be the momentum fluctuation as:
\begin{gather*}
\Braket{\varepsilon;s,q|\Delta P_i^2|\varepsilon;s,q} = \Braket{\varepsilon;s,q|P_i^2|\varepsilon;s,q} = \varepsilon.
\end{gather*}
Therefore, assuming 1), 2), and 3) in their original form might only be a valid approximation if the hadron at hand can be seen as a non-relativistic system.
\chapter{Clebsch-Gordan Series of $D(p,q)\otimes D(1,1)$}\label{app:Young}
As we have seen in \autoref{sec:GMO_formula}, the Clebsch-Gordan series of $\sigma\otimes 8$ for finite-dimensional multiplets $\sigma$ can be used to calculate the number of octets in $\sigma\otimes\bar{\sigma}$. Therefore, these Clebsch-Gordan series are of great importance for the derivation of the GMO mass relations. We want to compute these series now by using Young tableaux. Young tableaux were introduced in \autoref{sec:GMO_formula}. The rules on how to use Young tableaux to compute the Clebsch-Gordan series of tensor product representations are not presented in this thesis, but can commonly be found in literature, for instance in \cite{Lichtenberg} and in review \textit{46. $\text{SU}(n)$ Multiplets and Young Diagrams} in \cite{PDG}. For these calculations, we denote $8$ with $D(1,1)$ and $\sigma$ with $D(p,q)$. This notation was also introduced in \autoref{sec:GMO_formula}.\par
Before we start with the actual computations, we can derive one relation that simplifies the following calculations: Suppose that we have found the Clebsch-Gordan series of $D(p,q)\otimes D(1,1)$:
\begin{gather*}
D(p,q)\otimes D(1,1) = \bigoplus_{P,Q} n_{D(P,Q)}(D(p,q)\otimes D(1,1))\ D(P,Q),
\end{gather*}
where $n_{D(P,Q)}(D(p,q)\otimes D(1,1))$ denotes the multiplicity of $D(P,Q)$ in {${D(p,q)\otimes D(1,1)}$}. We then have:
\begin{align*}
D(q,p)\otimes D(1,1) &= \overline{D(p,q)}\otimes\overline{D(1,1)} = \overline{D(p,q)\otimes D(1,1)}\\
&= \overline{\bigoplus_{P,Q} n_{D(P,Q)}(D(p,q)\otimes D(1,1))\ D(P,Q)}\\
&= \bigoplus_{P,Q} n_{D(P,Q)}(D(p,q)\otimes D(1,1))\ \overline{D(P,Q)}\\
&= \bigoplus_{P,Q} n_{D(P,Q)}(D(p,q)\otimes D(1,1))\ D(Q,P).
\end{align*}
This means that the Clebsch-Gordan series of {${D(p,q)\otimes D(1,1)}$} directly implies the Clebsch-Gordan series of {${D(q,p)\otimes D(1,1)}$}. Let us now turn to the computations:\\\\
\textbf{Singlet (}$p=0$\textbf{ and }$q=0$\textbf{)}
\begin{align*}\ytableausetup{boxsize = 1.3em}
D(0,0)\otimes D(1,1) = \emptyset\otimes\ydiagram{2,1} = \ydiagram{2,1} = D(1,1)
\end{align*}
\newpage
\noindent\textbf{Totally symmetric multiplets (}$p\geq 1$\textbf{ and }$q=0$\textbf{ or }$p=0$\textbf{ and }$q\geq 1$\textbf{)}
\begin{align*}
D(1,0)\otimes D(1,1) &= \ydiagram{1}\otimes\ydiagram{2,1} = \ydiagram{1}\otimes\ytableaushort{aa,b} = \left(\ytableaushort{\, a}\oplus\ytableaushort{\,,a}\right)\otimes\ytableaushort{a,b}\\
&= \left(\ytableaushort{\, a,a}\oplus\ytableaushort{\, aa}\right)\otimes\ytableaushort{b}\\
&= \ytableaushort{\, a,a,b}\oplus\ytableaushort{\, a,ab}\oplus\ytableaushort{\, aa,b}\\
&= \ydiagram{1}\oplus\ydiagram{2,2}\oplus\ydiagram{3,1}\\
&= D(1,0)\oplus D(0,2)\oplus D(2,1)\\\\
D(p,0)\otimes D(1,1) &= \begin{ytableau}\, & \none[\dots] & \,\end{ytableau}\otimes\ydiagram{2,1}= \begin{ytableau}\, & \none[\dots] & \,\end{ytableau}\otimes\ytableaushort{aa,b}\\
&= \left(\begin{ytableau}\, & \none[\dots] & \, & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \,\\ a\end{ytableau}\right)\otimes\ytableaushort{a,b}\\
&= \left(\begin{ytableau}\, & \none[\dots] & \, & a & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & a\\ a\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ a & a\end{ytableau}\right)\otimes\ytableaushort{b}\\
&= \begin{ytableau}\, & \, & \none[\dots] & \, & a & a\\ b\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & a\\ a & b\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & a\\ a\\ b\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ a & a \\ b\end{ytableau}\\
&= \begin{ytableau}\, & \, & \none[\dots] & \, & \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \,\\ \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \,\\ \,\\ \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ \, & \, \\ \,\end{ytableau}\\
&= \begin{ytableau}\, & \, & \none[\dots] & \, & \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \,\\ \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \none[\dots] & \,\\ \,\end{ytableau}\\
&= D(p+1,1)\oplus D(p-1,2)\oplus D(p,0)\oplus D(p-2,1),\quad p\geq 2
\end{align*}
The Clebsch-Gordan series of $D(0,q)\otimes D(1,1)$ follows directly from the Clebsch-Gordan series of $D(p,0)\otimes D(1,1)$:
\begin{align*}
D(0,1)\otimes D(1,1) &= D(0,1)\oplus D(2,0)\oplus D(1,2)\\
D(0,q)\otimes D(1,1) &= D(1,q+1)\oplus D(2,q-1)\oplus D(0,q)\oplus D(1,q-2),\quad q\geq 2
\end{align*}\\
\textbf{Remaining multiplets (}$p \geq 1$\textbf{ and }$q \geq 1$\textbf{)}
\begin{align*}
D(1,1)\otimes D(1,1) &= \ydiagram{2,1}\otimes\ydiagram{2,1} = \ydiagram{2,1}\otimes\ytableaushort{aa,b}\\
&= \left(\ytableaushort{\,\,a,\,}\oplus\ytableaushort{\,\,,\,a}\oplus\ytableaushort{\,\,,\,,a}\right)\otimes\ytableaushort{a,b}\\
&= \left(\ytableaushort{\,\,aa,\,}\oplus\ytableaushort{\,\,a,\,a}\oplus\ytableaushort{\,\,a,\,,a}\oplus\ytableaushort{\,\,,\,a,a}\right)\otimes\ytableaushort{b}\\
&= \ytableaushort{\,\,aa,\,b}\oplus\ytableaushort{\,\,aa,\,,b}\oplus\ytableaushort{\,\,a,\,ab}\oplus\ytableaushort{\,\,a,\,a,b}\oplus\ytableaushort{\,\,a,\,b,a}\\ &\ \ \ \oplus\ytableaushort{\,\,,\,a,ab}\\
&= \ydiagram{4,2}\oplus\ydiagram{4,1,1}\oplus\ydiagram{3,3}\oplus\ydiagram{3,2,1}\oplus\ydiagram{3,2,1}\\ &\ \ \ \oplus\ydiagram{2,2,2}\\
&= \emptyset\oplus\ydiagram{2,1}\oplus\ydiagram{2,1}\oplus\ydiagram{3}\oplus\ydiagram{3,3}\oplus\ydiagram{4,2}\\
&= D(0,0)\oplus D(1,1)\oplus D(1,1)\oplus D(3,0)\oplus D(0,3)\oplus D(2,2)
\end{align*}
\begin{align*}
D(p,1)\otimes D(1,1) &= \begin{ytableau}\, & \, & \none[\dots] & \,\\ \,\end{ytableau}\otimes\ydiagram{2,1} = \begin{ytableau}\, & \, & \none[\dots] & \,\\ \,\end{ytableau}\otimes\ytableaushort{aa,b}\\
&= \left(\begin{ytableau}\, & \, & \none[\dots] & \, & a\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ \, & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \,\\ \,\\ a\end{ytableau}\right)\otimes\ytableaushort{a,b}\\
&= \left(\begin{ytableau}\, & \, & \none[\dots] & \, & a & a\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & a\\ \, & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & a\\ \,\\ a\end{ytableau}\right.\\ &\ \ \ \left.\oplus\begin{ytableau}\, & \, & \, & \, & \none[\dots] & \,\\ \, & a & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ \, & a\\ a\end{ytableau}\right)\otimes\ytableaushort{b}\\
&= \begin{ytableau}\, & \, & \, & \none[\dots] & \, & a & a\\ \, & b\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & a & a\\ \,\\ b\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \, & \none[\dots] & \, & a\\ \, & a & b\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & a\\ \, & a\\ b\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & a\\ \, & b\\ a\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \, & \none[\dots] & \,\\ \, & a & a\\ b\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ \, & a\\ a & b\end{ytableau}\\
&= \begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \,\\ \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \,\\ \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \, & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \,\\ \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \,\\ \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \, & \none[\dots] & \,\\ \, & \, & \,\\ \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ \, & \,\\ \, & \,\end{ytableau}\\
&= \begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \,\\ \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \, & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \,\\ \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \,\end{ytableau}\\
&= D(p+1,2)\oplus D(p+2,0)\oplus D(p-1,3)\\ &\ \ \ \oplus D(p,1)\oplus D(p,1)\oplus D(p-2,2)\oplus D(p-1,0),\quad p\geq 2\\
\Rightarrow D(1,q)\otimes D(1,1) &= D(2,q+1)\oplus D(0,q+2)\oplus D(3,q-1)\\ &\ \ \ \oplus D(1,q)\oplus D(1,q)\oplus D(2,q-2)\oplus D(0,q-1),\quad q\geq 2
\end{align*}
\begin{align*}
\hspace{-1.5cm} D(p,q)\otimes D(1,1) &= \begin{ytableau}\, & \none[\dots] & \, & \, & \none[\dots] & \,\\ \, & \none[\dots] & \,\end{ytableau}\otimes\ydiagram{2,1} = \begin{ytableau}\, & \none[\dots] & \, & \, & \none[\dots] & \,\\ \, & \none[\dots] & \,\end{ytableau}\otimes\ytableaushort{aa,b}\\
&= \left(\begin{ytableau}\, & \none[\dots] & \, & \, & \none[\dots] & \, & a\\ \, & \none[\dots] & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \,\\ \, & \none[\dots] & \, & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \none[\dots] & \,\\ \, & \, & \none[\dots] & \,\\ a\end{ytableau}\right)\\ &\ \ \ \otimes\ytableaushort{a,b}\\
&= \left(\begin{ytableau}\, & \none[\dots] & \, & \, & \none[\dots] & \, & a & a\\ \, & \none[\dots] & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & a\\ \, & \none[\dots] & \, & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \none[\dots] & \, & a\\ \, & \, & \none[\dots] & \,\\ a\end{ytableau}\right.\\ &\ \ \ \left.\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \,\\ \, & \none[\dots] & \, & a & a\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \,\\ \, & \, & \none[\dots] & \, & a\\ a\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \none[\dots] & \,\\ \, & \, & \, & \none[\dots] & \,\\ a & a\end{ytableau}\right)\\ &\ \ \ \otimes\ytableaushort{b}\\
&= \begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & a & a\\ \, & \none[\dots] & \, & b\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \none[\dots] & \, & a & a\\ \, & \, & \none[\dots] & \,\\ b\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \, & a\\ \, & \none[\dots] & \, & a & b\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & a\\ \, & \, & \none[\dots] & \, & a\\ b\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & a\\ \, & \, & \none[\dots] & \, & b\\ a\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \none[\dots] & \, & a\\ \, & \, & \, & \none[\dots] & \,\\ a & b\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \,\\ \, & \, & \none[\dots] & \, & a & a\\ b\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \,\\ \, & \, & \, & \none[\dots] & \, & a\\ a & b\end{ytableau}
\end{align*}
\begin{align*}
\Rightarrow D(p,q)\otimes D(1,1) &= \begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & \, & \,\\ \, & \none[\dots] & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \none[\dots] & \, & \, & \,\\ \, & \, & \none[\dots] & \,\\ \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \none[\dots] & \, & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \, & \none[\dots] & \, & \,\\ \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \, & \none[\dots] & \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \none[\dots] & \, & \,\\ \, & \, & \, & \none[\dots] & \,\\ \, & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \,\\ \, & \, & \none[\dots] & \, & \, & \,\\ \,\end{ytableau}\oplus\begin{ytableau}\, & \, & \, & \none[\dots] & \, & \, & \, & \none[\dots] & \,\\ \, & \, & \, & \none[\dots] & \, & \,\\ \, & \,\end{ytableau}\\
&= \begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & \, & \,\\ \, & \none[\dots] & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \none[\dots] & \, & \, & \,\\ \, & \none[\dots] & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \none[\dots] & \, & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \none[\dots] & \, & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \, & \,\\ \, & \none[\dots] & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \none[\dots] & \, & \,\\ \, & \none[\dots] & \,\end{ytableau}\\ &\ \ \ \oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \, & \none[\dots] & \,\\ \, & \none[\dots] & \, & \, & \,\end{ytableau}\oplus\begin{ytableau}\, & \none[\dots] & \, & \, & \, & \none[\dots] & \,\\ \, & \none[\dots] & \, & \,\end{ytableau}\\
&= D(p+1,q+1)\oplus D(p+2,q-1)\oplus D(p-1,q+2)\oplus D(p,q)\oplus D(p,q)\\ &\ \ \ \oplus D(p+1,q-2)\oplus D(p-2,q+1)\oplus D(p-1,q-1),\quad p,q\geq 2
\end{align*}
\chapter{Properties of $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$}\label{app:F-and_D-symbols}
In \autoref{sec:GMO_formula}, we found that there are at most two octets in the representation $\sigma\otimes\bar{\sigma}$ of \text{SU}(3) for multiplets $\sigma$ of \text{SU}(3). We parametrized these octets by introducing the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_k$ as elements of the vector space the representation $\sigma\otimes\bar{\sigma}$ acts on. However, we skipped some technical and rather lengthy calculations in the process of showing that these matrices actually form the octets in $\sigma\otimes\bar{\sigma}$. In this part of the appendix, we want to perform these computations. Some of the calculations can be found or follow computations in \cite{Lichtenberg}. Throughout this part, $\sigma$ always denotes a non-trivial, complex, finite-dimensional, and unitary multiplet of \text{SU}(3). Before we begin, let us quickly summarize the most important results and formulae given in \autoref{sec:GMO_formula}:
\begin{gather}
D^{(\sigma\otimes\bar{\sigma})}(A)\circ DD^{(\sigma)}\vert_\mathbb{1} = DD^{(\sigma)}\vert_\mathbb{1}\circ D^{(8)}(A)\quad\forall A\in\text{SU}(3),\label{eq:sigma8}\\
F^{(\sigma\otimes\bar{\sigma})}_k\coloneqq DD^{(\sigma)}\vert_\mathbb{1}(F_k)\quad\forall k\in\{1,\ldots, 8\}\label{eq:Fdef},\\
D^{(\sigma\otimes\bar{\sigma})}_k\coloneqq \frac{2}{3}\sum^{8}_{l,m=1}d_{klm} F^{(\sigma\otimes\bar{\sigma})}_l F^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k\in\{1,\ldots, 8\},\\
[F_k,F_l] = \sum^{8}_{m=1}if_{klm} F_m\quad\forall k,l\in\{1,\ldots, 8\},\label{eq:com_F}\\
[F^{(\sigma\otimes\bar{\sigma})}_k,F^{(\sigma\otimes\bar{\sigma})}_l] = \sum^{8}_{m=1}if_{klm} F^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k,l\in\{1,\ldots, 8\},\label{eq:com_F_sigma}\\
\text{Tr}(F_kF_l) = \frac{\delta_{kl}}{2}\quad\forall k,l\in\{1,\ldots, 8\},\label{eq:trace}\\
f_{klm} = -2i\text{Tr}\left([F_k,F_l]F_m\right)\quad\forall k,l,m\in\{1,\ldots, 8\},\label{eq:fklm}\\
d_{klm} = 2\text{Tr}\left(\{F_k,F_l\}F_m\right)\quad\forall k,l,m\in\{1,\ldots, 8\}.\label{eq:dklm}
\end{gather}
For definitions and notations, confer \autoref{sec:GMO_formula}.\par
Let us begin with the quadratic Casimir operator $C^{(\sigma\otimes\bar{\sigma})}$:
\begin{gather*}
C^{(\sigma\otimes\bar{\sigma})}\coloneqq \sum^8_{k,l=1}\delta_{kl}F^{(\sigma\otimes\bar{\sigma})}_kF^{(\sigma\otimes\bar{\sigma})}_l.
\end{gather*}
In \autoref{sec:GMO_formula}, we stated that the quadratic Casimir operator commutes with $F^{(\sigma\otimes\bar{\sigma})}_k$ for every $k\in\{1,\ldots, 8\}$. This can be shown with a straightforward calculation using \autoref{eq:com_F_sigma}:
\begin{align*}
\left[C^{(\sigma\otimes\bar{\sigma})}, F^{(\sigma\otimes\bar{\sigma})}_k\right] &= \sum^8_{l,m=1}\delta_{lm}\left[F^{(\sigma\otimes\bar{\sigma})}_lF^{(\sigma\otimes\bar{\sigma})}_m, F^{(\sigma\otimes\bar{\sigma})}_k\right]\\
&= \sum^8_{l,m=1}\delta_{lm}\left(F^{(\sigma\otimes\bar{\sigma})}_l\left[F^{(\sigma\otimes\bar{\sigma})}_m,F^{(\sigma\otimes\bar{\sigma})}_k\right] + \left[F^{(\sigma\otimes\bar{\sigma})}_l,F^{(\sigma\otimes\bar{\sigma})}_k\right]F^{(\sigma\otimes\bar{\sigma})}_m\right)\\
&= \sum^8_{l,m,n=1}\delta_{lm}\left(if_{mkn}F^{(\sigma\otimes\bar{\sigma})}_lF^{(\sigma\otimes\bar{\sigma})}_n + if_{lkn}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_m\right)\\
&= \sum^8_{l,n=1}if_{lkn}\left\{F^{(\sigma\otimes\bar{\sigma})}_l,F^{(\sigma\otimes\bar{\sigma})}_n\right\}\\
&= 0\quad\forall k\in\{1,\ldots,8\}
\end{align*}
The last line follows from the fact that $f_{lkn}$ is antisymmetric in $l$ and $n$ -- this follows directly from \autoref{eq:fklm} -- and $\left\{F^{(\sigma\otimes\bar{\sigma})}_l,F^{(\sigma\otimes\bar{\sigma})}_n\right\}$ is symmetric in $l$ and $n$. A contraction of an antisymmetric with a symmetric tensor always yields zero.\par
In \autoref{sec:GMO_formula}, we also stated that the quadratic Casimir operator $C^{(\sigma\otimes\bar{\sigma})}$ is a constant times the identity. The property that an operator commutes with all generators of a connected, simple, and compact Lie group is sufficient to show that the operator has to be constant for multiplets, nevertheless, we will show explicitly that $C^{(\sigma\otimes\bar{\sigma})} = c^{(\sigma\otimes\bar{\sigma})}\cdot\mathbb{1}$ for some $c^{(\sigma\otimes\bar{\sigma})}\in\mathbb{R}\backslash\{0\}$. In order to do this, we will show that $C^{(\sigma\otimes\bar{\sigma})}$ is a singlet under $\text{SU}(3)$. First, consider the transformation of $F_k\equiv F^{(3\otimes\bar{3})}_k$ under $A\in\text{SU}(3)$ (cf. \autoref{sec:GMO_formula}):
\begin{gather*}
\sum^{8}_{l=1}\left(D^{(8)}(A)\right)_{kl}F_l \coloneqq D^{(8)}(A)(F_k) = D^{(3\otimes\bar{3})}(A)(F_k) = AF_kA^\dagger\quad\forall k\in\{1,\ldots,8\},
\end{gather*}
where $\left(D^{(8)}(A)\right)_{kl}$ are the coefficients mediating the transformation of $F_k$. As the matrices $F_k$ are Hermitian, the coefficients $\left(D^{(8)}(A)\right)_{kl}$ are real:
\begin{align*}
\sum^{8}_{l=1}\left(D^{(8)}(A)\right)^\ast_{kl}F_l &= \left(\sum^{8}_{l=1}\left(D^{(8)}(A)\right)_{kl}F_l\right)^\dagger = \left(AF_kA^\dagger\right)^\dagger = AF_kA^\dagger\\
&= \sum^{8}_{l=1}\left(D^{(8)}(A)\right)_{kl}F_l\quad\forall k\in\{1,\ldots,8\}\\
\Rightarrow \left(D^{(8)}(A)\right)^\ast_{kl} &= \left(D^{(8)}(A)\right)_{kl}\quad\forall k,l\in\{1,\ldots, 8\}.
\end{align*}
Using the cyclicity of the trace and \autoref{eq:trace}, we also find that $D^{(8)}(A)$ is unitary:
\begin{align*}
\delta_{kl} &= 2\text{Tr}\left(F_kF_l\right) = 2\text{Tr}\left(F_kA^\dagger AF_lA^\dagger A\right) = 2\text{Tr}\left(AF_kA^\dagger AF_lA^\dagger\right)\\
&= \sum^8_{m,n=1}\left(D^{(8)}(A)\right)_{km}\left(D^{(8)}(A)\right)_{ln}2\text{Tr}\left(F_mF_n\right)\\
&= \sum^8_{m,n=1}\left(D^{(8)}(A)\right)_{km}\left(D^{(8)}(A)\right)_{ln}\delta_{mn}\\
&= \sum^8_{m=1}\left(D^{(8)}(A)\right)_{km}\left(D^{(8)}(A)^\text{T}\right)_{ml}\quad\forall k,l\in\{1,\ldots,8\}\\
\Rightarrow& \left(D^{(8)}(A)^{-1}\right)_{kl} = \left(D^{(8)}(A)^\text{T}\right)_{kl} = \left(D^{(8)}(A)^\dagger\right)_{kl}\quad\forall k,l\in\{1,\ldots,8\}.
\end{align*}
The fact that the coefficients $\left(D^{(8)}(A)\right)_{kl}$ are real was used in the last line.\par
Using \autoref{eq:sigma8} and \autoref{eq:Fdef}, it is easy to see that the coefficients $\left(D^{(8)}(A)\right)_{kl}$ also mediate the transformation of $F^{(\sigma\otimes\bar{\sigma})}_k$ under $A\in\text{SU}(3)$:
\begin{align*}
D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_k\right) &= \left(DD^{(\sigma)}\vert_\mathbb{1}\circ D^{(8)}(A)\right)\left(F_k\right)\\
&= \sum^{8}_{l=1} \left(D^{(8)}(A)\right)_{kl} F^{(\sigma\otimes\bar{\sigma})}_l\quad\forall k\in\{1,\ldots,8\}.
\end{align*}
Now consider the transformation of the quadratic Casimir $C^{(\sigma\otimes\bar{\sigma})}$ under $A\in\text{SU}(3)$:
\begin{align*}
D^{(\sigma\otimes\bar{\sigma})}(A)\left(C^{(\sigma\otimes\bar{\sigma})}\right) &=D^{(\sigma)}(A)\left(\sum^8_{k,l=1}\delta_{kl}F^{(\sigma\otimes\bar{\sigma})}_kF^{(\sigma\otimes\bar{\sigma})}_l\right)D^{(\sigma)}(A)^\dagger\\
&=\sum^8_{k,l=1}\delta_{kl} D^{(\sigma)}(A)F^{(\sigma\otimes\bar{\sigma})}_k D^{(\sigma)}(A)^\dagger D^{(\sigma)}(A)F^{(\sigma\otimes\bar{\sigma})}_l D^{(\sigma)}(A)^\dagger\\
&= \sum^8_{k,l=1}\delta_{kl} D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_k\right) D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_l\right)\\
&= \sum^8_{k,l,m,n=1}\delta_{kl}\left(D^{(8)}(A)\right)_{km}\left(D^{(8)}(A)\right)_{ln} F^{(\sigma\otimes\bar{\sigma})}_m F^{(\sigma\otimes\bar{\sigma})}_n\\
&= \sum^8_{k,l,m,n=1}\delta_{kl}\left(D^{(8)}(A)^\text{T}\right)_{mk}\left(D^{(8)}(A)^\text{T}\right)_{nl} F^{(\sigma\otimes\bar{\sigma})}_m F^{(\sigma\otimes\bar{\sigma})}_n\\
&= \sum^8_{k,l,m,n=1}\delta_{kl}\left(D^{(8)}(A^{-1})\right)_{mk}\left(D^{(8)}(A^{-1})\right)_{nl} F^{(\sigma\otimes\bar{\sigma})}_m F^{(\sigma\otimes\bar{\sigma})}_n\\
&= \sum^8_{m,n=1}\delta_{mn}F^{(\sigma\otimes\bar{\sigma})}_mF^{(\sigma\otimes\bar{\sigma})}_n\\
&= C^{(\sigma\otimes\bar{\sigma})}.
\end{align*}
This shows that $C^{(\sigma\otimes\bar{\sigma})}$ is a singlet in $\sigma\otimes\bar{\sigma}$. In \autoref{sec:GMO_formula}, we showed that there is exactly one singlet in $\sigma\otimes\bar{\sigma}$, namely the multiples of the identity. This means that $C^{(\sigma\otimes\bar{\sigma})} = c^{(\sigma\otimes\bar{\sigma})}\cdot\mathbb{1}$ for some $c^{(\sigma\otimes\bar{\sigma})}\in\mathbb{C}$. Furthermore, $C^{(\sigma\otimes\bar{\sigma})}$ is an Hermitian matrix, as it is the sum of squares of Hermitian matrices. This enforces $c^{(\sigma\otimes\bar{\sigma})}$ to be real. Moreover, $c^{(\sigma\otimes\bar{\sigma})}$ cannot be zero, since if it was zero, we would find:
\begin{gather*}
0 = v^\dagger C^{(\sigma\otimes\bar{\sigma})}(v) = \sum^8_{k=1} \left|F^{(\sigma\otimes\bar{\sigma})}_k(v)\right|^2\quad\forall v\in V,
\end{gather*}
where $V$ is the vector space $\sigma$ acts on. However, this equation can only be satisfied, if all $F^{(\sigma\otimes\bar{\sigma})}_k$ are zero. This contradicts the statement that the $F^{(\sigma\otimes\bar{\sigma})}_k$ form an octet. Therefore, we find $C^{(\sigma\otimes\bar{\sigma})} = c^{(\sigma\otimes\bar{\sigma})}\cdot\mathbb{1}$ for some $c^{(\sigma\otimes\bar{\sigma})}\in\mathbb{R}\backslash\{0\}$.\par
Let us now turn to the operators $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$:
\begin{gather*}
\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k\coloneqq \sum^{8}_{l,m=1}f_{klm}F^{(\sigma\otimes\bar{\sigma})}_lF^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k\in\{1,\ldots,8\}.
\end{gather*}
In \autoref{sec:GMO_formula}, we found that these operators are elements of the octet spanned by $F^{(\sigma\otimes\bar{\sigma})}_k$. We also claimed that these operators are linearly independent. We can convince ourselves that this is true by showing that the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ form an invariant subspace of the octet. For this, consider the transformation of the structure constants $f_{klm}$ under $A\in\text{SU}(3)$. With \autoref{eq:fklm}, we obtain:
\begin{align*}
f_{klm} &= -2i\text{Tr}\left(\left[F_k,F_l\right]F_m\right) = -2i\text{Tr}\left(\left[F_kA^\dagger A,F_lA^\dagger A\right]F_mA^\dagger A\right)\\
&= -2i\text{Tr}\left(\left[AF_kA^\dagger,AF_lA^\dagger\right]AF_mA^\dagger\right)\\
&= -2i\sum^8_{n,r,s=1}\left(D^{(8)}(A)\right)_{kn}\left(D^{(8)}(A)\right)_{lr}\left(D^{(8)}(A)\right)_{ms}\text{Tr}\left(\left[F_n,F_r\right]F_s\right)\\
&= \sum^8_{n,r,s=1}\left(D^{(8)}(A)\right)_{kn}\left(D^{(8)}(A)\right)_{lr}\left(D^{(8)}(A)\right)_{ms}f_{nrs}\quad\forall k,l,m\in\{1,\ldots,8\}.
\end{align*}
Using the fact that $D^{(8)}(A)$ is unitary and real, we can rewrite this as:
\begin{gather*}
\sum^8_{k=1}\left(D^{(8)}(A)\right)_{kn}f_{klm} = \sum^8_{r,s=1}\left(D^{(8)}(A)\right)_{lr}\left(D^{(8)}(A)\right)_{ms}f_{nrs}\quad\forall l,m,n\in\{1,\ldots,8\}.
\end{gather*}
Now consider the transformation of $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ under $A\in\text{SU}(3)$:
\begin{align*}
D^{(\sigma\otimes\bar{\sigma})}(A)\left(\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k\right) &= \sum^{8}_{l,m=1}f_{klm}D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_l\right) D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_m\right)\\
&= \sum^{8}_{l,m,n,r=1}f_{klm}\left(D^{(8)}(A)\right)_{ln}\left(D^{(8)}(A)\right)_{mr}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_r\\
&= \sum^{8}_{l,m,n,r=1}\left(D^{(8)}(A^{-1})\right)_{nl}\left(D^{(8)}(A^{-1})\right)_{rm}f_{klm}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_r\\
&= \sum^8_{n,r,s=1}\left(D^{(8)}(A^{-1})\right)_{sk}f_{snr}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_r\\
&= \sum^8_{s=1}\left(D^{(8)}(A)\right)_{ks}\tilde{F}^{(\sigma\otimes\bar{\sigma})}_s.
\end{align*}
As the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ are only mapped into each other under $A\in\text{SU}(3)$, they span an invariant subspace of the octet spanned by $F_k$. However, the octet is a multiplet, thus, the space spanned by $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ is either $\{0\}$ or the entire octet. The invariant space cannot be $\{0\}$, as if it was $\{0\}$, all $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ would have to be zero. Nevertheless, we find for $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_1$ with the help of the structure constants $f_{klm}$, which can be found, for instance, in \cite{Lichtenberg}:
\begin{gather*}
\tilde{F}^{(\sigma\otimes\bar{\sigma})}_1 = \frac{3i}{2}F^{(\sigma\otimes\bar{\sigma})}_1.
\end{gather*}
So if the span of the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ was $\{0\}$, $F^{(\sigma\otimes\bar{\sigma})}_1$ would be zero which is a contradiction, as the matrices $F^{(\sigma\otimes\bar{\sigma})}_k$ provide a basis for an octet in $\sigma\otimes\bar{\sigma}$. This implies that the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$ span the entire octet spanned by $F^{(\sigma\otimes\bar{\sigma})}_k$. Since there are 8 matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$, they have to be linearly independent concluding the proof.\par
In \autoref{sec:GMO_formula}, we aimed to compute the commutator of $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_l$. In order to do this, we introduced the following formula for $f_{klm}$ and $d_{klm}$:
\begin{gather*}
\sum^8_{m=1}f_{klm}d_{nrm} = -\sum^8_{m=1}\left(f_{nlm}d_{krm} +f_{rlm}d_{nkm}\right)\quad\forall k,l,n,r\in\{1,\ldots,8\}.
\end{gather*}
However, we skipped the derivation of this formula. Thus, we now want to proof it. The proof is actually just a straightforward calculation using \autoref{eq:com_F}, \autoref{eq:dklm}, and the cyclicity of the trace. Consider the following expression:
\begin{align*}
\frac{i}{2}\sum^8_{m=1}f_{klm}d_{nrm} &= i\sum^8_{m=1}f_{klm}\text{Tr}\left(\{F_n,F_r\}F_m\right) = \text{Tr}\left(\{F_n,F_r\}\sum^8_{m=1}if_{klm}F_m\right)\\
&= \text{Tr}\left(\{F_n,F_r\}\left[F_k,F_l\right]\right)\\
&= \text{Tr}\left(F_nF_rF_kF_l + F_rF_nF_kF_l - F_nF_rF_lF_k - F_rF_nF_lF_k\right)\\
&= \text{Tr}\left(F_rF_kF_lF_n + F_nF_kF_lF_r - F_kF_nF_rF_l - F_kF_rF_nF_l\right)\\
&= \text{Tr}\left(F_rF_kF_lF_n + F_nF_kF_lF_r - F_kF_nF_rF_l - F_kF_rF_nF_l\right.\\
&\ \ \ \ \ \,\left.+ F_kF_rF_lF_n - F_nF_kF_rF_l + F_kF_nF_lF_r - F_rF_kF_nF_l\right)\\
&= \text{Tr}\left(F_kF_rF_lF_n + F_rF_kF_lF_n + F_nF_kF_lF_r + F_kF_nF_lF_r\right.\\
&\ \ \ \ \ \,\left. - F_kF_rF_nF_l - F_rF_kF_nF_l - F_nF_kF_rF_l - F_kF_nF_rF_l\right)\\
&= \text{Tr}\left(\{F_k,F_r\}[F_l,F_n] + \{F_n,F_k\}[F_l,F_r]\right)\\
&= -\text{Tr}\left(\{F_k,F_r\}[F_n,F_l] + \{F_n,F_k\}[F_r,F_l]\right)\\
&= -i\sum^8_{m=1}\left(f_{nlm}\text{Tr}\left(\{F_k,F_r\}F_m\right) + f_{rlm}\text{Tr}\left(\{F_n,F_k\}F_m\right)\right)\\
&= \frac{-i}{2}\sum^8_{m=1}\left(f_{nlm}d_{krm} +f_{rlm}d_{nkm}\right)\quad\forall k,l,n,r\in\{1,\ldots,8\}.
\end{align*}
We also postponed the computation of the commutator of $F^{(\sigma\otimes\bar{\sigma})}_k$ and $D^{(\sigma\otimes\bar{\sigma})}_l$ itself to the appendix. For this, apply \autoref{eq:com_F_sigma} and the previous formula:
\begin{align*}
\left[F^{(\sigma\otimes\bar{\sigma})}_k,D^{(\sigma\otimes\bar{\sigma})}_l\right] &= \frac{2}{3}\sum^8_{m,n=1}d_{lmn}\left[F^{(\sigma\otimes\bar{\sigma})}_k,F^{(\sigma\otimes\bar{\sigma})}_mF^{(\sigma\otimes\bar{\sigma})}_n\right]\\
&= \frac{2}{3}\sum^8_{m,n=1}d_{lmn}\left(\left[F^{(\sigma\otimes\bar{\sigma})}_k,F^{(\sigma\otimes\bar{\sigma})}_m\right]F^{(\sigma\otimes\bar{\sigma})}_n + F^{(\sigma\otimes\bar{\sigma})}_m\left[F^{(\sigma\otimes\bar{\sigma})}_k,F^{(\sigma\otimes\bar{\sigma})}_n\right]\right)\\
&= \frac{2}{3}\sum^8_{m,n,r=1}d_{lmn}\left(if_{kmr}F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n + if_{knr}F^{(\sigma\otimes\bar{\sigma})}_mF^{(\sigma\otimes\bar{\sigma})}_r\right)\\
&= \frac{2i}{3}\sum^8_{m,n,r=1}\left(d_{lmn}f_{kmr}F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n + d_{lrm}f_{kmn}F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n\right)\\
&= \frac{2i}{3}\sum^8_{m,n,r=1}\left(d_{lmn}f_{kmr} + d_{lrm}f_{kmn}\right)F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n\\
&= \frac{2i}{3}\sum^8_{m,n,r=1}\left(f_{rkm}d_{lnm} + f_{nkm}d_{rlm}\right)F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n\\
&= -\frac{2i}{3}\sum^8_{m,n,r=1}f_{lkm}d_{rnm}F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n\\
&= \sum^8_{m=1}if_{klm}\left(\frac{2}{3}\sum^8_{n,r=1} d_{mrn}F^{(\sigma\otimes\bar{\sigma})}_rF^{(\sigma\otimes\bar{\sigma})}_n\right)\\
&= \sum^8_{m=1}if_{klm}D^{(\sigma\otimes\bar{\sigma})}_m\quad\forall k,l\in\{1,\ldots,8\}
\end{align*}
Lastly, we want to calculate the transformation of $D^{(\sigma\otimes\bar{\sigma})}_k$ under $A\in\text{SU}(3)$. This computation can be performed analogously to the calculation for the transformation behavior of the matrices $\tilde{F}^{(\sigma\otimes\bar{\sigma})}_k$. First, consider the transformation of $d_{klm}$ under $A\in\text{SU}(3)$. With \autoref{eq:dklm}, we obtain:
\begin{align*}
d_{klm} &= 2\text{Tr}\left(\left\{F_k,F_l\right\}F_m\right) = 2\text{Tr}\left(\left\{F_kA^\dagger A,F_lA^\dagger A\right\}F_mA^\dagger A\right)\\
&= 2\text{Tr}\left(\left\{AF_kA^\dagger,AF_lA^\dagger\right\}AF_mA^\dagger\right)\\
&= \sum^8_{n,r,s=1}\left(D^{(8)}(A)\right)_{kn}\left(D^{(8)}(A)\right)_{lr}\left(D^{(8)}(A)\right)_{ms}2\text{Tr}\left(\left\{F_n,F_r\right\}F_s\right)\\
&= \sum^8_{n,r,s=1}\left(D^{(8)}(A)\right)_{kn}\left(D^{(8)}(A)\right)_{lr}\left(D^{(8)}(A)\right)_{ms}d_{nrs}\quad\forall k,l,m\in\{1,\ldots,8\}.
\end{align*}
Using the fact that $D^{(8)}(A)$ is unitary and real, we can rewrite this as:
\begin{gather*}
\sum^8_{k=1}\left(D^{(8)}(A)\right)_{kn}d_{klm} = \sum^8_{r,s=1}\left(D^{(8)}(A)\right)_{lr}\left(D^{(8)}(A)\right)_{ms}d_{nrs}\quad\forall l,m,n\in\{1,\ldots,8\}.
\end{gather*}
Now consider the transformation of $D^{(\sigma\otimes\bar{\sigma})}_k$ under $A\in\text{SU}(3)$:
\begin{align*}
D^{(\sigma\otimes\bar{\sigma})}(A)\left(D^{(\sigma\otimes\bar{\sigma})}_k\right) &= \frac{2}{3}\sum^{8}_{l,m=1}d_{klm}D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_l\right) D^{(\sigma\otimes\bar{\sigma})}(A)\left(F^{(\sigma\otimes\bar{\sigma})}_m\right)\\
&= \frac{2}{3}\sum^{8}_{l,m,n,r=1}d_{klm}\left(D^{(8)}(A)\right)_{ln}\left(D^{(8)}(A)\right)_{mr}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_r\\
&= \frac{2}{3}\sum^{8}_{l,m,n,r=1}\left(D^{(8)}(A^{-1})\right)_{nl}\left(D^{(8)}(A^{-1})\right)_{rm}d_{klm}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_r\\
&= \frac{2}{3}\sum^8_{n,r,s=1}\left(D^{(8)}(A^{-1})\right)_{sk}d_{snr}F^{(\sigma\otimes\bar{\sigma})}_nF^{(\sigma\otimes\bar{\sigma})}_r\\
&= \sum^8_{s=1}\left(D^{(8)}(A)\right)_{ks}D^{(\sigma\otimes\bar{\sigma})}_s.
\end{align*}
\end{appendix}
\newpage
\pagenumbering{Roman}
\input{ms.bbl}
\newpage
\chapter*{Acknowledgments}
\addcontentsline{toc}{chapter}{Acknowledgments}
\markboth{}{Acknowledgments}
I thank Professor Hubert Spiesberger and Professor Jens Erler for supervision of my master's thesis, insightful conversations, and helpful comments. Furthermore, I would like to thank the members of THEP and, in particular, the members of the ``AG Spiesberger'' for their kind welcome, Professor a. D. J\"urgen K\"orner for his helpful literature recommendations, Christian P. M. Schneider for (often) enlightening conversations and proofreading my thesis, my former roommates Florian Stuhlmann and Alexander Segner for proofreading and technical support, and my family and friends for moral support.
\end{document}
| {
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{"url":"https:\/\/unitedwardrobe.com\/nl\/crop-tops-evenodd","text":"# Crop Tops Even&Odd\n\n## Andere gebruikers zochten ook naar: Longsleeves Even&Odd, Enkellaarzen Even&Odd, (Half) hoge laarzen Even&Odd\n\n45 resultatenSorteer op:\nS\u20ac\u00a08,99\nS\u20ac\u00a05\nS\u20ac\u00a07\nL\u20ac\u00a04\nM\u20ac\u00a04,50\nS\u20ac\u00a07,50\nS\u20ac\u00a05,50\nS\u20ac\u00a05,95\nM\u20ac\u00a03,15\nXS\u20ac\u00a02,25\nM\u20ac\u00a06\nS\u20ac\u00a04,20\nL\u20ac\u00a08,10\nS\u20ac\u00a06\nXS\u20ac\u00a05,99\nM\u20ac\u00a04,05\nM\u20ac\u00a08,50\n12\n\n### Gerelateerde zoekopdrachten\n\n(47)(66)(19)(10)(58)(32)(10)(22)(39)(113)(15)(34)(83)(84)(21)(46)(11)(27)(14)(34)(98)(8)(7)(9)(9)(5)(6)","date":"2020-07-10 04:27:14","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9924480319023132, \"perplexity\": 8799.614151447198}, \"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\/1593655902496.52\/warc\/CC-MAIN-20200710015901-20200710045901-00247.warc.gz\"}"} | null | null |
{"url":"https:\/\/math.stackexchange.com\/questions\/2879533\/how-do-you-notate-the-circumstances-of-functions-relevant-to-where-they-are-inve","text":"# How do you notate the circumstances of functions relevant to where they are invertible and defined?\n\nI want to state a particular claim such as\n\n\"functions in the form $f(x)=ag(x)^2-b$ have a solution in the form of $x=g^{-1}( \\pm \\frac{ \\sqrt{f(x)+b}}{ \\sqrt{a}})$\n\nPeople know that not literally all functions are always defined and invertible over all real numbers to satisfy that solution, but it should be beyond exceptionally obvious that this claim would never be referring to a function in a circumstance where it fails to be true.\n\nHow do I notate that the statement is always true for anywhere a function is defined and that it is only using a single branch in the event that such a function would otherwise fail to be invertible? I'm not even talking about any analytic continuation or calculus so the functions $f(x)$ and $g(x)$ wouldn't even need to be continuous in this circumstance.\n\n\u2022 Wow I didn't think this would be that difficult to answer, I thought it was pretty standard mathematical notation, guess I was wrong. \u2013\u00a0user561159 Aug 11 '18 at 18:58\n\u2022 I think the difficulty is in stating precisely what one means by \"...circumstances where it fails to be true.\" If you can write out a full paragraph or so stating in unequivocal ways a test clarifying when it is true and when it is not, then perhaps we can help. Otherwise, your request is simply ill-defined and cannot have an answer. \u2013\u00a0David G. Stork Aug 11 '18 at 19:18\n\u2022 It can very easily have an answer, it's already stated, the only thing left is how it's notated. It seems like you're intentionally being hostile for no reason, so I'll reiterate the facts: At the very least, the statement is true for functions over a domain such that they are defined for all values and invertible. What I have not seen conventionally notated in particular are the branches. The square root function has multiple branches, but despite that, the statement is still true for either one branch or the other at a time, just not both at once since such a map is no longer of a function. \u2013\u00a0user561159 Aug 11 '18 at 19:44\n\u2022 I don\u2019t see this as a matter of notation but of clarity and precision of language. \u2013\u00a0Lubin Aug 11 '18 at 19:54\n\u2022 But that's the problem, the question asking how to notate it precisely because there is very specific terminology that matches it. I haven't found myself and I don't specialize in functional analysis and I would have expected the math community here to be competent enough to answer what should be a trivial question, but it seems I was mistaken. \u2013\u00a0user561159 Aug 11 '18 at 20:53\n\nIf $f$ is a function of the form $f(x)=ag(x)^2+b$, then we can solve for $x$ in terms of $y=f(x)$ by taking any $x$ such that $g(x)=\\pm \\frac{ \\sqrt{y+b}}{ \\sqrt{a}}$.\nThis avoids the problematic use of $g^{-1}$ when you are not actually making any claim about the invertibility of $g$. It also makes it a lot clearer what you mean by \"solving a function\" (though that may be clear from context in what you are writing).","date":"2020-04-06 16:13:37","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\": 0.7759149670600891, \"perplexity\": 234.83911152742286}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371637684.76\/warc\/CC-MAIN-20200406133533-20200406164033-00493.warc.gz\"}"} | null | null |
Il parco nazionale della Foresta Nera (in tedesco: Nationalpark Schwarzwald) è un parco nazionale situato nel Baden-Württemberg, in Germania.
Altri progetti
Collegamenti esterni
Foresta Nera | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 742 |
Os resultados masculinos da Final da Copa do Mundo de Ginástica Artística de 2008, contaram com as provas individuais por aparelhos, como desde a edição de 1998.
Resultados
Solo
Barra fixa
|}
Barras paralelas
Cavalo com alças
|}
Argolas
Salto sobre a mesa
|}
Ver também
Seleções de ginástica artística
Biografias dos ginastas
Referências
Ligações externas
Final da Copa do Mundo de Ginástica Artística de 2008 | {
"redpajama_set_name": "RedPajamaWikipedia"
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You are here: Home / Fibromyalgia / Fibromyalgia Treatments / Is Your Doctor Over Medicating You?
There are so many medications used to treat the symptoms of illnesses such as CFIDS and FM. Of course, they mask the symptoms so that we can function better, but are they really doing us more harm than good? I believe our conditions will continue to spiral downwards due to a lot of these medications, yet we deserve to have the right to live a life free of pain and sickness. It is just sad to think of the cost we are paying to feel better for a little while and to have some relief from pain and suffering.
Another Catch 22 for most of us with CFIDS and FM is that our bodies are over sensitive to meds, so we have to be extra careful.
Nonsteroidal Anti-Inflammatory Drugs – NSAID – Used to relieve pain in some patients, and some can be purchased as over-the-counter meds: Naproxen (Aleve), Ibuprofen (Advil, Motrin, Nuprin), Celebrex, & Vioxx.
Side effects that can occur from using these drugs can be kidney damage, gastrointestional bleeding, abdominal pain, nausea and vomiting, but generally they are safe to use if you follow the usage directions.
I know for the pain that I have, the over-the-counter meds listed above did nothing for me.
Low-Dose Tricyclic Antidepressants – TCAs can be prescribed for CFIDS/FM patients to improve sleep and to relieve mild pain.
Some examples are amitriptyline (Elavil), imipramine (Tofranil), trazodone (Desyrel).
Dosages used to treat CFIDS/FM patients are usually a lot lower than prescriptions prescribed to treat depression.
Some side effects may include dry mouth, drowsiness, weight gain, elevated heart rate.
Other Antidepressants – SSRIs such as Prozac, Zoloft, Paxil, Wellbutrin, and Effexor can be effective in correcting the different brain-chemical imbalances that are a problem for many CFIDS patients.
Anxiolytic Agents – These are used to treat symptoms of anxiety in CFIDS/FM patients. Some examples include Xanax, Ativan, and Klonopin.
Use of these drugs, though, may be habit forming. Some reactions to these meds may include sedation & amnesia.
Before taking any meds, please have your doctor review with you the pros and cons. I myself have reduced my prescription medication intake to just a sleeping pill and I still have to take pain pills, and one med for my bladder (Elmiron). Do some research on more natural alternatives of meds that can relieve the same symptoms as some of the prescription meds. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,447 |
{"url":"https:\/\/deepai.org\/publication\/hybrid-adaptive-and-positivity-preserving-numerical-methods-for-the-cox-ingersoll-ross-model","text":"# Hybrid, adaptive, and positivity preserving numerical methods for the Cox-Ingersoll-Ross model\n\nWe introduce an adaptive Euler method for the approximate solution of the Cox-Ingersoll-Ross short rate model. An explicit discretisation is applied over an adaptive mesh to the stochastic differential equation (SDE) governing the square root of the solution, relying upon a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller's condition, we prove that such a scheme is strongly convergent, of order at least 1\/2. Under Feller's condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step schemes extant in the literature, both implicit and explicit, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive over the entire domain of Feller's condition.\n\n\u2022 3 publications\n\u2022 4 publications\n\u2022 1 publication\n08\/31\/2019\n\n### Strong convergence of an adaptive time-stepping Milstein method for SDEs with one-sided Lipschitz drift\n\nWe introduce explicit adaptive Milstein methods for stochastic different...\n12\/31\/2021\n\n### A Strongly Monotonic Polygonal Euler Scheme\n\nRate of convergence results are presented for a new class of explicit Eu...\n05\/20\/2021\n\n### A flexible split-step scheme for MV-SDEs\n\nWe present an implicit Split-Step explicit Euler type Method (dubbed SSM...\n09\/08\/2019\n\n### Semi-explicit discretization schemes for weakly-coupled elliptic-parabolic problems\n\nWe prove first-order convergence of the semi-explicit Euler scheme combi...\n09\/22\/2019\n\n### Contractivity of Runge-Kutta methods for convex gradient systems\n\nWe consider the application of Runge-Kutta (RK) methods to gradient syst...\n12\/25\/2019\n\n### Enforcing strong stability of explicit Runge\u2013Kutta methods with superviscosity\n\nA time discretization method is called strongly stable, if the norm of i...\n05\/08\/2019\n\n### Modern theory of hydraulic fracture modeling with using explicit and implicit schemes\n\nThe paper presents novel results, obtained on the basis of the modified ...\n\n## 1. Introduction\n\nThe Cox-Ingersoll-Ross (CIR) process is a short rate model used in the pricing of interest rate derivatives and is given by the following It\u00f4-type stochastic differential equation (SDE),\n\n (1) dX(t)=\u03ba(\u03bb\u2212X(t))dt+\u03c3\u221aX(t)dW(t),\u00a0t\u2208[0,T];\u00a0X(0)=X0>0,\n\nwhere is a Wiener Process and and are positive parameters, and for some fixed . Solutions of (1) are almost surely (a.s.) non-negative: in general they can achieve a value of zero but will be reflected back into the positive half of the real line immediately. Moreover, if , referred to as the Feller condition, solutions will be a.s. positive. No closed form solution of (1) is available, though it is known that has, conditional upon for\n\n, a non-central chi-square distribution with\n\nand Var.\n\nFor Monte Carlo estimation, exact sampling from the known conditional distribution of\n\nis possible but computationally inefficient and potentially restrictive if the innovating Wiener process of (1) is correlated with that of another process: see [1, 5, 9, 19]. Consequently a substantial literature has developed on the efficient numerical approximation of solutions of (1); we now highlight the parts which are relevant to our analysis.\n\nAn approach that seeks to directly discretise (1) using some variant of the explicit Euler-Maruyama method leads to schemes of the form\n\n (2) Vn+1=g0(Vn+\u0394t(\u03ba(\u03bb\u2212g1(Vn))+\u03c3\u221ag2(Vn))\u0394Wn\n\nfor given functions , and . These functions are selected to ensure that the diffusion coefficient remains real-valued (so that (2) is well defined), and in some cases to preserve the positivity of solutions. This approach seeks to accommodate the non-Lipschitz (square-root) diffusion, which facilitates overshoot when the solutions are close to zero, but it introduces additional bias to the approximation. A survey of choices common in practice may be found in [19], and we summarise them in Table 1 using the convention .\n\nAn alternative approach is to transform (1) before discretisation to make the diffusion coefficient globally Lipschitz. For example, applying the Lamperti transform yields an auxiliary SDE in with a state independent and therefore globally Lipschitz diffusion, but a drift coefficient that is unbounded when solutions are in a neighbourhood of zero. This approach is effective: a fully implicit Euler discretisation over a uniform mesh that preserves positivity of solutions was proposed in [1]\n\nand shown to have uniformly bounded moments. A continuous time extension interpolating linearly between mesh points was shown to have strong\n\norder of convergence (up to a factor of ) in [8] when , and a continuous-time variant based on the same implicit discretisation was shown to have strong convergence of order one when in [2].\n\nIn this article we will show that a strongly convergent numerical scheme can be constructed by an application of the Lamperti transform to (1) followed by an explicit Euler-Maruyama discretisation over a procedurally generated adaptive mesh. The purpose of the adaptivity is to manage the nonlinear drift response of the discrete tranformed system, a framework for which was introduced in [13] for SDEs with one-sided Lipschitz drift and globally Lipschitz diffusion and extended to allow for monotone coefficients and a Milstein-type discretisation in [14, 15] respectively. This framework imposes maximum and minimum timesteps and in a fixed ratio and requires the use of a backstop numerical method in the event that the timestepping strategy attempts to select a stepsize below . As in [15], we will use here path-bounded strategies, this time designed to increase the density of timesteps when solutions approach zero, and we additionally require the backstop method to retake a step when the adaptive strategy overshoots zero. This latter is carried out without discarding samples from the Brownian path (preserving the trajectory), and without bridging (preserving efficiency).\n\nWe prove, under parameter constraints that imply the Feller condition (specifically, ), that the order of strong convergence in is at least . This parameter constraint is technical, and ensures sufficiently many finite conditional inverse moments of solutions of (1) (as described by [4]). We separately prove that, under exactly the Feller condition, the probability of invoking the backstop method to avoid negative values can be made arbitrarily small by choosing sufficiently small, and provide a practical method for doing so given a user defined tolerance level. The proof relies upon a finite partitioning of the sample space of trajectories induced by and\n\n, which allows us to handle the randomness of the number of timesteps via the Law of Total Probability.\n\nNumerically we compare the convergence and efficiency of our hybrid adaptive method with a semi-implicit adaptive variant, the fixed-step explicit method due to [10], and the transformed implicit fixed-step method proposed and analysed in [1, 8, 2], examining the parameter dependence of the numerical order of convergence in each case. The numerical convergence rates of adaptive methods are seen to outperform those of fixed-step methods over the entire domain where Feller\u2019s condition holds.\n\nThe structure of the article is as follows. In Section 2 we give the form of the SDE governing the Lamperti transform of (1), specify the constraints placed upon the parameters for the main strong convergence theorem, and examine the availability of conditional moment and inverse moment bounds under these constraints. In Section 3 we set up the framework for our random mesh, characterise the class of path-bounded timestepping strategies and define our adaptive numerical method. In Section 4 we present the two main theorems on strong convergence and positivity, providing illustrative examples in the latter case. Finally in Section 5 we numerically compare convergence and efficiency of several commonly used methods.\n\n## 2. Mathematical Preliminaries\n\nThroughout this article we let be the natural filtration of . By using It\u00f4\u2019s Formula and applying the transformation we get,\n\nBy then setting,\n\n \u03b1=4\u03ba\u03bb\u2212\u03c328,\u00a0\u03b2=\u22124\u03ba8,\u00a0\u03b3=\u03c32,\n\nwe can write,\n\n (3) dY(t)=(\u03b1Y(t)+\u03b2Y(t))dt+\u03b3dWt,\u00a0t\u2208[0,T];Y(0)=\u221aX0\u2208R+,\n\nwhere is not globally Lipschitz continuous, but when it satisfies a one-sided Lipschitz condition with constant :\n\n [f(x)\u2212f(y)](x\u2212y)\u2264\u03b2(x\u2212y)2,\u00a0for all\u00a0x,y\u2208R+,\n\nwhich can be seen by noting that\n\n f(x)\u2212f(y)=(x\u2212y)[\u03b2\u2212\u03b1xy].\n\nMeanwhile the diffusion coefficient is constant and is therefore globally Lipschitz continuous. The SDE (3) has integral form\n\n (4) Y(t)=Y(0)+\u222bt0(\u03b1Y(s)+\u03b2Y(s))ds+\u222bt0\u03b3dWt,t\u22650.\n\nIn order to ensure the a.s. positivity of solutions of (3) and the boundedness of certain inverse moments of solutions of (3), we will also need the following assumption:\n\n###### Assumption 1.\n\nSuppose that\n\n (5) \u03ba\u03bb>2\u03c32.\n\nEq. (5) implies the Feller Condition (; see, for example [20]), which ensures that solutions of (1), and therefore (3), remain positive with probability one:\n\n P[Y(t)>0,\u00a0t\u22650]=1.\n\nAssumption 1 provides inverse moment bounds as follows:\n\n###### Lemma 2.\n\nLet be a solution of (3), where Assumption 1 holds, and let . For any , and for , there exists such that\n\n (6) E[1Y(s)p\u2223\u2223\u2223Ft]\u2264C(p,T)Y(t)p,a.s.\n###### Proof.\n\nLet be a solution of (1) where Assumption 1 holds. By Lemma A.1 in Bossy & Diop\u00a0[4],\n\n (7) E[1X(t)]\u2264e\u03batX0% andE[1X(t)p]\u2264C(p,T)Xp0,\n\nfor some and any such that . Assumption 1 ensures that , and since , (6) follows by Lemma A.1 in [4] as it applies to conditional expectations, the former requiring an additional application of Jensen\u2019s inequality to the first inequality in (7). \u220e\n\nWe also need the following bounds on positive moments of solutions of (3), which apply under Feller\u2019s condition and in particular under Assumption 1.\n\n###### Lemma 3.\n\nLet be a solution of (3), where Assumption 1 holds, and let . For any and any , there exist constants , such that\n\n (8) E[supu\u2208[0,T]Y(u)p\u2223\u2223\u2223Ft]\u2264M1,p(1+Y(t)p),a.s.,\n\nand\n\n (9) E[supu\u2208[0,T]Y(u)p]\u2264M2,p.\n###### Proof.\n\nThe proof of (8) is an application of [4, Lemma 2.1] to conditional expectations requiring an invocation of Jensen\u2019s inequality when . Eq. (9) is provided by [8, Lemma 3.2]. \u220e\n\nFinally, we will make frequent use of the following elementary inequalities: for and and ,\n\n (10) \u221a|a1+a2| \u2264 \u221a|a1|+\u221a|a2|; (11) |a1a2| \u2264 12(a21+a22); (12) (a1+\u2026+an)p \u2264 np(|a1|p+\u22ef+|an|p).\n\n## 3. An Adaptive Numerical Method\n\n[13] provided a framework within which to construct timestepping strategies for an adaptive explicit Euler-Maruyama numerical scheme applied to nonlinear It\u00f4-type SDEs of the form\n\n (13) dX(t)=f(X(t))dt+g(X(t))dB(t),t\u2208[0,T],\n\nover a random mesh on the interval given by,\n\n (14) Yn+1=Yn+hn+1f(Yn)+g(Yn)(W(tn+1)\u2212W(tn)),\n\nwhere is a sequence of random timesteps and with . The random time step is determined by .\n\nFor completeness, we present here the essential elements of that framework, before discussing the dynamical considerations specific to the transformed CIR model (3) corresponding to (13) with\n\n (15) f(y)=\u03b1y+\u03b2y,g(y)=\u03b3,\n\nwhich lead to our proposed timestepping strategy.\n\n### 3.1. Framework for a random mesh\n\n###### Definition 4 ([18]).\n\nSuppose that each member of the sequence is an -stopping time: i.e. , where is the natural filtration of . We may then define a discrete time filtration by\n\n Ftn={A\u2208F:A\u2229{tn\u2264t}\u2208Ft},n\u2208N.\n###### Assumption 5.\n\nEach is -measurable, and is a random integer such that,\n\n N=max{n\u2208N:tn\u22121\n\nand the length of maximum and minimum stepsizes satisfy , and\n\n (16) hmin\u2264hn+1\u2264hmax\u22641.\n###### Remark 6.\n\nThe lower bound ensures that a simulation over the interval can be completed in a finite number of timesteps, and the upper bound\n\nprevents stepsizes from becoming too large. The latter is used as a convergence parameter in our examination of the strong convergence of the adaptive method. The random variable\n\ncannot take values outside the finite set , where and .\n\nis a Wiener increment over a random interval the length of which depends on , through which it depends on . Therefore is not independent of\n\n; indeed it is not necessarily normally distributed. Since\n\nis a bounded -stopping time and -measurable, then is -conditionally normally distributed, by Doob\u2019s optional sampling theorem (see for example\u00a0[22]), and for all there exists such that\n\n E[|W(tn+1)\u2212W(tn)||Ftn] = 0,a.s.; E[|W(tn+1)\u2212W(tn)|2|Ftn] = hn+1,a.s.; (17) E[\u2223\u2223\u2223\u222bstndW(r)\u2223\u2223\u2223p\u2223\u2223\u2223Ftn] = \u03c5p|s\u2212tn|p2,a.s.\n\nTo ensure strong convergence, our strategy will be to reduce the size of each timestep if discretised solutions attempt to enter a neighbourhood of zero. If we wish to control the likelihood of invoking the backstop to avoid negative values, we will also reduce the timestep when solutions grow large.\n\n###### Definition 7 (A path-bounded time-stepping strategy).\n\nLet be a solution of (14). We say that is a path-bounded time-stepping strategy for (14) if the conditions of Assumption 5 are satisfied and there exist real non-negative constants (where may be infinite if ) such that whenever ,\n\n (18) Q\u2264|Yn|\n\nWe now give two examples of path-bounded strategies that are valid for (14), the first with infinite (which is sufficient to ensure strong convergence), and the second with finite (which is useful if we also wish to minimise the use of the backstop to ensure positivity).\n\n###### Lemma 8.\n\nLet be a solution of (14), and let be a time-stepping strategy that satisfies Assumption 5. If satisfies, for some ,\n\n (19) hn+1:=hmax\u00d7min{1,|Yn|r},n\u2208N,\n\nor\n\n (20) hn+1=hmax\u00d7min(|Yn|r,|Yn|\u2212r),\n\nthen it is path-bounding for (14).\n\n###### Proof.\n\nSuppose that (19) holds. When ,\n\n hn+1\u2264hmax|Yn|r\u21d41|Yn|r\u2264hmaxhn+1\u2264hmaxhmin=\u03c1,n\u2208N,\n\nand when it is obvious that , so we also have . Hence, when using the strategy defined by (19),\n\n |Yn|\u22651\u03c11\/r,1|Yn|\u2264\u03c11\/r,n\u2208N,\n\nso that (18) holds with and .\n\nWe can similarly show that (20) is path-bounding for (14), with and . \u220e\n\nNote that for the strategies defined by (19) and (20), solutions of (14) cannot enter the neigbourhood , and therefore terms of the sequence are uniformly bounded from above. This has the effect of controlling inverse moments of the solutions of (14). Moreover for (19), when (15) holds,\n\n f(Y2n)=\u03b1|Yn|2+\u03b2|Yn|2\u2264\u03b1\u03c11\/r+\u03b2|Yn|2,\n\nand therefore that strategy is admissible in the sense of [13, Definition 2.2] with and . Similarly, (20) is admissible with and .\n\n### 3.3. The adaptive numerical method with backstop\n\nWe consider an adaptive scheme based upon the following explicit Euler-Maruyama discretisation of (3) over a random mesh given by,\n\n (21) Yn+1=Yn+hn+1(\u03b1Yn+\u03b2Yn)+\u03b3\u0394Wn+1.\n\nwhere the timestep sequence is constructed according to (19). For , the continuous version is given by\n\n (22) \u02dcY(s)=Yn+\u222bstn(\u03b1Yn+\u03b2Yn)dr+\u03b3\u222bstndW(r),\n\nso that for each .\n\nWe combine this scheme with a positivity-preserving backstop scheme that is to be applied if the timestepping strategy attempts to select a timestep below (in which case we choose ) or if the current selected timestep and subsequently observed Brownian increment would result in the approximation becoming negative:\n\n###### Definition 9.\n\nDefine the map such that\n\n \u03b8(y,z,h):=y+h(\u03b1y+\u03b2y)+\u03b3z,\n\nso that, if is defined by (21), then\n\n Yn+1=\u03b8(Yn,\u0394Wn+1,hn+1),\u00a0n\u2208N.\n\nThen we define the continuous form of an adaptive explicit Euler scheme to be the sequence of functions obeying\n\n (23) \u00afY(s)=\u03b8(\u00afYn,\u222bstndW(r),\u222bstndr)I{hmin0}+\u03c6(\u00afYn,\u222bstndW(r),\u222bstndr)I{hn+1=hmin}\u2229{Yn+1>0}+\u03c6(\u00afYn,\u222bstndW(r),\u222bstndr)I{hmin\n\nfor , , where , satisfies the conditions of Assumption 5, and the map satisfies\n\n (24) E[\u2223\u2223\u03c6(\u00afYn,\u0394Wn+1,hmin)\u2212Y(tn+hmin)\u2223\u2223p\u2223\u2223Ftn]\u2212\u2223\u2223\u00afYn\u2212Y(tn)\u2223\u2223p\u2264C1\u222bstnE[|\u00afY(r)\u2212Y(r)|p|Ftn]dr+C2hp\/4+1min,\u00a0n\u2208N,\u00a0a.s.,\n\nfor some non-negative constants and , independent of N, and\n\n (25) \u03c6(y,\u0394Wn+1,h)>0a.s.,y\u2208R+,h\u2208[hmin,hmax].\n###### Remark 10.\n\nSince the events and are -measurable but not -measurable, if a negative value of is observed following a step of length we must retake the step using the backstop method, which will ensure positivity over that step by (25). This introduces an element of backtracking into the algorithm, but as long as the originally computed stepsize and Brownian increment are retained we can stay on the same trajectory while avoiding the use of a Brownian bridge. Theorem 15 in Section 4.2, illustrated by Example 16, demonstrates that it is always possible to choose to ensure that this particular use of the backstop can be avoided with probability , for arbitrarily small , on each trajectory.\n\n## 4. Main Results\n\nIn this section, we first demonstrate strong convergence of solutions of (23) to those of (3) under Assumption 1 and a path-bounded timestepping strategy. Second, we investigate the likelihood that the adaptive part of the method generates a negative value (triggering the use of the backstop to ensure positivity) and show how may be chosen to control the probability of this occurring.\n\n### 4.1. Strong convergence of the adaptive method with path-bounded timestepping strategy\n\n###### Lemma 11.\n\nLet be a solution of (3) and let be a random mesh such that each is an -stopping time. Fix and suppose that , where . Then, for any , we have\n\n E[|Y(s)\u2212Y(tn)|p\u2223\u2223Ftn]\u22642p\u03b3p\u03c5p|s\u2212tn|p\/2+\u00afLn,p|s\u2212tn|p,\u00a0a.s.,\n\nwhere\n\n \u00afLn,p:=22p(\u03b1pC(p,T)Y(tn)p+|\u03b2|pM1,p(1+Y(tn)p))\n\nis an -measurable random variable with finite expectation, and , are the constants defined by (6) and (8) in the statements of Lemmas 2 and 3 respectively .\n\n###### Proof.\n\nSolutions of (3) satisfy the integral equation\n\n Y(s)=Y(tn)+\u222bstn(\u03b1Y(u)+\u03b2Y(u))du+\u222bstn\u03b3dW(u),tn\u2264s\u2264T,\n\nand therefore\n\n Y(s)\u2212Y(tn)=\u222bstn(\u03b1Y(u)+\u03b2Y(u))du+\u03b3(W(s)\u2212W(tn)),tn\u2264s\u2264T.\n\nUsing the triangle and Cauchy-Schwarz inequalities, and the elementary inequality (12) with ,\n\n |Y(s)\u2212Y(tn)|p \u2264 2p\u2223\u2223 \u2223\u2223\u222bstn(\u03b1Y(u)+\u03b2Y(u))du\u2223\u2223 \u2223\u2223p+2p\u03b3p|W(s)\u2212W(tn)|p \u2264 2p|s\u2212tn|p\u22121\u222bstn\u2223\u2223\u2223\u03b1Y(u)+\u03b2Y(u)\u2223\u2223\u2223pdu+2p\u03b3p|W(s)\u2212W(tn)|p \u2264 22p|s\u2212tn|p\u22121(\u222bstn\u03b1pY(u)pdu+\u222bstn|\u03b2|pY(u)pdu) +2p\u03b3p|W(s)\u2212W(tn)|p,\n\nfor . Now apply conditional expectations on both sides with respect to and (17) to get,\n\nwhere we have used (6) and (8) from the statement of Lemma 2 at the last step. Therefore\n\n E[|Y(s)\u2212Y(tn)|p\u2223\u2223Ftn]\u22642p\u03b3p\u03c5p|s\u2212tn|p\/2+22p(\u03b1pC(p,T)Y(tn)p+\u03b2M1,p(1+Y(tn)p))|s\u2212tn|p,\u00a0a.s,\n\nas required.\n\n###### Lemma 12.\n\nLet be a solution of (3) and let Assumption 1 hold. Let arise from the adaptive timestepping strategy satisfying (18) in Definition 7 for some , and formulate the Taylor expansion of around , where is as given in (15), as\n\n (26) f(Y(s))=f(Y(tn))+Rf(s,tn,Y(tn)),s\u2208[tn,tn+1],\n\nwhere\n\n (27)\n\nFor any , the conditional moment of\n\nsatisfies\n\n (28) E[|Rf|p\u2223\u2223Ftn]\u2264Kn,php\/2n+1,a.s.,\n\nwhere is an a.s. finite and -measurable random variable given by\n\nMoreover, there exists independent of such that\n\n (29) Kp:=E[Kn,p]<\u221e.\n###### Proof.\n\nBy direct substitution of from (15) into (27), and taking the -moment conditional upon , we get\n\n E[|Rf|p\u2223\u2223Ftn]=E[\u2223\u2223 \u2223\u2223(Y(s)\u2212Y(tn))(\u03b2\u2212\u03b1Y(s)Y(tn))\u2223\u2223 \u2223\u2223p\u2223\u2223\u2223Ftn].\n\nUsing the triangle inequality and (12) we get\n\n E[|Rf|p\u2223\u2223Ftn]\u22642p|\u03b2|pE[|Y(s)\u2212Y(tn)|p\u2223\u2223\u2223Ftn]+2p\u03b1pY(tn)pE[\u2223\u2223 \u2223\u2223((Y(s)\u2212Y(tn))\u22c51Y(s))\u2223\u2223 \u2223\u2223p\u2223\u2223\u2223Ftn]\n\nNext apply Lemma 11 followed by the Cauchy-Schwarz inequality to get\n\n E[|Rf|p\u2223\u2223Ftn] \u2264 2p|\u03b2|php\/2n+1(2p\u03b3p\u03c5p+\u00afLn,php\/2n+1) +2p\u03b1pY(tn)p\u221aE[|Y(s)\u2212Y(tn)|2p|Ftn]\ue001\ue000 \ue000\u23b7E[1|Y(s)|2p\u2223\u2223\u2223Ftn] \u2264 2p|\u03b2|php\/2n+1(2p\u03b3p\u03c5p+\u00afLn,php\/2n+1) +2p\u03b1p\u221aC(p,T)Y(tn)2p\u221aE[|Y(s)\u2212Y(tn)|2p|Ftn]\n\nAgain applying Lemma 11 and the elementary inequality (10) this becomes\n\n E[|Rf|p\u2223\u2223Ftn]\u22642p|\u03b2|php\/2n+1(2p\u03b3p\u03c5p+\u00afLn,php\/2n+1) +2php\/2n+1\u03b1p\u221aC(p,T)Y(tn)2p(2p\u03b3p\u03c51\/22p+\u00afL1\/2n,2php\/2n+1),\n\nfrom which the statement of the Lemma follows when we observe that the a.s. finiteness of is ensured by Assumption 1, and (29) is ensured by Lemmas 2 & 3. \u220e\n\n###### Lemma 13.\n\nLet be the solution of (3) with initial value . Let be a solution of (22) over the interval and be a sequence of random timesteps defined by (3) and with . Then for , there exists an -measurable random variable with finite expectation such that\n\n (30) E[E(tn+1)2\u2223\u2223Ftn]\u2212E(tn)2\u2264\u222btn+1tnE[E(r)2|Ftn]+\u00afKnh2n+1,a.s.\n\nwhere the error , .\n\n###### Proof.\n\nFor , we subtract (22) from (4) to get\n\n (31) E(s) = Y(s)\u2212\u02dcY(s) = [Y(tn)+\u222bstnf(Y(r))dr+\u03b3\u222bstndW(r)] \u2212[Yn+\u222bstnf(Yn)dr+\u03b3\u222bstndW(r)] = E(tn)+\u222bstn~f(Y(r),Yn)dr,\n\nwhere is defined as in (15) and . Applying the It\u00f4 formula and setting , we can write,\n\n E(tn+1)2=E(tn)2+2\u222btn+1tnE(r)~f(Y(r),Yn)dr.\n\nBy (26) in the statement of Lemma 12, , where is defined in (27). This, and an application of (11) gives\n\n (32) E(tn+1)2\u2212E(tn)2 = 2\u222btn+1tnE(r)Rf(r,tn,Y(tn))dr+2\u222btn+1tnE(r)~f(Y(tn),Yn)dr \u2264 \u222btn+1tnE(r)2dr+\u222btn+1tnRf(r,tn,Y(tn))2dr +2\u222btn+1tnE(r)~f(Y(tn),Yn)dr.\n\nConsider the third term on the RHS of (32), and substitute (31) into the integrand:\n\n (33) \u222btn+1tnE(r)~f(Y(tn),Yn)dr = E(tn)~f(Y(tn),Yn)\u222btn+1tndr+~f(Y(tn),Yn)2\u222btn+1tn\u222brtndudr +~f(","date":"2022-08-09 13:03:59","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.874878466129303, \"perplexity\": 699.8140728203106}, \"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-2022-33\/segments\/1659882570977.50\/warc\/CC-MAIN-20220809124724-20220809154724-00537.warc.gz\"}"} | null | null |
namespace Microsoft.Azure.Management.Search.Models
{
using System;
using System.Linq;
using System.Collections.Generic;
using Newtonsoft.Json;
using Microsoft.Rest;
using Microsoft.Rest.Serialization;
using Microsoft.Rest.Azure;
/// <summary>
/// Defines all the properties of an Azure Search service.
/// </summary>
public partial class SearchServiceReadableProperties
{
/// <summary>
/// Initializes a new instance of the SearchServiceReadableProperties
/// class.
/// </summary>
public SearchServiceReadableProperties() { }
/// <summary>
/// Initializes a new instance of the SearchServiceReadableProperties
/// class.
/// </summary>
public SearchServiceReadableProperties(SearchServiceStatus? status = default(SearchServiceStatus?), string statusDetails = default(string), ProvisioningState? provisioningState = default(ProvisioningState?), Sku sku = default(Sku), int? replicaCount = default(int?), int? partitionCount = default(int?))
{
Status = status;
StatusDetails = statusDetails;
ProvisioningState = provisioningState;
Sku = sku;
ReplicaCount = replicaCount;
PartitionCount = partitionCount;
}
/// <summary>
/// Gets the status of the Search service. Possible values include:
/// 'running', 'provisioning', 'deleting', 'degraded', 'disabled',
/// 'error'
/// </summary>
[JsonProperty(PropertyName = "status")]
public SearchServiceStatus? Status { get; private set; }
/// <summary>
/// Gets the details of the Search service status.
/// </summary>
[JsonProperty(PropertyName = "statusDetails")]
public string StatusDetails { get; private set; }
/// <summary>
/// Gets the state of the last provisioning operation performed on the
/// Search service. Possible values include: 'succeeded',
/// 'provisioning', 'failed'
/// </summary>
[JsonProperty(PropertyName = "provisioningState")]
public ProvisioningState? ProvisioningState { get; private set; }
/// <summary>
/// Gets or sets the SKU of the Search Service, which determines price
/// tier and capacity limits.
/// </summary>
[JsonProperty(PropertyName = "sku")]
public Sku Sku { get; set; }
/// <summary>
/// Gets or sets the number of replicas in the Search service. If
/// specified, it must be a value between 1 and 6 inclusive.
/// </summary>
[JsonProperty(PropertyName = "replicaCount")]
public int? ReplicaCount { get; set; }
/// <summary>
/// Gets or sets the number of partitions in the Search service; if
/// specified, it can be 1, 2, 3, 4, 6, or 12.
/// </summary>
[JsonProperty(PropertyName = "partitionCount")]
public int? PartitionCount { get; set; }
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,537 |
Q: HTML CSS: How to add Sliding Header? In Boostrap Slide show carousel, how would I add my descriptions to the title above instead of below at the captions sliding? I am trying to find Bootstrap functionality to conduct this, but I cannot locate it. It is sliding in the below pictures, I want to place it at the top where green box is.
Does Bootstrap allow this functionality?
.carousel-inner {
border-style: solid;
border-color: black;
border-right-width: 1px;
border-left-width: 1px;
border-bottom-width: 1px;
}
.carouselheader {
text-align: center;
background-color: green;
color: white;
border-top-left-radius: 8px;
border-top-right-radius: 8px;
overflow: hidden;
height: 32px;
border-color: black;
border-style: solid;
border-top-width: 1px;
border-right-width: 1px;
border-left-width: 1px;
}
.carousel slide {
border-top-left-radius: 8px;
border-top-right-radius: 8px;
overflow: hidden;
}
.carouselleftarrow {
font-family: Material Icons;
font-size: 36px;
position: absolute;
bottom: 5px;
content: "\e408";
color: black;
}
.carouselleftarrow:hover {
color: black;
}
.carouselrightarrow {
font-family: Material Icons;
font-size: 36px;
position: absolute;
bottom: 5px;
content: "\e409";
color: black;
}
.carouselrightarrow:hover {
color: black;
}
.carousel-indicators li {
text-indent: 0 !important;
width: 24px !important;
height: 24px !important;
margin: 3px !important;
border-radius: 50px !important;
bottom: 3px;
font: 10px;
display: inline-flex;
justify-content: center;
align-items: center;
flex-direction: row;
color: #999999;
background-color: blue;
font-size: 14px;
}
.carousel-indicators li.active {
color: white;
background-color: black;
}
.carousel-indicators li:hover {
color: white;
background-color: black;
}
.left .carousel-control {
border-radius: 8px;
background-image: none;
}
.right .carousel-control {
border-radius: 8px;
background-image: none;
display: inline-flex;
}
.carousel-control.left {
background-image: none;
}
.carousel-control.right {
background-image: none;
}
<!-- Latest compiled and minified CSS -->
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.0/css/bootstrap.min.css">
<!-- jQuery library -->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.4.1/jquery.min.js"></script>
<!-- Latest compiled JavaScript -->
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.0/js/bootstrap.min.js"></script>
<link href="https://fonts.googleapis.com/icon?family=Material+Icons" rel="stylesheet">
<ipts-carousel>
<div class="carousel slide" data-ride="carousel" id="Carouselid9a15830fddaa4c479b693696f4c9690c" style=" width: 500px;
height: 516px;">
<div class="carouselheader">Place Descriptions in Top Here</div>
<ol class="carousel-indicators">
<li data-slide-to="0" data-target="#Carouselid9a15830fddaa4c479b693696f4c9690c" class="">1</li>
<li data-slide-to="1" data-target="#Carouselid9a15830fddaa4c479b693696f4c9690c" class="">2</li>
<li data-slide-to="2" data-target="#Carouselid9a15830fddaa4c479b693696f4c9690c" class="active">3</li>
</ol>
<div class="carousel-inner" style="height: 500px;">
<div class="item carousel" style="width: 500px;
height:500px; height: 100%;"><img class="imgcarousel mCS_img_loaded" src="https://img1.10bestmedia.com/Images/Photos/352450/GettyImages-913753556_54_990x660.jpg" style="width: 500px;
height:500px; width: 100%; height: 100%;">
<div class="carousel-caption">
<h3>Ocean sea</h3>
<p>Ocean Sea has nice view</p>
</div>
</div>
<div class="item carousel" style="width: 500px;
height:500px; height: 100%;"><img class="imgcarousel mCS_img_loaded" src="https://amp.businessinsider.com/images/5b75a356e199f336008b528b-750-563.jpg" style="width: 500px;
height:500px; width: 100%; height: 100%;">
<div class="carousel-caption">
<h3>Nice Houses</h3>
<p>Neighboor houses in Chicago</p>
</div>
</div>
<div class="item carousel active" style="width: 500px;
height:500px; height: 100%;"><img class="imgcarousel mCS_img_loaded" src="https://www.mcpl.us/sites/default/files/styles/large/public/bookstack.jpg?itok=pHICdzg9" style="width: 500px;
height:500px; width: 100%; height: 100%;">
<div class="carousel-caption">
<h3>Reading Books</h3>
<p>Stack of library books</p>
</div>
</div>
</div><a class="left carousel-control" data-slide="prev" href="#Carouselid9a15830fddaa4c479b693696f4c9690c"><span class="carouselleftarrow">navigate_before</span></a><a class="right carousel-control" data-slide="next" href="#Carouselid9a15830fddaa4c479b693696f4c9690c"
style="display: inline-flex;"><span class="carouselrightarrow">navigate_next</span></a></div>
</ipts-carousel>
A: After some adaptations...
Live example:
.carousel-inner {
border-style: solid;
border-color: black;
border-right-width: 1px;
border-left-width: 1px;
border-bottom-width: 1px;
display: inline-block;
border-top-left-radius: 8px;
border-top-right-radius: 8px;
}
.carousel slide {
border-top-left-radius: 8px;
border-top-right-radius: 8px;
overflow: hidden;
}
.carouselleftarrow {
font-family: Material Icons;
font-size: 36px;
position: absolute;
bottom: 5px;
content: "\e408";
color: black;
}
.carouselleftarrow:hover {
color: black;
}
.carouselrightarrow {
font-family: Material Icons;
font-size: 36px;
position: absolute;
bottom: 5px;
content: "\e409";
color: black;
}
.carouselrightarrow:hover {
color: black;
}
.carousel-indicators li {
text-indent: 0 !important;
width: 24px !important;
height: 24px !important;
margin: 3px !important;
border-radius: 50px !important;
bottom: 3px;
font: 10px;
display: none;
justify-content: center;
align-items: center;
flex-direction: row;
color: #999999;
background-color: blue;
font-size: 14px;
}
.carousel-indicators li.active {
color: white;
background-color: black;
}
.carousel-indicators li:hover {
color: white;
background-color: black;
}
.left .carousel-control {
border-radius: 8px;
background-image: none;
}
.right .carousel-control {
border-radius: 8px;
background-image: none;
display: inline-flex;
}
.carousel-control.left {
background-image: none;
border-radius: 8px;
}
.carousel-control.right {
background-image: none;
border-radius: 8px;
}
.carousel-caption {
width: 100%;
top: 0;
padding: 8px !important;
text-align: center;
background: green;
color: white;
border-top-left-radius: 8px;
border-top-right-radius: 8px;
overflow: hidden;
height: 69px;
border-color: black;
border-style: solid;
border-top-width: 1px;
border-right-width: 1px;
border-left-width: 1px;
right: 0 !important;
left: 0 !important;
}
.carousel-caption h3 {
margin: 0px;
}
.carousel-caption p {
margin: 0px;
}
.carousel-inner>.item>a>img,
.carousel-inner>.item>img {
top: 69px;
position: absolute;
}
<!-- Latest compiled and minified CSS -->
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.0/css/bootstrap.min.css">
<!-- jQuery library -->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.4.1/jquery.min.js"></script>
<!-- Latest compiled JavaScript -->
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.0/js/bootstrap.min.js"></script>
<link href="https://fonts.googleapis.com/icon?family=Material+Icons" rel="stylesheet">
<ipts-carousel>
<div class="carousel slide" data-ride="carousel" id="Carouselid9a15830fddaa4c479b693696f4c9690c" style=" width: 500px;
height: 516px;">
<ol class="carousel-indicators">
<li data-slide-to="0" data-target="#Carouselid9a15830fddaa4c479b693696f4c9690c" class="">1</li>
<li data-slide-to="1" data-target="#Carouselid9a15830fddaa4c479b693696f4c9690c" class="">2</li>
<li data-slide-to="2" data-target="#Carouselid9a15830fddaa4c479b693696f4c9690c" class="active">3</li>
</ol>
<div class="carousel-inner" style="height: 100%;">
<div class="item carousel" style="width: 500px;
height:500px; height: 100%;"><img class="imgcarousel mCS_img_loaded" src="https://img1.10bestmedia.com/Images/Photos/352450/GettyImages-913753556_54_990x660.jpg" style="width: 500px;
height:500px; width: 100%; height: 100%;">
<div class="carousel-caption">
<h3>Ocean sea</h3>
<p>Ocean Sea has nice view</p>
</div>
</div>
<div class="item carousel" style="width: 500px;
height:500px; height: 100%;"><img class="imgcarousel mCS_img_loaded" src="https://amp.businessinsider.com/images/5b75a356e199f336008b528b-750-563.jpg" style="width: 500px;
height:500px; width: 100%; height: 100%;">
<div class="carousel-caption">
<h3>Nice Houses</h3>
<p>Neighboor houses in Chicago</p>
</div>
</div>
<div class="item carousel active" style="width: 500px;
height:500px; height: 100%;"><img class="imgcarousel mCS_img_loaded" src="https://www.mcpl.us/sites/default/files/styles/large/public/bookstack.jpg?itok=pHICdzg9" style="width: 500px;
height:500px; width: 100%; height: 100%;">
<div class="carousel-caption">
<h3>Reading Books</h3>
<p>Stack of library books</p>
</div>
</div>
</div><a class="left carousel-control" data-slide="prev" href="#Carouselid9a15830fddaa4c479b693696f4c9690c"><span class="carouselleftarrow">navigate_before</span></a><a class="right carousel-control" data-slide="next" href="#Carouselid9a15830fddaa4c479b693696f4c9690c"
style="display: inline-flex;"><span class="carouselrightarrow">navigate_next</span></a></div>
</ipts-carousel>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,476 |
\section{Introduction}
Low-dimensional systems offer enormous potential for stabilising and controlling novel magnetic states and textures~\cite{Burch:2018, Huang:2020}. However, magnetic fluctuations are known to destabilise long-range order in two-dimensional (2D) systems (the famous Mermin-Wagner theorem \cite{Mermin:1966}). While magnetic anisotropy can allow long-range order to develop again~\cite{Onsager:1944, Gibertini:2019}, fluctuations can still be expected to play a crucial role. This necessitates their fundamental study, and potentially provides new opportunities in which to control magnonic excitations for spintronic technologies. 2D magnetic systems can be realised in a top-down approach, by exfoliating few/single layers from a bulk van der Waals magnet~\cite{Burch:2018,Gong:2017, Huang:2017}, or can be realised in a bottom-up approach, by realising magnetic order in a thin-film or surface geometry. The delafossite oxide, PdCoO$_2$, has recently been demonstrated as a model host of the latter~\cite{Mazzola:2018}.
While it is a non-magnetic Pd $d^9$ metal in the bulk \cite{Mackenzie:2017}, the polar crystal structure (Fig.~\ref{f:overview}a) leads to pronounced charge carrier doping at the surface ~\cite{Sunko:2017,Mazzola:2018, Kim:2009, Noh:2009}. For the Pd-termination, the resulting self-doping is electron-like, which acts to shift a large peak in the unoccupied density of states towards the Fermi level. This in turn triggers a Stoner instability, generating a 2D ferromagnetic surface layer, as predicted by density-functional theory~\cite{Kim:2009,Mazzola:2018} and confirmed from electronic structure~\cite{Mazzola:2018} and anomalous Hall measurements~\cite{Harada:2019}. This thus provides a model environment, accessible to spectroscopic probes, in which to study the influence of magnetic excitations on the electronic structure of a 2D magnet. Here, we use angle-resolved photoemission (ARPES) and scanning-tunnelling microscopy and spectroscopy (STM/S) to investigate this system, finding evidence for a strong and highly-tuneable electron-magnon coupling.
\begin{figure*}[ht]
\includegraphics[width=\textwidth]{Fig_1_ARPES_STM.pdf}
\caption{{\bf Electronic structure of the Pd-terminated ferromagnetic surface of PdCoO$_2$.} (a) Schematic of the crystal structure (side view) of PdCoO$_2$. After sample cleaveage, a CoO$_2$ surface termination (left) and a Pd termination (right) are both present. (b) ARPES spectra ($h\nu=80$~eV along $\bar{\Gamma}$-\={M} and $h\nu=82$~eV along $\bar{\Gamma}$-\={K}) show the superposition of spectral weight arising from both terminations. The labelled $\alpha$-$\beta$ and $\gamma$-$\delta$ bands derive from the Pd-terminated surface, and represent exchange-split pairs by a surface ferromagnetism. (c) STM topographic image of a Pd-terminated region. (d) The STM quasiparticle interference map (bottom) from a Pd-terminated region is in good agreement with the Fermi surface measured by ARPES (top), determined by the $\gamma$ and $\delta$ bands. Signatures of such electron-like bands are also visible in energy-dependent QPI measurements (e) for intra-$\gamma$ and intra-$\delta$ band scattering, reported together with the corresponding fits. Additional spectral weight at 0.1~eV binding energy can be attributed to intra-$\alpha$ band scattering, in good agreement with the flat top of the $\alpha$ band observed by ARPES.}
\label{f:overview}
\end{figure*}
\section{Results}
\subsection{Electronic structure of the Pd-terminated ferromagnetic surface}
Figure~\ref{f:overview} shows an overview of the surface electronic structure of PdCoO$_2$ as measured by angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM). The crystal hosts two distinct surface terminations (Fig.~\ref{f:overview}a): a Pd-terminated surface and a CoO$_2$-terminated one. These would be expected with approximately equal probability, with a typical cleaved sample having both types present. Consistent with this, our measured ARPES spectra shown in Fig.~\ref{f:overview}b exhibit signatures of the electronic states associated with both surface terminations. A pair of heavy hole-like bands around the Brillouin zone centre derive from the CoO$_2$-terminated surface ~\cite{Sunko:2017, Noh:2009}, and will not be considered further here. Several additional low-energy states are observed, which we have previously attributed as deriving from the Pd-terminated surface~\cite{Mazzola:2018}. Of particular relevance here are the two pairs of exchange-split ferromagnetic surface states labelled ($\alpha$,$\beta$) and ($\gamma$,$\delta$) in Fig.~\ref{f:overview}b~\cite{Mazzola:2018}. The very flat band top of the spin-majority $\alpha$-band is visible $\approx100$~meV below $E_\mathrm{F}$. Its high associated density of states would sit at the Fermi level in the non-magnetic state; this is responsible for triggering a Stoner transition to itinerant surface ferromagnetism,~\cite{Mazzola:2018, Kim:2009} pushing this flat band below $E_\mathrm{F}$ as observed here.
To confirm that these key electronic states are indeed derived from the Pd-terminated surface, we have performed STM measurements to selectively probe a single surface termination (Fig.~\ref{f:overview}c) \cite{Yim:2021}. While the intrinsic defect density in the bulk is extremely low \cite{Sunko:2021}, we find a number of defects are present in our STM measurements. We tentatively attribute these to Pd vacancies at the cleaved surface ~\cite{Foot1}. Clear quasiparticle interference (QPI) patterns are visible around these, permitting a local measurement of the surface electronic structure. The Fourier transform of the QPI patterns, shown in Fig.~\ref{f:overview}d, exhibits two clear concentric circles, whose wavevector is in good agreement with intra-band scattering of the $\gamma$ and $\delta$ Fermi surfaces observed by ARPES. An energy-dependent cut through the QPI data (Fig.~\ref{f:overview}e) confirms this assignment from the electron-like character of these pockets, while an additional intense QPI signal at 100~meV below the Fermi level, peaked around $\mathbf{q}=0$ and with little evident dispersion, is consistent with the flat top of the $\alpha$ band observed in the ARPES.
\subsection{Quasiparticle dynamics}
Having confirmed the electronic structure of the ferromagnetic Pd-terminated surface of PdCoO$_2$, we will focus now on spectroscopic signatures of marked electronic interactions evident for these surface states. Fig.~\ref{f:magnon}a shows in detail the $\gamma$ and $\delta$ bands. While the minority-spin $\delta$-band appears to be well described as a simple parabolic band, the $\gamma$-band exhibits several anomalies, or `kinks', in its dispersion. These are visible in the raw data, and in fits to momentum distribution curves (MDCs) shown as blue markers in Fig.~\ref{f:magnon}a. Typically, such spectral kinks are signatures of an electron-boson coupling, which can be described via an electronic self-energy. To investigate this further, we have extracted the self-energy from a self-consistent analysis of our ARPES measurements (see Methods), and report the Kramers-Kronig-consistent real and imaginary parts of the electron-boson self-energy in Fig.~\ref{f:magnon}b \cite{Foot2}
Two steep rises in the real-part of the electron-boson self-energy are visible at $\hbar\omega_{1}\approx50$~meV and at $\hbar\omega_{2}\approx130$~meV. These are in approximate agreement with phonon energies reported in the literature for PdCoO$_2$ \cite{Cheng:2017, Kumar:2013, Takatsu:2007}. The low-energy kinks of our measured data can thus be satisfactorily attributed to electron-phonon coupling. However, there is an additional broad hump in the extracted real part of the electron-boson self energy which extends out to $>300$~meV. This is an implausibly high energy to be due to a phonon mode (well above the highest phonon mode energies of $\approx 150$~meV expected for PdCoO$_2$ \cite{Cheng:2017, Kumar:2013, Takatsu:2007}). Indeed, a model self-energy calculation including only coupling to phonon modes fails to capture the high-energy part of the extracted experimental self energy for any electron-phonon coupling strength (the best fit is shown as the grey lines in Fig.~\ref{f:magnon}b; see also Supplementary Fig.~2). We thus conclude that there must be a third boson mode which is active here, with a characteristic energy scale of $>200$~meV, and which exhibits a marked coupling to the electronic sub-system. The high associated energy scale of this boson mode suggests that it may have a magnetic origin: in other itinerant ferromagnets, the spin-wave spectra are known to extend to similarly high energies of, for example, $>300$~meV in Fe, $>500$~meV in Co and $\approx500$~meV in Ni \cite{Halilov:1998, kubler:2009}.
\begin{figure}
\includegraphics[width=\columnwidth]{Fig_2_bosons.pdf}
\caption{{\bf Electron-boson coupling.} (a) Electronic structure measurements ($h\nu=90$~eV, measured along the $\bar{\Gamma}$-\={K} direction) showing the $\gamma$ and $\delta$ bands. The non-interacting, bare band, dispersion is taken as a $\mathbf{k}\cdot\mathbf{p}$ band, set to match the Fermi wavevectors of the experimental dispersions as described in the Methods. (b) The real and imaginary parts of the electron-boson self energy extracted from the data and from a Migdal-Eliashberg calculation as described in the Methods. A two-phonon model (grey lines) fails to describe the experimentally determined self energy, while a three-mode model (red line, with coupling to two phonons plus a magnon mode) is in much better agreement. The arrows in (b) show the characteristic mode energies. The real and imaginary parts of the self-energy retain causality through Kramers-Kronig transformation.}
\label{f:magnon}
\end{figure}
\begin{figure*}
\includegraphics[width=0.8\textwidth]{Fig_3_topo_Arpes.pdf}
\caption{{\bf Surface-dependent variation of electron-magnon coupling strengths.} (a) Electronic structure measurements of the $\gamma$ and $\delta$ bands as function of hole-doping. A pronounced increase in the band renormalisation is observed from the left to the right measurement, which were performed on different patches of the same sample (S1 and S2) and on different samples (S3 and S4). (b) Fits of MDCs (shown here as a function of $k_y-k_\mathrm{F}$; shifts in $k_\mathrm{F}$ of the $\gamma$ band between the samples are not resolvable experimentally)} show a strong increase in effective mass at the Fermi level (decrease in the Fermi velocity). By tracking the binding energy of the flat portion of the $\alpha$ band (see Supplementary Fig.~3), we find that this is correlated with an increasing hole doping (c). Fitting the extracted self energy for the these and additional samples, we find that the increased Fermi velocity (red) results due to a large increase in electron-magnon coupling strength, while the electron-phonon coupling strength does not vary significantly. (d) Large-scale topography measured by STM shows a large concentration of impurities (likely Pd vacancies) distributed across the cleaved Pd-terminated surface, which likely give rise to the variable hole doping determined above.
\label{f:variation}
\end{figure*}
Previous ARPES measurements have observed signatures of the coupling of electrons with such magnetic excitations in elemental magnetic metals~\cite{Hayashi:2013, Mynczak:2019, Cui:2007, Schneider:2016}, including for their two-dimensional surface electronic states~\cite{Hofmann:2009, Claessen:2004}, with spectroscopic signatures and energy scales very similar to those observed here. This can be understood via a process similar to the observation of electron-phonon coupling in ARPES: upon photoemission from a majority-state band, the resulting photohole can be filled by an electron with opposite spin accompanied by the emission or absorption of a magnon. Nonetheless, the different dispersion relations of phonons and magnons leads to differences in the functional form of the corresponding self-energy (see Methods). While we cannot describe our measured self-energies solely using electron-phonon-based models, extending them to include electron-magnon coupling leads to excellent agreement with our experimentally-determined self-energies (Fig.~\ref{f:magnon}b). The determined characteristic mode energy (left as a free parameter in fits to our experimental self-energies) is at 245~meV, while it exhibits a moderate coupling strength, $\lambda_{\mathrm{el-mag}}=0.55\pm0.05$, similar to, but slightly higher than, the coupling strength to the two phonon modes at lower energies: $\lambda^{(1)}_{\mathrm{el-ph}}=0.30\pm0.02$ and $\lambda^{(2)}_{\mathrm{el-ph}}=0.25\pm0.03$. These values are in rather good agreement with calculated electron-paramagnon and electron-phonon coupling constants, respectively, in bulk elemental Pd \cite{Savrasov:1996, Bose:2008, Pinski:1979}, which is itself thought to be close to a ferromagnetic instability.
Intriguingly, we find a strong sample-to-sample and spatial variation of the strength of the electron-magnon coupling. These are summarised in Fig.~\ref{f:variation}a (see also Supplementary Fig.~3). The electronic structure remains qualitatively the same as that shown in Fig.~\ref{f:overview}. However, the measured Fermi velocity extracted from fits to momentum distribution curves (Fig.~\ref{f:variation}b) decreases by a factor of $\approx3$ across these samples. At the same time, we observe a shift of the binding energy of the flat portion of the $\alpha$-band towards the Fermi level by $\approx{40}$~meV (Supplementary Fig.~4). The latter likely arises due to surface Pd vacancies. These are readily apparent in our STM measurements (Fig.~\ref{f:variation}d), and would act to partially counteract the polar surface charge, in turn leading to a reduction of the electron-like doping at the surface with respect to the hypothetical bulk-like surface termination.
For the STM measurements presented in Fig.~\ref{f:variation}d, we estimate a surface Pd vacancy concentration of $\sim\!2\%$. This would lead to a nominal surface carrier density of 0.48 electrons/unit cell (el/u.c.), in good agreement with the Luttinger count of the ARPES-measured Fermi surfaces from sample S1 (Fig.~\ref{f:magnon}a). Assuming a rigid band shift with increasing Pd vacancy concentration, we estimate from tight-binding analysis (see Supplementary Fig.~5) that the variations observed across our samples here correspond to variations in the surface electron concentration of a further 0.02 el/u.c., corresponding to a further 2\% of vacancies in the surface Pd layer. While high concentrations of Pd vacancies would lead to exposed areas of CoO$_2$-terminated surface, the low concentrations of Pd vacancies observed here are thus best considered as isolated point defects, consistent with our STM measurements and providing a natural source of charge carrier doping.
\subsection{Tuneable electron-magnon coupling}
Taken together with the change in Fermi velocity, the above findings indicate a dramatic enhancement of the many-body renormalisations of the surface electronic structure with hole doping from surface Pd vacancies. From model fits to the electronic self energy from our ARPES measurements, we find that it is the magnon mode identified above which exhibits a strongly varying coupling strength, while the electron-phonon coupling remains unchanged within experimental error (Fig.~\ref{f:variation}c, see also Supplementary Fig.~6). The enhancement of this electron-magnon coupling is dramatic, reaching coupling strengths on the order of $\lambda\approx3$, where a weak-coupling Migdal-Eliashberg picture would be expected to break down.
\begin{figure}
\includegraphics[width=\columnwidth]{Fig_4_plot.pdf}
\caption{{\bf Magnonic polarons.} (a) Measured $\gamma$-band dispersion for the sample with strongest electron-magnon coupling (S4, same data of Fig.~\ref{f:variation}a) but with incresed saturation and different color scale). The $\gamma$ band is now strongly renormalised, with the coherent quasiparticle band having an occupied bandwidth of only $\approx100$~meV (measurement taken at $h\nu=80$~eV). (b) A replica of this band is evident, shifted to higher binding energies by the characteristic magnon energy of $\approx240$~meV, indicative of polaron formation. (c) Energy distribution curves taken at the momentum value indicated by the green line in (a) show how this replica band feature is independent of photon energy, pointing to a strong intrinsic electron-magnon coupling regime.}
\label{f:pola}
\end{figure}
Indeed, for the sample marked S4 in Fig.~\ref{f:variation}, our weak-coupling models fail to adequately describe the extracted dispersion from our ARPES measurements. We show the measured dispersion again in Fig.~\ref{f:pola} with a different contrast. It is clear that the quasi-particle band is strongly renormalised, having an occupied band width of only $\approx100$~meV, significantly reduced compared to the $>400$~meV occupied band width for sample S1 seen in Fig.~\ref{f:magnon}. Within a Migdal-Eliashberg approach, this would imply a quasiparticle residue $Z=m_0/m^*<0.25$, a rather low value, and outside the regime of applicability of this approximation. Consistent with this, instead of simply generating a kink in the dispersion, we note that this strong electron-magnon coupling leads to a replica of the quasiparticle band, evident in our measurements as a weak dispersive feature with the same dispersion as the quasiparticle band, but shifted to higher binding energies by $\approx230$~meV (Fig.~\ref{f:pola}b). This is similar to the replica features observed due to strong coupling with phonons~\cite{Moser:2013, Lee:2014, Chen:2015, Wang:2016, Cancellieri:2016} or plasmons~\cite{Rliley:2018, Caruso:2021}, generating polaronic states. The separation between the quasiparticle band and the replica band observed here is equal to the characteristic mode energy of the magnon mode determined from fitting our data for lower coupling strengths (Fig.~\ref{f:variation}), and so we attribute this as a shake-off replica due to the strong electron-magnon coupling here. Its spectroscopic signatures are largely independent of photon energy (Fig.~\ref{f:pola}c). Together with the large quasiparticle mass renormalisation observed, this rules out that the replica band results from extrinsic photoelectron energy loss~\cite{Li:2018}, and instead indicates that the intrinsic electron-magnon coupling becomes strong enough to drive the system into a polaronic regime.
\section{Discussion}
Our measurements above indicate a dramatic enhancement of electron-magnon coupling due to increased disorder (vacancy concentration) of the cleaved surface. This can impact the coupling of magnons to the itinerant electrons in multiple ways. First, with the concomitant increased hole doping, the flat top of the $\alpha$ band is shifted towards the Fermi level (Fig.~\ref{f:variation}c). Correspondingly, the onset of the Stoner continuum from finite $\mathbf{q}$ spin-flip electron-hole excitations will be shifted to lower energies. The collective magnon mode will thus enter the Stoner continuum more rapidly, becoming Landau-damped and leading to a decreased quasiparticle lifetime, and thus enhanced coupling strength, as we observe here. We also note that the magnetism predicted in this surface layer by density-functional theory is only stabilised by a surface relaxation which increases the surface Pd-O bond length~\cite{Mazzola:2018}. The stability of the magnetic order, and as a consequence the strength of the electron-magnon coupling, is thus likely to be extremely sensitive to structural distortions and defects. Indeed, we find that the linewidth at the Fermi level also increases with hole doping here, a clear indicator of an enhanced electron-impurity scattering pointing to increased surface disorder which accompanies the increased electron-magnon coupling strength.
The detailed mechanisms by which the electron-magnon coupling becomes enhanced here require further detailed theoretical study. Nonetheless, our experimental results point to a surprisingly-large response of this coupling to small changes in carrier concentration and surface disorder. This firmly establishes the surface states of PdCoO$_2$ as a model system in which to investigate the coupling of itinerant electrons to spin excitations, of fundamental importance to understand the limits of stability of magnetic order in 2D. We hope that the findings presented here motivate future studies aimed at gaining true deterministic control over the large changes in electron magnon coupling which we have observed. In this respect, we note the enormous recent progress on the growth of thin films of PdCoO$_2$ \cite{Harada:2018, Ok:2020, Sun:2019, Brahlek:2019}, in some of which signatures of the ferromagnetism of the Pd-terminated surface have already been reported in transport measurements~\cite{Harada:2019}. Such thin films would be ideally suited for studies of gate-tuning of surface doping (or even surface disorder) via, {\it e.g.}, ionic liquid gating. Our results above show that for changes in the surface carrier concentration accessible to such gating techniques, the electron-magnon coupling at PdCoO$_2$ surfaces can be driven from a weak- to a strong-coupling polaronic regime, opening tantalising new possibilities for studying the creation and control of spin-polarons.
\section{Materials and Methods}
{\bf Angle-resolved photoemission:} Single-crystal samples of PdCoO$_2$ were grown via a flux method in sealed quartz tubes \cite{Tanaka:1996}. These were cleaved \textit{in situ} at the measurement temperature of $\approx10$~K. ARPES measurements were performed at the I05 beamline of Diamond Light Source, using a Scienta R4000 hemispherical electron analyser, and photon energies between 70 and 90~eV. All measurements used linear-horizontal ($p$-) polarised light. The lateral spot size of the photon beam on the sample is $\approx 50$~$\mu$m.
\
{\bf STM measurements:} The STM and tunneling spectroscopy experiments were performed using a home-built low temperature STM that operates at a base temperature down to 1.8 K \cite{White:2011}. Pt/Ir tips were used, and conditioned by field emission on a gold target. Differential conductance ($dI/dV$) maps and single point spectra were recorded by means of standard lock-in technique (see also Supplementary Fig.~7), with the frequency of the bias modulation set at 413 Hz. The STM/S results reported here were obtained at a sample temperature of 4.2 K.
\
{\bf Self-energy analysis:} The electron-boson self-energy was extracted through a self-consistent analysis. The bare band is described by a nearly-parabolic $\mathbf{k} \cdot \mathbf{p}$ band, whose Fermi wavevectors are fixed to the experimental value. The parameters that describe the non-interacting band have been iteratively determined such that the causality connection between the extracted real and imaginary parts of the self energy enforced by Kramers-Kronig transformation is preserved. The electron-electron contribution to the self energy has been subtracted from the self-energies shown in Fig.~\ref{f:magnon}b to aid comparison of the electron-boson contribution. The full extracted self energy, including electron-electron contribution, is shown in Supplementary Fig.~1.
The real and imaginary parts of the self-energy have been modeled using a Migdal-Eliashberg approach. The electron-phonon interaction was described by a conventional two-phonon Debye-model with characteristic cut-off energies. Finite temperature was also included in the model, as described in Ref. \cite{Mazzola:2013a, Mazzola:2017a}. For the electron-magnon coupling, the magnon dispersion relation, $\omega_{mgn}\propto q^2$, yields a density of states proportional to the square root of the energy, which means $D_{mgn}\propto \omega^{\frac{1}{2}}$. This manifests in a different shape of both real and imaginary parts of the self energy compared to the one expected for phonons (($D_{ph}\propto \omega$)). This approach is similar to what has been used in Ref. \cite{Hofmann:2009}. Under the Migdal-Eliashberg approach, the contribution of electron-boson coupling to the imaginary part of the self-energy is calculated as:
\begin{align}
&\Im\Sigma^{electron-boson}(\omega,T)=\pi\int_{0}^{\omega_\mathrm{max}}\alpha^2(\omega^\prime)F(\omega^\prime)[1+2n(\omega^\prime) \nonumber \\
&\quad +\textit{f}(\omega+\omega^\prime)-f(\omega-\omega^\prime)] d{\omega^\prime} \nonumber
\end{align}
where $\omega_{\mathrm{max}}$ is the highest energy allowed for the boson mode, $f(\omega)$ and $n(\omega)$ are the fermion and boson distribution, respectively, and $T$ is the temperature. For the phonons, $\alpha^2 F(\omega)=0$ if $\omega>\omega_\mathrm{max}$ and $\alpha^2 F(\omega)=\lambda(\omega/\omega_\mathrm{max})^2$ for $\omega<\omega_\mathrm{max}$. In an analogous way, for magnons, $\alpha^2 F(\omega)=0$ if $\omega>\omega_\mathrm{max}$ and $\alpha^2 F(\omega)=\lambda_{mgn} (\omega/\omega_{mgn})^{\frac{1}{2}}$ for $\omega<\omega_\mathrm{max}$, where $\lambda_{mgn}$ represents the electron-magnon coupling strength of the system~\cite{Hofmann:2009}.
\section{Acknowledgements}
We thank C.~Hooley, T.~Frederiksen, G. van der Laan, G. Panaccione, H. Rosner and G. Siemann for useful discussions. We gratefully acknowledge support from the European Research Council (through the QUESTDO project, 714193), the Royal Society, the Max-Planck Society, and the UKRI Engineering and Physical Sciences Research Council (Grant No.~EP/S005005/1). We thank Diamond Light Source for access to Beamline I05 (Proposals SI12469, SI14927, and SI16262), which contributed to the results presented here. V.S., O.J.C., and L.B. acknowledge the EPSRC for PhD studentship support through Grants EP/L015110/1, EP/K503162/1, and EP/G03673X/1, respectively. I.M. and D.C. acknowledge studentship support from the International Max-Planck Research School for Chemistry and Physics of Quantum Materials.
\linespread{0.9}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,265 |
module MiqReport::ImportExport
extend ActiveSupport::Concern
module ClassMethods
VIEWS_FOLDER = File.join(Rails.root, "product/views")
def import_from_hash(report, options = nil)
raise _("No Report to Import") if report.nil?
report = report["MiqReport"] if report.keys.first == "MiqReport"
if !report["menu_name"] || !report["col_order"] || !report["cols"] || report["rpt_type"] != "Custom"
raise _("Incorrect format, only policy records can be imported.")
end
user = options[:user] || User.find_by_userid(options[:userid])
report.merge!("miq_group_id" => user.current_group_id, "user_id" => user.id)
report["name"] = report.delete("menu_name")
rep = MiqReport.find_by_name(report["name"])
if rep
# if report exists
if options[:overwrite]
# if report exists delete and create new
if user.admin_user? || user.current_group_id == rep.miq_group_id
msg = "Overwriting Report: [#{report["name"]}]"
rep.attributes = report
result = {:message => "Replaced Report: [#{report["name"]}]", :level => :info, :status => :update}
else
# if report exists dont overwrite
msg = "Skipping Report (already in DB under a different group): [#{report["name"]}]"
result = {:message => msg, :level => :error, :status => :skip}
end
else
# if report exists dont overwrite
msg = "Skipping Report (already in DB): [#{report["name"]}]"
result = {:message => msg, :level => :info, :status => :keep}
end
else
# create new report
msg = "Importing Report: [#{report["name"]}]"
rep = MiqReport.new(report)
result = {:message => "Imported Report: [#{report["name"]}]", :level => :info, :status => :add}
end
_log.info("#{msg}")
if options[:save] && result[:status].in?([:add, :update])
rep.save!
_log.info("- Completed.")
end
return rep, result
end
# @param db [Class] name of report (typically class name)
# @param current_user [User] User for restricted access to reports
# @param options [Hash]
# @option options :association [String] used for a view suffix
# @option options :view_suffix [String] used for a view suffix
# @param cache [Hash] cache that holds yaml for the views
def load_from_view_options(db, current_user = nil, options = {}, cache = {})
filename = MiqReport.view_yaml_filename(db, current_user, options)
yaml = cache[filename] ||= YAML.load_file(filename)
view = MiqReport.new(yaml)
view.db = db if filename.ends_with?("Vm__restricted.yaml")
view.extras ||= {} # Always add in the extras hash
view
end
def view_yaml_filename(db, current_user, options)
suffix = options[:association] || options[:view_suffix]
db = db.to_s
role = current_user.try(:miq_user_role)
# Special code to build the view file name for users of VM restricted roles
if %w(ManageIQ::Providers::CloudManager::Template ManageIQ::Providers::InfraManager::Template
ManageIQ::Providers::CloudManager::Vm ManageIQ::Providers::InfraManager::Vm VmOrTemplate).include?(db)
if role && role.settings && role.settings.fetch_path(:restrictions, :vms)
viewfilerestricted = "#{VIEWS_FOLDER}/Vm__restricted.yaml"
end
end
db = db.gsub(/::/, '_')
role = role.name.split("-").last if role.try(:read_only?)
# Build the view file name
if suffix
viewfile = "#{VIEWS_FOLDER}/#{db}-#{suffix}.yaml"
viewfilebyrole = "#{VIEWS_FOLDER}/#{db}-#{suffix}-#{role}.yaml"
else
viewfile = "#{VIEWS_FOLDER}/#{db}.yaml"
viewfilebyrole = "#{VIEWS_FOLDER}/#{db}-#{role}.yaml"
end
if viewfilerestricted && File.exist?(viewfilerestricted)
viewfilerestricted
elsif File.exist?(viewfilebyrole)
viewfilebyrole
else
viewfile
end
end
end
def export_to_array
h = attributes
["id", "created_on", "updated_on"].each { |k| h.delete(k) }
h["menu_name"] = h.delete("name")
[{self.class.to_s => h}]
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,963 |
\section{Introduction}
Speech technology has traditionally been divided into automated speech recognition (ASR) and speech synthesis. Hearing humans, however, perform both tasks --- speech production and speech perception --- with a high degree of mutual influence (the so-called production-perception loop; \cite{vihman15}).
This paper proposes that the two principles should be modeled simultaneously and argues that a GAN-based model called ciwGAN/fiwGAN \cite{begusCiw} learns linguistically meaningful representations for both production and perception. In fact, lexical learning in the architecture emerges precisely from the requirement that the network for production and the network for perception interact and generate data that is mutually informative. We show that with only the requirement to produce informative data, the models not only produce desired outputs (as argued in \cite{begusCiw}), but also learn to classify lexical items in a fully unsupervised way from raw unlabeled speech.
\subsection{Prior work}
Most of the existing models of lexical learning focus primarily on either ASR/speech-to-text (perception) or text-to-speech/speech synthesis (production; see \cite{wali22} for an overview). Variational Autoencoders (VAEs) involve both an encoder and decoder, which allows unsupervised acoustic word embedding as well as generation of speech, but these proposals only use VAEs for either unsupervised ASR \cite{chung16,chorowski19,baevski20,niekerk20} or for speech synthesis/transformation (e.g.~\cite{hsu17}). Earlier neural models replicate brain mechanisms behind perception and production \cite{guenther11}, but they do not focus on lexical learning or classification and do not include recent progress in performance of deep learning architectures. GAN-based synthesizers are mostly supervised and get text or acoustic features in their input \cite{kumar19,kong20,binkowski20,cong21}. \cite{donahue19} propose a WaveGAN architecture, which can generate any audio in an unsupervised manner, but does not involve a lexical classifier --- only the Generator and the Discriminator, which means the model only captures synthesis and not classification (the same is true for Parallel WaveGAN; \cite{yamamoto20}). \cite{begusCiw} proposes the first textless fully unsupervised GAN-based model for lexical representation learning, but evaluates only the synthesis (production) aspect of their model by only evaluating outputs of the Generator network.
\subsection{New challenges}
Here, we model lexical learning with a classifier network (the Q-network) that mimics perception and lexical learning and is, crucially, trained from another network's production data (the Generator network). Using this architecture, we can \textit{both} generate new words in a controlled causal manner by manipulating the Generator's latent space \textit{as well as} classify novel words from unobserved test data with a classifier that never directly accesses the training data.
This paper also introduces some crucial new challenges to the unsupervised acoustic word embedding and word recognition paradigm \cite{dunbar17}. First, the architecture enables extremely reduced vector representations of lexical items. In fiwGAN, the network needs to represent $2^n$ classes with only $n$ variables. To our knowledge, no other proposal features such dense representation of acoustic lexical items. Second, the models introduce a challenge to learn meaningful representations of words without ever directly accessing training data. The lexical classifier network is twice removed from training data. The Q-network learns to classify words only from the Generator's outputs and never accesses training data directly. But the Generator never accesses the training data directly either --- it learns to produce words only by maximizing the Discriminator's error rate.
Why are these challenges important? First, representation learning with highly reduced vectors is more interpretable and allows us to analyze the causal effect between individual latent variables and linguistically meaningful units in the output of the synthesis/production part of the model (Section \ref{flip}). We can also examine the causal effect between linguistically meaningful units in the classifier's input and the classifier's output in the perception/recognition part of the model (Section \ref{flr}).
Reduced vectors also enable analysis of the interaction between individual latent variables. For example, each element (bit) in a binary code (e.g., [1, 0], [0, 1], [1, 1]) can be analyzed as a feature $\phi_n$ (e.g.~[$\phi_1$, $\phi_2$]). Such encoding allows both holistic representation learning and featural representation learning. We can test whether each unique code corresponds to unique lexical semantics and how individual features in binary codes ([$\phi_1$, $\phi_2$]) interact/represent sublexical information (e.g.~the presence of a phoneme; Section \ref{sublexical}).
Second, humans acquire speech production and perception with a high degree of mutual influence \cite{vihman15}.
Modeling production (synthesis) and perception (recognition) simultaneously will help us build more dynamic and adaptive systems of human speech communication that are closer to reality than current models which treat the two components separately.
Third, the paper tests learning of linguistically meaningful representations in one of the most challenging training settings. Results from such experiments test the limits of deep learning architectures for speech processing.
Fourth, unsupervised ASR \cite{baevski21} and ``textless NLP'' \cite{lakhotia21} have the potential to enable speech technology in a number of languages that feature rich phonological systems. Most deep generative models for unsupervised learning focus exclusively on either lexical (see above) or phonetic learning \cite{eloff19,shain19} and do not model phonological learning. Exploring how the two levels interact will be increasingly important as speech technology becomes available in languages other than English.
Finally, speech technology is shifting towards unsupervised learning \cite{baevski21}. Our understanding of how biases in data are encoded in unsupervised models is even more poorly understood than in supervised models. The paper proposes a way to test how linguistically meaningful units self-emerge in fully unsupervised models for word learning.
Speech carries a lot of potentially harmful social information \cite{holliday21}; a better understanding of how linguistically meaningful units self-emerge and get encoded and how they interact with other features in the data is the first step towards mitigating the risks of unsupervised deep generative ASR models.
\begin{figure}\centering
\includegraphics[width=.5\textwidth]{FiwganFigure.pdf}
\caption{The fiwGAN network \cite{begusCiw} in training and test tasks.}
\label{architecture}
\end{figure}
\section{Models}
We use Categorical InfoWaveGAN (ciwGAN) and Featural InfoWaveGAN (fiwGAN) architectures (\cite{begusCiw}; based on WaveGAN in \cite{donahue19} and InfoGAN in \cite{chen16}). In short, the ciwGAN/fiwGAN models each contain three networks: a Generator $G$ that upsamples from random noise $z$ and a latent code $c$ to audio data using 1D transpose convolutions, a Discriminator $D$ that facilitates the minimization of the Wasserstein distance between the distribution of the generated outputs $G(z, c)$ and real outputs $x$ using traditional 1D convolutions, and a Q-network $Q$ mirroring the Discriminator architecture that aims to recover $c$ given generated output $G(z, c)$. As in the traditional GAN framework, the Generator and Discriminator operate on the same loss in a zero-sum game, forcing the Generator to create outputs similar to the training data. However, the Generator (along with the Q-network) is additionally trained to minimize the negative log-likelihood of the Q-network, forcing the Generator to maximize the mutual information between the latent code $c$ and generated output $G(z, c)$ and the Q-network to recover the relationship between $c$ and $G(z, c)$. CiwGAN models $c$ as a one-hot vector of several classes, while fiwGAN models $c$ as a vector using a binary encoding.
Previous work on ciwGAN and fiwGAN \cite{begusCiw} has focused on the ability of the Generator to learn meaningful representations in $c$ that encodes phonological processes and lexical learning, with no exploration of the Q-network. In this paper, we focus on the Q-network's propensity for lexical learning. Towards this end, we maintain the architecture of a separate Q-network (in contrast to the original InfoGAN proposal, where $Q$ is estimated by appending additional hidden layers after the convolutional layers of the Discriminator). This allows us to simultaneously model speech recognition using the Q-network and speech synthesis using the Generator.
\section{Experiments}
We train three networks: one using the one-hot (ciwGAN) architecture on 8 lexical items from TIMIT, one with the binary code (fiwGAN) architecture on 8 lexical items from TIMIT. To test how the proposed architecture scales up to larger corpora, we also train a fiwGAN network on 508 lexical items from LibriSpeech \cite{librispeech}.\footnote{Checkpoints and data: \url{doi.org/10.17605/OSF.IO/NQU5W}.\label{fn}}
\subsection{Data}
The lexical items used in 8-words models are: \textit{ask}, \textit{dark}, \textit{greasy}, \textit{oily}, \textit{rag}, \textit{year}, \textit{wash}, and \textit{water}. A total of 4,052 tokens are used in training (approximately 500 per each word). The words were sliced from TIMIT and padded with silence into 1.024s .wav files with 16kHz sampling rate which the Discriminator takes as its input.
In the LibriSpeech experiment, 508 words were chosen. We discarded the 78 most common lexical items in the LibriSpeech train-clean-360 dataset \cite{librispeech} because of their disproportionate high frequency (5,290 to 224,173 tokens per word). We then arbitrarily choose the 508 next most common words for training, resulting in a total of 757,120 tokens. The individual counts for each word in the training set ranges from 571 to 5,113 tokens.
\subsection{Perception/classification}
To test if the Q-network is successful in learning to classify lexical items without ever accessing training data, we take the trained Q-network from the architecture (in Figure \ref{architecture}) and feed it novel, unobserved data. In other words, we test if the Q-network can correctly classify novel lexical items by assigning each lexical item a unique code.
Altogether 1,067 test data in raw waveforms from unobserved TIMIT were fed to the Q-network (both in the ciwGAN and fiwGAN architectures). The raw output of this experiment are pairs of words with their TIMIT transcription and the unique code that the Q-network outputs in its final layer. We test the performance of the models using inferential statistics rather than comparison to existing models due to the lack of models with similarly challenging learning objectives.
To perform hypothesis testing on whether lexical learning emerges in the Q-network, we fit the word/code pairs to a multinomial logistic regression model with the \textit{nnet} package \cite{nnet}.
In the ciwGAN setting (one-hot encoding), AIC of a model with $c$ as a predictor is substantially lower ($2129.1, df=56$) than the empty model ($4448.2, df=7$). Figure \ref{fiwganFigure} gives predicted values for each code/word. The figure suggests that most words (with some exceptions especially in the fiwGAN model) have a clear and substantial rise in estimates for a single unique code. This suggests that the Q-network learns to classify novel unobserved TIMIT words into classes that correspond to lexical items.
Lexical learning emerges in the binary encoding (fiwGAN) as well, but the code vector is even more reduced in this architecture (3 variables total), which makes error rates higher compared to the ciwGAN architecture (Figure \ref{fiwganFigure}).
\begin{figure}\centering
\includegraphics[width=.27\textwidth]{qZciwfiwEff.pdf}
\caption{Estimates of a multinomial regression model.}
\label{fiwganFigure}
\end{figure}
\subsection{Production/synthesis}
To test the production (synthesis) aspect of the model, we generate 100 outputs for each unique latent code $c$ both in the ciwGAN and fiwGAN setting (1,600 outputs total). According to \cite{begus19,begus2020identity,begusCiw}, setting latent codes to marginal values outside of the training range while keeping the rest of the latent space constant reveals the underlying value of each latent code, which is why we generate data with code variables set at 3 (e.g.~[0, 0, 3], [0, 3, 3], etc). One hundred outputs per each code for each model (ciwGAN and fiwGAN) were analyzed by a compensated trained phonetician who was not a co-author on this paper. The annotator annotated generated outputs as either featuring the eight lexical items, deviating from the eight items (annotated as \textit{else}), or as unintelligible outputs (also \textit{else}). For coding of annotations, see online data in fn.~\ref{fn}.
Code variables are significant predictors of generated words according to the AIC test in both models. The learned representations are very similar and mostly match across the Q-network and in the Generator. One advantage of the Generator network is that we can force categorical or near categorical outputs by manipulating latent variables to marginal values outside of training range (e.g.~in our case to 3). For example, \textit{greasy} has 100\% success rate in ciwGAN; \textit{water} 99\% in fiwGAN and 96\% in ciwGAN.
\section{Holistic and featural learning}
\label{sublexical}
Binary encoding allows simultaneous holistic encoding of lexical semantic information (unique code = lexical item) as well as featural learning, where features (bits) correspond to sublexical units such as phonemes (e.g.~[s] or [\textipa{S}]). This paper proposes a technique to explore lexical and sublexical learned representations in a classifier network. To test whether evidence for sublexical learning emerges in the perception aspect of the proposed model, we annotate inputs to the Q-network for any sublexical property and use regression analysis with each feature (bit) as a predictor to test how individual features correspond to that property.
\subsection{TIMIT}
We focus on one of the the most phonetically salient sublexical properties in the training data: presence of a fricative [s], [\textipa{S}]. We include the word for \textit{dark} among the words containing [s] because a high proportion of \textit{dark} tokens feature [s] frication (due to \textit{dark} standing before \textit{suit} in TIMIT). The data were fit to a logistic regression linear model with presence of [s] in the input test data as the dependent variable and the three features ($\phi_1$, $\phi_2$, $\phi_3$) as predictors. Estimates of the regression model suggest that the network encodes a sublexical phonemic property (presence of frication noise of [s]) with $\phi_3 = 0$ ($\beta=-0.5,z= -2.9, p=0.004$ for $\phi_1$, $\beta= 0.2,z= 1.0,p= 0.3$, for $\phi_2$ and $\beta= -2.5,z= -13.8,p<0.0001$ for $\phi_3$).
\subsection{LibriSpeech}
To test how the proposed technique of unsupervised lexical and sublexical learning extends to larger corpora, we test the Q-network trained on 508 lexical items from LibriSpeech. The model has 9 latent feature variables $\phi$ which yields $2^9=512$ classes. Altogether 10,914 test tokens (withheld from training) of the 508 unique words were fed to the Q-network in fiwGAN architecture trained for 61,707 steps.
\subsubsection{Holistic representation learning}
First, raw classification of outputs suggest that holistic lexical learning in the Q-network emerges even when the training data contains a substantially larger set (508 items and a total of 757,120 tokens) and a more diverse corpus. The training data here too is twice removed from the Q-network and the test data was never part of the training. Figure \ref{wellstill} illustrates four chosen words and the codes with which they are represented. Each word features a peak in one unique code. To verify that this particular code indeed represents that particular word, we also analyze which other words are classified with the most frequent code for each of the four chosen word. There too, each code represents one word more strongly.
\begin{figure}[t]\centering
\includegraphics[width=.3\textwidth]{wellstill.pdf}
\includegraphics[width=.3\textwidth]{interpolationQ6Ggplot.pdf}
\caption{(top left) Raw counts of code distribution per each of the four chosen tested words (from unobserved test data). The code with highest count is color-coded in red. (top right) Raw counts of all words classified with the code that has the highest count for each word from the left graph. Words that were never classified with this code are not on the graph. The word with highest count is color-coded in red. (bottom)
Outputs of the Generator network (waveforms) when $\phi_2$, $\phi_3$, and $\phi_5$ are simultaneously interpolated from 0.0 to 0.8 while all other latent variables are held constant. }
\label{wellstill}
\end{figure}
To test how common such well-learned representations are, we randomly selected 20 out of the 508 words from LibriSpeech, which includes words that occur extremely infrequently (e.g.~$\mathrm{N}=7$) in both the train and test sets. Of the 20 randomly selected words, 4 (20\%) have representations where the code most frequently assigned to a word (peak in Figure \ref{wellstill} top left) also has the highest count (peak in Figure \ref{wellstill} top right) of that same word when compared to all words labeled with that code (e.g.~011001110 is most common code given to \textit{well}, and \textit{well} is the most common word that is labelled as 011001110; Figure \ref{wellstill}).
In 5 further cases (25\%), two or more peaks have the same, but not higher counts than the word/code peak pair (for a total of 45\% of successful outcomes if both groups are counted as successful). In the remaining 55\% (11 items), the peaks do not match across the word/code pairs. We counted one case with all counts equal across the word/code pair as unsuccessful.
These counts are fully deterministic and therefore conservative. The distribution of code variables per each word are, however, not independent. For example, the second most common code for \textit{mister} in Figure \ref{wellstill} differs from the most common one in only one feature (bit). Violation in a single feature value is equally treated as violation in multiple feature values in our counts.
Likewise, there is substantial amount of phonetic similarity in words classified by a single code. For example, the word most commonly classified with [100000001] is indeed \textit{still}, but other frequent words for this classification code are \textit{state}, \textit{stand}, \textit{stood}, \textit{story}, etc. (Figure \ref{wellstill}).
\subsubsection{Featural representation learning}
\label{flr}
These similarities suggest that the network encodes sublexical properties using individual features in the binary code. To quantitatively test this hypothesis, we test how the network encodes presence of word-initial [\#s]. Frication noise of [s] is a phonetically salient property and restricting it to word-initial position allows us to test featural and positional (temporal) encoding.
Librispeech word/Q-network code pairs are annotated for presence of word-initial [s] (dependent variable) and fit to a logistic regression linear model with the nine feature variables $\phi_{1-9}$ (bits) as independent predictors.
Three features ($\phi_2$, $\phi_3$, and $\phi_5$) correspond to presence of initial [\#s] substantially more strongly than other features. It is reasonable to assume that the network encodes this sublexical contrast with the value of the three features ($\phi_2$, $\phi_3$, $\phi_5$) at 0. It would be efficient if the network encodes word-initial [\#s] with 3 features, because there are approximately 54 s-initial words. The 6 feature codes remaining besides $\phi_2$, $\phi_3$, $\phi_5$ allows for $2^6=64$ classes.
To verify this hypothesis, the presence of [\#s] in input words (dependent variable) is fit to a logistic regression model with only one predictor: the value of the three features $\phi_2$, $\phi_3$, $\phi_5$ with two levels: 0 and 1. Only 5.0\% [4.6\%, 5.5\%] of words classified with $\phi_2$, $\phi_3$, and $\phi_5$ = 1 contain word-initial [\#s], while 47.9\% [44.3\%, 51.5\%] of words classified as $\phi_2$, $\phi_3$, and $\phi_5$ = 0 contain word-initial [\#s].
\subsection{Featural learning in production}
\label{flip}
The fiwGAN architecture allows us to test both holistic and featural learning in both production and perception. Value 0 for $\phi_2$, $\phi_3$, $\phi_5$ has been associated with word-initial [\#s] in the Q-network (perception). To test whether the Generator matches the Q-network in this sublexical encoding, we generate sets of outputs in which all other $\phi$ variables (except $\phi_2$, $\phi_3$, and $\phi_5$) and all $z$-variables are held constant, but the $\phi_2$, $\phi_3$, and $\phi_5$ variables are interpolated from 0 to 3 in intervals of 0.2. We analyze 20 such outputs (where the other $\phi$ variables and $z$-variables are sampled randomly for each of the 20 sets).
In 11 out of the 20 generated sets (or 55\%), word-initial [\#s] appears in the output for code $\phi_2$, $\phi_3$, $\phi_5$ = 0 and then disappears from the output as the value is interpolated (annotated by the authors because presence of [s] is a highly salient feature). Additionally, in the majority of these cases (approximately 8), the change from [\#s] to some other word-initial consonant is the only major change that happens as the output transitions from [s] to no [s] with interpolation. In other words, as we interpolate values of the three features representing [\#s], we observe a causal effect in the generated outputs as [\#s] gradually changes into a different consonant with other major acoustic properties remaining the same in the majority of cases. Figure \ref{wellstill} illustrates this causal effect: the amplitude of the frication noise of [\#s] gradually attenuates with interpolation, while other acoustic properties remain largely unchanged. The sublexical encoding of word-initial [\#s] is thus causally represented with the same code both in the Generator network and in the Q-network.
\section{Conclusion}
This paper demonstrates that a deep neural architecture that simultaneously models the production/synthesis and perception/classification learns linguistically meaningful units --- lexical items and sublexical properties --- from raw acoustic data in a fully unsupervised manner.
We also argue that we can simultaneously model holistic lexical representation learning (in the form of unique binary codes) and sublexical (phonetic and phonological) representations in the form of individual feature codes (bits) in the fiwGAN architecture.
\bibliographystyle{IEEEtran}
\section{Introduction}
Speech technology has traditionally been divided into automated speech recognition (ASR) and speech synthesis. Hearing humans, however, perform both tasks --- speech production and speech perception --- with a high degree of mutual influence (the so-called production-perception loop; \cite{vihman15}).
This paper proposes that the two principles should be modeled simultaneously and argues that a GAN-based model called ciwGAN/fiwGAN \cite{begusCiw} learns linguistically meaningful representations for both production and perception. In fact, lexical learning in the architecture emerges precisely from the requirement that the network for production and the network for perception interact and generate data that is mutually informative. We show that with only the requirement to produce informative data, the models not only produce desired outputs (as argued in \cite{begusCiw}), but also learn to classify lexical items in a fully unsupervised way from raw unlabeled speech.
\subsection{Prior work}
Most of the existing models of lexical learning focus primarily on either ASR/speech-to-text (perception) or text-to-speech/speech synthesis (production; see \cite{wali22} for an overview). Variational Autoencoders (VAEs) involve both an encoder and decoder, which allows unsupervised acoustic word embedding as well as generation of speech, but these proposals only use VAEs for either unsupervised ASR \cite{chung16,chorowski19,baevski20,niekerk20} or for speech synthesis/transformation (e.g.~\cite{hsu17}). Earlier neural models replicate brain mechanisms behind perception and production \cite{guenther11}, but they do not focus on lexical learning or classification and do not include recent progress in performance of deep learning architectures. GAN-based synthesizers are mostly supervised and get text or acoustic features in their input \cite{kumar19,kong20,binkowski20,cong21}. \cite{donahue19} propose a WaveGAN architecture, which can generate any audio in an unsupervised manner, but does not involve a lexical classifier --- only the Generator and the Discriminator, which means the model only captures synthesis and not classification (the same is true for Parallel WaveGAN; \cite{yamamoto20}). \cite{begusCiw} proposes the first textless fully unsupervised GAN-based model for lexical representation learning, but evaluates only the synthesis (production) aspect of their model by only evaluating outputs of the Generator network.
\subsection{New challenges}
Here, we model lexical learning with a classifier network (the Q-network) that mimics perception and lexical learning and is, crucially, trained from another network's production data (the Generator network). Using this architecture, we can \textit{both} generate new words in a controlled causal manner by manipulating the Generator's latent space \textit{as well as} classify novel words from unobserved test data with a classifier that never directly accesses the training data.
This paper also introduces some crucial new challenges to the unsupervised acoustic word embedding and word recognition paradigm \cite{dunbar17}. First, the architecture enables extremely reduced vector representations of lexical items. In fiwGAN, the network needs to represent $2^n$ classes with only $n$ variables. To our knowledge, no other proposal features such dense representation of acoustic lexical items. Second, the models introduce a challenge to learn meaningful representations of words without ever directly accessing training data. The lexical classifier network is twice removed from training data. The Q-network learns to classify words only from the Generator's outputs and never accesses training data directly. But the Generator never accesses the training data directly either --- it learns to produce words only by maximizing the Discriminator's error rate.
Why are these challenges important? First, representation learning with highly reduced vectors is more interpretable and allows us to analyze the causal effect between individual latent variables and linguistically meaningful units in the output of the synthesis/production part of the model (Section \ref{flip}). We can also examine the causal effect between linguistically meaningful units in the classifier's input and the classifier's output in the perception/recognition part of the model (Section \ref{flr}).
Reduced vectors also enable analysis of the interaction between individual latent variables. For example, each element (bit) in a binary code (e.g., [1, 0], [0, 1], [1, 1]) can be analyzed as a feature $\phi_n$ (e.g.~[$\phi_1$, $\phi_2$]). Such encoding allows both holistic representation learning and featural representation learning. We can test whether each unique code corresponds to unique lexical semantics and how individual features in binary codes ([$\phi_1$, $\phi_2$]) interact/represent sublexical information (e.g.~the presence of a phoneme; Section \ref{sublexical}).
Second, humans acquire speech production and perception with a high degree of mutual influence \cite{vihman15}.
Modeling production (synthesis) and perception (recognition) simultaneously will help us build more dynamic and adaptive systems of human speech communication that are closer to reality than current models which treat the two components separately.
Third, the paper tests learning of linguistically meaningful representations in one of the most challenging training settings. Results from such experiments test the limits of deep learning architectures for speech processing.
Fourth, unsupervised ASR \cite{baevski21} and ``textless NLP'' \cite{lakhotia21} have the potential to enable speech technology in a number of languages that feature rich phonological systems. Most deep generative models for unsupervised learning focus exclusively on either lexical (see above) or phonetic learning \cite{eloff19,shain19} and do not model phonological learning. Exploring how the two levels interact will be increasingly important as speech technology becomes available in languages other than English.
Finally, speech technology is shifting towards unsupervised learning \cite{baevski21}. Our understanding of how biases in data are encoded in unsupervised models is even more poorly understood than in supervised models. The paper proposes a way to test how linguistically meaningful units self-emerge in fully unsupervised models for word learning.
Speech carries a lot of potentially harmful social information \cite{holliday21}; a better understanding of how linguistically meaningful units self-emerge and get encoded and how they interact with other features in the data is the first step towards mitigating the risks of unsupervised deep generative ASR models.
\begin{figure}\centering
\includegraphics[width=.5\textwidth]{FiwganFigure.pdf}
\caption{The fiwGAN network \cite{begusCiw} in training and test tasks.}
\label{architecture}
\end{figure}
\section{Models}
We use Categorical InfoWaveGAN (ciwGAN) and Featural InfoWaveGAN (fiwGAN) architectures (\cite{begusCiw}; based on WaveGAN in \cite{donahue19} and InfoGAN in \cite{chen16}). In short, the ciwGAN/fiwGAN models each contain three networks: a Generator $G$ that upsamples from random noise $z$ and a latent code $c$ to audio data using 1D transpose convolutions, a Discriminator $D$ that facilitates the minimization of the Wasserstein distance between the distribution of the generated outputs $G(z, c)$ and real outputs $x$ using traditional 1D convolutions, and a Q-network $Q$ mirroring the Discriminator architecture that aims to recover $c$ given generated output $G(z, c)$. As in the traditional GAN framework, the Generator and Discriminator operate on the same loss in a zero-sum game, forcing the Generator to create outputs similar to the training data. However, the Generator (along with the Q-network) is additionally trained to minimize the negative log-likelihood of the Q-network, forcing the Generator to maximize the mutual information between the latent code $c$ and generated output $G(z, c)$ and the Q-network to recover the relationship between $c$ and $G(z, c)$. CiwGAN models $c$ as a one-hot vector of several classes, while fiwGAN models $c$ as a vector using a binary encoding.
Previous work on ciwGAN and fiwGAN \cite{begusCiw} has focused on the ability of the Generator to learn meaningful representations in $c$ that encodes phonological processes and lexical learning, with no exploration of the Q-network. In this paper, we focus on the Q-network's propensity for lexical learning. Towards this end, we maintain the architecture of a separate Q-network (in contrast to the original InfoGAN proposal, where $Q$ is estimated by appending additional hidden layers after the convolutional layers of the Discriminator). This allows us to simultaneously model speech recognition using the Q-network and speech synthesis using the Generator.
\section{Experiments}
We train three networks: one using the one-hot (ciwGAN) architecture on 8 lexical items from TIMIT, one with the binary code (fiwGAN) architecture on 8 lexical items from TIMIT. To test how the proposed architecture scales up to larger corpora, we also train a fiwGAN network on 508 lexical items from LibriSpeech \cite{librispeech}.\footnote{Checkpoints and data: \url{doi.org/10.17605/OSF.IO/NQU5W}.\label{fn}}
\subsection{Data}
The lexical items used in 8-words models are: \textit{ask}, \textit{dark}, \textit{greasy}, \textit{oily}, \textit{rag}, \textit{year}, \textit{wash}, and \textit{water}. A total of 4,052 tokens are used in training (approximately 500 per each word). The words were sliced from TIMIT and padded with silence into 1.024s .wav files with 16kHz sampling rate which the Discriminator takes as its input.
In the LibriSpeech experiment, 508 words were chosen. We discarded the 78 most common lexical items in the LibriSpeech train-clean-360 dataset \cite{librispeech} because of their disproportionate high frequency (5,290 to 224,173 tokens per word). We then arbitrarily choose the 508 next most common words for training, resulting in a total of 757,120 tokens. The individual counts for each word in the training set ranges from 571 to 5,113 tokens.
\subsection{Perception/classification}
To test if the Q-network is successful in learning to classify lexical items without ever accessing training data, we take the trained Q-network from the architecture (in Figure \ref{architecture}) and feed it novel, unobserved data. In other words, we test if the Q-network can correctly classify novel lexical items by assigning each lexical item a unique code.
Altogether 1,067 test data in raw waveforms from unobserved TIMIT were fed to the Q-network (both in the ciwGAN and fiwGAN architectures). The raw output of this experiment are pairs of words with their TIMIT transcription and the unique code that the Q-network outputs in its final layer. We test the performance of the models using inferential statistics rather than comparison to existing models due to the lack of models with similarly challenging learning objectives.
To perform hypothesis testing on whether lexical learning emerges in the Q-network, we fit the word/code pairs to a multinomial logistic regression model with the \textit{nnet} package \cite{nnet}.
In the ciwGAN setting (one-hot encoding), AIC of a model with $c$ as a predictor is substantially lower ($2129.1, df=56$) than the empty model ($4448.2, df=7$). Figure \ref{fiwganFigure} gives predicted values for each code/word. The figure suggests that most words (with some exceptions especially in the fiwGAN model) have a clear and substantial rise in estimates for a single unique code. This suggests that the Q-network learns to classify novel unobserved TIMIT words into classes that correspond to lexical items.
Lexical learning emerges in the binary encoding (fiwGAN) as well, but the code vector is even more reduced in this architecture (3 variables total), which makes error rates higher compared to the ciwGAN architecture (Figure \ref{fiwganFigure}).
\begin{figure}\centering
\includegraphics[width=.27\textwidth]{qZciwfiwEff.pdf}
\caption{Estimates of a multinomial regression model.}
\label{fiwganFigure}
\end{figure}
\subsection{Production/synthesis}
To test the production (synthesis) aspect of the model, we generate 100 outputs for each unique latent code $c$ both in the ciwGAN and fiwGAN setting (1,600 outputs total). According to \cite{begus19,begus2020identity,begusCiw}, setting latent codes to marginal values outside of the training range while keeping the rest of the latent space constant reveals the underlying value of each latent code, which is why we generate data with code variables set at 3 (e.g.~[0, 0, 3], [0, 3, 3], etc). One hundred outputs per each code for each model (ciwGAN and fiwGAN) were analyzed by a compensated trained phonetician who was not a co-author on this paper. The annotator annotated generated outputs as either featuring the eight lexical items, deviating from the eight items (annotated as \textit{else}), or as unintelligible outputs (also \textit{else}). For coding of annotations, see online data in fn.~\ref{fn}.
Code variables are significant predictors of generated words according to the AIC test in both models. The learned representations are very similar and mostly match across the Q-network and in the Generator. One advantage of the Generator network is that we can force categorical or near categorical outputs by manipulating latent variables to marginal values outside of training range (e.g.~in our case to 3). For example, \textit{greasy} has 100\% success rate in ciwGAN; \textit{water} 99\% in fiwGAN and 96\% in ciwGAN.
\section{Holistic and featural learning}
\label{sublexical}
Binary encoding allows simultaneous holistic encoding of lexical semantic information (unique code = lexical item) as well as featural learning, where features (bits) correspond to sublexical units such as phonemes (e.g.~[s] or [\textipa{S}]). This paper proposes a technique to explore lexical and sublexical learned representations in a classifier network. To test whether evidence for sublexical learning emerges in the perception aspect of the proposed model, we annotate inputs to the Q-network for any sublexical property and use regression analysis with each feature (bit) as a predictor to test how individual features correspond to that property.
\subsection{TIMIT}
We focus on one of the the most phonetically salient sublexical properties in the training data: presence of a fricative [s], [\textipa{S}]. We include the word for \textit{dark} among the words containing [s] because a high proportion of \textit{dark} tokens feature [s] frication (due to \textit{dark} standing before \textit{suit} in TIMIT). The data were fit to a logistic regression linear model with presence of [s] in the input test data as the dependent variable and the three features ($\phi_1$, $\phi_2$, $\phi_3$) as predictors. Estimates of the regression model suggest that the network encodes a sublexical phonemic property (presence of frication noise of [s]) with $\phi_3 = 0$ ($\beta=-0.5,z= -2.9, p=0.004$ for $\phi_1$, $\beta= 0.2,z= 1.0,p= 0.3$, for $\phi_2$ and $\beta= -2.5,z= -13.8,p<0.0001$ for $\phi_3$).
\subsection{LibriSpeech}
To test how the proposed technique of unsupervised lexical and sublexical learning extends to larger corpora, we test the Q-network trained on 508 lexical items from LibriSpeech. The model has 9 latent feature variables $\phi$ which yields $2^9=512$ classes. Altogether 10,914 test tokens (withheld from training) of the 508 unique words were fed to the Q-network in fiwGAN architecture trained for 61,707 steps.
\subsubsection{Holistic representation learning}
First, raw classification of outputs suggest that holistic lexical learning in the Q-network emerges even when the training data contains a substantially larger set (508 items and a total of 757,120 tokens) and a more diverse corpus. The training data here too is twice removed from the Q-network and the test data was never part of the training. Figure \ref{wellstill} illustrates four chosen words and the codes with which they are represented. Each word features a peak in one unique code. To verify that this particular code indeed represents that particular word, we also analyze which other words are classified with the most frequent code for each of the four chosen word. There too, each code represents one word more strongly.
\begin{figure}[t]\centering
\includegraphics[width=.3\textwidth]{wellstill.pdf}
\includegraphics[width=.3\textwidth]{interpolationQ6Ggplot.pdf}
\caption{(top left) Raw counts of code distribution per each of the four chosen tested words (from unobserved test data). The code with highest count is color-coded in red. (top right) Raw counts of all words classified with the code that has the highest count for each word from the left graph. Words that were never classified with this code are not on the graph. The word with highest count is color-coded in red. (bottom)
Outputs of the Generator network (waveforms) when $\phi_2$, $\phi_3$, and $\phi_5$ are simultaneously interpolated from 0.0 to 0.8 while all other latent variables are held constant. }
\label{wellstill}
\end{figure}
To test how common such well-learned representations are, we randomly selected 20 out of the 508 words from LibriSpeech, which includes words that occur extremely infrequently (e.g.~$\mathrm{N}=7$) in both the train and test sets. Of the 20 randomly selected words, 4 (20\%) have representations where the code most frequently assigned to a word (peak in Figure \ref{wellstill} top left) also has the highest count (peak in Figure \ref{wellstill} top right) of that same word when compared to all words labeled with that code (e.g.~011001110 is most common code given to \textit{well}, and \textit{well} is the most common word that is labelled as 011001110; Figure \ref{wellstill}).
In 5 further cases (25\%), two or more peaks have the same, but not higher counts than the word/code peak pair (for a total of 45\% of successful outcomes if both groups are counted as successful). In the remaining 55\% (11 items), the peaks do not match across the word/code pairs. We counted one case with all counts equal across the word/code pair as unsuccessful.
These counts are fully deterministic and therefore conservative. The distribution of code variables per each word are, however, not independent. For example, the second most common code for \textit{mister} in Figure \ref{wellstill} differs from the most common one in only one feature (bit). Violation in a single feature value is equally treated as violation in multiple feature values in our counts.
Likewise, there is substantial amount of phonetic similarity in words classified by a single code. For example, the word most commonly classified with [100000001] is indeed \textit{still}, but other frequent words for this classification code are \textit{state}, \textit{stand}, \textit{stood}, \textit{story}, etc. (Figure \ref{wellstill}).
\subsubsection{Featural representation learning}
\label{flr}
These similarities suggest that the network encodes sublexical properties using individual features in the binary code. To quantitatively test this hypothesis, we test how the network encodes presence of word-initial [\#s]. Frication noise of [s] is a phonetically salient property and restricting it to word-initial position allows us to test featural and positional (temporal) encoding.
Librispeech word/Q-network code pairs are annotated for presence of word-initial [s] (dependent variable) and fit to a logistic regression linear model with the nine feature variables $\phi_{1-9}$ (bits) as independent predictors.
Three features ($\phi_2$, $\phi_3$, and $\phi_5$) correspond to presence of initial [\#s] substantially more strongly than other features. It is reasonable to assume that the network encodes this sublexical contrast with the value of the three features ($\phi_2$, $\phi_3$, $\phi_5$) at 0. It would be efficient if the network encodes word-initial [\#s] with 3 features, because there are approximately 54 s-initial words. The 6 feature codes remaining besides $\phi_2$, $\phi_3$, $\phi_5$ allows for $2^6=64$ classes.
To verify this hypothesis, the presence of [\#s] in input words (dependent variable) is fit to a logistic regression model with only one predictor: the value of the three features $\phi_2$, $\phi_3$, $\phi_5$ with two levels: 0 and 1. Only 5.0\% [4.6\%, 5.5\%] of words classified with $\phi_2$, $\phi_3$, and $\phi_5$ = 1 contain word-initial [\#s], while 47.9\% [44.3\%, 51.5\%] of words classified as $\phi_2$, $\phi_3$, and $\phi_5$ = 0 contain word-initial [\#s].
\subsection{Featural learning in production}
\label{flip}
The fiwGAN architecture allows us to test both holistic and featural learning in both production and perception. Value 0 for $\phi_2$, $\phi_3$, $\phi_5$ has been associated with word-initial [\#s] in the Q-network (perception). To test whether the Generator matches the Q-network in this sublexical encoding, we generate sets of outputs in which all other $\phi$ variables (except $\phi_2$, $\phi_3$, and $\phi_5$) and all $z$-variables are held constant, but the $\phi_2$, $\phi_3$, and $\phi_5$ variables are interpolated from 0 to 3 in intervals of 0.2. We analyze 20 such outputs (where the other $\phi$ variables and $z$-variables are sampled randomly for each of the 20 sets).
In 11 out of the 20 generated sets (or 55\%), word-initial [\#s] appears in the output for code $\phi_2$, $\phi_3$, $\phi_5$ = 0 and then disappears from the output as the value is interpolated (annotated by the authors because presence of [s] is a highly salient feature). Additionally, in the majority of these cases (approximately 8), the change from [\#s] to some other word-initial consonant is the only major change that happens as the output transitions from [s] to no [s] with interpolation. In other words, as we interpolate values of the three features representing [\#s], we observe a causal effect in the generated outputs as [\#s] gradually changes into a different consonant with other major acoustic properties remaining the same in the majority of cases. Figure \ref{wellstill} illustrates this causal effect: the amplitude of the frication noise of [\#s] gradually attenuates with interpolation, while other acoustic properties remain largely unchanged. The sublexical encoding of word-initial [\#s] is thus causally represented with the same code both in the Generator network and in the Q-network.
\section{Conclusion}
This paper demonstrates that a deep neural architecture that simultaneously models the production/synthesis and perception/classification learns linguistically meaningful units --- lexical items and sublexical properties --- from raw acoustic data in a fully unsupervised manner.
We also argue that we can simultaneously model holistic lexical representation learning (in the form of unique binary codes) and sublexical (phonetic and phonological) representations in the form of individual feature codes (bits) in the fiwGAN architecture.
\bibliographystyle{IEEEtran}
| {
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Directors, Drama, Plays, Politics, Society, Theatre
Fancasting British Literature: Sarah Kane's Blasted
Blasted was Kane's debut play. Set in an upmarket hotel room in Leeds (England), the play is about the violence that befalls a dying journalist and his rampant rape of Cate, a mentally troubled young woman suffering from fits. Its exploration of domestic violence is broadened into a bloody spectacle when the room is moulded into a warzone.
Main themes: war, sexual violence, race, gender, society
Ironically, critics blasted this play at its first outing at the Royal Court in 1995. The criticism towards it makes reviews for films like Batman V Superman, Ghostbusters (2016) and The Emoji Movie look pleasant.
Respected newspapers, television shows and anyone else with an opinion took pleasure in tearing it to shreds:
"This disgusting feast of filth" – Daily Mail
"A sordid little travesty of a play" – The Spectator.
"A gratuitous welter of carnage" – The Telegraph
Imagine if the same criticism occurred today but with social media. Social media changed the game for reviews, especially Twitter. With something like this, Twitter would have a field day!
In essence, the response to this play was like throwing a fifty-inch television from a hotel window that's twenty stories from the ground.
My Cast
Director: Ben Wheatley (Free Fire)
Screenwriter: Ben Wheatley & Amy Jump (Sightseers)
Producer: Simon Pegg (Hot Fuzz)
Cinematographer: Laurie Rose (Kill List)
Musical Score: Dan Jones (Lady Macbeth)
Distributor: Film4
After Sarah Kane
After the fact, now over twenty years later, Sarah Kane's Blasted is now seen in a more favourable light. It's a play that changed the landscape for theatre and is now being taught to students up and down the United Kingdom.
It continues to be on the BA Creative Writing syllabus in Britain, and used when discussing gender (both male and female) and Marxism. Let's not forget to mention history. The history of humanity is the history of war. And Blasted is an analysis of humanity, through horror.
Though respected abroad, she remained a misunderstood playwright in Britain. The dust began to settle with Crave in 1998. But with her sudden death, her work began to become target practice for journalists and others to probe and speculate about her personal life.
Despite the sheering reception at home, Sarah Kane was given a home in European theatre. Blasted was soon seen as one of the most important British plays of the 1990s. Her plays were produced throughout the Continent. She then commited suicide in February 1999 at twenty-eight years old.
"I'm simply trying to tell the truth about human behaviour as I see it"
BlastedSarah Kane
Fancasting British Literature: Daphne du Maurier's Rebecca
Fancasting British Literature: Martin Amis' Money | {
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Haggard este o formație germană de metal simfonic. În primii ani (1991-1994) a abordat stilul death metal, urmând ca apoi să se axeze pe stilul care i-a consacrat, un stil destul de greu de definit (metal simfonic cu puternice influențe clasice, medievale, dar și de death metal și doom metal).
Discografie
Demo-uri
Introduction (1992)
Progressive (1994)
Once... Upon A December´s Dawn (1995)
Albume
And Thou Shalt Trust... the Seer (1997)
Awaking the Centuries (2000)
Awaking the Gods: Live In Mexico (2001)
Eppur Si Muove (2004)
Tales of Ithiria (2008)
DVD-uri și casete video
In A Pale Moon's Shadow (VHS) (1998)
Awaking the Gods: Live In Mexico (DVD/VHS) (2001)
Legături externe
Site oficial
Haggard Pe Myspace
Haggard in concert la Bucuresti (18.04.2010)
Formații rock germane
Formații symphonic metal | {
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Retaining its endearing coastal charm throughout with fresh contemporary enhancements, this warm and bright beach house is an idyllic family retreat. Placed on generous 539sqm with new swimming pool bathed in Northerly sunlight, it is peacefully nestled in the heart of Chirn Park, moments from cafes, restaurants and shops, the Broadwater is only a short breath away.
Promising family comfort and privacy, and designed for relaxed living and entertaining, this home offers renovated interiors opening to the all weather outdoor living and pool area adjoining a bonus self-contained granny flat / pool house / retreat. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/math.stackexchange.com\/questions\/2530861\/split-conormal-sequence-in-kahler-differentials-where-is-the-flaw-in-reasoning","text":"# Split conormal sequence in Kahler differentials. Where is the flaw in reasoning?\n\nConsider lemma 10.130.10 from this Stacks Project site.\n\nLemma. Let $\\alpha:R\\to S,\\pi:S\\to T$ be two ring maps and assume that $\\pi$ is surjective. If there is a $R$-linear $\\iota:T\\to S$ such that $\\pi\\circ\\iota = 1_T$, then there is a following short split exact sequence of $T$-modules $$0\\to I\/I^2\\to\\Omega_{S\/R}\\otimes_S T\\to\\Omega_{T\/R}\\to 0,$$ where $I=\\ker(\\pi).$\n\nAssume that we have ring $K$ and two $K$-algebras $A,B$ and $K$-algebra homomorphism $\\phi:A\\to B.$ (Everything here is commutative, associated and unitial). In addition define $$\\pi:B\\otimes_K A\\to B,\\quad\\iota:B\\to B\\otimes_K A$$ respectively by $$\\pi(b\\otimes a)=\\phi(a)b,\\quad \\iota(b)=b\\otimes 1.$$ We will now apply lemma to two cases:\n\nFirst: Let $\\alpha:K\\to B\\otimes_K A$ be natural map form $K$-algebra structure of $B\\otimes_K A.$ Then $$0\\to I\/I^2\\to\\Omega_{(B\\otimes_K A)\/K}\\otimes_{(B\\otimes_K A)} B\\to\\Omega_{B\/K}\\to 0$$ is short split exact seqence of $B$-modules.\n\nSecond: Let $\\alpha:A\\to B\\otimes_K A$ be given by formula $\\alpha(a)=\\phi(a)\\otimes 1.$ Then $$0\\to I\/I^2\\to\\Omega_{(B\\otimes_K A)\/A}\\otimes_{(B\\otimes_K A)} B\\to\\Omega_{B\/A}\\to 0$$ is short split exact seqence of $B$-modules.\n\nEverything looks ok, but then we put $B=K$ then the first exact sequence degenerates to $$0\\to I\/I^2\\to\\Omega_{A\/K}\\otimes_A K\\to 0$$ and the second one to $$0\\to I\/I^2\\to 0.$$\n\nHowever $\\Omega_{A\/K}\\otimes_A K$ does not have to vanish in general.\n\nQuestion. Were is the flaw in the above reasoning?\n\nThe second exact sequence is incorrect because in case $B = K$ the map $\\alpha : A \\to A$ is not the identity $\\operatorname{id}_{A}$ but rather the composite $A \\to K \\to A$ where the first map is $\\phi$. Thus $\\Omega_{A\/A} \\simeq \\Omega_{A\/K}$.\n\u2022 You are right. Due to the misleading notation $\\Omega_{S\/R}$ I recklessly canel out some objects using algebraic properties of tensor. $\\Omega_{S\/R}$ should be rather denoted by $\\Omega_\\alpha.$ For example the second sequence is correct if you interprate $\\Omega_{(B\\otimes_K A)\/A}$ correctely. Don't you have problems with that? Or maybe you think in terms of $\\Omega_\\alpha$? Nov 21, 2017 at 18:55","date":"2022-06-29 11:03:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9877136945724487, \"perplexity\": 149.31192080288318}, \"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-2022-27\/segments\/1656103626162.35\/warc\/CC-MAIN-20220629084939-20220629114939-00139.warc.gz\"}"} | null | null |
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In 3 months, he went from having 100% over bite, which means when he bites together, you can't see any of his lower teeth. In 3 months, it opened all the way up to where his teeth are almost touching perfectly. He sleeps through the night. He stopped wetting his bed in 3 months at 4 years old. It's an amazing appliance. I know it works.
Ortho-Tain® can do a lot more than improve your child's smile.
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Due in part to this improved sleep, other children have reported that they stopped wetting the bed. Parents have also reported improvements in behavior and better performance at school.
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To improve the precision and overall experience of the treatment, we have also invested in state-of-the-art dental technology, including digital panoramic X-rays and a cephalometric imaging unit, both of which help us to diagnose airway issues.
If you would like to learn more about what Ortho-Tain South Bend, IN can do for your child, call Tulip Tree Dental Care at 574-208-5668 for an appointment. You can also contact us using our convenient online form.
Dr. Nicole Hurcomb graduated at Indiana University School of Dentistry, she became a dentist in two separate dental practice before starting her own dental practice, Tulip Tree Dental Care in 2014. Dr. Nicole Hurcomb is a member of various professional organization such as The American Dental Association, The Indiana Dental Association, The North Central Dental Society, andThe Academy of General Dentistry. | {
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\section{Introduction}
\label{sec:introduction-1}
A \emph{conformal symplectic structure} on a manifold $M$ is a generalization of a symplectic structure. Locally, a conformal symplectic manifold is equivalent to a symplectic manifold, but the local symplectic structure is only well-defined up to scaling by a constant, and the monodromy of the local symplectic structure around curves may induce these rescalings. To our knowledge, the notion was first introduced by Vaisman in \cite{Vaisman_lck} and \cite{Vaisman}. They also appeared in work of Guedira and Lichnerowicz in \cite{Guedira_lichnerow}. It was later studied by Banyaga \cite{Banyaga_propertieslcs}, among many others.\footnote{In previous literature, this structure is often called a \emph{locally} conformal symplectic structure, which is a more accurate term. But also more cumbersome.} More recently a first general existence result was given in \cite{Apo_Dlou} for complex surfaces with odd first Betti number, it was later proved in \cite{E_M_Symp_Cob} that an almost symplectic manifold with non zero first Betti number admits a conformal symplectic structure, providing a large class of examples of such structures.
More formally, we can take a number of equivalent definitions. We can say that a conformal symplectic structure on $M$ is an atlas of charts to $\mathbb{R}^{2n}$, so that the transition maps $\psi_{ij}$ satisfy $\psi_{ij}^*\omega_\text{std} = c_{ij}\omega_\text{std}$, for constants $c_{ij} > 0$. Equivalently, let $E \to M$ be a flat, orientable, real line bundle. Then a conformal symplectic structure is a $2$-form on $M$ taking values in $E$, which is closed and non-degenerate.
Taking a connection on $E$ leads to the most tractable definition. A \emph{conformal symplectic structure} on $M$ is a pair $(\eta, \omega) \in \Omega^1M \times \Omega^2M$, so that $d\eta = 0$, $d\omega = \eta \wedge \omega$, and $\omega^{\wedge n} \neq 0$. Because the choice of connection is non-canonical, $(\eta, \omega)$ defines the same conformal symplectic structure as $(\eta + df, e^f\omega)$, for any $f \in C^\infty M$. (The choice of a specific $\eta$ representing $[\eta]$ is roughly analogous to a choice of contact form for a given contact structure, $\eta$ is called a \textit{Lee form} of the conformal symplectic structure since such a form appeared in the work of Lee \cite{Lee_evendim}.)
For an example of a conformal symplectic manifold, let $\beta$ be a closed $1$-form on a manifold $Q$, let $\lambda = p\cdot dq$ be the tautological $1$-form on $T^*Q$, let $\eta \in \Omega^1(T^*Q)$ be the pullback of $\beta$ under the projection, and let $\omega = d\lambda - \eta \wedge \lambda$.
Conformal symplectic manifolds enjoy many of the properties that make symplectic manifolds interesting. The definitions of Lagrangian, isotropic, etc. are exactly the same, and there is a natural notion of exact conformal symplectic structures, and exact Lagrangians inside them. They have a natural Hamiltonian dynamics (a smooth function defines a flow preserving the structure). They satisfy a Moser-type theorem, which implies that Darboux's theorem and the Weinstein tubular neighborhood theorems hold in this context (a small neighborhood of any Lagrangian is equivalent to the example above). When restricted to a coisotropic, the kernel of $\omega$ is a foliation, and in the case the leaf space is a manifold it inherits a conformal symplectic structure. The Poisson bracket on Hamiltonians intertwines the Lie bracket.
However, many of the more modern methods in symplectic geometry cannot be easily generalized, and often the theorems fail to hold true. For example, suppose that $\beta \in \Omega^1 Q$ is a $1$-form which never vanishes. Then in the conformal symplectic manifold $(T^*Q, \eta, \omega)$ defined above, the zero section $Z = \{p=0\}$ is displaceable by Hamiltonian isotopy. In fact, if $\varphi_t$ is the Hamiltonian isotopy generated by the Hamiltonian $H = 1$, then $\varphi_t(Z) \cap Z = \varnothing$ for any $t>0$.
From the point of view of Floer theory, the problem with conformal symplectic structures is that we cannot even get started. The set of almost complex structures $J$ compatible with $\omega$ is still a non-empty contractible space, and $\overline\partial_J$ is still an elliptic operator, but because $\omega$ is not closed we have no bounds on energy, and therefore we do not expect Gromov compactness to hold, even for conformal symplectic manifolds which are both closed and exact. We discuss explicit examples suggesting failure of Gromov compactness in Section \ref{sec:failure-compactness}. Whether Gromov compactness can be generalized to this context by defining a more sophisticated compactification remains to be seen.
The main theorem of this paper proves the following an analogous of Laudenbach-Sikorav's Theorem \cite{Lau_Sik} in the context of conformal symplectic manifolds. We let $H^\text{Nov}_*(L, [\eta]; \mathbb{F})$ be the Novikov homology of $L$ in the homology class $[\eta] \in H^1(L; \mathbb{R})$.
\begin{thm}\label{thm:rigidprinc}
Let $\eta$ be a closed $1$-form on an orientable manifold $Q$, and let $\mathbb{F}$ be a field. Let $\phi$ be an Hamiltonian diffeomorphism of the conformal symplectic manifold $(T^*Q,\eta,d\lambda-\eta\wedge\lambda)$ (where $\lambda$ is the canonical form) such that $\phi(Q_0)$ intersect $Q_0$ transvesally, then $\#\{\phi(Q_0) \cap Q_0\}\geq \operatorname{rk} H^\text{Nov}_*(Q, [\eta]; \mathbb{F})$. If $Q$ is non-orientable then $\#\{\phi(Q_0) \cap Q_0\}\geq \operatorname{rk} H^\text{Nov}_*(Q, [\eta]; \mathbb{Z}_2)$
\end{thm}
\begin{coro}\label{cor:C0 close}
Let $(M, \eta, \omega)$ be a conformal symplectic manifold, and let $L \subseteq M$ be a Lagrangian. If $\varphi_t$ is any $C^0$ small Hamiltonian isotopy, then $\#\{\varphi_1(L) \cap L\} \geq \operatorname{rk} H^\text{Nov}_*(L, [\eta]|_L; \mathbb{Z}_2)$. If $L$ is oriented we may replace $\mathbb{Z}_2$ with any field $\mathbb{F}$.
\end{coro}
In order to prove Theorem \ref{thm:rigidprinc} we prove an analogous to Sikorav's Theorem \cite{Sikorav_famgen} about persistence of generating families which allows to provide bounds for intersection of Lagrangian submanifolds in conformal contangent in terms of stable $\eta$-critical points of function.
The layout of the paper follows. Section \ref{sec:introduction} introduces the basic definitions and theorems in conformal symplectic geometry. We include a number of propositions which are not necessary for the proof of Theorem \ref{thm:rigidprinc}, with the hope of giving a large-scale introduction to the theory, particularly for symplectic and contact geometers.
Section \ref{sec:overv-morse-novik} gives a brief overview of Morse-Novikov homology needed for the proof of Theorem \ref{thm:rigidprinc}. Finally, in Section \ref{sec:rigid-lagr-inters} we complete the proof.
\section*{Acknowledgements}
\label{sec:aknowledgments}
The authors thank Vestislav Apostolov and François Laudenbach for inspiring discussions. The first author benefited from the hospitality of several institutions and wishes to thank the institute Mittag-Leffler in Stockholm, CIRGET in Montréal and MIT in Cambridge for the nice work environment they provided. The second author would like to thank Universit\'e de Nantes and the Radcliffe Institute for Advanced Study for their pleasant work environments. B. Chantraine is partially supported by the ANR project COSPIN (ANR-13-JS01-0008-01) and the ERC starting grant G\'eodycon. E. Murphy is partially supported by NSF grant DMS-1510305 and a Sloan Research Fellowship.
\section{Main definitions}
\label{sec:introduction}
\subsection{Conformal symplectic manifolds.}
\label{sec:conf-sympl-manif}
\begin{prop}\label{prop:equiv def}
Let $M$ be a $2n$-manifold. The following are equivalent:
\begin{itemize}
\item An atlas of charts $M = \bigcup U_i$, $\varphi_i:U_i \to \mathbb{R}^{2n}$, so that the transition maps $\psi_{ij} = \varphi_i \circ \varphi_j^{-1}$ preserve the standard symplectic form up to scaling by a positive local constant: $\psi_{ij}^*(\omega_\text{std}) = c_{ij} \omega_\text{std}$. Two atlases are considered equivalent if they admit a common refinement.
\item A flat, real, orientable line bundle $E \to M$, and a $2$-form $\sigma \in \Omega^2(M, E)$, so that $\sigma$ is non-degenerate (as a map $TM \to T^*M \otimes E$) and closed (as a form with values in a flat line bundle). Two such structures $(E_1, \sigma_1)$, $(E_2, \sigma_2)$ are considered equivalent if there is an isomorphism $\varphi: E_1 \to E_2$ covering the identity map, so that $\varphi^*\sigma_2 = \sigma_1$.
\item A pair $(\eta, \omega) \in \Omega^1M \times \Omega^2M$, so that $d\eta = 0$, $d\omega = \eta \wedge \omega$, and $\omega^{\wedge n} \neq 0$. $(\eta, \omega)$ is equivalent to $(\eta + df, e^f\omega)$ for any $f \in C^\infty M$.
\end{itemize}
Any of these structures is called a \emph{conformal symplectic structure} on $M$.
\end{prop}
\begin{proof}
Given an atlas, the association of the positive number $c_{ij}$ to every intersection $U_i \cap U_j$ can be thought of as the clutching function for a principal $GL^+(1, \mathbb{R})$-bundle on $M$, with the discrete topology. That is, the numbers $c_{ij}$ define an orientable flat line bundle, $E$, and the form $\sigma =\varphi_i^*\omega_\text{std}$ is well defined as a $2$-form taking values in $E$. It is closed and non-degenerate, since these are local conditions.
Given a pair $(E, \sigma)$, we can identify $E \cong M \times \mathbb{R}$ as smooth vector bundles globally, since $E$ is a real, orientable line bundle. A choice of flat connection on $E$ then is simply a closed $1$-form $\eta \in \Omega^1(M)$. In this way we can identify $\sigma$ as an ordinary $2$-form $\omega \in \Omega^2(M)$. Then to say that $\omega$ is closed as a $2$-form with values in $E$ is equivalent to the statement $d\omega - \eta \wedge \omega = 0$. The choice of connection $\eta$ is not canonical; gauge symmetries of $E$ act by $\eta \mapsto \eta + df$ for any $f \in C^\infty(M)$, and this gauge symmetry acts on $\omega$ by $\omega \mapsto e^f\omega$.
Given a pair $(\eta, \omega)$ as above, consider the covering space $\pi: \widetilde M \to M$ associated to $[\eta]$. Then $\pi^*\eta = d\theta$ for some $\theta \in C^\infty \widetilde M$, and for any covering transformation $g \in \pi_1M / \ker [\eta] \subseteq \operatorname{Diff}\widetilde M$, we have $\theta \circ g = \theta + \langle[\eta], g \rangle$. Let $\widetilde\omega = e^{-\theta}\pi^*\omega \in \Omega^2\widetilde M$. Then $\widetilde\omega$ is a symplectic form, and $g^*\widetilde\omega = e^{-\langle[\eta], g \rangle}\widetilde\omega$. Therefore by taking a Darboux atlas on $(\widetilde M, \widetilde\omega)$, we get a conformal symplectic atlas on $M$.
\end{proof}
\begin{definition}\label{def:deta}
Let $\eta$ be a closed $1$-form on a manifold $M$. The \emph{Lichnerowicz-De Rham differential} on $\beta \in \Omega^*(M)$ is defined as $d_\eta \beta=d\beta-\eta\wedge\beta$.
\end{definition}
From the fact that $\eta$ is closed and of odd degree we get that $d_\eta^2=0$ and from the fact that $\eta$ has degree $1$ we see that $d_\eta$ is degree $1$ as well. Perhaps the most essential difference between $d_\eta$ and $d$ is that $d_\eta$ does not satisfy a Stokes' theorem. We note two important formulas:
$$d_{\eta + df}\beta = e^f d_\eta(e^{-f}\beta)$$
$$\mathcal{L}_X\beta = X \, \lrcorner \, d_\eta\beta + d_\eta(X \, \lrcorner \, \beta) + \eta(X)\beta$$
where $\mathcal{L}_X$ is the Lie derivative along the vector field $X$. By passing to the covering space of $M$ defined by $\ker [\eta]$, it follows that the homology $H^*(\Omega^*M, d_\eta)$ is isomorphic to $H^*(M; [\eta], \mathbb{R})$, the homology of $M$ with local coefficients defined by the homomorphism $[\eta]: \pi_1M \to \mathbb{R}$.
\begin{note}
For concreteness, we will state definitions and prove propositions in the setup where a conformal symplectic structure is a pair $(\eta, \omega)$, and then when relevant show that the definitions are invariant under gauge equivalence $\eta \rightsquigarrow \eta + df$. By working directly with the complex $\Omega^*(M, E)$ for the flat line bundle $E$, the arguments are more elegant, but also they are likely more opaque.
\end{note}
\begin{definition}\label{def:basics}
Let $(M, \eta, \omega)$ be a conformal symplectic structure. We say that a submanifold $L \subseteq M$ is \emph{isotropic} if $\omega|_L = 0$, \emph{coisotropic} if $TL^{\perp \omega} \subseteq TL$, and \emph{Lagrangian} if it is isotropic and coisotropic.
We say that $(\eta, \omega)$ is \emph{exact} if $\omega = d_\eta\lambda$ for some $\lambda \in \Omega^1M$. In this case we say that $\lambda$ is a \emph{Liouville form} for $(\eta, \omega)$, and the vector field $Z_\lambda$ defined by $Z_\lambda \, \lrcorner \, \omega = \lambda$ is called the \emph{Liouville vector field}.\footnote{This usage is somewhat different from the standard symplectic case, where a Liouville form is required to be a primitive of $\omega$ and also satisfy a convexity condition near the boundary/infinite portion of $M$. The same conditions on $Z_\lambda$ make sense in this context, so they could easily be imposed.} If $L \subseteq M$ is a Lagrangian in $(M, \eta, d_\eta\lambda)$, we say that $L$ is \emph{exact} if $\lambda|_L = d_\eta h$ for some $h \in C^\infty L$.
\end{definition}
Notice that all of the above definitions are well defined up to gauge equivalence, because if we replace $(\eta, \omega)$ with $(\eta + df, e^f\omega)$, we can also replace $\lambda$ and $h$ with $e^f\lambda$ and $e^fh$. In particular, the Liouville vector field $Z_\lambda$ is well defined independent of gauge, it is \emph{not} rescaled by $e^f$. Note also that there is nothing precluding the existence of a closed, exact conformal symplectic manifold.
\begin{ex}\label{ex:confsymplectisation}
Let $(Y,\lambda)$ be a manifold with a $1$-form $\lambda$. We denote by $S^1_T$ the quotient $\mathbb{R}/T\mathbb{Z}$ and parametrise it with the coordinate $\theta$. On $S_T^1\times Y$ let $\eta = -d\theta$. Then $\omega =d_\eta\lambda =d\lambda + d\theta\wedge\lambda$, so $\omega$ is non-degenerate if and only if $\lambda$ is a contact form. Furthermore, given another contact form defining the same cooriented contact structure, $e^f\lambda$, we have that the conformal Liouville manifold $(S_T^1 \times Y, \eta, \lambda)$ is gauge equivalent to $(S_T^1 \times Y, \eta+df, e^f\lambda)$, which is conformal symplectomorphic to $(S_T^1\times Y, \eta, e^f\lambda)$ under the coordinate change $(\theta, y) \mapsto (\theta - f(y), y)$.
The minimal cover which makes $\eta$ exact is $\mathbb{R}\times Y\rightarrow S^1\times Y$, and the conformal Liouville structure $(\eta, \lambda)$ pulls back to $(-dt, \lambda)$, which is gauge equivalent to the exact symplectic structure $e^t\lambda$, known as the symplectization of $(Y,\ker\lambda)$. Hence, the conformal symplectic manifold will be called the \textit{conformal symplectization} of $(Y,\xi)$. We denote this manifold by $S^\text{conf}_T(Y, \ker\lambda)$. (Notice that $\langle[\eta], S^1 \times \{\text{point}\}\rangle = -T$, so the choice of $T$ affect the conformal symplectomorphism type.) If the choice of $T$ is irrelevant, we will use the notation $S^\text{conf}(Y, \ker\lambda)$.
Now, let $\Lambda \subseteq Y$ be a Legendrian submanifold. Then the cylinder $\mathbb{R}\times \Lambda$ is an exact Lagrangian in the symplectization of $(Y, \ker \lambda)$, and furthermore it is invariant under the covering transformations of $\mathbb{R}\times Y\rightarrow S^1\times Y$. Therefore it descends to an exact Lagrangian submanifold $S^1 \times \Lambda$ of $S^\text{conf}(Y, \ker\lambda)$. More generally, if $\Sigma$ is a Lagrangian cobordisms from $\Lambda$ to itself which is cylindrical outside of $(0,T)\times Y$ then $\Sigma$ descends to a Lagrangian submanifold of $S^\text{conf}_T(Y, \ker\lambda)$.
\end{ex}
This paper will focus primarily on the following example:
\begin{ex}\label{ex:cotangent}
Let $\beta$ be a closed $1$-form on a smooth manifold $Q$, let $\pi$ be the projection $T^*Q\rightarrow Q$, and let $\lambda_\text{std}$ be the tautological form on $T^*Q$. Define $\eta:=\pi^*\beta$. Then $(T^*Q,\eta, \lambda_\text{std})$ is a conformal Liouville manifold (since $\eta\wedge \lambda_\text{std} \wedge d\lambda_\text{std}^{n-1}=0$). We will denote this conformal Liouville manifold by $T^*_\beta Q$.
Let $\alpha:Q \rightarrow T^*Q$ be a $1$-form. Then $\alpha^*\lambda_\text{std}=\alpha$ and thus $\alpha(Q)$ is Lagrangian in $T_\beta^*Q$ if and only if $d_\beta \alpha = 0$. Furthermore this Lagrangian is exact if and only if $\alpha$ is $d_\beta$-exact.
\end{ex}
\subsection{Conformal symplectic transformations.}
\label{ssec:conf-sympl-transf}
A diffeomorphism between conformal symplectic manifolds $\varphi: (M_1, \eta_1, \omega_1) \to (M_2, \eta_2, \omega_2)$ is called a \emph{conformal symplectomorphism} if $\varphi^*\eta_2 = \eta_1 + df$ and $\varphi^*\omega_2 = e^f \omega_1$ for some $f \in C^\infty M_1$. We denote by $\operatorname{Symp}(M,\eta,\omega)$ the group of conformal symplectomorphisms of $(M,\eta,\omega)$. The Lie algebra $\mathfrak{S}\operatorname{ymp}(M,\eta,\omega)$ of this group is given by vector fields $X$ satisfying $\mathcal{L}_X\omega=f\omega$ and $\mathcal{L}_X\eta=df$ for some $f$, and using Cartan's formula we see this is equivalent to
\begin{align*}
d_\eta(X \, \lrcorner \, \omega) + \eta(X)\omega &= \mathcal{L}_X \omega = f \omega\\
d(\eta(X)) &= \mathcal{L}_X\eta =df
\end{align*}
Therefore $X$ is a conformal symplectic vector field if and only if $d_\eta(X\, \lrcorner \, \omega) = c\omega$ for a constant $c \in \mathbb{R}$.
The subalgebra of $\mathfrak{S}\operatorname{ymp}(M,\eta,\omega)$ which are $\omega$-dual to $\eta$-closed $1$-forms (i.e. vector fields $X$ with $d_\eta(X\, \lrcorner \, \omega) = 0$) will be called \textit{divergence free conformal symplectic vector fields}. Notice the following fact: a conformal symplectic manifold is exact if and only if it admits a conformal symplectic vector field which is not divergence-free. In this case, after choosing a Liouville form $\lambda$, we see that every conformal symplectic vector field is of the form $X + cZ_\lambda$, where $X$ is divergence-free.
\begin{definition}
The conformal symplectic vector fields which are dual to $\eta$-exact forms is called \emph{Hamiltonian vector fields}. For any function $H \in C^\infty M$, the Hamiltonian vector field $X_H$ associated to $H$ is defined by $X_H \, \lrcorner \, \omega = - d_\eta H$. A conformal symplectomorphism of $M$ which is the time-$1$ flow of a path of Hamiltonian vector fields will be called a \textit{Hamiltonian diffeomorphism}.
\end{definition}
Notice that the association $H \rightsquigarrow X_H$ depends on $\eta$, but the algebra of Hamiltonian vector fields (and therefore the group of Hamiltonian diffeomorphisms) does not. Using $X_H^\eta$ to denote this dependence, we immediately have $X_H^\eta = X_{e^f H}^{\eta + df}$. To phrase the dependence in a gauge free way, let $E$ be the flat line bundle determined by $[\eta]$, then Hamiltonians are taken to be $H \in \Omega^0(M, E)$. Since $\omega \in \Omega^2(M, E)$, the equation $X_H \, \lrcorner \, \omega = - d_E H$ defines $X_H$ unambiguously.\footnote{The reader experienced with contact geometry will be familiar with this situation: the contact vector field $X_H$ associated to a Hamiltonian $H \in C^\infty M$ depends on a choice of contact form, but the correct way to define the relationship without reference to a contact form is to take contact Hamiltonians $H \in \Omega^0(M, TM/\xi)$.}
\begin{definition}
Given a conformal symplectic structure with a chosen connection $\eta$, we define the \emph{Lee vector field} to be the Hamiltonian vector field generated by $H = 1$. We sometimes denote this vector field as $R_\eta = X_1$.
\end{definition}
\begin{ex}
Given a contact manifold $(Y, \ker \alpha)$, we defined the conformal symplectization in Example \ref{ex:confsymplectisation} by $(S^1 \times Y, \eta = -d\theta, \omega = d_\eta\alpha)$. In this case, the Lee vector field of $\eta$ is equal to the Reeb vector field of $\alpha$.
\end{ex}
Note that this exhibit a first difference with the symplectic case: this flow has no fixed point for small time. This is an example of a more general phenomenon: given a conformal symplectic manifold $(M, \eta, \omega)$, if $\eta$ can be chosen to have no zeros, then $R_\eta$ is a Hamiltonian vector field with no zeros, and therefore we can construct many Hamiltonian diffeomorphisms with no fixed points. Another place where this condition is relevant is about Lagrangian displaceabality:
\begin{ex}
Given a cotangent bundle $T_\beta^*Q$ with Liouville structure given by a closed one form $\beta$ on $Q$ as in Example \ref{ex:cotangent}, the Liouville vector field $Z_\lambda$ is given by the standard Liouville vector field on $T^*Q$. The Lee vector field $R_\eta$ corresponds to the symplectic vector field dual to $\eta$ (in the standard symplectic sense), i.e. its flow acts fiberwise, and translates the vertical fiber $T_q^*Q$ in the direction $\beta_q$. In particular, if $[\beta] \in H^1Q$ can be represented by a non-vanishing closed $1$-form, then there exists an autonomous Hamiltonian flow on $T^*_\beta Q$ so that $\varphi_H^t(Q) \cap Q = \varnothing$ for \emph{any} $t>0$.
\end{ex}
Given an exact conformal symplectic manifold $(M, \eta, d_\eta \lambda)$, there is some interplay between the vector fields $Z_\lambda$ and $R_\eta$. Always, we have $$\lambda(R_\eta) = \omega(Z_\lambda, R_\eta) = - \eta(Z_\lambda).$$ Furthermore, suppose that $X \in \ker d\lambda$. Since $\omega = d_\eta\lambda = d\lambda - \eta \wedge \lambda$, we get that $X \, \lrcorner \, \omega = \lambda(X)\eta - \eta(X)\lambda$, and therefore $X = \lambda(X)R_\eta - \eta(X)Z_\lambda$ since $\omega$ is non-degenerate. It follows that, if $d\lambda$ has non-zero kernel, it must be equal to the span of $R_\eta$ and $Z_\lambda$. Plugging either $R_\eta$ or $Z_\lambda$ into the original equation, we see that $d\lambda$ has non-zero kernel if and only if $\eta(Z_\lambda) = -1$, if and only if $\lambda(R_\eta) = 1$.
In the literature, exact conformal symplectic manifolds satisfying $\eta(Z_\lambda) = -1$ everywhere are often referred to as conformal symplectic manifolds \emph{of the first kind}. Notice that this condition is not gauge invariant, and therefore it can be difficult to tell when a given conformal symplectic manifold is gauge-equivalent to one satisfying this property. However, this condition has strong implications, as shown in \cite[Theorem A]{Bazzoni_conf}: if $M$ is compact and $\eta(Z_\lambda) = -1$ everywhere, and if $\{\eta = 0\}$ has at least one compact leaf, then $M$ is conformal symplectomorphic to the suspension of a strict contactomorphism of a contact manifold $Y$.
\subsection{Moser's theorem and tubular neighborhoods}
\label{ssec:moser}
We have the following version of Moser's Theorem (first proved in \cite{Banyaga_propertieslcs}).
\begin{thm}\label{thm:Moser}
Let $(M, \eta)$ be a closed smooth manifold equipped with a closed $1$-form, and let $\{\omega_t\}_{t \in [0,1]}$, be a path of conformal symplectic structures in the same homology class, i.e. $\omega_t = \omega_0 + d_\eta\lambda_t$. Then there is an isotopy $\varphi_t: M \to M$ and a path of smooth functions $f_t \in C^\infty M$ so that $\varphi_t^*\eta = \eta + df_t$ and $\varphi_t^*\omega_t = e^{f_t}\omega_0$. That is, an exact homotopy of conformal symplectic structures is always induced by an isotopy (and gauge transformations)
\end{thm}
\begin{proof}
We find a vector field $X_t$ suitable for $X_t = \dot{\varphi_t} \circ \varphi_t^{-1}$, and then using compactness of $M$ integrate $X_t$. Differentiating the desired equations gives
$$\varphi_t^*\left(\mathcal{L}_{X_t}\eta\right) = d\dot f_t$$
$$\varphi_t^*\left(\dot\omega_t + \mathcal{L}_{X_t}\omega_t\right) = \dot f_t e^{f_t}\omega_0 = \dot f_t \varphi_t^*\omega_t.$$
Letting $\mu_t = \dot f_t \circ \varphi_t^{-1}$, we have
$$d(\eta(X_t)) = \mathcal{L}_{X_t}\eta = d\mu_t$$
$$\dot\omega_t + d_\eta(X_t \, \lrcorner \, \omega_t) + \eta(X_t)\omega_t = \mu_t\omega_t.$$
Defining $X_t$ by $X_t \, \lrcorner \, \omega = \dot\lambda_t$, and taking $\mu_t = \eta(X_t)$, we see that this system has a solution.
\end{proof}
Just as in the symplectic case, the Moser theorem for conformal symplectic structures is used to prove a Weinstein neighborhood theorem for Lagrangians in conformal symplectic manifolds, showing that Example \ref{ex:cotangent} is a universal model. More precisely we have the following result, which appears in the recent paper \cite{Otiman_Stanciu_darboux}. It also follows from a more general version for coisotropic manifolds, appearing in \cite[Theorem 4.2]{VanLe_Oh_cositrop}.
\begin{thm}\label{thm:lag nbhd}
Let $(M,\eta,\omega)$ be a conformal symplectic manifold and $L\subset M$ a compact Lagrangian. Let $\beta=\eta|_L$. Then there exists a neighborhood $\mathcal{N} \subseteq M$ of $L$ so that $\mathcal{N}$ is conformally symplectomorphic to a neighborhood $U$ inside $(T^*_\beta L, \pi^*\beta, d_{\pi^*\beta}\lambda_\text{std})$, sending $L \subseteq \mathcal{N}$ to the zero section $\{p=0\} \subseteq U$.
\end{thm}
\begin{proof}
Letting $\mathcal{N}$ be a tubular neighborhood of $L$, we have a diffeomorphism $\psi: T^*L \to \mathcal{N}$ sending the zero section $Z \subseteq T^*L$ to $L\subseteq \mathcal{N}$, so that $\psi^*\omega|_{T_ZT^*L} = d_{\pi^*\beta}\lambda_\text{std}|_{T_ZT^*L}$. $\psi^*\eta$ and $\pi^*\beta$ are homologous, so by taking a gauge transformation on $\mathcal{N}$ we can assume that they are equal.
Since $\psi^*\omega_1|_Z = 0$, therefore $\psi^*\omega = d_{\pi^*\beta}\lambda_1$ is exact (the Lichnerowicz-De Rham complex is homotopy invariant). We can furthermore assume that $\lambda_1|_{T_ZT^*L} = 0$ by addition of a $d_{\pi^*\beta}$-closed $1$-form. Let $\lambda_t = t\lambda_1 + (1-t)\lambda_\text{std}$, and $\omega_t = d_\eta\lambda_t$. By taking a smaller neighborhood if necessary, we may assume that $\omega_t$ is non-degenerate for all $t \in [0,1]$.
We then run the Moser method, as above. The vector field $X_t$ obtained is zero along $Z$, and therefore by shrinking $\mathcal{N}$ further if necessary, we get a conformal symplectomorphism from a neighborhood of $Z \subseteq T^*_\beta L$ to $\mathcal{N}$.
\end{proof}
\subsection{Flexibility and rigidity}\label{ssec:formal}
Given a closed manifold $M^{2n}$, we would like to know when $M$ admits a conformal symplectic structure. If so, how many distinct structures are there up to isotopy. There are some basic obstructions/invariants. For ease of notation we focus on the exact case.
\begin{definition}
Let $M^{2n}$ be a closed manifold. An \emph{exact almost conformal symplectic structure} (or EACS structure) is a pair $(a, w)$, where $a \in H^1(M; \mathbb{R})$, and $w \in \Omega^2M$ is a non-degenerate $2$-form.
\end{definition}
Any exact conformal symplectic structure $(\eta, d_\eta\lambda)$ defines an almost conformal symplectic structure with $a = [\eta]$, $w = d_\eta\lambda$. Therefore, for a manifold to admit an exact conformal symplectic structure, it must also admit an EACS structure. Similarly, for two conformal symplectic structures to be isotopic, they necessarily must be homotopic through EACS structures (where $w$ is homotoped but $a$ is fixed). The benefit of passing to EACS structures is that they are classified purely by algebraic topology.
We show that that the classification of exact conformal symplectic structures is strictly more subtle than their almost conformal counterpart.
\begin{prop}\label{prop:nonflexible}
There exists a manifold $M$ of any dimension and and $ [\eta] \neq 0 \in H^1M$, which admits two exact conformal symplectic structures $(\eta, d_\eta\lambda_1)$ and $(\eta, d_\eta\lambda_2)$ which are homotopic through non-degenerate $2$-forms, but they are not conformally symplectomorphic.
\end{prop}
\begin{proof}
We take $M = S^1 \times S^{2n-1}$. Let $\xi_\text{ot} = \ker \alpha_\text{ot}$ be an overtwisted contact structure (see \cite{BEM}) on $S^{2n-1}$, which is homotopic through almost contact structures to the standard contact structure $\xi_\text{std}$.
As the symplectization of $(S^{2n-1},\xi_\text{ot})$ is different form the one of $(S^{2n-1},\xi_\text{std})$ (actually the second cannot embed in the first as $(S^{2n-1},\xi_\text{std})$ is fillable) we know that those two structure are not equivalent.
Moser's theorem therefore implies that they are not homotopic through exact conformal symplectic structures. But they are homotopic through non-degenerate $2$-forms, since the original contact structures are homotopic through almost contact structures.
\end{proof}
\begin{remark}
As shown in \cite{Murphy_closedlagsymplectisation}, when $n>2$ the symplectization $(\mathbb{R} \times S^{2n-1}, e^r\alpha_\text{ot})$ contains an exact Lagrangian, $L \subseteq \mathbb{R} \times S^{2n-1}$. Let $R > 0$ be a constant so that $L \subseteq (0, R) \times S^{2n-1}$. Then by identifying $S^1 = \mathbb{R} / R\mathbb{Z}$ $L$ embeds as an exact Lagrangian into the conformal symplectization $(M, \eta = -d\theta, \lambda_1 = \alpha_\text{ot})$.
However, there can be no nulhomologous exact Lagrangian in the conformal symplectization of the standard contact structure, $(S^1 \times S^{2n-1}, \eta, \lambda_2 = \alpha_\text{std})$. For if there was, it would lift to a compact exact Lagrangian in the universal cover, which is conformally symplectomorphic to $\mathbb{C}^n \setminus \{0\}$, which then contradicts Gromov's theorem. This shows that those two conformal symplectic structure can be distinguished by their Lagrangian submanifolds.
\end{remark}
For $2n = 2$ we know that conformal symplectic structures are equivalent to almost conformal symplectic structures, since all $2$-forms are exact when $[\eta] \neq 0$.
On the question of existence, a nearly complete answer was established in \cite{E_M_Symp_Cob}:
\begin{thm}
Let $M$ be a closed manifold with an EACS structure $(a, w)$, where $a \neq 0$ is in the image $H^1(M; \mathbb{Z}) \to H^1(M, \mathbb{R})$. Then for all sufficiently large $C \in [1, \infty)$, there is an exact conformal symplectic structure $(\eta, d_\eta\lambda)$, with $[\eta] = Ca$ and $d_\eta\lambda$ being homotopic to $w$ through non-degenerate $2$-forms.
In particular, given any closed manifold $M$ satisfying $H^1(M; \mathbb{R}) \neq 0$ and having a non-degenerate $2$-form, $M$ admits an exact conformal symplectic structure.
\end{thm}
This follows indeed from the existence result of symplectic structure on almost symplectic cobordisms by cutting along an hypersurface Poincaré dual to $[\eta]$ which is non-separating by hypothesis. Making then the cobordism symplectic between the same contact structure on the ends leads to a symplectic cobordisms whose ends can be identified to give a conformal symplectic manifold.
It is an open question whether we can in general construct conformal symplectic structures where $[\eta]$ is a prescribed $1$-form, either in the case where $[\eta]$ is small, or when $[\eta]: \pi_1M \to \mathbb{R}$ has non-discrete image. Note that in \cite{Apo_Dlou}, there is one example that shows that after fixing a complex structure J, the set of $[\eta]$ for which there exists a conformal symplectic structure compatible with $J$ consists of a single point.
\begin{remark}
From this theorem, it is easy to construct non-exact conformal symplectic structures: given an exact symplectic structure $(\eta, d_\eta\lambda)$ and a class $b \in H^2(M; [\eta], \mathbb{R})$, choose a $d_\eta$-closed $2$-form $B \in \Omega^2M$ representing $b$. Then for sufficiently large $C$, $\omega = Cd_\eta\lambda + B$ is a conformal symplectic structure.
\end{remark}
\begin{remark}
The same strategy also works for constructing conformal symplectic manifolds with convex contact boundary: any contact manifold which admits an almost complex filling admits a conformal symplectic filling. Indeed from such a formal filling the main theorem in \cite{E_M_Symp_Cob} allows us to construct a symplectic cobordisms from $(S^{2n-1},\xi_\text{ot})$ to $(S^{2n-1},\xi_\text{ot})\sqcup (Y,\xi)$, and gluing the two $(S^{2n-1},\xi_\text{ot})$ leads to a conformal symplectic filling of $(Y, \xi)$.
As stated there, the theorem only applies to cobordisms with tight convex boundary when $n > 2$. However, after an examination of the proof there it is clear that the result applies when $n=2$ as well, as long as a single component of the convex boundary is overtwisted.
\end{remark}
\subsection{Pseudo-holomorphic curves and a ``failure'' of Gromov compactness.}
\label{sec:failure-compactness}
Inspired from successes in symplectic and contact geometry, we might try to develop a theory of pseudo-holomorphic curves for conformal symplectic manifolds. Similar to the symplectic case, in a conformal symplectic manifold the space of compatible almost complex structures $J$ is contractible, and the equation $\overline{\partial}_J u = 0$ is elliptic. But when trying to prove a compactness theorem, we run into an immediate problem: the Lichnerowicz-De Rham complex does not satisfy Stokes' theorem, and without this we have no way to give a $C^0$ bound on holomorphic energy. We give here two examples of holomorphic curves arising naturally which suggest bad compactness properties.
Consider the conformal sympectisation of an overtwisted sphere $Y = S^{2n-1}_\text{ot}$ with a generic contact form. When equipped with a cylindrical almost complex structure $J$, there is a holomorphic plane which is asymptotic to one of the Reeb orbits, as shown in \cite{Albers_Hofer}. In the conformal symplectization $S^\text{conf}(Y)$ this gives a holomorphic plane, which looks something like a leaf of a Reeb type foliation. Though such curves are handled in the symplectization setting by SFT compactness \cite{Bourgeois_&_Compactness} and \cite{Abbas_book}, this sort of compactness strongly relies on having contact asymptotics and a cylindrical complex structure, which is not a natural notion in general conformal symplectic manifolds.
A second phenomena is given by the conformal symplectic filling of $S^3_{\text{ot}}$ described in the previous section. The bishop family near the elliptic point of an overtwisted disk leads to a family of holomorphic curves which cannot be compactified in a standard way. We suspect that such a family breaks into curves involving half-plane such as those described before.
\subsection{Contactisation, reduction, and generating functions}
\label{ssec:cont-red-gen}
\begin{definition}\label{def:contactization}
If $(M,\eta,\lambda)$ is a conformal Liouville manifold, we define the \emph{contactisation} of $(M, \eta, \lambda)$ to be the manifold $M\times \mathbb{R}$ equipped with the contact structure given by the kernel of $\alpha=dz-z\eta-\lambda=d_\eta z-\lambda$ (note that $\alpha$ is a contact form if and only if $d_\eta \lambda$ is non-degenerate).
\end{definition}
Notice that an exact Lagrangian submanifold $L$ lifts to a Legendrian submanifold $\widetilde{L}= \{(q,f(q)); q \in L\}$ where $f$ is such that $d_\eta f=\lambda|_L$.
In the cotangent case, the space $\mathcal{J}^1_\beta(Q):=T^*_\beta Q\times \mathbb{R}$ with the contact form $d_\beta z-\lambda$ is called the \emph{$\beta$-jet bundle}, natural Legendrian submanifolds arise as $\beta$-jets of functions $f$, i.e. $j^1_\beta f(q) = (q,d_\beta f_q,f(q))$. Note that $\mathcal{J}^1_\beta(Q)$ is actually contactomorphic to $\mathcal{J}^1(Q)$ by the map $\varphi(q,p,z) = (q,p-z\beta_q,z)$. However, this contactomorphism is not compatible with the projection to $T^*Q$ and therefore does not identify the symplectic properties of $T^*Q$ with the conformal symplectic properties of $T^*_\beta Q$. Rather, it maps $\beta$-jets of functions to (regular) jets of functions.
The Reeb vector field of $\alpha$ is given by $R_\alpha=(1+\lambda(R_\eta))\partial_z-R_\eta$.
\begin{ex}
Let $(M,\alpha)$ be a contact manifold. Then the contact form on $M\times S^1\times\mathbb{R}$ is $dz-zd\lambda-\alpha$ the Reeb vector field is just the original Reeb vector field seen as vector fields invariant by $S^1\times\mathbb{R}$.
The contactization of a conformal symplectization is a contact manifold associated to $(M,\ker \alpha)$, this manifold is the product of the original contact manifold and $T^*S^1$ with the Liouville structure given by $pd\theta-dp$.
\end{ex}
Let $(M,\eta,\omega)$ be a conformal symplectic manifold, and let $N\subset M$ be a coisotropic submanifold (i.e. such that $TN^{\perp \omega}\subset TN$). Since $d\omega=\eta\wedge \omega$ it follows from the Fröbenius integrability theorem that the distribution $\ker(\omega|_N)$ is integrable. Note that for a vector field $X$ in $\ker \omega|_N$ we have that
\begin{align}\label{eq:3}
\mathcal{L}_X(\omega|_N)&=\eta(X)\omega \\
\mathcal{L}_X\eta&=d(\eta(X)).\nonumber
\end{align}
Let $N^\omega$ be the leaf space of the associated foliation and assume that $N^\omega$ is a manifold. It follows from equation \eqref{eq:3} that on charts of $N^\omega$ there is a well define conformal symplectic structure and thus that $N^\omega$ is a conformal symplectic manifold. This conformal symplectic manifold is called the \textit{conformal symplectic reduction} of $N$.
\begin{ex}
Let $Q,Q'$ be two manifolds and $\beta,\beta'$ closed one forms on $Q,Q'$. And let $T^*_{\beta\oplus\beta'}(Q\times Q')\simeq T^*_\beta Q\times T_{\beta'}Q'$ be the conformal symplectic manifold associated to $\pi^*\beta\oplus(\pi')^*\beta'$. Then $T^*_\beta Q\times Q'_0$ is cositropic and its reduction gives back $T^*_\beta Q$.
\end{ex}
Consider a Lagrangian submanifold $L$ of $(M, \eta, \omega)$ and assume the coisotropic $N$ intersects $L$ transversely. Let $\varphi$ be the projection $N\rightarrow N^\omega$. Then the manifold $L_N=\varphi(L\cap N)$ is an immersed Lagrangian submanifold of $N^\omega$. (This is linear algebra and thus equivalent to the symplectic case and follows from\cite[Section 5.1]{Bates_Wein} for instance).
Now given a map $F:Q\times Q'\rightarrow \mathbb{R}$, we can take the graph to get a Lagrangian section $d_{\beta\oplus\beta'}F$ of $T^*_{\beta \oplus \beta'}(Q \times Q')$, and apply symplectic reduction as above to the coisotropic $T^*_\beta Q \times Q'_0$ to obtain an immersed exact Lagrangian submanifold of $T_{\beta}Q$. When $Q'=\mathbb{R}^m$ (and $\beta'=0$) and $F$ is quadratic at infinity we denote the corresponding Lagrangian $L_F$. $F$ is called a \textit{$\beta$-generating family} for $L_F$.
Note that $L_F$ lifts to a Legendrian submanifold $\Lambda_F$ of $\mathcal{J}_\eta^1(Q)$ and through the contactomorphism $\phi$ defined above we get a Legendrian submanifold of $\mathcal{J}^1(Q)$, and $F$ is a generating family (in the standard sense) for $\phi(\Lambda_F)$. Chekanov persistence's Theorem \cite{Chekanov_Quasifunctions_Generating} therefore implies that if $L$ in $T_\beta^*Q$ admits a generating family and $L_t$ is an isotopy of $L$ through exact Lagrangians then $L_1$ admits a $\beta$-generating family as well.
We remark that if $L$ is given by a $\beta$-generating family $F:Q\times \mathbb{R}^{m}$ then zeros of $d_\eta F$ (called $\eta$-critical points of $F$) corresponds to intersection points between $L$ and the zero section. All together, this implies
\begin{thm}\label{stablecrit}
Let $\beta$ be a closed $1$-form on a closed manifold $Q$, and define $\operatorname{Stab}_{\beta}(Q) = \min\limits_F \#\{q \in Q | d_\beta F(q)=0\}$, where $F$ is taken among all functions $F:Q\times \mathbb{R}^m\rightarrow \mathbb{R}$ which are quadratic at infinity, for all $m \in \mathbb{N}$. That is, $\operatorname{Stab}_\beta(Q)$ is the stable $\beta$-critical number of $Q$. Let $L \subseteq T^*_\beta Q$ be a Lagrangian which is Hamiltonian isotopic to the zero section $Z$. Then the number of intersections between $L$ and $Z$ is at least $\operatorname{Stab}_{\beta}(Q)$.
\end{thm}
Form the previous theorem we find that in order to have a lower bounds on the number of intersection points of a deformation a the $0$-section in a cotangent bundle we need to estimate the number of $\beta$-critical points of a generating family $F$. To get such estimates we study the Morse-Novikov homology of a deformation of the function $\ln |F|$.
\section{Overview of Morse-Novikov homology}
\label{sec:overv-morse-novik}
In this section we present the basics of the construction of the Morse-Novikov complex associated to a closed $1$-forms on $Q$. We also recall its essential properties we need to prove Theorem \ref{thm:rigidprinc}. We refer the reader to more comprehensive references like \cite{Farber_oneforms} and \cite{Pajitnov_circledvalued} for more details.
We assume that $Q$ is connected. A closed $1$-forms $\eta$ is \textit{Morse} if near each of its zeros it is the derivative of a non-degenerate quadratic form $h$. It follows from Morse's Lemma that any $1$-forms whose graph intersect the $0$-section transversely is Morse, therefore closed $1$-forms are generically Morse. We refer to a zero of a closed $1$-form $\eta$ as a \textit{critical point} of $\eta$. Given a critical point $q$ of $\eta$ the number of negative eigenvalues of the quadratic form $h$ such that near $q$, $\eta=dh$ is independent of the coordinate system, this number is called the \textit{index} of $q$ written $I_\eta(q)$.
Given a metric $g$ on $Q$ we define the vector field $\nabla\eta$ to be the dual of $\eta$ with respect to $g$. We denote by $x_t^\eta$ the flow of $\nabla\eta$. We will always assume that there is a compact subset $K$ such that
\begin{itemize}
\item The frontier of $K$ is a manifold with corners.
\item All critical points of $\eta$ are in $K$ (in particular there are finitely many critical points),
\item $\nabla\eta$ is complete, so $x_t^\eta$ is defined for all $t \in \mathbb{R}$,
\item For every component $V$ of $Q\setminus K$ either for all $q\in V$ and $t\geq 0$ $x_t^\eta(q)\not\in K$ or for all $q\in V$ and $t\leq 0$ $x_t^\eta(q)\not\in K$.
\end{itemize}
Any critical points $q$ have stable and unstable manifolds $W^s(q)$ and $W^u(q)$ defined by
$W^s(q)=\{q'|\lim_{t\rightarrow \infty}x_t^\eta(q')=q\}$ and $W^u(q)=\{q'|\lim_{t\rightarrow -\infty}x_t^\eta(q')=q\}$.
Note that $W^u(q)$ is a disk of dimension $I_\eta(q)$ and $W^s(q)$ is a disk of dimension $n-I_\eta(q)$. We say that the pair $(\eta,g)$ is \textit{Morse-Smale} if for any critical points $q,q'$ of $\eta$ the disks $W^u(q)$ and $W^s(q')$ intersects transversely. In this situation $W^s(q)\cap W^u(q')$ are open manifolds of dimension $I_\eta(q)-I_\eta(q')$ with an action of $\mathbb{R}$ on them which is free (unless $q=q'$).
For a ring $R$ we denote by $\Lambda_\eta(Q,R)$ the completion of the group ring $R[\pi_1(Q)]$ given by series of the form $\sum_ia_ig_i$ where $\lim_{i\rightarrow\infty}\int_{g_i}\eta=-\infty$.
Now for any critical points $q$ of $\eta$ we choose
\begin{enumerate}
\item A path $g_q$ from $q$ to the base point.
\item An orientation of the tangent space of $q$ (we do not assume $Q$ is orientable).
\end{enumerate}
Given two critical points $q,q'$ of $\eta$ such that $I_\eta(q)-I_\eta(q')=1$ then any component $\gamma$ of $W^s(q)\cup W^u(q')$ is copy of $\mathbb{R}$ which compactifies to a path from $q$ to $q'$ which, together with the capping paths $g_q$ and $g_{q'}$, gives an element $g_\gamma$ of $\pi_1(Q)$.
This gives the series $u_{q,q'}=\sum_\gamma\pm g_\gamma$, which is in $\Lambda_\eta(Q,R)$ since we follow negative trajectories of $\nabla\eta$. The sign $\pm$ is determined by whether or not the chosen orientation of $T_qQ$ transports to the one of $T_{q'}Q$ along the path $\gamma$.
The Morse-Novikov complex of $(\eta,g)$ is given by $C_k(\eta,R)=\oplus_{I_\eta(q)=k}\Lambda(Q,R)\langle q\rangle$ with differential $d(q)=\sum_{I_\eta(q')=k-1}u_{q,q'}q'$. We have that $d^2=0$ and the homology of $(C_*,d)$ is called the Morse-Novikov homology of $\eta$, denoted $H^{\text{Nov}}_*(\eta)$.
When $Q$ is compact it follows from a Theorem of Sikorav in \cite{Sik_Thesis} (see also \cite{Poz_Thesis}) that this homology is the homology of $Q$ with local coefficients in $\Lambda_\eta(Q, R)$. Notice that if $Q$ is compact with boundary and if $\nabla \eta$ points inward the boundary then the same results holds.
When $R$ is a principal ring, $\Lambda_\eta (Q,R)$ is a principal ring as well and we call the $k-th$ \textit{Novikov-Betti number} $b^k_\eta$ of $\eta$ the rank of the free part of $H^{\text{Nov}}_k(\eta)$. For compact oriented closed manifold they satisfy $b^k_\eta=b^{n-k}_\eta$ (see \cite[Corollary 2.9]{Pajitnov_circledvalued} and \cite[Section 1.5.3]{Farber_oneforms}), for non-oriented on the same is true if $R=\mathbb{Z}_2$.
Note that $\Lambda_\epsilon(Q,R)$ is a module over $R[\pi_1(Q)]$, the Morse-Novikov homology of $\eta$ is isomorphic to the homology with local coefficient in $H_*(K,\partial_- K;\Lambda_\epsilon(K,R))$ (see \cite{Latour}) where $\partial_- K$ (resp $\partial_+ K$) denotes the points where $\nabla \eta$ points outward (resp. inward) $K$.
\section{Rigidity of Lagrangian intersection}
\label{sec:rigid-lagr-inters}
We are now ready to prove Theorem \ref{thm:rigidprinc}.
\begin{proof}[Proof of Theorem \ref{thm:rigidprinc}]
From Theorem \ref{stablecrit} it remains to find a lower bound of the number of zeroes of $dF-F\beta$ where $F$ is a function on $Q\times\mathbb{R}^m$, so that there is a compact ball $B \subseteq \mathbb{R}^m$ such that outside of $Q\times B$, $F(x,\xi)=q(\xi)$ where $q$ is a non-degenerate quadratic form of index $k$. Because we can stabilize $F$ we can assume that $k>0$ without losing generality. Up to a small perturbation not affecting the zeroes we can also assume that $\beta$-critical values of $F$ are not $0$. Fix a metric $g$ on $Q$ inducing the product metric $g\oplus g_0$ on $Q\times \mathbb{R}^m$, we denote by $\nabla: T^*(M\times \mathbb{R}^m)\rightarrow T(M\times \mathbb{R}^m)$ the map induced by this metric.
Choose $\varepsilon>0$ such that for all $(x,\xi)$ such that $|F(x,\xi)|<2\varepsilon$ we have $d_\beta F(x,\xi)\not=0$ and $dF(x,\xi)\not=0$. Let $X_+:=\{(x,\xi)|F(x,\xi)\geq\varepsilon\}$, $X_-:=\{(x,\xi)|F(x,\xi)\leq-\varepsilon\}$ and $X_0:=\{(x,\xi)|-\varepsilon\leq F(x,\xi)\leq \varepsilon\}$.
Let $G$ be a function on $Q\times \mathbb{R}^N$ so that $G(x,\xi)=|F(x,\xi)|$ outside of $X_0$, depends only on $\xi$ outside of $Q\times B$ and $G(x, \xi) > 0$ everywhere. Note that outside of $Q\times B$ the gradient vector field of $d_\beta|F|$ is transverse to $\partial X_\pm$ so we can ensure that $\nabla(dG - G\beta) \pitchfork \partial X_0$.
We then let $H := \log G$, and notice that the zeros of $\gamma = dH - \beta$ are the same as the zeros of $G(dH - \beta) = dG - G\beta$. Therefore, the zeros of $dF - F\beta$ are the same as the zeros of $\gamma$ on $X_+ \cup X_-$, though $\gamma$ will also have additional zeros in $X_0$. Since $\gamma$ is closed and $[\gamma] = [\beta]$ we can now relate this to $H^\text{Nov}_*(Q, [\beta])$.
At first, we work with $H^\text{Nov}_*$ as homology with local coefficients in $\Lambda_\beta(Q, \mathbb{F})$, rather than Morse-Novikov homology. Throughout the next section, $H_*^{\text{Nov}}$ will always be taken in the homology class $[\beta]$. Let $r = \operatorname{rk}(H_*^\text{Nov}(Q, [\beta]))$, and consider the exact sequence of the pair $(X_0, X_0 \setminus Q \times B)$.
$$
\xymatrix{H_*^\text{Nov}(X_0, X_0 \setminus Q \times B) \ar[r]^{[-1]} &H_*^\text{Nov}(X_0 \setminus Q \times B) \ar[d]\\
&H_*^\text{Nov}(X_0) \ar[ul]&}
$$
Outside of $B$, we have that $F(x, \xi) = q(\xi)$, and therefore $X_0 \setminus Q \times B \cong Q \times S^{k-1} \times S^{m-k-1} \times (-\varepsilon, \varepsilon) \times (0, \infty)$. Thus
$$\operatorname{rk}(H_*^\text{Nov}(X_0 \setminus Q \times B)) = \operatorname{rk}(H^{\text{Nov}}_*(Q) \otimes H_*(S^{k-1}\times S^{m-k-1})) = 4r,$$
and it follows that either $H_*^\text{Nov}(X_0, X_0 \setminus Q \times B)$ or $H_*^\text{Nov}(X_0)$ must have total rank at least $2r$. We consider each case separately, with the goal of proving the inequality
\begin{equation}\label{eq:1}
\operatorname{rk}(H_*^\text{Nov}(X_+\partial X_+)) + \operatorname{rk}(H_*^\text{Nov}(X_-, \partial X_-)) \geq r.
\end{equation}
\emph{Case (1) \quad $\operatorname{rk}(H_*^\text{Nov}(X_0)) \geq 2r$:}
Consider the Mayer-Vietoris sequence for $(X_+ \cup X_0) \cup (X_- \cup X_0)$, which is
$$
\xymatrix{H_*^\text{Nov}(Q \times \mathbb{R}^m) \ar[r]^{[-1]} & H_*^\text{Nov}(X_0) \ar[d]\\
&H_*^\text{Nov}(X_+ \cup X_0) \oplus H_*^\text{Nov}(X_- \cup X_0)\ar[ul]&}
$$
This implies that $$\operatorname{rk}(H_*^\text{Nov}(X_+ \cup X_0)) + \operatorname{rk}(H_*^\text{Nov}(X_- \cup X_0)) \leq \operatorname{rk}(H_*^\text{Nov}(X_0)) + r.$$
By looking at the exact sequence of the pair $(X_+ \cup X_0, X_0)$, we get
$$\operatorname{rk}(H_*^\text{Nov}(X_+ \cup X_0, X_0)) \geq \operatorname{rk}(H_*^\text{Nov}(X_0)) - \operatorname{rk}(H_*^\text{Nov}(X_+ \cup X_0)),$$
and we get a similar inequality for the pair $(X_- \cup X_0, X_0)$. Rearranging these three inequalities gives
\begin{align*}
\operatorname{rk}(H_*^\text{Nov}&(X_+ \cup X_0, X_0)) + \operatorname{rk}(H_*^\text{Nov}(X_+ \cup X_0, X_0)) \geq \\
& 2 \operatorname{rk}(H_*^\text{Nov}(X_0)) - \operatorname{rk}(H_*^\text{Nov}(X_+ \cup X_0)) - \operatorname{rk}(H_*^\text{Nov}(X_- \cup X_0)) \geq \\
& \operatorname{rk}(H_*^\text{Nov}(X_0)) - r \geq r,
\end{align*}
which proves the inequality \ref{eq:1} after applying excision.
\emph{Case (2) \quad $H_*^\text{Nov}(X_0, X_0 \setminus Q \times B) \geq 2r$:}
Consider the Mayer-Vietoris sequence:
\begin{equation}\label{eq:2}
\xymatrix{H_*^\text{Nov}(Q \times \mathbb{R}^m, Q \times \mathbb{R}^m \setminus Q \times B) \ar[r]^{[-1]} & H_*^\text{Nov}(X_0, X_0 \setminus Q \times B) \ar[d]\\
& \hspace{-2.2 in} H_*^\text{Nov}(X_+ \cup X_0, (X_+ \cup X_0)\setminus Q \times B) \oplus H_*^\text{Nov}(X_- \cup X_0, (X_- \cup X_0)\setminus Q \times B)\ar[ul]&}
\end{equation}
Notice that the pair $(X_+ \cup X_0, (X_+ \cup X_0) \setminus Q \times B)$ is homeomorphic to $(X_+, X_+ \setminus Q \times B)$, since $\nabla F$ presents $X_0$ as a collar neighborhood of $\partial X_+ \cup X_0$. Using excision we have that $$H_*^\text{Nov}(X_+, X_+\setminus Q \times B) \cong H_*^\text{Nov}(X_+ \cap Q \times B, X_+ \cap \partial (Q \times B)).$$ Furthermore since we are using field coefficients, and using the hypothesis of either orientability of $Q$ or $\mathbb{F} = \mathbb{Z}_2$ we can apply Poincar\'e duality to obtain $$H_*^\text{Nov}(X_+\cap Q \times B, X_+ \cap \partial (Q \times B)) \cong H_{m+n-*}^\text{Nov}(X_+ \cap Q \times B, (\partial X_+)\cap Q \times B).$$
We also note that the pair $(X_+ \cap Q\times B, (\partial X_+) \cap Q\times B)$ is homeomorphic to $(X_+, \partial X_+)$, since $F$ is standard outside $Q \times B$. In summation we obtain an isomorphism
$$H_*^\text{Nov}(X_+ \cup X_0, (X_+ \cup X_0)\setminus Q \times B) \cong H_{m+n-*}^\text{Nov}(X_+, \partial X_+),$$
and also a similar isomorphism for $X_-$. Using excision and K\"unneth we also see that $$H_*^\text{Nov}(Q \times \mathbb{R}^m, Q \times \mathbb{R}^m \setminus Q \times B) \cong H^\text{Nov}_*(Q) \otimes H_*( \mathbb{R}^m, \mathbb{R}^m \setminus B) \cong H^\text{Nov}_{*+m}(Q).$$
Putting these ingredients together, the exact sequence \ref{eq:2} becomes
$$
\xymatrix{H_{*+m}^\text{Nov}(Q) \ar[r]^{[-1]} & H_*^\text{Nov}(X_0, X_0 \setminus Q \times B) \ar[d]\\
& H_{m+n-*}^\text{Nov}(X_+, \partial X_+) \oplus H_{m+n-*}^\text{Nov}(X_-, \partial X_-)\ar[ul]&}
$$
This immediately implies the inequality \ref{eq:1}.
We now return to Morse-Novikov homology. By our construction of $\gamma = dH - \beta$, we have that $-\nabla \gamma$ is transverse to $\partial X_+$, and pointing out of $X_+$. Furthermore $-\nabla \gamma$ is transverse to $X_+ \cap \partial (Q \times B)$, pointing into $X_+ \cap \partial (Q \times B)$. Therefore the Morse-Novikov homology of $\gamma$ on the manifold $X_+ \cap Q \times B$ is computing the homology $H^\text{Nov}(X_+ \cap Q \times B, (\partial X_+) \cap Q \times B, [\beta])$, which is isomorphic to $H^\text{Nov}_*(X_+, \partial X_+, [\beta])$. Similarly the Morse-Novikov homology of $\gamma$ on the manifold $X_- \cap Q\times B$ computes $H^\text{Nov}_*(X_-, \partial X_-, [\beta])$. Inequality \ref{eq:1} therefore implies that $\gamma$ must have at least $r$ critical points on $X_+ \cup X_-$, which completes the proof.
\end{proof}
\bibliographystyle{plain}
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final_BOOK_bloomsbury_USA:Layout 3 4/28/09 12:35 PM Page 1
LOGICOMIX
APOSTOLOS DOXIADIS
CHRISTOS H. PAPADIMITRIOU
ART
ALECOS PAPADATOS
COLOR
ANNIE DI DONNA
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BY THE UNITEP KINGDOM'S
HAVING SIGNED THE "MUNICH
PACT"OF NON-AGGRESSION
WITH HITLER.
...THREE PAYS AFTER
THE INVASION, BERTRANP
RUSSELL, THEN MORE
WIPELY KNOWN AS A PUBLIC
THINKER, IS SCHEPULEP
TO GIVE A TALK AT AN
AMERICAN UNIVERSITY
ON THE "ROLE OF
LOGIC IN HUMAN AFFAIRS".
ANOTHER HISTORIC PATE,
...FOR ON THAT PAY,
THE UNITEP KINGDOM
DECLARED WAR ON
GERMANY.
SEPTEMBER 4, '1359.
16
W ARSAW
NOW LISTEN..,
AND IT'S ON THAT PAY THAT
OUR STORY BEGINS™
read A ll
4 & ouT it /
THERE'LL BE A LOT
TO SAY OF RUSSELL'S
REACTION TO THE NEW
SITUATION IN WHAT
FOLLOWS™
A NUMBER OF
AMERICANS, THE
SO-CALLED
"ISOLATIONISTS",
MOBILIZED TO
WARN AGAINST
THE POSSIBILITY
OF U.S.
INVOLVEMENT IN A
EUROPEAN WAR.
THEY WERE A MIXED CROWD: FROM U.S. COMMUNIST PARTY
MEMBERS TO NAZI SYMPATHIZERS, FROM IDEALIST PACIFISTS
TO COMMON CITIZENS, NATURALLY CONCERNED ABOUT THE
CONSEQUENCES OF A MAJOR WAR.
\--\--\---\--\--\---
\--\---\---\---
—
\---\--\--
T T
\----\----\---
— j
\--\---\--\---\--\----
\----------\-------------
17
k t i O d W O R L P '
" WA^>-
Reap ai> it(
Vs. / \ g . o u r I T -
...BUT BEFORE THAT; WE TAKE
A LOOK AT ANOTHER REACTION!
^ OH, I KNOW WHAT THE
"ISOLATIONISTS" WERE!
BUT WHAT ARE THEY POINCt
IN THE "QUEST"? >
r WAIT! NOW X
AS RUSSELL 1
ARRIVES AT THE
UNIVERSITY... j
18
" ... A CROUP OF >
ASSORTEP ACAPEMIC
PIGNITARIES AWAIT
. T O GREET HIM. >
BUT THEY ARE NOT ALONE.
WE ttAvE
A... £fR".£>U<5HT
P R O & ^« ... ,
^ ? o f N
W HAT
s 3 o RT?
T h E/YEW \
in te r ^ a tio w al
PEVELOPA1EHT5-
o'—s
... H/lve cseArep
$ o m £ R z A c n o r t f !
MORE SPECIFICALLY,,.
,"A GROUP OF "ISOLATIONISTS"
IS PICKETING THE ENTRANCE
TO THE BUILPING WHERE HE
WILL BE GIVING HIS TALK!
THEY HAVE A VERY
SPECIFIC REQUEST
OF THE SPEAKER...
19
Y p O / v l 'T ^ T H i ^ 1
' T A t^ p R o f e s s 0^ ^ 1
t f A l Hffc£\AlfTHU£!
k Io ia / our. P K.re£r! /
NOW, THERE IS A VERY GOOP REASON
THE "ISOLATIONISTS" ARE MARINO
THIS PEMANP, PARTICULARLY OF
BERTRANP RUSSELL!
r i | KNOW: ^
RUSSELL WAS
FAMOUS FOR HIS
PACIFIST
.ACTIVISM!
y m! e v e n N ,
GOING TO TAIL
FOR IT! BUT IN THE
RUST WORLP WAR!
YoO AP£ A
M a o ! o f R £ A $ 0 ^
P k o F e ^ o e .! j
'A t A 1 /
W O W ? j
'^ o T A R e
one. sipej^.
^ -W H Y ^ o n 't Y o iA
o w e iw5ij>r
ANj) usreW to
. n i U t T U R E ? /
^ N o\ N
lo V R . P lA C £ IS
OUT HeS-E,
MllTff Uj! y
" ■ ^ s , a
Pe o p l e W ,
OF COURSE,
RUSSELL IS
NO EASY
APVERSARY.
& u t i \tJ H i e e
SPfAioMj
A$our...
t y L A & H , It J ITS HU3H£^r
FoKH-. lob\c\ A 6t& X
lATT&OjWCrUM/TO A ,
chat A&oUT THf WAR-'-
Wf P*J>N
TO
C o m €
H e ^ £
OK,
H £ A P ,T A U 5
this
C H AT
A 6 T i< ^
T ilA E
TDK,
LOOK AT THAT.
SO ? PO TH E '
"ISOLATIONISTS"
A C C E P T HIS j
v INVITATION? /
W / OH, THEY >
¥ COULPN'T MISS T H E
' CHANCE TO AIR TH EIR
VIEWS, WITH SUCH A STAR
PACIFIST PRESIPINCi!
AH...
...oo you
EVER. MISS
ATHENS?
HOW
CO U LP I N O T ?
IT 'S W HERE
I GREW UP...
...T H O U G H
W HERE I L IV E IS
NO HELL-H O L E
E IT H E R !
ee
BY TH E \
WAY, WHEN ARE '
YOU FLYING BACK
V TO BERKELEY?y
TOMORROW!
SO, W E'VE O N LY C O T
TOPAY TO TA K E YOU
THROUGH TH E FIRST
PART OF T H E .
STORY!
S WHICH, N
QUITE FRANKLY,
IS T H E C R A Z IE S T
T H IN G I'V E EVER
HEARP! V
I UNPERSTANP YOUR PASSION FOR
T H E "QUEST". BU T WHY IN..,
(Z C O M IC S ?
f TH E FORM I S ^ ^ >
P E R F E C T FOR STORIES
OF HEROES IN SEARCH
v OF GREAT GOALS!
RIGHT! FROM
PONALP PUCK TO
BERTRANP RUSSELL,
VIA SUPERMAN! .
TH E HEROES OF
T H E "QUEST" ARE
FASCINATING PEOPLE.
PASSIONATE...
Zz
...TORTUREP.
IN FACT, TR U E
S U P E R H E R O E S !
■i/YAY
T X i x O u c r r
io u a -
^ YOU PO
W ELL T O START
WITH RUSSELL:
HE IS ONE OF TH E
STORY'S BIG
V STARS! >
/ BUT N
' TH A T 'S
NOT TH E O N L Y
REASON WE
CHOSE HIM. HE
ALSO HAS TH ESE
O T H E R
\ S IP E S ...
A
AH YES... N
POLITICAL
ACTIVIST,
PHILOSOPHER,
LAPIES'
MAN! j
1 \i
BUT MORE:
HIS C O M P L E X IT Y 'fe ss > S
AS A C H AR A CTE R !/ y CS
7/,, t" ''
v ^ Nl
A T YOU KNOW,
T H E "FOUNPATIONAL N
QUEST" IS ESPECIALLY
FASCINATING T O ME, FOR MY
WORK. MOST OF T H E GREAT
IPEAS IN MY FIELP HAVE
TH E IR SEEP IN T H E R E
s. SOMEWHERE! v f
> WELL, WE
FOCUS ON T H E
P E O P L E !
r TH E IR IPEAS ^
INTEREST US ONLY TO
T H E E X T E N T T H A T TH E Y
SPRING FROM TH E IR
V PASSIONS. ^
L 2 / BY T H E \
—f WAY, I READ \
f GIAN-CARLO \
( ROTA'S ARTICLE
YOU SENT, ON TH E
CURIOUSLY HIGH
I R A TE OF PSYCHOSIS
I IN T H E LIVES OF
\ T H E FOUNDERS y
\ OF LOGIC. /
y S SO,
r WHAT'S TH E V
ALTERNATIVE?
"T HE Y BECAME
LOGICIANS FR O M
j. MADNESS?
' D O N ' T ! Y
DON'T YOU
REVEAL
"WHODUNIT"
V YET! y
Y DOESN'T \ =
/ T H A T MARE Y
YOU THINK.? '
ESPECIALLY SINCE,
CONTRARY TO
POPULAR LEGEND,
MOST O T H E R
MATHEMATICIANS i
\ ARE N O T J
\ MADI A t
^ T H A T ' S \
r CLOSER T O
T H E TRU TH, IF
YOU REPHRASE
] V IT AS - A
A so, w hy
SUCH A HIGH Y
RATE OF MADNESS V
AMONG L O G IC IA N S , |
PARTICULARLY? I
S . MIND YOU...
' ...I THINK. T H E >
CUCHi. "THEY WEN T
MAD FROM T O O
M U C H L O G IC "
. WON'T HOLD A
\ W ATER! ^
' \ V ^ f / l A t
* Apostolos' d o g is n o /1, tro w ed o f+ e r Japanese cow ie s. "M a n ga " is a s la n g w ord in G re ek, w ea n in g
sow eA h in g lik e ,"c o o l du d e "C U.S.) o r " J 'a c < "f-h e .-la d "C u .k.) .
WAIT... WHERE ARE WE?
I'VE NEVER COME THIS
WAV BEFORE. ^
THAT'S RIGHT...
HENRY MILLER SAIP TH AT
T R U E ATHENIANS NEVER
COME NEAR THE
v ACROPOLIS. y
THIS
IS IT...
/ IF OUR \
] "aUEST" IS ^
' HALF AS CUTE
AS T H E STUPIO,
WE WON'T
li HAVE A .
^ P R O B L E M ! /
LE A P TH E WAV,
M A N G A!*
* Annie, is French.
£6
p
HI
THERE!
OUR "LOGICAL
EXPERT" IS HERE
AT LAST!
I'M ALECOS.
I HOPE YOU'LL LIKE
OUR PROTECT. .
ANNIE.
SENKS TOR
Z E HELP! *
PON'T
ASSUME
VOU'LL
NEEP
IT!
I'M ANNE. BEING
T H E RESEARCHER,
I'M AL L FOR HELP.
COFFEE
ANYONE?
.SO, 6ERTR A N P RUSSELL
ARRIVES A T THIS
UNIVERSITY TO GIVE HIS
LECTURE, ANP - A
7 I'VE ALREAPY >
TOLP CHRISTOS OF
TH E PROTESTERS!
£7
AH, OK! SO, AFTER SOME HAGGLING, T H E "ISOLATIONISTS"
V FLOCK IN T H E HALL, T O HEAR T H E TA LK . A
I THIN K I'L L TOIN TH EM, TOO...
^ IN T H A T CASE, ^
PLEASE, EMPLOY YOUR
IMAGINATION TO GIVE OUR
PRAWINGS C O L O U R !
1. PEMBROKE LODGE
,,,Gfea\ Philosopher av\d
above, all, great Logician!
Ladies and 0len+lemem,,,
Lord Bertrand Russe.ll!!!
L q /ip a ^ 4 » c ih lc^ !
31
And so, Ladies and
C\entlewen, it is wy great
honour to present to you
our speaker, a great
Mathematician ,"
Thank, you,,,
Wall, the- Peon has asked we
to speak, on " The Role of Logic
in Hui/vtam Affairs". Of course, if I
take the injunction literally,,,
,"You shall hear the
shortest lecture In
recorded history!
3£
V /Aansi
so-called ^
"great events" are great only
in their irrationality. And
none is more irrational
L. than War!
lo tine protesters who "welcomed"we
here, I say: you brought to my wind other
protests, in which I also fooK part.
^ I totally ^
agree: people should
have a say in the
momentous decisions
affecting their
^ lives! A
y Anal certainly ^
nothing Is wore
mowentous for humanity
at the present i
moment,,, /A
...Than the ^
terrifying possibility
of another VJor\d War!,
" This is the question that poses
itself: Should you join-in. In England's
war against Nazism? Should you be
your "brother's
V Keeper"?
^ N o !\
We should ]
k n o t /
Shhh!
Be patient,
please... j
For, frrat I wort to ask7l
by what method should
your decision be wade?'j
IRefleot
on this.
Well, to short with, I hope^
you agree, that rational tools
should be employed! A
But what are these?
What are the special
tools of Reason?
~ To answer this ~
wtean'ngfully We wiust,
like the Greeks, 50
further baok"._
...And ask: ^
What is Logic?]
' It is 1
exactly this
question
I want to
address
today, a
' Aristotle said that
" in order to understand something
you wiust go to its origins." .
Oh,great! Now we'll
hear ancient history!
r But wiy way
of telling you
the story of
^ Lo^ic..,
...Will be
through the
tale of one of
its iMOst ardent
^ fans. .
... Myself!
34
I was a young child, and thus
by no weans yet a logician, when
I arrived at "Pembroke Lodge".
Here
it is.
...This was the howie. of
wy paternal Grandfather,
Lord John Russell.
And of wy Ciramd/i/uother, whowi
everyone. called "Lady John''!
—
The day of i/uy arrival
is rwy first d ear
wiewory.
It w a s ra in in g ,
MIND YOU
DON'T SOIL
YOUR. SHOES,
MASTER
BERTIE! .
And the uwbrellas
Were. blaolc.,
My Grandfather
Was an im p orta n t
iMai/w In -fact,
he had been
Prime Minister
of Great Britain.
HELLO, CHAPPIE!
SEE THE COIN? I'LL
MAKE IT DISAPPEAR!
But he was no t
Prinne M in is te r
o{ Pembroke Lodge.!
NOT
NOW,
JOHN!
r FOLLOW '
ME, YOUNOi
l MAN! a
Lady John'' ltd we. into
a waz-t. of corridors,
up and (douin stairs,
through countless aloors,.
All th e w h ile expounding
on th e rules th a t Would
3ouern wy new existence
YOU MUST
NOT RAISE
YOUR VOICE,
FOR ANY
REASON... .
...YOUR 1
HAIR
p r o p e r s
BRUSHED!
YOO S^ 00^ °
W ELL" ^ 0 O '-
^ A L
y o u i/ v r 1
S H A L L g j
T A L A
s ^ ° 4
Rules, rules, rules...
36
You MUst
Know th is: Logic is ^
all about rules, In fact, it'
begins with definitions
, j and continues w ith i
rules.
r And in this sense^
— and this alone —
I think that I aw
m y G ra nd m o th e r's
disciple. A
. . . In ^
a sense, her
most devoted
pupil.
Oh, m'l Grandmother
loved rules and,,,
A &EPROOM IS A
PLACE WHERE VOU
GOTO SLEEP AT NIGHT!
,,. Pefinltions!
Even the most ordinary
concepts had to be
redefined, by her own,
strict si^darcls.
I can still feel, as I speak
to you now, the sense of
profound desolation
I felt as darkness fell
on that first day,,,
The wind sighing through
the trees echoed
iM'f
own
■feelings.
A
m m
".And then, suddenly,
I heard an unearthly
Ido an !
4lthoigh it contained
all the unbridled ewiof'on
0/ the animal, the moan
was, in an uncanny sort
of way, hvwian.
In n*y new abode,
the hooting of the owl
resounded with a new
oi/winousness.
38
T h e provenance, of th is
ghastly moan became one of the
. first mysteries of wiy life...
Er» \
excuse me,
Lord Russell.
You haven' t
mentioned your
parents! .
r Ah, yes! That, "N
of course, was the g r e a t e s t
y mystery of all! ^
r I had no
idea where
they were,
at this
^ time. A
My f ather ^
had hold me that
my mo+her had
gone on a "very
^ long -trip", a
So, after he too
disappeared, I assumed
t-haf he had gone to
L- join her! ^
...Yet, I was receiving such
conf licting messages on the
matter, it was impossible for
me to fathom the truth.
But let me return to the
ghostly experience of that
first night at the Lodge...
f . . . Or ^
rather to the
, following
^ d a y .
This was my -firsf
ta s te o f th e . pro blem atic 1
n a tu re o f knowledge.;
Why was everyone
denying S om e th in g
I h a d so d e a r ly
h e a r d ?
Was I mistaken? Or was
everyone else? Or were they
all lying? &ut" a fourth...
' ".M i»ch m o re
S in is ter a lte rn a tiv e
also presented its e lf
a n a lte rn ativ e th a t
fille d we w ith
L . d re a d . ^
39
I was in -fo r a s u rp ris e .
ER... ^
EXCUSE ME,
MISS... j
' yes,
MASTER
BERTIE?
DOES '
PEMBROKE
LODGE HAVE
A
ftHOST?
... I SHOULD
THINK NOT!
D i D ^ S
YOU HEAR
A TERRIBLE
. HOWL? .
CERTAINLY N O T !
...IT MUST
'AVE BEEN
THE WIND!
Was the howl a
hallucination? Could I
Wave, heard something that
had not actually occurred?
^A n d did this wean,.,
,"l was
M a d '!
.Thankfully, wy
Curiosity did not
let we ruminate
too IMUOh...
IT CAME
FROM UP
THERE!
THAT'S
IT !
TH E LORD
PRESERVE ME...
r m
As on iwany
future
occasions, wiy
eagerness
to know was
stronger
than iwy
■fear.
LOCKED!
HO
,..&ut drove we
to investigate.
Pespite its heavy share of
restrictions, it was only in the
old Mansion's garden that
I acquired wy first, rare
experiences of freedow.
^ /iv
In it, there was
always something
interesting to do.
In the first few worths of wy stay a t PewibroKe
Lodge I drew a plan of ihe house. I discovered
it recently in a wouldy suitcase,,.
LooK at It!
Full of prohibitions
and dark, secrets.,,
I remember in particular a day in
early spring, a fine day Made- even
finer by one of Grandmother's
rare trips to London.
TUM TE
TUfATUM
E M
Grandfather's
study was near
the hop of..,
□
" Grandmother s omnipotent
'forbidden areas" list.
Her absence presented
Me, with a rare chance
for exploration.
f SO,
DESPITE AAALL
TEMPTA-AAATIONS
|
TO BELONG J
TO OTHER NAAA-TIONS
j. TA TE-TE-TE-TE
J TE-TUM-TUM... n
43
HE REMAINS AN E N 6 U S H M A A A N ...
HE-EE REMAAAIN5 AN E-EEEE... ^
UNLESS YOU
COME OUT, I
CAN'T SHOW YOU
MY LI&RARY!
WHOA I
IS TH A T YOU
IN THERE,
BERTIE?
The tight reins
G r a n d m o th e r K ep t me
under hod never ^iven
me the oppo rtu nitY
for on un-choperoned
discussion with
Grandpa.
COME ON, N
THEN! SEE WHERE
MY SO-CALLEP
"COLPEN YEARS"
. ARE SPENT. y
CHOP CHOP!
WHAT YOUR
GRANDMAMMA
DOESN'T
KNOW... a
...WON'T
HURT HER!
Oh, what a treasure
trove this was!
HAVE YOU
READ ALL
THESE BOOKS,
GRANDFATHER?
DON'T BE
ABSURD, OLD
BEAN!
~ MOST ^
BELONGED TO
M Y GRANDFATHER!
?
YET,
I ENJOY BEING
IN TH E MIDST OF
GREAT IDEAS!
HE, HE!
45
I'M SORRY,
CHAPPIE, THIS IS A
FORBIPPEN BOOK!
If I load any doubts
that Knowledge, was
a dangerous affair,
row they were dispelled.
"FOMIPPEN"
BOOK?
YOUR NT
GRANDMOTHER 1
IS A GREAT BELIEVER
IN NOT EATING OF
THE FRUIT OF
k KNOWLEDGE! A
~ But; actually,
I t h i n k t h e ro le o f
t h e s e r p e n t w a s
w o r e a p p ro p ria te !
AND ~
I HAVE TO
PLAY SOME
KINO OF
CERBERUS...
NOVELS, ON
THE LEFT WALL,
ARE CONSIDERED
O U TRE
, BETTER
PLAY IT SAFE AND
STAY AWAY...
r SOCIAL ^
THEORISTS ANP
PHILOSOPHERS,
IN THE TOP SHELVES
TO
MV
RIGHT, ARE
DEFINITELY
N. N O -N O 'S ! A
AND THERE
r IS YET ANOTHER
CATEGORY, OF T O TA LLY
FORBIDDEN BOOKS!
L UP THERE... A
Though he, didn't
actually offer ut
anything, Oiramdpa
wiade an enticing
description of the,
gradations of evil!
...NOW, NATURE BOOKS,
UNLESS THEY CONTAIN
MATERIAL ON REPROPUGTION,
ARE KOSHER.
...APPROPRIATELY
KEP T UNDER LOCK
ANP KEY!
P ro h ib itio n w o rk e d its
cu sto m ary a ttra c tio n .
H6
Y & O
r ...M a n y y e a rs y
w e re to pass u n til
I re - e n te r e d ,
L t h e lib ra ry . J
< You see, as I lay am Ke in >
My bed that night, the prospect
of investigating a mew world A
^ filling wy wind,,,
". Cawe the
same horrible
houol I had heard
on wy first night
at Pembroke
Lodge,,,
Yet, after the
first few seconds
of paralyeing fear,,.
,"The resolve
to investigate
its source grew
irresistible!
£>ut as I was
about to venture
into exploration,,,
?
48
OH DOCTOR,
PLEASE,
HURRY!
I waited.,,
And waited™
Until,,,
DOCTOR! ^
IS SOMETHING,
WRONG, WITH
GRANPFATHER?
I'M TERRIBLY SORRY,
OLD CHAP,,.
I'M
AFRAID...
...HE'S GONE.
"G ONE"?
Ill
Ar\ ©nge/" whose
OTiuse I f hought I Knew/.
W hen
the stonw
coitue, fhat
evening,.,
The next
day, I paid
wy last
respects to
Grandfather.
Sitting beside hiwi, I
Couldn't he-lp thinking that
Grandwother's reaction
was
wore, anger than grief,
49
,"My very wors\ fears
were deafening!^
confirn/ed.
Co randfeather had been
punished, -for giving
w e o f fh e F ru if o f
fh e T re e o f Know ledge.
Now [ was being warned,
and warned mgs+ clearly
Indeed:
ve n tu re on
a n o th e r
exp e d ition in
■ forb idden
te r rito r y ,.,
a v e rtin g d e ity 's
th u n d e r would
s tr ik e m y w o rld
k f l a t ! j
An expert in dead languages
©TT(T>fe$
(S O rT tiurm
®tt\t )a&u £
©7T\T)££
©TtvoaSu ^
©TTYDfe/)
dared
D a n g who
s h e w as,
G ra n d m o th e r
c o u ld n o t t r u s t
iiuy e d u c a tio n
to a s c h o o l.
5 o s h e
h ire d a
tu to r .
NOW
YOU SAY
IT !
B u t ioo
o u ts id e r
Could be
tr u s te d
w ith w y
religious
e d u ca tio n .
r " TME HAND OF THE ^
LORD WAS UPON ME, AND
CARRIED ME OUT, AND SET
ME DOWN IN THE MID OF
THE VALLEY..." S
f IN THE
MIDST OF
THE VALLEY,
u BOY!
5£
"...IN THE MIDST
OF THE VALLEY WHICH
WAS FULL OF BONES.
AND THEY WERE VERY
DRV..." ___________^
GRANDMOTHER?
PLEASE
TELL ME
WHERE MY
PARENTS
ARE!
WELL, LET US
SAY THEY ARE
OUT OF HARM'S
WAY. a
YOU MEAN
... WHERE NO
FURTHER HARM
CAN COME
T O THEM?
6
o O
' ■
2
Mysteries H
proliferated... |
...WHO AR T IN ^
HEAVEN, HALLOWED
BE TH V NAME,
THY KINOiPOM
^ COME... A
I MEAN ^
WHERE
T H E Y
CAN DO NO FURTHER
HARM TO OTHERS!
Tine years passed
but r/y first questions
regained unanswered.
Mysteries which,
looking knowledge,
I could only address
through Faith,,.
with
no great
success
53
NO.
54
Yet, a5 I grew, iwy situation
considerably improved.
One spring
morning.,,
&ERTRANP?
0
a
... Brought IMS pleasant surprises!
I AM
VOUR NEW,,,
... GERMAN
TEACHER!
The wash
p le a s a n t o f
a ll being,,.
A young man...
...Whose beliefs somehow
had escaped Grandmother's
stern controls!
Y I'M GOING TO
DEMONSTRATE EUCLIP'S
PROPOSITION TH A T IF TW O
ANGLES OF A TRIANGLE...
L ANY TRIANGLE... .
It was he who introduced we to a very old gewHewan.
55
...THERE EXISTS A
POINT D ON SIDE AB,
SUCH THAT AD = AC.
...ARE EQUAL,THEN,
OF NECESSITY, THE TWO
ADJACENT SIPES ARE
ALSO EQUAL...
WHAT DO
YOU MEAN "O F
N E C E S S IT Y " ?
... OF LOGICAL
NECESSITY!
NOW, ASSUME THE
PROPOSITION IS N O T
TRUE... THEN SIDE AS
IS GREATER THAN
SIDE AC AND THUS...
r ...WHICH ^
CONCLUDES THE
PROOF. dUO PERAT
PEMONSTRANPUM.
What oi
da\f f h a f
was!
56
WORK IT OUT
FOR
YOURSELF!
...M A G IC !
57
Nothing in wiy life was quite ^
tine same after that first- Meeting
with Euclid. In his works, I found what
I had vainly sought for in
Grandma's faith!
_____
T G e o m e try ~
showed ims the only way
towards reality. Reason.
In it, I e n c ounte re d f o r th e
f ir s t tim e t h e delicious
e x p e rie n ce o f Know ing
so m e th in g w ith to ta l
C &rfain iy !
* Proof i
thus became
My Royal
Road to i
L Truth!
This encounter began to permeate
my whole world-view,,.
...Especially
as wy new
teacher did
not stop at
Euclid.
r r e l ia b l e n ,
KNOWLEDGE
ABO UT THE
WORLP CAN
O N LY
BE GIVEN TO US
, BY SCIENCE. /
AND
PHYSICAL
SCIENCE
DERIVES
ITS POWER FROM
M ATHE M ATICS!
DOES?
Suck statements 'Here,
ax call ent comfort on
suwwer days.
'W l
But when winter
nigkts came,.,
58
/ OPTICAL ^
PHENOMENA,
ELECTRICITY, THE
MOVEMENT OF THE
PLANETS, ALL CAN
BE EXPLAINED
v BY SCIENCE! /
CAN ^
SCIENCE
EXPLAIN
THUNPER,
w TO O? J
OH, IT CAN
PARTICULARLY
WELL EXPLAIN
THUNDER, LAD!
'S C IE N C E
IS OUR ONLY
HOPE.
59
...W ith a ll th e , b a d th in g s aiwaK ening...
",A n d w y
a rc h -e n o w y ,
thunder; on
th e prowl,..
",l th e n
s ou g h t
out.,,
...M ore
tra d itio n a l
forws of
coiMfort\
rR A U L E /N
MULLER?,
<Shall9comjKar/ zee? to ct/w ium '& day/?
%owcuY>mro lovely/and/m n ten fiem to
dbouylv wuul&do-diakez&daduty/ lads* o f May/ ,
And&imned&leaso... ^ 0
HAPPY
NOW,
t o r LITTLE BERTIE?
60
/And th e n if c o m e : os
I slept1 one toiglnt, I received
on unusual m essage. i
I coin only S urm is e
■the. identity o f th e
m e ssenge r,,.
," B u t I oio
to ta lly ce rtain o f
its e ff e c t on me.
M y c u rio s ity b a re ly e xceeded t h e f e a r o f g e ttin g caught.
61
T h a t s a m e d a y, I e sc a p e d G ra n d m o th e r's a tte n tio n ,
t o fo llo w t h e r o u te in d ic a te d in th e m ysterious n o te ,
6£
I had
arrived,..
63
U e re w y
o r i g i n s lay
b u r ie d , . .
dub
I w as
n o t
alone..
;
y^S. NATIONS NOT
so
& i£ v r as 'tme.e j
MygT IN THElfc TU R N S
>Q TV ^ A nts FAaALL...
RULE, EWTANHa ,
r RULE THE WAVes r l
BRITONS NEE£££\/eR
1
WILL
BE
SLAVEsm f t ^
\ j3
_______
__
r r /
COME
BACK. HERE,
YOU FILTHY
M O S T '.
k '
(A
... WHILE TbtOuSHAN'T
puOURSH OREa T ANO
pp.EE THE DRS.AO ANO
EN\IY OF Th E N \ Auu'-
ANCiELS ANO
MINISTERS
OF GRACE!
I SEE A
GHO ST!!!
IT IS MY
FAMILY'S
VAULT,
SIR. ,
YOU
SEE I -
His legs
were left at
Sebastopol,
CriiM&a1.
65
I ran again info poor
"Old RarKer", a -few years
later; af the village.
r This old ^
invalid was w\y first
encounter with
k the evils of War.
I now Knew where iwy parents
were. But though I had no idea
how they ended up there,,, ,
J Knew/
where to
find out.
F A GOOD ^
THING I SAW
GRANDMOTHER
HIDING THE
k KEYS! d
"TO TALLY
FORBIDDEN
BOOKS"!
This
was it.
66
/4t long
last, I could
put faces fo
iwy fauwily.
Rachel Russell,
my sister.
Katharine Russell,
nde. Stanley...
Her death from
diphtheria, just
after she turned
six, sparhed-off
the. chain...
...IAy mother
died of the
sa*ue disease.,.
..Their deaths
killing my father,
who had lost all
mil) to live.
T he family
albuiAA revealed
the cause of
the deaths,
but not the
terrible crime
they had
committed to
deserve the
wrath of rwy
Grandmother.
Who, always
keen to explain...
I < t3 3 ° ^ . . . ^ h J ' ■ . , v w =s- \
o t -
RESPECTFULLY
GRANDMA , I KNOW
WHAT I'M DOING!
B u t I was, to o sh o c k ed ,
to b e iio tlw 'id a te d .
I'M TRYING T O FIND OUT
EVERYTHING TH AT YOU
K E P T FROM ME!
67
Bagged in to th e sc e n e
o f th e criwie.
WHAT DO YOU
T H IN K YOU ARE
DOING, YOUNG
. MAN? A
AND THAT'S PRECISELY
WHAT I INTEND TO DO!!!
G ra nd w io th ej'" w a s
to o s h o c k e d t o
re a c t.
68
A n h ou r Ia te r
I'd learned, a t
long long last,
w y pare nts'
" te rrib le secret",,.
...Which was nothing
wore than that
they'd lived in a
Mtmge a trois with
a sickly young wan.
Unusual, certainly! But nothing
terrible enough to werit cutting off
a young boy frow the wewory of
his parents.
But there
were also
hints of
a darker
truth,,.
I HAVE
AN U N C L E ?
69
T h a t year,
I began iuy f irst
philosophical work:
an intiwiate diary!
But as I '
was now
at war with
Grandmother,
It was in code.
5iwply, a transliteration
into 6 reek, a language
she did not know.
This reads: "Most huwans behave
irrationally. All the wore reason
to pursue the study of Logic...
Of course I aw also huwan and thus
aw no stranger to fits of non-logical
thinking. Bufalso lean discern these
tendencies in wyself and thus aw wore
able to resist thew,""
"Creek Exercises",
as wy notebook
W as called for
reasons of
subterfuge,
beoawe a haven
fo r all wy secret,
forbidden
thoughts.
THE FACT T H A T THROUGH A POINT
OUTSIDE A LINE, ONLY ONE PARALLEL
TO THE LINE PASSES.
— » BUT WE
HAVEN'T YET
PROVEN TH A T!
r WELL, EVEN ^
OLD EUCLIP
HAS TO TAKE
SOMETHING FOR
GRANTED!
T h is iMon/e/it
M arked a
h rr'i ble,
disappoW wem t
B u t ig w e d
th e r e s t o f
w y life..
70
I h a d a lo t to p u t \r\ i t !
WHICH IS?
...THEREFORE, AS WE KNOW BY
THE PARALLEL POSTULATE...
TH AT'S BECAUSE
IT IS AN AXIOM,
MY LAD!
BUT YOU SAID IN
GEOMETRY WE MUST
PROVE EVERYTHING
WE SAY!
WHAT'S THE VALUE
OF A PROOF IF IT RESTS
ON THE U N - PROVEN ? ,
71
Or\e cold
evening, in my
lois+ year at
the. Lodge...
,"As I was waiting wy thoughts
in "Cwk Exercises", cawe
•the. old, terrible moan.
This time I was
determined
to get to the
truth!
BLESSED ^
ARE THE POOR.
IN SPIRIT... .
...FO R THEIRS IS THE
KIN6POM OF HEAVEN.
BLESSED ARE THEY T H AT
MOURN, FOR THEY SHALL
BE GOMFORTEP.
BLESSED ARE T H E ...
Here then was
iwy uncle., wy
■fovfhcr's bro+Ktr.
7£
BLESSED
ARE THEY
WHICH DO
HUNGER
AND
.THIRST,.,
In fact, wadmess
alwosh foot; hold
I of wty wind ah
hhah MOMemi.
I would have
ended wy life
hhere and fhern.,,
Were, ih nor
fo r hhe, hope
of Reason,,,
...The vision of a ho+ally logical world
^ / , I had glimpsed in Mahhewahios.
73
In his face I saw hhe
ewbodiiMenf1 of wha+
was ho becowie wy
greahesh nighhi/uore,,,
M ad n e s s !
2. THE SORCERER'S APPRENTICE
77
MM...
Z Z Z
WELL?
N N N N
I CAN SEE HOW
YOU1 RE BUILDING
TOWARDS YOUR
"LOGIC AND
MADNESS"
THEME. / .
IT'S LIFE
Z A T IS
BUILDING
Z A T ! .
BUT - DIDN'T RUSSELL
HAVE A BROTHER.?
YES, MUCH ^
OLDER. HE WENT
TO BOARDING
SCHOOL.
BUT WE DECIDED
TO CUT HIM!
"COMIC
LICENCE"...
*
1
j would Kill M y s e lf."
* * "Yes! mow, if you oloiVf imiho/, the story continues/,/"
YES, OR I'D BE
DRAWING TILL
KINGDOM CO M E !
AS FOR THE
FEAR OF
MADNESS...
OH, WE ^
CERTAINLY
DIDN'T CUT
T H A T I
I MEAN, ISN'T IT
FAR-FETCHED
TO SAY RUSSELL
WOULD COM MIT
SU IC ID E...
' ...WERE >
IT NOT FOR
MATHEMATICS?
IT'S TRUE! HE WRITES
IT IN "CREEK
-----
r
EXERCISES"! .
YOU MEAN HE
W RITES: "A l ^ouXS
kl\ X jjoC ceX tp?*.
...WE CAN
USE AN
ALGEBRAIC
TECHNIQUE
This was the -first
step towards -fulfilling
wy dream of becoming
a Mathematician.
79
Ladies and Gentlewen,
please iwag'ine we row
in a hall very much liKe
this one... ^
,., B u t th is '
tim e a s a m e m b e r
o f t h e a u d ie n c e !
f A n a u d ie n c e , ^
W ind you, c o n s is tin g
exclusively o f y o u n g
w e n ! .
I see Ainm...
Third row!
»,Mr. Bertie
R u s s e l l!
It's my firs t
y e a r a t
Cambridge
Um'i/ersity.
THEREFORE,
IF WE CONSIDER
e TO BE
INFINITESIMAL...
NO! NEWTON
IS DISPUTING EUCLID'S
INVENTION OF
R IG O U R !
i
80
EXCUSE ME,
PROFESSOR.!
B u t wty in tn o d u c tib n to
t h e "Q u e e n o f th e / S c ie n c e s "
w a s a s h e e r\- d is a p p o in tiM e n t.
r I WANTED TO ASK >
y o u now yo u d e f in e
. "INFINITESIMAL"? V
WHy, OBV IO US LY ,
AS "T H AT WHICH IS
INFINITELY SMALL"!
BUX T H AT
IS C IR C U L A R !
y o u C A N 'T DEFINE
A TERM BY USING IT
IN THE PEFINITION!
OH? ARE YOU
DISPUTING NEWTON'S
INVENTION OF THE
C A L C U L U S?
MATHEMATICS IS THE LAST RECOURSE
OF R E A S O N ! WE CANNOT UNDERMINE IT,
\--\---\--\---
-
\----\----\-----
. WITH SLOPPY THINKING!
NOR CAN WE
C H A N G E IT,
TO SUIT A
PRESUMPTUOUS
YOUNG MAN!
'TH E R E ARE N O
PRINCIPLES,' BAZAROV SAIP.
'I LIKE TO P E N Y ! MV BRAIN
IS MAPE ON T H A T P LA N !1"
T h e d elicio usly wicK.ee! wew no ve ls
he lpe d wie g ra d u a lly lose w y fe a r o f
G ra n d m o th e r's s te m coiMwa^ds.
81
S tu d y in g M a th e m atics I h o d hoped to
pe n e tra te th e es s e nc e o f tru th ...
S /G H
...But all I was learning
was cheap calculating
tric -K s !
Still, iwy thirst fo r
Knowledge did not
diminish.
EXCUSE ME, '
WHICH WAY TO
THE UNIVERSITY,
LIBRARY? A
A t C a w b rid g e ,
I d is c o v e re d
new w o rld s .
New o p tio n s .
AS ^
f THOUGH A
TIGHT IR O N
RING, WAS BEING
S CR E W E D ROUND
MY NECK. !l!
I BEGAN TO
FEEL THE MOST
V IO L E N T P A IN S
IN M Y HEAP...
f\v\d
the. new drawia ^ave
uwe Keys fo unlock, dm k
secrets,..
Secrets of
inheritance.
" THERE IS SOMETHING
W O R M -E A T E N ABO UT YOU
SINCE B IR TH !" THE SINS OF
TH E FATHERS ARE YISITEP
UPON T H E C H IL D R E N !
OHHHH!
8£
THE
DOCTOR
TOLD ME THE
T R U T H !!!
WH... WHAT?
G H O S T S ! WE
ARE A L L G H O S T S !! !
W itln v is io n
Caiwe- p a in .
...WHICH WE
M U S T GET
RID OF !
HIS MESSAGE IS ANNOYING
ANP THU S T R U E : " WE ARE
TRAVELLING WITH DEAD
" WEIGHT ON BOARD."
Bur pain was
transformed
into courage,.
I was mow i
re ad y to battle
against my old
en em y.,, ^
," Irrationality,
' in its highest form! ^
To my enlightened mind,
madness was a disease
of weak spirits, pulling
them away from the
A natural harmony /
of Reason.
Pt/ni^g a vacation in wales,
the lines of Shelley's great
poem "Alastor" accompanied
a journey to an inner-,
beauteous land.
83
F IL T H !
AN OPEN
DRAIN!
A B S O L U T E L Y
LOATHSOME!
WELL DONE,
MR. IBSEN!
84
^ O N E D A R K E S T G LE N S E N T FR O M ITS WOODS
U OF M U S K -R O S E TW IN ED WITH JA SM IN E..."^
« V *.k AM V I
II
" A SO U L-D IS S O LV IN G ODOUR TO, IN VITE
ju TO SOM E M O R E LO VEL Y M Y S T E R Y ,,," .
^ > .h— M d . . /_n -
k d i f p C ® cD0M?F
In na tu re ., I S aw th e .
ewbodiiA/ient of a
new freedow,,,
...The, freedom
I needed ho ge-h
rid of iwy own
"dead weight".
I w a s s t r o n g
e n o u g h f o
c r y o u t .
EARTH, OCEAN, AIR.,
BELOVED BROTHERHOOD!
MOTHER OF THIS UNFATHOMABLE WORLD!
FAVOUR MV SOLEMN SONG, FOR I HAVE
LOVED THEE EVER AND THEE ONLY!!!"
A t la s t, I c o u ld
t u r n w y b a c k ,
o n iwy d o r k
le g a c y .
86
During another excursion,
I ran into a perfect symbol
o f Grandmother 's faith.
"A HOUSE
B U ILT ON
S A N D "...
...SINNING!
Until them, churches had
inspired in me fear of
an all-powerful Being.,,
Not th is tim e .
I found -this
total emptiness
so comforting.
A new experience of
ecstasy was the best
antidote to any lurking
residue of fear.
"P RO FE S S IO N AL MATHEMATICIAN" ? WHAT
DOES A PROFESSIONAL MATHEMATICIAN DO?
_ L O N G SUMS? >
\------\---------\---------\---------\-------
' YOU ARE N
NOT FAR OFF
TH E M ARIA
v T H E R E! .
Yet our friendship
at firs t progressed
sanely...
...In fact, a bit
rather 'too sanely
fo r i/uy ta s te !
^ —■ ■ ■
87
In those years, I was
often accompanied by
extreme inner tension.
My near-i/wania passion
fo r certain, absolute
Knowledge,,,
Was doubtless
wade wore intense
by loneliness.
It was then that '
I wet the woiwan
who was later to
k become my
wife.
Alys Pearsall 5with.
LiKe we, she came f row a
sternly religious
family,,, j
__
r
Which, naturally, also
contained an insane-
streaK!
HOW C R U C IA L
T H E Y /ARE!
r IF ONLY YOU >
KNEW HOW M U C H
PEPENPS ON
TH ESE
v QUESTIONS... y
I SO WISH
M A T H E M A T IC IA N S
HAP B U T A T IN Y
B IT O F T H E
PASSION FO R
T R U T H T H A T
A N IM A TE S T H E
PHILOSOPHER!
88
A T CA M BRID G E,
NO ONE T A L K S
A B O U T T H E R E A L
ISSUES O F
M A T H E M A TIC S .
T H E
REAL
ISSUES?
LIKE W H A T IS
T H E N A T U R E OF
MATHEMATICAL
v TRUTH? .
r ... A N D 1
H O W CAN
W E KN OW
!T ?
HOW
INPEEP?
' AH... >
PHILOSOPHY
IS CLOSER TO
MY HEART!
THEN MAYBE I SHOULD PURSUE IT...
AW well.,.
NOT ON M V ACCOUNT]
YOU SHOULDN'T!
I wets an absolute,
beginner In c ou rtsh ip
89
W ith n o t th e - s lig h te s t
b it o f b e g in n e r's luck.!
Yet there was
no doubt,,,
GENTLEMEN,
YOU MAY NOW
. BEGIN!
J was
infatuated!
Euem d u rin g
t h e " T r ip o s "
t h e d re a d e d
fin a l e xa w s,,.
A ly s d om inate d
wy th o u g h ts !
FIRST CLASS
HONOURS,..
NOT B A P !
[
M ath e m a tica l
Triposjl
90
T h o u g h fo r tu n a te ly !
b e n e fic ia l
e ffe c t's ! i
WELL DONE,
MR. RUSSELL!
0
I HOPE SUCCESS N
HAS HELPED IM P R O V E
YOUR OPINION OF
k MATHEMATICS! y
r ON THE CONTRARY, T
I AM NOW CONVINCED,
PROFESSOR! T H E R O T T E N
F O U N D A T IO N S WILL /
V GIVE WAY.
N ow , I c o u ld a f f o r d
+o s p e a h w y w in d.
TH E EDIFICE ,
OF MATHEMATICS
WILL C O L L A P S E !
OH? A R EN'T YOU CONCERNED
THAT ITS F A LL WILL
C R U S H YOU TOO? .
r NO! YOU SEE, 1
I DON'T PLAN T O BE
. IN S IP E IT. a
j
A n d f o M ake
M \j d e c is io n s .
w as h u n g ry f o r
tr u e K no w le d g e -
9'1
Having earned
a Fellow ship,
I c o u ld now
pursue a new,
alternative
itin e r a ry .
I re a d w ith t h e passio n o f S h e lle y 's
" in s p ir e d a n d d e s p e r a te a lc h e w is t".
M e an w h ile , I p ers is te d in
wty sie_ge o f E ly s 's h e a rt.
SO, HOW ^
FARES YOUR
NEW
PARAMOUR?
r ^ MM, ™
I'M NOT SO
^ SURE... ^
^ OH? IS
MASTER BERTIE^
A L S O DISPLEASED
WITH P H IL O SO P H Y?
AT LE A ST MATHEMATICIANS
s. T R Y NOT TO CONTRAPICT,
V
___
ONE A N O T H E R ! . /
N O T SO PHILOSOPHERS!
TH EY ARE A L L "OREAT"
...AND A L L IN T O T A L
L D IS A G R E E M E N T ! S
9£
"STUDYING PHILOSOPHY"
R E A L L Y MEANS GORGING
YOURSELF ON A STEW OF
E V E R Y I PEA IMAGINABLE!
A PLATONIST >
THINKS APPEARANCE
IS BUT A BAP COPY OF
REA L REALITY...
WHILE AN ARISTOTELIAN PUTS ALL
HIS FAITH IN ITS OBSERVATION!
ARE MENTAL CONCEPTS
INNATE OR I
\--\---\----\---\---\---
ACQUIRED? J
"INNATE", SAYS
THE GREAT KANT!
IS THERE AN OPPOSITION
BETWEEN MIND r
------
-
AND MATTER? )
YES, SAYS
DESCARTES!
N O , SAYS
SPINOZA...
TA K E YOUR
P IC K , MISS
SMITH!
' AND WWAT
ABOUT T H E
MATERIAL WORLD
AROUND M E?
3 E 3 E
...RATHER EXT R EM E VIEW, IF
— II
-\----
77777!
\---\--\---
.YOU ASA ME
"WHY, >
IT 'S A LL IN
T H E MIND,"
BERKELEY
SAYS.
WHICH, ACTUALLY,
IS ABOUT FINDING
YOUR O T H E R >
\H A L F... y — ^
' NOW I'M
READING
PLATO'S
'S YM POSIUM
' JU IC Y
APPLES, SIR
ACTUALLY,
I THINK I BETTER
BE GOING!
Euc-lid load taught mie
to abhor contradiction
l'd turned to Philosophy
loo King for truth,
but also guidance,,,
",Of practical
value for life.
93
...A
RATH
...N ot alw ays I
s uc c e s s fu lly !
OH DEAR.!
TH E T R A N S IT IO N FROM T H E CATEGORY OF
CONTRADICTION SHOWS TH A T THE EXCLUSIVE
REFLECTION OF TH E STABLE OPPOSITION MAKES IT A
NEGATIVE, AND THUS TH E REFLECTION DEGRADES ITS
PREVIOUS STABLE OF PETERMINATIONS T O TH E LEVEL OF
BEING O N LY PETERMINATIONS. AND SINCE THE POSITION
HAS BEEN MADE POSITION, IT HAS GONE BACK
TO UNITY WITH ITSELF.
With wiy friend
Moore, I sought
enlightenment at
f ha feet of tha
latest fashionable
Hegelian.
T H E Y C A L L THIS TRASH
P H IL O SO P H Y? I W AN T TO
FIND MY WAY TO REALITY,
MAN! I W A N T A M E TH O D
TO ACQUIRE C E R T A IN
. KNOW LED G E ! .
M oore, u nd e rs to od w e .
9A
W ELL, HEGEL IS
OBVIOUSLY NOT
YOUR MAN!
...BUT
WHO IS?
IF ONLY
PHILOSOPHY HAD
. A E U C L ID ! a
YOUR N
BOWLER IS
READY, SIR!
THAT'S EXACTLY
WHAT LEIBNIZ PIP...
SOMEONE TO GIVE IT STRONG
FOUNDATIONS AND A L O G IC A L L Y
P R E C IS E LANGUAGE!
In a hatter's shop,
I fo u n o t a t la s t w h a t
I w a s looking f o r .
B U T F O R T H A T T O HAPPEN, LOGIC
W OULD H AVE T O BECOM E A N E X A C T
S C IE N C E !
T H E
STRUGG LE
H AS T U S T
BE G U N ,,.
95
f ...WITH HIS ^
"C A L C U L U S 1
R A T IO C IN A T O R i \
"R A TIO C IN A T O R "?
YES! A WAY
TO MAKE
THINKING
AS CLEAR AS
GEOMETRY!
S O CLEAR TH A T WHEN
A DISAGREEMENT
ARISES, WE TU S T
L HAVE TO S AY ... >
"C A L C U L E M U S !"
"...L E T US
C A L C U L A T E ."
A
WHAT LEIBNIZ DREAMT OF WITH THE
'C A L C U L U S R A T lO C IN A T O R ,GEORGE BOOLE
CONTINUED WITH HIS " L A W S " .
_____
PERHAPS
YOU'D LIKE TO
TRY IT ON, SIR?
%
O
WHY DON'T I
K N O W THIS?
because ^
PHILOSOPHERS THINK
IT'S M A T H E M A T I C S
AND MATHEMATICIANS
P H IL O S O P H Y ! A
LOOK...
|
THE ATOMS
OF LOGIC ARE THE
PROPOSITIONS.
WHICH WE C O M B IN E
THROUGH CERTAIN
. LAWS. ^
My
f ir s t
weetlmg
with
E u c lid h a d
p la n te d a s eed ...
...But hearing about
Le/bniz's dreaiw was
th e actual calling.
-4-f+er th a t day
I Knew: I was a,"
^ AHA... AND >
NOW THE PLOT
T H IC K E N S !
T h e m a m d f h e r e , M o o re
in tr o d u c e d wte to a neoy
e x tr a o rd in a ry w o rld .
ALL THAT LOGIC
REALLY IS, 15 USING
COMBINATIONS OF THE
KNOWN, TO REACH
THE U NKNO W N.
"A TAUTOLOGY
APPEP T O ITSELF
IS A TAUTOLOGY.'
"A TAUTOLOGY IS A STATEM ENT
WHICH IS NECESSARILY TRUE BY
VIRTUE OF ITS LOGICAL
FORM, AS IN A l l re d
AN TS ARE RED1."
97
T H E ^
GREEKS
KNEW A L L
T H A T !
B U T LEIBNIZ.
USED A FORMAL,
S Y M B O L IC LANGUAGE
TO SAY THINGS
. LIKE THIS... 4
T IM E O U T !
"A TAUTO LO GY ADPEP TO IT S ELF IS A
TAU TO LO G Y ." IS N 'T THIS A LIT T L E
TO O TECHN ICAL?
I K N O W WHAT
A TAUTOLOGY
IS, THAN K YOU!
r BUT DOES >
THE AVERAGE
N. REAPER j
f A KNOW f I
THIS? / J f
IS THERE
SUCH A
BEING? J
98
I THINK. TH AT A T THIS POINT
YOU SHOULP INTROPUCE
p"7
\----\------\-----
.S O ME LOGIC...
^ ...S O M E
ELEMENTARY
NOTIONS. ^
I'M T A LK IN G
T O YOU,
OLP BOY!
HE C A N 'T
H EAR YOU, YOU
MUST T E L L US!
TH IS IS TH E
STORY OF LO G IC ,
. RIG HT? .
NO! IT'S THE
STORY OF ITS
P E O PLE!
YOU C AN 'T U N P E R STA N P
T H E PEOPLE W ITH OU T
T H E IPEAS^ r ^ ? v
C A N 'T YOU?
WELL, T H A T ALSO
PEPENPS ON WHICH
WAY TH E STORY IS
v___ G O IN G . _____
r IT 'S GOING ^
TH E WAY OF A L L
STORIES, PASSIONS
. LEAPING TH E
^ WAY...
...A T R A G E P Y
WITH LOGICIANS
. A S HEROES!
IF THEY
f WERE PAINTERS,
W OULPN'T YOU SHOW
THEIR PA IN TIN G S? .
99
OK, W HAT PO YOU HAVE INMINP,
E X A C T L Y ?
\----\----\----
/ S I
' PEFINE
YOUR,
TERMS AT
LEAST'
WHAT ^
IS THIS
"LOGIC"?
W ELL, IT'S '
, AH... E R ... >
...A >
METHOP?
...A ^
SYSTEM!
^ YOU TAKE
A BIG THING.
^ A N P . .
NO...
f YOU T A K E M AN Y A
k L I T T L E TH IN G S ..^
NO, N O ! ^
YOU TAK E SOME
S IM PLE FACTS
. ANP... a
f ALRIGHT] >,
( LIS TEN NOW
TO ARISTOTLE'S
\P E F IN IT IO N .m v
> "LO G IC IS N
NEW ANP
NECESSARY
. r e a s o n in g : '.
"N E W "?
"N E C E S S A R Y "?
N E W BECAUSE YOU LEARN WHAT YOU
PO N 'T KNOW...
AN P "
N E C E S S A R Y .,,
100
...BECAUSE
CONCLUSIONS ARE
INESCAPABLE!
G E T I T ?
S T A K E i
T H E F A M O U S
E X A M P LE
W HICH EVERY
SCHOOL KID .
V lK N O W S . . . ^
"A L L MEN ARE M O R TA L
"SOCRATES
IS A MAN."
"THEREFORE
SOCRATES IS
M O R T A L !" ,
S EE ? FROM TWO S T A TE M E N T S '
ALREADY KNOWN, YOU PRODUCED
. A N E W A N D N E C E S S A R Y y
C O N C L U S IO N !
Q U O P E R A T
PEM ON S TRA N PU M ,
. M A NC A ! A
... ANP SO, IT WAS THI5 TYPE OF
REASONING T H A T LEIBNIZ TURNED
INTO A SYMBOLIC SYSTEM.
' <AND ^
BOOLE?
' S H ALL
WE CO ON
WITH TH E
, STORY?
... BOOLE TOOK
THIS FURTHER
ANP—
y L E T 'S ASK OUR ^
PROTAGONIST TO
INTRODUCE US TO BOOLE!
101
Frow tine. d a y w hen I fir s t \
le a d e d o f tH e dreaw f o r a purely
logical calculus, I w as F o o te d .
T h is new fa s c in a to r com p letely
to o k o v e r wiy life...
- ...Well, 1
a / w s t
completely!
BERTIE?
E ven in a n ideal C ity o f
R e a s o n th e irra tio n a lity
o f E ro s will c ree p in .
IS THE MAN
YOU ARE READING
SO FA SCIN A TIN G ?
FASCINATING
ENOUGH...
...TO BE THE
HERO OF THE
MAN YOU ARE
READING!
OH?
r INDEED! \
" LEWIS CARROLL", \
a.R.a. MR. POPGSON,
. IS AN EX PE R T
\ IN BOOLE'S /
A IDEAS! A _
T H E SAID >
BOOLE BEING T H E
MAN WHO HAS
M ADE LOGIC AS
C LE A R AS .
. ALGEBRA! /
r YOU N
PON'T
SAY!
CONTRARIWISE, IF IT WAS SO, IT M IG H T BE.
AND IF IT WERE SO, IT W OU LD BE. BUT
AS IT ISN'T, IT AIN'T.
T H A T 'S LOGIC!"
10£
f D O N 'T YOU
REM EM BER
TW EEPLEPEE'S
V WORDS? >
oh, is it; >
R E A L L Y ? .
IF YOU WANT T O KNOW, FOLLOW M E ,
LIT TL E ALICE, / c t
-\--\---
BUT I DON'T LIKE
TW EEPLEPEE.
in
OR \
TWEEPLEPUM
FOR T H AT
V M A T T E R ! y
SPELLED
A-L-Y-S
OF COURSE!
H E Y !
M EEEO W
/ OH, ^
/ HI THERE
CHESHIRE CAT!
T E L L ME,
i WHICH WAY DO
\ I HAVE TO
> \ 0 0 FROM i
HERE? A
TH A T DEPENDS
ON W H E R E YOU
WANT T O 0 0 !
HM...
I DON'T
MUCH
C A R E
WHERE
100!
THEN IT DOESN'T
REALLY M A T T E R
WHICH W AY YOU CO!
103
YOU SEE, LOGIC
IS R E A L L Y A T O O L . }
YOU MUST CHOOSE WHAT ,
USE TO PUT IT IN! I
SIMPLICITY
ITSE LF.
. AN D... J
...NOW, I'LL BE THE CATERPILLAR!
W A IT ! I HAVE
THE IDEAL PART
FOR YOU!
TH E M A P H A T T E R !
TH E TIM E 1
HAS COME,
TH E WALRUS
v SAID,.. a
HELP!
...FOR LIT TLE ALYS
T O S H U T HER EYES!
YOU A R E MAD,
B E R T IE ! ,
^ I'M MAD!
Y O U 'R E MAD!
WE' RE A L L MAD!
ONE.,.
TWO.,.
THREE,,,
104
SIXTEEN...
SEVENTEEN...
T W E N T Y !
G O T
YOU!
WMER.E
IS ME?
105
BERTIE?
WHERE A R E YOU,
BERTIE?
NO "BERTIE" H E R E ,
LIT TL E ALYS!
I
' I'L L \
CATCH \
YOU, 1
MISTER
HATTER!
106
HA HA
HA HA!
I
CATCH M E IF YOU C A N ,
L IT T L E ALYS! ^
\---\---
, r
OH, I WILL, YOU ABOMINABLE
'\--------\-------------.C REA T URE!
CUCKOO!
GOT YOU THIS TIM E!
The Hampton Court
wiaae< was ideally suited
fo r im froduci^ Boolean
Logic!
To navigate it, you have to
decide if certain paths have
Value 1, weaning "this path
leads to the exit,,,"
" .0 r 0 ,
weaning
"it doesn't!"
107
ALRIGHT, I GIVE U P ! ^
WHICH WAY DO I GO NOW?
T H A T A LL DEPENDS ON
W H E R E YOU WANT T O GO,
LIT T LE ALYS!
So, if a path X has value 1
to a certain point and then
forks into Y and Z
r^",W ethen write ^
our choice fo r a correct
< continuation this way.i
...Signifying that X continues'
-
-------
asY an Z...
J ...And retains v a lu e ^ ^
4 if either Yar Z is "1, or if
both are. But takes value 0
^ i f both Y ana/ Z are 0!
108
Ye+, as wy sweefbeart-bo-be was, for
sowe sfrange- reason, uninferesfed in +he
nioe-bres of algedrafc Logic...
____
^
...I ended up ~
navigating bine
waze wyself!
Bub b hen I -found Alys
engaged in anobber f onu
of binary invesbfgabion.
ME HOLES ME.,.
HE LOVES
ME NOT...
ALVS...
HE LOVES ME...
OH, HELLO BERTIE!
SORRY,
OLD OilRL...
I JUST WANTED '
TO ILLUSTRATE THE
CONJUNCTIVES
"AND" AND "OR" ASA
SERIES OF DECISIONS
HELPING. YOU...
HE 1
LOVES
ME
NOT... .
HE LOVES ME... HE LOVES ME NOT...
...TO SOLVE
THE PUZZLE
OF THE MAZE,..
109
...WITH METHODS
OF A LOGICAL
CALCULUS.
HE ^
LO V E S...
AND HOW
ABOUT M Y
P U Z Z L E !
E H ...
...WHICH
PUZZLE
IS T H A T ?
DOES HE L O V E M E ?
E H ... HUM...
DOES WHO
LOVE YOU?
TA K E
A G U ES S!
And s o ,,,
,"M y iM pro M p i-u a tte m p t
a t a le s s or end ed w ith m e
as t h e s tu d en t.
B e in g no e x p e rt in
fem inine psychology,
I de cid e d th e n Meeting
w a s a hu g e success.
'MO
MMMMMMmm...
A n d th o u g h
u n p re p a re d , I d id
r a t h e r w e ll!
Soon after, I took.
Ays to Pembroke
Lodge..
GRANDMAMMA,
I PRESENT MISS
ALVS SMITH!
" OH, BERTIE,
I THINK. SHE
H A T E D
k M E ! >
r DON'T
BE SILLY,
OLD GIRL.
SHE
A D O RES
YOU! j
DO YOU ■'
THINK SO?
AND
SO
DO !!!!
B E R T IE !
BUT I L O V E
HER., G R A N D M AM M A .
^ "MARRY" THE SMITH
WOMAN? PO PPYC O C K ! SHE'S
. T O T A L L Y UNSUITABLE, BERTIE!
T "LOVE" H E R ?
W H AT U T T E R NON SENSE
YOU M E R ELY L U S T
^ A F T E R HER!!!
WELL,
T H A T TOO!
m
I've- said if already:
with Alys I encountered
■for A he. firs t tivue...
,"That vviost illogical
of passions hui/uan
beings call "love".
And being a
neophyte, —
I saw but f l
one course R
.of action.^Bl
AN P H E A P f^
T H IS : I WAS
IN FO RM E P
TH E R E IS
INSANITY IN
H E R BROOP!
AS
T H E R E
IS IN
OURS...
A L L >
T H E M O R E
REASON FOR
YOU N O T T O
M ARR Y
^ H ER ) y
This was a rare instance when
the bugbear of wiadness had
no effect on wie.
,"5o pressing was wy
ut^e to consuwiwate
^ o u r relationship! >
OH/
P O N 'T LOOK SO
G LU M , O LD CHAP,
YOU NOW HAVE AN
A IM : TO LE A RN
. LOGIC! ,
5ut though my love life was at last
Making headway, my career as a
thinker was stalling.
VE BEEN LEARNING LOGIC'
FOR T H E PAST YEAR..,
&ut Science depended on Mathematics,
which was a tofo/ <wess, plagued by unproven
assumptions and circular definitions. To
repair it, a powerful Logic was needed... ^
",&ut there
wasn't one!
And so we
came to on
im passe-.
ME
...BUT IT 'S HARDLY
E N O U G H TO COVER
MY NEEDS!
L E A R N M O R E
T H EN !
T H E R E
IS N 'T
ANY...
^ To understand tmiy predicament,
I remember that my profound, underlying
I aim had never changeal: to acquire
^ ^ c & r ta in knowledg& about the world.../
r ... Knowledge
which could only
come from
h. Science. .✓
toy now. I'd come to realiee that Mathematics >
resembled the Cosmos of Indian Myth : its apparent
solidity really depended on the reptilian whims of
its carrier. Mathematics rested... /
WE ARE APPROACHING TH E
DEMOCRITEAN VISION, TH E
DISCOVERY OF THE A TO M S
^ OF M A T T E R ! ^
TH E WORK OF THOMSON
AND RUTHERFORD IS
TRU LY REVOLUTIONARY!
The sorry
state of the
"Queen of
the Sciences"
was wade
even worse by
+he successes
of Physics.
...BUT OUR ^
POOR MATHEMATICS
LAGS LAMELY
. BEHIND! ^
^ AND ^
WHAT'S WORSE IS
THAT MATHEMATICIANS
P O N 'T FACE UP TO
k TH E PROBLEM! V
WE M U S T ^
MARE THEM REALIZE
TH E TERRIBLE
v MESS! X
Wi+W Moore, I
dreawed of ^reoit
discoveries.
ORDER
AWFULLY
K SORRY TO BARGE IN,
CHAPS, B U T YOU ARE
TA LK IN G SALPERPASH!
M ATHEM ATICS IS
IN PERFECT
The situation
shocked we: i/uost
iwathewaticians
were painfully
| unaware of the
fliwisiness of the
foundations.
M3
r"- ONLY T H E N C A N ^ N ,
WE BEGIN T O S E T T H E
HO USE OF MATHEMATICS
v IN ORDER!!! v
^ T H E Y AR E T
FO O L S , MO O R E !
r P O N 'T ^
LO SE
HEA R T,
O LP M A N !
In A lfr e d W h ife h e a d ,
I fo u n d a s tr o n g a n d
k in d r e d s p iW f.
' OH, T H E R E N ;
v ARE PEO PLE WHO
CAN SEE T H E SITU ATIO N
CLEARLY, R USSELL.
BU T ALL OF T H E M
. AR E, ALAS, ON T H E /
\ CONTINENT. A
m A wentor.
T O A C H IEV E A N Y KIND OF CERTAINTY IN MATHEMATICS,
WE M U S T R E -E X A M IN E IT S BASIC ASSUMPTIONS, WE
M U ST B E G IN A T T H E B E G IN N IN G ,.
Proposition XV. If *
fy,
then
yfx.
P roposition XVI. If z 4 *y, then z 4 x,
2
4 y, z 4 x + y.
P ropo sitio n XV II.
\fzfxy,
then x y 4 z, x \+
y
4 z.
P rop osition X V III. If z ^ x + y, then z ^x , z ^ y, z^xy-
P rop osition XIX. Ifz ^ x \+ y , then
xy^z,
x + ,y fz .
P ro p o s itio n XX. Ifx z 4y . and x ^y + z, then x^.y-
Yet wy
despair
drd not last.
For finally
I w e-f a w a n
refreshingly
rigorous
in his
approach.
HEAR
HEAR!
j T IF WE
' UNITE T H E X
H E AL T H Y PARTS
OF MATHEMATICS AND
T H E C O N C E P TU AL
SOPHISTIC ATIO N O F
T H E NEW LOGIC,
WE CAN LAUNCH
A P O W ER F UL ,
A T T A C K . A
A " N E W " ? \
LOGIC
HAS
N O T ADVANCED
ONE BIT AF TER
. BOOLE!
j
' SURE,
IT HAS C O M E
A CERTAIN WAY
SINCE OLD
ARISTOTLE...
^ B U T ^
IS IT STRONOi
ENOUCiH YET
T O PEAL WITH
M A T H E M A T IC S ?
f T H E R E
f IS AN OLD \
G ER M A N S AYIN G :
"IF YOU W AN T T O
LEARN SO M ETHING,
. GO ON A
Ls T O U R N E Y !" A
1+ was Whitehead, More. +han
anyone else, who helped 1we see
beyond +he provincialfsw of the
English wiaifhewa+'cal esfoibfohwient.
I suffered the. 'silent- prayer'...
With on ly minor,,,
Alys and I we/e married a t th e
"Mee+'ng P lace"of th e Quakers,
which hen family attended. I
115
It was A«s encouragement'
that made uwe set out- on
a grand voyage of
k intellectual discovery,,, J
But before setting th e house of
Mathematics in order,.,
r ... In the ^
first part of
which, I mef
the new stars
. of Logic. A
,"l made
th e f irst
move to
create my
own!
I was very brave.
," Signs of e nn ui.
And I was so glad when if
was over!
3. WANDERJAHRE
The Continent
was for wie a garden
of rare intellectual
^ delights. ,
I crossed
it as I would
an enchanted
land!
Every day I was learning [,
something new, something
which led me deeper into
a magical kingdom,,,
r IN E E P A H A N D >
WITH THE LUGGAGE,
V B E R TIE . y
A Kingdom fre e
o f th e errors and
confusion which
plague th e world
o f material reality
S iin f HUnuten B n b altl
At Cambridge, I had
chanced upon the
enigmatic German text
called "degriffsschrifi
The "co/tc&pf script"
it introduced Has in line
with Leibniz's vision of
a fully logical language.
T he auth o r lived
In
a
sm all G erm an town,
fam ous f o r
Its
philosophers,,.
W
However, neither In a nor his worh
was wall Known. But that didn't
Make we doubt its potential
importance. Truth is.,.
...Tha "concapt
Script" died not
look van/ inviting!
".Yet, ones fha abstruse
surface was penetrated,
a lot of sense could be
found underneath.
-ISO
^ EXCUSE M E,
IS THIS PROFESSOR
FREGE'S HOUSE?
r NO,
THIS IS HIS
GARPEN!
F HIS ^
HOUSE IS IN
T H E R E !
I was not yet aware
of the odd habits
of Logicians,,,
IS THE
PROFESSOR
AT HOME?
NO, HE
IS IN THE
GARPEN.
<1, Principal among which is that they
always wean exactly what they say!
I AM THE
PROFESSOR.
WHO ARE
y ou '? .
' IT IS ~
A G REA T
HONOUR T O
M E E T VOU.
MY N A M E IS
RUSSELL.
Gottlob Frege was
a true giant.
,..Though, of course,
only in the metaphorical
sense!
IT'S SO SURPRISING TO HEAR
A MATHEMATICIAN SAY T H A T !
^ ...AND THIS IS '
MRS. ALYS RUSSELL,
MY WIFE. ^
r DELIGHTED, T
HERR PROFESSOR.
HM. YOU GO INSIDE. \
HELP THE O T H E R
WIFE, MAKE THE TEA!
j
WOMEN ARE
SUCH ILLOGICAL
CREATURES.
r I TRY TO >
EXPLAIN
TH E FACT TO
MY WIFE, j
^ ...BUT "
SHE C A N N O T
U N P E R S T A N P !
SO T E L L M E WHY YOU
ARE HERE... WHAT SAY
YOU OF MY WORK?
MY " B E G R IF F S S C H R IF T " ?
FRANKLY, I FINP IT
DIFFICULT GOING. IT'S
S O DIFFERENT FROM
\ B O O L E ! ^ /
r Y E S' ^
r BUT MY A IM '
IS S O D IFFERENT!
BOOLE W ANTS A
CALCULATING .
K TOOL. A
BUT THE AIM OF LOGIC
IS N O T CALCULATION.
r IT IS ^
T O MODEL
REALITY!
"S UR PR IS IN G "?
...WHAT
IS SURPRISING
ABOUT A RATIONAL
BEING TE L L IN G
k T H E TRUTH?
WE MUST WRITE A BOOK, F R A U FREGE:
"THE ORDEAL OF BEING M — —
. MARRIED TO A L O G I C I A N "! 1 ^ , g
r OH, MY GOTTLOB
IS A GOOD HEART. BUT
SOMETIMES HE TRIES
. MY NERVES! ^
ALWAYS THE
" EXACTITUDE
THE "RIGOUR".
,..THE ORDINARY LANGUAGE IS NOT
SUITED TO
-
----------
SCIENCE! a
THAT'S
E N O U G H
ROSES!!!
...IT 15
GOOD ONLY
FOR TH E
K IT C H E N !
<7^
M S
^ ADD TO THAT ^
A B S E N T \- M I N P E P N E S S !
G O T T L O B 1.
G E T INSIDE, YOU FOOL, OR.
YOU'LL D ESTRO Y M Y GARDEN!!!
...SO, IN ORDER T O
UNDERSTAND REALITY, WE
MUST FIRST C R E AT E .,.
...A LANGUAGE
TH AT IS
C O M P L E T E L Y
. LOGICAL! >
QUITE.,.
^ ...BUT, IN T H E
MEANTIME, SHOULD
WE PERHAPS TOIN
T H E LADIES?
P R E C IS E L Y
MY REASON
FOR W ANTIN G T O
LE A RN M O R E
A B O U T IT !
O N L Y SUCH A LA N G UAG E
C A N D E A L W ITH T H E
FOUNDATIONS OF
S. MATHEMATICS! ^
YOU A R E
A CENTAUR H E R R
RUSSELL: H A L F
MATHEMATICIAN, AND
H A L F PHILOSOPHER!
YOU A R E LIK E ME
IN TH IS DICHO TOM Y!
INDEED, YOU A N D I
A R E...
..K IN P R E P
S P IR IT S !
FR O M ARISTOTLE
T O BOOLE, LO G ICIANS EM PLOY
SYLLOGISMS OF T H E T Y P E "SOCRATES
IS A MAN". B U T IF WE AR E TO STUDY
LO G ICALLY MATHEMATICS IT S E L F
^ WE S H A L L N EED M O R E!
^ HM... LIK E >
W H AT E X A C T L Y ?
A t The.
o f F rege 's
/ie w lang uag e,
la y a
Simple. ide^.
Ye+, it was
enough Yo
open up fo r
US new, virgin
territory.
TH E S E
THREE
C O O K -
A CH !
W OM AN !
...WHICH IS TRUE IF, ~
FOR EXAM PLE, X IS
E Q U A L T O "RUSSELL" B U T
FALSE IF IT IS O NE OF...
W E N E ED T O IN TRO D U C E
V A R IA B LE S! W E HAVE T O BE
A BL E T O SAY T H IN GS L IK E
" X IS A MAN"...
1£3
THERE 15 A SIMPLE
SOLUTION TO THE
..."MYSTERY OF THE
THIRD COOKIE": >
r BUT
GOTTLOB,
I PUT
T H R E E -
SHE IS CONSTANTLY LACKING IN UNDERSTANDING
OF MY RIGOUR, HERR OORTOR
1S4
' T ' WHERE TT
r ARE THE THR EE
COOKIES OF MINE?
THE T H R E E
COOKIES I REQUIRE
FOR MY T E A ! y
MAYBE
YOU A T E
. ONE? .
OF COURSE I DID N O T ! I N E V E R EAT A COOKIE
BEFORE 5:00 AND IT IS ONLY 4:A8!
__
_______
DO YOU THINK I'M S E N IL E ?
' NO,
GOTTLOB,
^ B U T -
ER... ^
PROFESSOR...
THEN WHY ARE
YOU IM P LY IN G I T ?
ER, '
PROFESSOR
FREGE.,.
ER...
I... E R ...
AT E IT!
r HOW
UNUSUAL!
r NO ONE ~
E V E R EATS
MY COOKIES!
S ' THAT ^
IS BECAUSE
NO ONE IS E V E R
.HERE, DEAR!
ANYWAY, LE T 'S WAIT A BIT BEFORE
WE PASS TU D G M E N T ON T H E
H A BIT S OF G IA N TS . ^
T H IS T A L K
OF "GIANTS"™
LIK E FAIR YTALES
1£5
F i'oim J e y \a ,
we
Moved
■bo our irexf
desbirabion.
WILL YOU BE LIKE
T H A T SOME DAY?
HM?
YOU MEAN LIKE FREGE?
I CERTAINLY H O P E SO!
YOU S E E HIM A S AN OLD
E C C E N T R IC , B UT HE IS
A G R E A T M A N !
B U T W HAT IF HIS
SMALL auiRKS ARE
T H E O TH E R SIDE OF
. HIS G E N IU S ?
W H A T IF HIS RIGOUR IN BIG
THINGS IS TH E EXTENSION
OF HIS PASSION FOR
**w
_______
EXACTITUDE...
r I W OULDN'T
WANT TO BE TH E
"GREAT MAN"'S
. WIFE! ^
...IN 1
L IT T L E
ONES?
YOU CANNOT
C O U N T TH E
^ INFINITE!
TOMORROW I'M M E E TIN G
A TRUE MYTHICAL HERO! CALL
HIM "T H E MAN WHO A T E OF
T H E TR E E OF KNOWLEPGE
L O F T H E IN F IN IT E !"
T H E G RE AT GAUSS HAP
WARNEP M ATHEM ATICIAN S
" P O N 'T P E A L PIRECTLV
W ITH INFINITY..." .
"...N EV E R
LOOK. A T IT
FACE TO
FACE!"
B U T GEORG CANTOR PISOBEYEP!
ANP SO HE PISCOVEREP TH E
AMAZING FACT THAT TH ER E ARE
D E G R E ES OF INFINITY! ANP HE
EVEN FOUNP WAYS T O j
^ COUNT THEM...
1£6
SOUNPSM O S T
BLASPHEMOUS!
W ELL,
IT IS, IN A
WAY...
LOOK...
RAIN.
BEFORE CANTOR,
WE SAW INFINITY...
...THROUGH
A CLASS.
DARKLY.
f ISAY... 1
THINK, OF A
HOTEL WHICH
HAS A F IN IT E
NUMBER OF
k ROOMS. .
r is i
THERE
ANOTHER.
L KIND? "
QUESTION: WHAT HAPPENS
IF IT'S FULL AND A
NEW GUEST
ARRIVES? NO ROOM-SHARING,,
^ MIND YOU!
A
r T H E \ |
GUEST WILL
BE SHOWN
THE DOOR!,
YOUR HOTEL "HIMMELGARTEN
M E IN E H E R R SC H A FTE N .
THAN K
GOD IT'S
FINITE!
V /h e n p o e .fi
are. in love.
fHe.y re o ife
verse s f o
fh e ir
beloved,.,
1£7
BUT CONSIDER NOW AN INFINITE HOTEL:
EVEN IF IT IS FULL,,. i M
,"A ROOM CA/V
■ BE FOUND! ^
H E Y !
1£8
goo p ^
EVENING!
MR. ANP MRS.
RUSSELL...
REMEMBER: >
OUR IPEAL HOTEL
. IS F U L L ! v
INPEEP, X
WE ARE SIR, '
BUT WE HAVE YOUR
RESERVATION.
LET'S CALL TH E GUEST
IN ROOM NUMBER "X",
^ "GUEST X". >
So, i+!s onk;
noifural...
TO FIND
A ROOM FOR A
N E W GUEST...
...W E'LL T R A N S P O S E
A L L GUESTS IN A
SPECIFIC WAY.
r WE TARE >
GUEST 1, AND
MOVE HIM FROM
v R O O M .
^ THERE IS T
NO "ROOM 'I"
IN THIS HOTEL,
M EIN HER R! .
,"A iMa+hetwattCian in love. will
h v w h is o w n b ra n d o f po e.h'y!
TO ROOM 2 ,
THUS EMPTYING
ROOM 1!
Y
...ANP OF COURSE GUEST 2 TO ROOM 3
TO EMPTY ROOM 2 FOR GUEST 1!
r BUT WE N
ALREAPY H AVE
A ROOM FOR
. YOU, SIR! .
n -
__
n
, n
n n
IMAGINE MOVING E V E R Y GUEST,
FROM ROOM 1 ON, O N E R O O M POW N...
THEN ROOM 1 IS F R E E !
THE POINT IS TH AT IN INFINITY
THERE IS ALW AYS SOMETHING MORE!
1£9
AND 5 0 ON, GUEST 3 TO
ROOM 4, SO AS TO EMPTY
ROOM 3 FOR GUEST 2....
...AND GUEST4 TO
ROOM 5, SO AS
v TO EMPTY-
B E R T IE ?
' I'VE NO
IDEA WHAT
YOU'RE
TALKING
. ABOUT! y
' t h is W A V
OLD GIRL!
V SEE? IN \
A F IN IT E HO TEL >
THIS W OULDN'T BE
POSSIBLE. BUT IN AN
IN F IN IT E ONE... .
TH E Y ARE
CRAZY, THESE
BRITONS!
130
Leh we nom wake/ a general
rework regarolirig th is rather
sensitive subject,,,
,"Mat hematics.
All of you have some experience from school.
Those who dislike it, see if as sheer drudgery.
Those who don't, see if perhaps as a game,,.
And there is an
element of that
k. in if.
B u t th e re is also
a n o th e r sid e to
M athem atics,.,
,"A side which you can't
sense unless you start
thinking of,,.
,"The
Infinite!
A great wan once said th a t no
other idea has so inspired the
human wind. /Maybe so. One
thing however is certain,,,
No other idea has so pushed th e human mind, to
the absolute limits of its powers!
in j also,
no o th e r con cept
ha s exposed
to a sim ila r e xte nt
t h e in n e r f r a i lt y
o f m athem atica l
K now le dge!
GEORG
FR1E1DR1ICH
HANDEL
On th e way, I ran into
afeJlow/ who had wade. [tj
the. trip in the. opposite. "
direction.
H e ra I w as, a B r it o n w h o h a d s a t o u t
in s e a r c h o f G e rm a n w is d o m .
'13'!
It w as f o r t h is
re a so n , re ally,
th a t G auss
w arn ed a g a in s t
fr o n t a l a tta cK s
On In fin ity .
V e t, h is s te rn
W arn in g s d id n o t
e n te r m y w in d
o n t h a t m o rn in g
As I set out to meet
Georg Cantor, th e Maigus
o f the Infinite!
I h e a d e d f o r w y
d e s tin a tio n in high
s p ir its .
...Perfect
credentials
■for trouble!
Mine! you, Halle's University
had recently incorporated
Wittenberg's and thus could
lay claii/vt to being th e Aiw a
M a te r of both Hamlet and
Doctor Faust. And so it had,..
—-
13£
I did not find
th e atmosphere
exactly welcoming.
I'M SORRY.
I'M LOOKING FOR
HERR PROFESSOR
CANTOR?
A T Z IS
ADDRESS
ME IS...
I headed
for the
new location
in d ic a te d ,
certain that
Cantor had
left the
University,,,
So, as I reached
a group of dark
buildings,,,
Try and imagine a young
painter being received by
Michelangelo.
A composer
meeting
Beethoven.
That is how I felt,
as I knocked on
Cantor's door.
I
133
,"For a higher
academic position.
,.,l S u rm ised th e y
be longed to
an in s titu tio n
f o r t h e New
Mathematics.
THIS WAY FOR. H E R R *
PROFESSO R CANTOR?
J A , T A : FOR ALL
T H E "PROFESSORS".
If this was so, the
decrepit state of
the interior,,,
...Spoke badly of the state
of the New Mathematics!
134
HERR ~
PROFESSOR
CANTOR?
J A ?
In awe, I faceJ fhe
craofor of Se-h Theory.
I AM R U S S E L L ,^
SR... AN ENGLISH
MATHEMATICIAN...
...AND A M O S T ARDENT
S T U D E N T OF YOUR
W OR K!
\----\------\----\--
\--\--
r AH... SO >
THE ENGLISH
A R E READING
M Y W O R K ? y
WELL, EVEN IN STUFFY OLD
ALBION, THERE ARE A FEW
ENLIGHTENED SOULS!
"A FEW"?)
WELL, IT'S A
BEG INNING ^
\ M O !
ALL /MUST
^BELIEVE!!'
i
135
WELL, SIR, S E T THEORY ISN' T
EXA C TL Y FOR EVERYONE!
WHAT?
' WHO
CARES OF
SET THEORY?
A L L T H A T M A TT E R S
IS MY N EW WORK!
A t f ir s t ', I th o u g h t I h a d
h is s e d o u t o n s o w a
great new discovery.
A F T E R UNMASKING
TH E PLAGIARIST
SHAKESPEARE,
k I NOW HAVE...
...COMPLETEP
MY MAGNUM
v OPUS.
THE TIME HAS COME FOR
THE C H E A T.
-\---\---
\--\--\--\--\----
(
v T R U TH ! J V
JESUS CHRIST WAS
IN RE ALITY T H E
"
-\---\----
-
SON OF...
' SOMETHING
IS TERRIBLY
WRONG HERE!
...J O S E P H O F A R IM A TH E A U !
Madness had
always terrified
Mt. But to see it
take over a
great wi'nd, was
devastating.
1
136
AH... ER...
I... I REALLY MUST GO,.,
r T H E ^
CONSPIRACY
IS
E X P O S E P !
YOU MUST CIO TO
THEGlUEEN!!! / T
SHE MUST.,, j y
...PROTEST MY
IMPRISONMENT!
r NOW, BE
A GOOD BOY
PROFESSOR!
I AM HELP CAPTIVE
"
\--\--\---\---
« A G AINST
\ /WM WILL!
I SPEA K THE WORPS OF
THE PROPHET!!! >
",..l W ILL
BLOW IN T H E
FIRE OF MY
, W R A TH!" ,
I fle d + H e o s y lu w wi+h a d o rk , le itm o tif
fro w i Msf cW ldhood ro a rin g for+issiw o,..
137
Later, liwet-Alys.
BERTIE...
WON'T YOU
TELL ME?
WHAT WAS THE
"g ia n t " l i r e ?
S I G H
But Caniov's rowings would rot leaue my miinol.,,
I w as to o c on fu se d to
a tte m p t a r y e x p lo ira tio r.
I
OPEN UP!
YOU ARE
W A N T E P !
W HAT
'g u a r d ia n s "?
T H E G U A R P IA N S
HAVE S E N T FOR
V VOU! A
T H E
GUARPIANS
OF... OF...
I N F I N I T Y !
WHY?
^ THEY ^
ARE IN THE
BASEMENT
RIGHT NOW!
TO ^
INSPECT OF
COURSE!!!
WHA." WHAT HAVE I P O N E ? !
BUT... T H A T 'S ...
N O T hA nP E L , T H A T'S
T H A T 'S ...
C a U U U S S ! ! !
r , c \ "
M y e/icoun+'et' w ith
G e o rg C a n to r s h o u ld
h a u e — i f n o th in g else. \-
w a d e w e aw are o f th e
po ssib ility t h a t t h e
jo urney I h a d em ba rke d
on w a s -frau g ht w ith
d a ng e rs .,,
...P
<M\opr&
fo r which The
apt epithet is ' [spiritual".
m
HUSH, '
MY PEAR...
JU S T A
PREAM .
WAS
. IT?
WHAT ARE YOU POING?
TH E STATUE IS S TILL THERE!
OF COURSE IT IS...
STATUES PON'T CO
FOR W ALKS!
...ANP SO IS THE
STORM.
Here-, converged
the visions of a
new, hopeful
huwanify.
r OH, I WANT
TO SEE THAT!
The abstract, spare
siwplioity of the
Eiffel Tower was a
perfect symbol of
what was best in if.
Science and
Technology were
fhe new fools
wifh which ho
realize an old
dreawi,,,
VISITEZ. LE C IN E M A T O G R A P H E H !j
dSunvCete
".A dreoiw o f to ta l triui/vipln over nature-
M w w i.
Logic and insanif y...
strange bedfellows.
Yef quite
.frequent.
' But lef we now ^
wove us on fo wore ]
k pleasanf things! A
The year was 1900. This was a time of change,
a time fo r new beginnings. And nowhere was this
optiwiswi wore apparent,,, ■
,"Than in our next destination,
the site of the International .
Exhibition!
P a r is !
The visitors to the.
Exhibition were like,
children, excited with
the toys of a brave
nem world,,,
BERTIE, '
IT LOOKS
SO H E A L !
MIMCOLO/.V
W ha t ha d b r o u g h t w e w as,,.
,"loys which ployed in unexpected ways!
tour we, had not cowe
to Par's for the Exhibition.
O r, r a th e r, I p ersonally h a d n 't!
THAT'S
MITTACi- LEFFLER
THE CiREAT
v ANALYST!
,"The
International
Congress, the
world'sgreatest
renafec-Hous of
Mathematicians.
I SAY,..
W H tTEH E A P !
1A3
MAMMA
M IA!!!
HEEEELP '
BERTIEEEE!!!
MON
O IE U H !
AAAAAAH!!!! j
LETS GO UP THE "MECHANICAL
STAIRCASE" ONCE AGAIN!
GOLLY!
Everybody who was anybody
In Mathematics was there!
RUSSELL!
I SEE...
AND LAST BUT NOT V
LEAST, HERR PROFESSOR
\----\------\---
, DEDEKIND, r -
MONStEUR L E PROFESSEUR HERMITE
HERR PROFESSOR MINKOWSKI...
HERR PROFESSOR
FELIX KLEIN... .
ENCHANTED!
...DEEPLY
HONOURED!
...A ll M'f new Heroes u n d e r on e r o o f !
' COME, \
SHE WILL
BE WAITING AT
T H E COLONIAL
PAVILION. .
F TH IS IS N
SO A M A Z IN G !
EVERY SINGLE
O N E OF THESE
PEOPLE IS A
MATHEMATICAL
. LEGEND! a
KLEIN CREATED A NEW
GEOMETRY. DEDEKIND A —
I D ON'T
THINK
YOU'VE
MET...
MRS. EVELYN WHITEHEAD,
_
___
MY WIFE! _
HOW A R E YOU,
MR. RUSSELL?
ER ... I'M,.,
AH well.,,
...T he new
lo g ical language,
flo a t w ould g ive
/M athem atics s o lid
fo u n d a tio n s .
H
It was clean th a t a
centnal nole in th is
would be played by the
Theon/ of Sets.
Henri Rjincare, th e great
French genius, a strong
believer in Ike. importance
of human Intuition.
145
^ .. . Not oil ^
encounters In
P a r is were
mathematical!
But I was too e*cited by
new Ideas +o pav serious
attention to mew emotions.
fmel there was mo shortage of mew'
ideas: mew theor/iss, mew i e c h m i e i u e s ,
mew we/hoc/s. The/e was even,,, i
,,,-A host of
new f r e J d s !
Yet w'/ own interest was focused In a single direction,,,
,"0n whose value th e Congress's two
greatest stars vehemently/ disagreed!
Pavid Hilbert, no
less great, the
Cierwiam apostle
of th e rigorous
exactness of
logical proof.
T h e Czech
progenitor of
sets was called
Bernhard
Bolzano.
/And th o s e o f you w ho f i n d sig n ific a nce In
Such th in g s w a v be tic k le d b y t h e f a c t t h a t
th is w an, w ho p la n te d -th e se e d o f "th e g re a te s t,
o f m a th e m a tical blasphem ies,.
But what exactly were ^
'Isets1'? What was this new
fashion, this mathematics.,,,
m&i /a
Cantor?
^ From the "
time of the Oireeks,
mathematicians had
looked a t individual
objects, such as... .
numben.
...A
shape.
...But In iaaIoI- 19^ century, a Czech
mathematician started looking
instead at coH&ctbms of objects,
defined by a common
f c x / f y r v ^ op£" +v- ,F or
[ ® / Y V exam ple,,, |
^ ". rAII ^
r numbers ^
greater than ? , 1
"all right triangles",
"all trigonometric j
k funotions". A
From A
this simple,
everyday
. notion,,, l
,"(neorg Cantor developed
th e majestic, wondrous
^ e d ific e of Set Theory!
^ ' T W a s ^ ^ '" " ''
also a Roman
Catholic priest.
/And,
sure enough,
the fru it of his
seed bred ,
\d is c o rd ! y
W
^ I LOVE T H IS ! N !
MATHEMATICIANS ARE,
A T LA ST, IN SERIOUS
v C O N F LIC T A BO U T y
k . A TH EOR Y!
' ...MAKING T
SPACE FOR US
LOGICIANS T O
AIR OUR VIEWS!
Puring th e Congress, every
restaurant and cafe. was
host to th e new ideas.
I WONPER IF PINERS
SIT AS " P R O -" OR
v '' C O N TR A - SETS "!
S 'lL VOUS P L A IT ,
^ M E S S IE U R S ! a
O f course, what wiade Set Theory
so controversial was its centrality
in th e guest fo r secure foundations.
... ACGORPING
TO POINGARIi...
SHH, OR HE LL 1
HEAR YOU... HE'S
OVER THERE!
W
M'SIEUR L E M
V PROPESSEUR, \
' HILBERT'S
PROBLEM IS TO O
MUCH GERMAN
k EFFICIENCY! ,
' NO! IT'S 1
TUST T H A T
T H E H ER R
PROFE SSO R
LIKES TO O
MUCH T H E
S A U C IS S E S t
HOW T
EXACTLY'
m
-IE W ANTS A MACHINE, TO 1 |
FEEP IT AXIOMS ANP MAKE
THEOREMS, LIKE ONE WHERE
A PIO ENTER S THE ONE SIPE...
...ANP TH E S A U S A G E S
COME OUT FROM THE
^ O T H E R !!!___
r HA H A ! "...THE \
S A U S A G E AT THE
N OTHER EN P"!
-1 HERR, PROFESSOR HILBERT'S IPEAS,
VICH YOU INANELY RIPICULE, ARE Z E
MATHEMATIC OF ZE FUTURE! y -
^ OH, I THINK
MY FRIENP HILBERT
HAS A 0 OOP EAR FOR
METAPHORS.
T O H ER R HILBERT,
ANP "ZE MATHEMATIC
-
\--\---\--
OF ZE FUTURE"!
O U , MAY IT CIVE A W U N D E R SA R
SAUSAGE!
\--\----\---\---\----\---\---\----
"
r FOR.
SHAME!
MEINE ^
HERREN!
B IT T E N IC H T !
r VES, GIVE
THE &OCHES A
GOOD WALLOP!
/H A IS C 'E S T ^ i
U \ [ C O M P LE T EM E N T
RID IC ULE! A
DEAR
ME!
1A9
Whoever thinks
wathewatfcians are
cold fish, was not
at th e International
Congress!
But let iMe give you an
Idea of the new intellectual
cliwiate we found ourselves
^ ^ ^ i n , 'n 1 9 0 0 ...
I work point "A"on the board,
and then I draw a straight
line rotgoing through it...
r Now, let w e ask
a que&tion,,,^™ ^
f,"You
L S ir!
",Say, this one.
r How wiany straight lines \
can we draw through "A"
th a t are parallel to this line?
O n e , obviously!
In de ed !
E u ciid would have use d th e e x a c t
Same w ord , a s would h ave a // j
l w ia th e w ia tid an s , f o r o v e r t w o ^ /
th ou s a n d y e a rs ! ^
But now,
suddenly, this
word, "obviously",
had become
very suspect! >
The advent of the mew, non-Eudidean geometries had
subverted -the notion of axioms as "obvious truths".
In fa c t, I t h a d sup p la n te d t h e V ery n o tio n o f "obvio usness"!
WHO WOULP
NOT LIKE TO LIFT
THE VEIL ANP SEE THE
FUTURE, WITH THE
NEW FACTS ANP
METHOPS IT WILL
DISCLOSE'!
INTUITIO N SHALL
N O M O R E HAVE PLACE
. IN OUR PROOFS!
\ \ ) r T H E NEW X j j
' M ATHEMATICS WILL >
N O T APM IT ANYTHING AS
" IN T U ITIVELY OBVIOUS"!
N O T R U T H EXISTS FOR US
OUTSIPE T H E CRUCIBLE
OF R IG O R O U S A
P R O O F !
AS FOR THE
AXIOM S OF A
THEORY...
This spirit was
nowhere better
expressed than in
Hilbert's wiuch-
expected talk on
"the Problems of
Mathematics
His stated aiwi was to give a
birdls-eye-view of the future
by way of twenty-three
major unsolved problems.
Yet, his speech was
above all a plea fo r
a new wientwlity,
which annulled a
hast of existing
preconceptions.
...T H E Y ARE T H E ^
STARTIN G POINTS FOR
T H E LOGICAL PROCESS.
B U T W E M U S TA BAN P O N
. ANY SENSE...
SQDooTnnnnnni
...OF THEIR
"N ATU R A L" TRUTH.
A LL WE CAN ASK OF
AXIOMS IS THAT THEY
BE L O G IC A L L Y
C O M P A T IB L E !
150
AS FOR. M A T H E M A TIC A L PROOF, WE
MUST REPUCE r ~
IT TO A...
FOR US, \
' T H E PEVILS ARE N
C O N T R A C TIO N ANP
PARAPOX! SO, FOR
M A THEM A TICS TO
CONTINUE TO REIGN AS
GUJEEN OF T H E
SCIENCES, WE MUST
v BANISH FROM IT... >
... PROCESS
SO P R E C IS E
TH A T IT CAN
BE EXECUTEP
BY,,.
~ I BET YOU
POINCARE IS
NOW THINKING
SAUSAGES.
... A M A C H IN E EG.UIPPEP
WITH T H E REQUIREP M
INSTRUCTIONS FOR M
PROOFS! y ^ M
...A L L T H A T WHICH
IS NOT PURELY ANP
STRICTLY LOGICAL
S o w e o f " H ilb e r ts Problem s" o f 1900
keep mathematicians busy even today.
But one o f th e m be co m e t h e ta rg e t
o f my own d re o w s .
y SO, T O M AKE \
7 M ATHEMATICS \
IMPREGNABLE T O
POUBT WE MUST FIRST
BUILP A RITH M E T IC ON
A GROUNP OF T O T A L .
\ C E R T A IN T Y ! A
■ T NUMBER \ "
r IS AT TH E CORE \
' OF EVERY BRANCH '
OF M ATHEMATICS, ANP
THUS A RIT H M ETIC IS T H E
ROCK UPON WHICH A L L
v OUR T R U TH S MUST
L ULTIM ATELY A
\ BE BASEP! /
...A f last, a o y a r A an d w o rth y ^ o a l!
151
^ ...RESTS ON T H E
PRINCIPLE T H A T T H E
WORLD IS T O T A L L Y
U N D E R ST A N D A B L E
^ BY REASON... .
Z ' ...T H A T F A \
QUESTION CAN BE N
RIGOROUSLY STATED,
IT CAN BE L O G IC A LLY
V A N S W E R E D ! >
IT IS IN TH IS \
y SPIRIT T H A T WE N®
' FACE T H E NEW CENTURY \
OF PROGRESS, SCIENCE
AND HOPE! WE HEAR WITHIN
US T H E C A L L : " T H E R E IS
T H E PROBLEM, SEE K ITS
SO LUTION, FOR IT CAN BE
FOUND!" FOR IN OUR SCIENCE .
y THERE IS N O " IT SHALL i
s \ N O T BE KNOWN!"
...IN ^
MATHEM ATICS
THERE IS N O
'IG N O R A B IM U S "
As Wordsworfk
loos said of a/i ear lien
revolufiom in France..,
"Bliss was if in fhatdawn fo be.
alive. Buf fo be young
\mos
ve/y
Heaven!
^ OUR PROFOUND ^
CONVICTION TH A T
A LL TH ES E
GREAT
PROBLEMS
ARE
S O LV A B LE ... .
15£
*L a fin for "we shall nof know."
I crossed th e Channel with vuy heart f irwly set on the
course it was to -follow henceforth. But really, I had
conue. full circle, to wiy firs t intellectual frustrations.
153
A t la st, I ha d to fa c e w y
disillusion w ith E u clid 's
"obvious" axioiws h e a d on.
PENNY
FOR. YOUR
THOUGHTS?...
OH... I D ON'T THINK THEY ARE
WORTH TH A T MUCH,,, Y E T !
TRY
ME...
AHEM ... W ELL, LE T ME SEE,..
\ FR E G E AND TH E ^
\ ITALIAN PEANO... /
...C R E A TE PA T
THEORY ABOUT
NUMBERS... £ R ...
L E T ME SEE HOW
TO PUT IT... .
IS TH ERE A BLA C KBOARD
ON BOARD? ,
\--\---\--\---\---
;
154
. . . M Y N O T E S !
A Mathematician's treasures luckily
reside in the. wind: they can't be
lost. So, as long as I possessed wy
senses I could venture forward..,
In to t e r r a incognita,,,
,"The dank,
basement o f
Arithmetic.
4. PARADOXES
/ This was ^
"The Principles o f '
Ma+hewial-ics " my firs t
go at becoming a
v new and 9 neater i
A Euclid!
H a rd w ork, w as
all I re e d e d t o
re a c h wiy 90a!.
/A fter wty return from Paris, I set out with fiery, though
rather misjudged, optimism, to write th e book th a t would
solve all foundational problems — and then some! ..
I built on th e ground created
by Frege in volume one of his
"FooncbHons o f Ari'thnuel'ic
u s e d a n e le g a n t n o ta tio n
in v e n te d b y P ean o .
was convinced I was
on th e riaht tra c k.
157
BERTIE?
WILL YOU
REQUIRE
ANYTHING?
FREEDOM FROM FURTHER
INTERRUPTIO NS. ,
GOODNIGHT.
T h e treasures of Logic came at a price.
Though som& m o v e s' B b
affairs I found wore
interesting flown others]
158
A s I becam e/ w o re
a n d w o r e a b s o rb e d
in w y work.™,
...M y w o rld s h ra n k to
fin e is s u e s s tu d ie d
in f h e " P rin a p k A "■
I d r if t e d fa r t h e r a n d f a r t h e r
aw ay fro m h um a n ity's
c o n ce rn s , sm all o r la i^ e .
N E W S P A P E R S //!
BOERS ARE GETTING
S Q U A S H E D ! READ ALL
^ ABOUT IT !
I didn't even
pay notice
+o its wars.
Mortal affa irs did
not concern we.
THE PROFESSOR
IS AT COLLEGE,
^ SIR. 'A
I WILL '
CALL AGAIN
LA T E R .
159
BERTIE !
I TRUST '
I A M NOT
INTRUDING...
NOT AT A LL!
HOW'S ALYS?
TOLLY WELL,
I S'PPOSE...
COME TOIN M E!
WAS ALFRED
EXPECTING YOU?
WELL,
YES...
I'M NOT SURPRISED
IF HE FORG OT...
T H A T'S
ALRIGHT,,.
HE IS SO '
\--\---\--
OTHERWORLDLY,
\ AT TIM ES...
■ H i '
AND SO OLD FOR
YOU, MY DEAR!
HE W O N'T SPEAR FOR DAYS,
AND THEN H E'LL R A G E A T ME
FOR SOME TRIVIALITY...
OH
DEAR!
160
HE T E L L S M E IT'S
WORKING W ITH LO G IC T H A T
MAKES HIM SO.
BU T I'M NOT SURE,,.
ARE YOU TH A T
WAY TOO?
I DEFINITELY WOULD
NOT RAGE A T YOU, NO
MATTER WHAT!
N O T EVER...
MY LIFE IS
SO STRANGE,
. BERTIE!
BUT
S T ILL,,.
It iYas a t th a t
tii/ue that I cawe
closer to the
Whitehead
■family.
161
...THERE ARE
G O O P MOMENTS,
ARE THERE NOT?
&EE-TOO.
<7
BEE-TO O!
WHAT ARE
YOU SAYING,
OLD CHAP?
AH, YES, BEE-TLES!
bEL-TOOL
A -fte r t h a t da y,
I called little Eric.
Whitehead "Beetle".
I lo*/ed hiwi dearly.
16£
,"l n/ here, to sp&aK >
about wy a ffa ir with
Logic. So I'll stioK. to
th a t — as w/uch os life
. will let iw e. .✓
In wiy research,
11M*c<e wwch
u s e of t h e siMple
id e a o f t h e
p rie s t Bolzano™
r S E T S , you SAY?^
I THOUGHT YOU
WERE INTERESTED
IN NUMBERS!
I AM! BUT SETS
ARE A T T H E B ASIS
OF NUMBER!
r WHAT IS "3" BUT
THE SET OF ALL SETS
WITH T H R E E
, ELEMENTS?
"THREE-NESS" IS ^
THE CO M M ON P R O P E RTY
OF T H R E E UMBRELLAS,
T H R E E HORSES,,, ;
OH?
TH R EE HATS,,,
T H R E E
COOKIES.
SETS >
HAVE M O S T
INTERESTING
PROPERTIES!
REALLY? AND I
THOUGHT THEM
B O R IN G !
163
FOR EXAMPLE,
A SET CAN CONTAIN
OTHER SETS OR..,
EVEN ITSELF!
HOW ^
CAN IT
CONTAIN
IT S E L F ?
TH E SET OF ALL IDEAS IS AN IDEA...
...THEREFORE, IT CONTAINS
IT S E L F AS AN ELEMENT.
BUT NOT A L L SETS CONTAIN THEMSELVES?
NO! THE
SET OF
ALL BIRDS
IS N O T A
BIRD!
I SAV... THAT'S AN ^
INTERESTING DICHOTOMY:
THE SET OF SETS WHICH
CONTAIN THEMSELVES...
' ...AND THE SET N
OF SETS WHICH DON'T.
ABOUT
WHICH,
WE CAN
WELL
ASK ...
DOES IT
C O N T -
W AIT '
A
M IN U T E !
164
In w\y life +o date.i I have written doeens
of books and hundreds o f artides... I've
_ given thousands
of lectures.
£>uf I suspect wy
mai/ue will survive,
if it does at all,,.
".Fbr a confo un d ed paradox
I discovered t h a t year.
paradox
th a t brought
Logic upside
v down,,, ^
I' 11 glue you a
taste of it.
Iwtag'ne a town
with a strict law
on showing.
&y it, every adult
wiale is required
to shave daily.
E>ut it's not obligatory
to shave yourself..,
",For those
who don't want
to, th e re is a
barber.
In fact, th e law decrees:
"Those who don't shave
themselves are shaved
by the barber."
He obviously cannot choose
ho shave himself, for,,.
Those who don't shave thewiselves are shaved
by th e barber." It sounds innocuous.,, However; if
taken literally, it leads straight to paradox!
Who will shove
fhe barber?"
165
For, you see, th e
qu e stio n arises:
,"Beiha the barber, it would mean
th a t he is shaved by the warn who
shaves only,,, .
," Those who
doin'i" shave
themselves!
But he can't '|go to th e barber", '
for, again, th a t will mean he'll shave
hiwiself, which the barber isn't fo r!
'P 'Y O L l >
SEE TH E
PROBLEM?,
'I 'M NOT
SURE!
r IT 'S VERY
MUCH LIRE
THE PARAPOX
OF TH E LIAR!
VICH >
"UAH"?]
THINK OF IT : IF HE IS L Y IN G , THEN HE IS IN
FACT TELLING T H E T R U T H ! AND IF HE IS
to-— TELLING THE TRUTH...
HE IS
L Y IN G !
O f cou rse, L O G IC O M IX to also s e lf-r e fe re n tia l.
NO, NO! BOOKS THAT INCLUDE REFERENCE TO
THEMSELVES, LIKE STERNE'S " T R IS T R A M
S H A N P Y " CALVINO'S " IF O N A W IN T E R 'S
iN /G H T A T R A V E L L E R "...
166
...THE FAMOUS
PRONOUNCEMENT
OF EUBOULIDES!
THE MAN
WHO SAID...
"MY FELLOW CITIZENS...
I AM NOW LY IN G TO YOU!'1
WHEN SOMETHING REFERS TO
IT S E L F , PARADOX IS NEARBY.
T A K E SELF-REFERENTIAL
BOOKS, FOR E X A M P LE ...
"REFERENCE"
. BOOKS?
\/T ...OR. V
V KURT \
VONNEGUT'S
"B REA KFAST O F
^ C H A M P IO N S " .*!
167
SUPPOSE NOW YOU MAKE A
COMPLETE CATALOGUE OF ALL BOOKS
THAT ARE NOT SELF-REFERENTIAL!
SURE! BUT THE QUESTION IS...
IT WILL BE A
5/iG CATALOGUE,
ZIS...
WILL IT N
CONTAIN
ITSELF?
GOT
IT!
IF IT POES, ZEN IT DOES NOT!
AND IF IT DOES NOT, IT POES!!!
...THE
STUDENT GETS
AN HI-PLUS!
SO? WHAT'S THIS GOT TO DO WITH
RUSSELL'S PARADOX?
LIKE THESE -
EXAMPLES, IT HAS
SELF-REFERENCE
. AT ITS CORE!
"... OF SETS
WHICH DO NOT
THE PROPERTY '5 BELONGS
TO S ' AND CONSIDER ITS
v NEGATION AS DEFINING
\--\---\-----
.THE SET...",—
GLORY BE
TO A L M IG H T Y
k GO P!!!
A
The. publication of iuy paradox wade we
am overnight celebrity in international
wathewatical circles.
UstC J .
Some f reefed
it with joy,,.
W HA, HA! 1
7 THIS RUSSELL
HIT TWO BIRDS
WITH ONE STONE:
LOGIC AND S E T
THEORY ARE BOTH
PESTROYEP!
I ...Like Fbincare, who saw/ in +he
paradox strong arguments qgainst
f amy attempt to create purely logical
$ foundations fo r Mathematics.
His oft-repeated credo that
"Logic is barrem"mow found
a perfect justification,,.
Rather surprisingly, Cantoris rtosAhn was also quite positive.
^ ^ " I t HEREFORE, IF WE TAKE H T T ^ i W W K )
168
r 11 Does th e set o f all
sets which do not 1
contain themselves contain ]
itself?" To which th e J
answer is,,,
If it does, then it
doesn't. And if it ^
doesn't, th e n it does!"
ACTUALLY, IT'S N O T BARREN: IT BREEDS
_____________
_ CO N TR ACTIO NS!
7 Voild,
"Russell s Paradox"!!
It sounds like a parlor
witticism. But it subverts
the nofionof "set"asa
collection defined by a
. coi/uwon property,,,
,,, A n d
with it,
Logic!
But in the "pro-set11 comp
there was bewilderment and
consternation. Logicians
were devastated.
Giuseppe Peamo
in Turin,.,
DAMNED,
UPSTART
BRIT!
David Hilbert
in Gottingen
J A ,T A
THERE
MUST
BE!
~ THERE ^
M U S T BE
SOME WAY
AROUND THIS,
HERR
PR O FESSOR...
And o f course,,.
169
I'M A MAN A T LAST!
DON'T YOU UNDERSTAND???
T H E ENGLISHMAN PROVED
TH E "S ET OF ALL SETS"
IS AN IM P O S S IB IL IT Y !
MV M O N S T E R , TH E >
USURPER OF GOD'S
ABSOLUTE GREATNESS
THUS NO LONGER j
V EXISTS!!! y
Given th e right
amount o f
irrationality, one
can read religion
even in Logic.
I'M ^
SAVEP...
G o ttlo b Frage,
in U en a .
He. read vuy paradox
on fine very day when
he was to.give the
go-ahead +0 print
voluwe two.,.
.Of Inis "Foundations o f Arifhvuetic
PON'T BE
LATE FOR.
PINNER.,
GOTTLOB!
A n d n o w h e h a d s e e n t h a t th is g ro u n d w as r o tte n — it h a d g iv e n w ay.
By m planting sets into Logic, he
had injected a lethal canAer into
Its body. So: the "Foundations o f
ArifhiM&tic were,,, unfounded.
' C7
170
In an instant,
he realieed th e
iwiport o f iwy
discovery.
Frege, too, had built
his edifice on th e
ground o f Boleano's
simple idea of s e t
In t h e en d , he d id publish volume, tw o o f
t h e "F o u m d afio n s o f A rF In iM & iic ". B u t w ith
a n addendui/w .
O f all th e acts of
intellectual honesty I have
witnessed in wiy life,
none compares with
(nottlob Frege's reaction
to my paradox.
T h e r e c a n n o t b e g re a te r in te lle ctual
c o u ra g e t h a n th is ,,.
WH\- WHAT.,. DESTROY
THE PRINTING PLA T E S ?
IMMEDIATELY!
PON'T YOU SEE?
-j IT'S WRONG!
IT'S A DISGRACE!
A G RO TE S Q UE
SHAM!
HERR PROFESSOR, \
WE SLAVEP ON THIS FOR >
YEARS ON END! IF YOU
PONT TARE PITY ON YOUR
" OWN WORK, THEN AT
A . LEAST CONSIPER ,
— T V . M IN E ! /
I IMPLORE YOU, SIR,
PON'T PO IT!
,.,7b p u t tine. T r u th above, o il e ls e.
17£
r L a s tly , you w a y
A/ell a s k : w h a t w a s w y
ow n re a c tio n t o wiy
^ v n \ \ p a ra d o x ? A
H im,,.
Well, I fe lt somewhat
as would a devout
Catholic journalist,,.
",ro r having ^
exposed the doings
of a wicked Pope!
MERCY,
S E R V E !
BRAVO '
BERTIE!!!
The joy of acclaim was severely
tempered by wiy knowledge of
th e e ffe c t of wiy discovery,.,
ALYS, YOUR
HUSBAND
TR O UN CE P
, ME!
S DON'T \
COMPLAIN,.,THAT'S \
N O T H IN G COMPARED
TO WHAT HE DID )
V TO LOCilC! y
<DA(
P E A R .
,"A ia e ffe c t I wasn't allowed to forget!
173
TRULY RUSSELL,
YOUR PARADOX
IS D E V A S T A T IN G !
' I WAS ABOUT TO \
BEGIN VOLUME TW O OF
MY "U N IV E R S A L
A LG EB RA "... ANP
THEN I REAP YOUR A
N BOOK!
" NEVER CLAP CONFIPENT
^ MORNINC AGAIN
WELL, IF IT'S
ANY COMFORT
TO YOU...
...I'VE ALSO CIVEN UP ON VOLUME TWO
OF MY "P R IN CIPLE S O F M A T H E M A T IC S "!
FOR US WORKING.
ON FOUNDATIONS,
IT'S BACK T O
SQUARE ONE!
TH E PROBLEM >
YOU EXPOSED 0 0 ES
TOO D E E P L Y . IT
DESTROYS... >
' ...OUR >
FINEST TOOL!
UNLESS, OF
COURSE... J
C
...I PO N'T KNOW ABOUT " Q V I N g T \
BACK". B UT THE PARAPOX T U S T )
M G H T BE CIR CUM VENTEP! J
m
-
--\---
...YOU HAVE
ANV BRIGHT IDEAS
FOR GIVING IT
B A C K TO US!
WELL...
OH?
TA R E THE "W HO
SHAVES THE
BARBER?"
V PROBLEM. ^
NOW, IMAGINE
THE BARBER'S
VILLAG E TO BE
SITUATEP... .
...IN A SOCIETY WITH A CASTE SYSTEM .
A CALL ITS >
CASTES 1 ,2,3,4
WHERE CASTE A
IS HIGHER THAN 5,
3 HIGHER THAN 2,
. 2 THAN 1. >
^ NOW, L E T 1 S SUPPOSE
A LOCAL PEITY DECREES
>
-\--\---\--
T H A T . ..
_________
■
SO A "A" CAN BE SHAVE 0
BY A "3"
A 'X
. . . A 3
BY A "2"
AND A "1 ",
ETC...
BUT T H AT'S
T H E IP E A :
SOME SETS
C A N 'T CONTAIN
OTHERS!
175
YOU SEE? BY FORBIPPING
INTRA-CASTE SHAVING
YOU ALSO RULE OUT
S E L F -SHAVING!
HM.
f IN "SET LANGUAGE", '
THIS MEANS A SET OF ONE
TYPE CAN O N L Y INCLUPE
SETS OF A LOWER! NO J
SELF-INCLUSION...
r ...N O
P A R A D O X !
/ INTERESTING! 1
BUT I WONPER:
TUST H O N MUCH OF
i SET THEORY...
' ...YOU T H R O W \
O U T TOGETHER
WITH T H E
i PARAPOX ,
\ ITSELF? L
x ALSO, ^
IN THIS NICE
VILLAGE OF
YOURS...
...YOU GET VERY H A IR Y "1"S!
HM... X
' WELL... YOUR
"TYPES" ARE WORTH
EXPLORING. ANP
V ANYWAY... y /
TH E Y'RE N
A L L W E'VE
GOT, FOR TH E
MOM ENT! 1
I PO N'T
KNOW...
EVERY MORNING, I WAKE UP
AN OPTIMIST, BUT AFTER A
^ PAY'S WORK, I PESPAIR.
TH E VERY S IZ E OF ^
TH E PROBLEM MAKES ME
. LOSE HEART. .
RUSSELL?
," Beginning
with a plain +o
improve/ my
mew "Theory
of Types",..
We were hoping our mew
Temple of Logic would
be complete in two years.
We started
by drafting
an outline
of our work,,.!
176
WHAT SAY YOU
WE J O IN FORCES?
YOU MEAN... WRITE TOGETHER THE SECONP
VOLUME OF YOUR "U N IV ER SA L A L G E B R A " ?
S NO! \
WRITE '
TOGETHER
A B RAN P
N E W B O O K !
To rebuild Logic from scratch
is not a project to be
embarked-upon lightly,,.
,..Yet, it only took.
Whitehead and
me a few minutes
to decide!
At lunch, we toasted
our fu tu re brainchild.
T O
PRINCIPIA C
MATHE/AATICA
"!!!,
HIP,., HIP,,,
HU&RAAAH!.1!
Yet th e
year t903
went by,,,
And
so did
490k...
S p rin g r)90 5
co m z. o *r\d
w en t, m d
o u r w o rK 's
c o m p le tio n
w«5 no w here,
in s ig ht...
HEY,
WHITEHEAD
...OPEN UP!
~ WE WORKED 1
FOR TWO YEARS
ON "SIMPLE"
k TYPES... a
...ONLY TO
PUMP THEM
FOR THE
"RAMIFIED".
BERTIE... T
WHAT A PLEASANT
SURPRISE! A
r THOSE 1
CiAVE US SOME
HOPE, AT FIRST,
BUT... .
177
MY PEAR
FELLOW...
ARE YOU
FEELING
ALRIGHT?
m
I t a S m
~ ...AFTER "
A NIGHT OF
COMPLETELY
RE-WRITING
CHAPTER
. THREE!
r THE ^
THEORY
OF TYPES
. IS ROT! i
r c u t
OUR WHOLE
ARGUMENT
IS BUILT ON
v TYPES! >
r ANP? IF ^
THE PREM ISE IS
WEAK, SO IS ALL
V th e REST! J
' ...SO, LET'S
PROP IT\
OH,
RUSSELL...
178
HARDLY A T
"SURPRISE",
MY DEAR! THIS IS
HIS /V -T H PRE
BREAKFAST VISIT
THIS MONTH...
r WITH N
DE FIN ITE LY
GREATER
THAN 3!
HI,
BERTIE!
HULLO
BEETLE!
7 l e t m e i
GUESS... NOW YOU
WANT TO DROP
"RAMIFIED
.T Y P E S " , TO 0 ?
YOUR FIRST INSTINCT WAS RIGHT:
"TYPES" ARE A R T IF IC IA L - NOT
UNIVERSAL ENOUGH!
BLOW
RINGS,
BERTIE!
T H A T IS
A PROBLEM OF
COURSE!
THE "P R IN C IPIA "
IS ADDRESSED TO
ERIC, YOU SEE!
r ITS ARGUMENTS
SHOULD BE SO
SIMPLE A C H IL D
CAN UNDERSTAND
V TH EM! .
AS
SIMPLE
A S .,.
The "Pmcipia"could not diverge from tlnis
principle. Yet, as you can see from some lines
o f the book, randomly/ picked, our understanding
of Simplicity was a wee b it idiosyncratic.
Whitehead and
I spent th a t
summer
re-examining
our premises.
,,, SO
IF WE TA K E
PR.EPIG.4TE " P 1
T O STANP
S. FOR.,. .
And by th e time
autumn came,
we had changed
course again.
Our new trick,
was as old as
Euclid: a new
set of axioms!
Once again,
we rebuilt from
th e bottom.
179
Even before
I knew this was
called "Occam's
fkaaor" I fe lt
th at th e simpler
a theory was,
th e greater was
its value.
ANPSO
REPUCES
TO ,.. ,
lo gain more time fo r work, rniy wife and
I moved in with th e Whiteheads.
We found fh e change
most pleasing - well, at
least half o f us did,1
Yet, living under the
Same roof was not a
cure for our problems,,,
180
T h e endless
hours spent
oif our desks
resulted in a
stronger
language.
But the project's
central problem
was always there.
THIS i
C A N 'T GO ON,
RUSSELL! ,
T h e . d e epe r
w e .g ot in to
o u r Quest,,.
",T he wore
I doubted its
prewiises.
K ISN 'T IT
0 3 V I0 U S T O YOU?
WITHOUT SECURE
FOUNDATIONS, WE
C A N N O T BUILD OUR
SYSTEM !
I KNOW, BUT
I CAN'T STOP
MYS ELF FROM
A SK IN G —
r OUR " n |
WORK IS NOT
ABOUT A S K IN G ,
. MAN... >
...IT 'S ABOUT
A N S W E R IN G !
OH
MY GOD...
WHEN WE S T A R TE D ,
ERIC COULD BARELY COUNT
v - T O THREE... AND NOW HE
A DOES THR EE-DIG IT
\ MULTIPLICATION! /
r AND >
THIS STORM
W O N 'T S T O P !
^ SIMPLY,
RUSSELL HAS
CONE O F F H IS
R O C K E R !
The strain of having to advance
in constant self-doubt was too iwch",
• BLOW, WINPS,
ANP C R A C K
YOUR. CHEEKS'.
R A G E !!!
S L O W !!!
r A L L THIS ^
E X A M IN IN G ANP
R E -E X A M IN IN G
TH E BASES OF
OUR WORK. M UST
V S T O P ! a
r I HAP A ^
NIGHTMARE THE
OTHER NICHT... I
WAS LOST IN THE
UNPERWORLP...
PRECISELY! THE UNPERWORLP
OF M A T H E M A T IC S ! .
r I'VESAIP T
"YES" TO YOUR
EVERY W H IM
UNTIL NOW! J
IACREEP
TO "T AK E IT
FROM T H E
T O P " F O U R
T IM E S !
r
. " A I W ^
NOW YOU
ARE POUBTINOi
OUR BASIC ,
^AXIOMS!!! A
W HAT'S ~
GOINC ON
HERE?
BERTIE!
r \ say, ^
T H AT'S QUITE!
A CHANGE,1 YOU
USED TO SWEAR
BY WHITEHEAD'S
L NAME! A
B LA S T IT A LL! B L A S T ! FREGE WAS RIGHT,
ONLY "WE CENTAURS" ARE FIT TO ACHIEVE
TH IS TASK! I SHOULD NEVER HAVE TRUSTED
• a A M E R E MATHEMATICIAN!!!
18£
THIS IS NOT TUST
A N Y BOOK, DOESN'T
HE REALIZE
^ TH A T? a
IN SU FFE R AB LE
OLD MAN!
ALFRED?
"O LP"?
HE'S OLD
TO M E !
I SHOULD NEVER HAVE AGREED
TO WORK WITH TH E OLD GOAT!
B E R T IE ...
HE'S GOING
S O F T IN T H E
B R AIN ! .
ARE YOU SURE
YOUR PROBLEM IS
REALLY,,. A L F R E P ?
WHO '
ELSE?
OH, DON'T YOU SEE?
IT 'S T H AT W OMAN!
S H E 'S DOING A LL a
THIS TO YOU!
F S N IFF,,, >
IT'S H E R .,, T H A T
N E V E L Y N ! J
183
r T H A T WOMAN
B E W IT C H EP YOU! SHE
PLAYS THE SAINT... BUT
PEEPPOWN SHE'S
^ A P E V /L !
OH,
REALLY?
T H EN T H A T
M A K ES
YOU...
...A TO T A L
A S S !!!
OOOUHH
' m s t C K
ANP T IR E P
OF YOU!
PLEASE.,. '
I N E E P YOU!
A P O C T O P
IS WHO YOU
NEEP! -
YOU BEAST
H i
M O N S T E R !!!
J WeJ|," I can't saY
I tu proud of w/y social
behavior at the. tiwe.
i But the. "
P riin d p io i
"
really taken its
foil on w/
J nerves. j
Even wore taxiing than
its difficulties, was its
colossal awbition. Peep
down, you see, I Knew
it was a job fo r
...Giants! i
But, alas. Fate had
assigned it to two
were msy): Alfred
i Whitehead and,,, .
Only by being
'Istupid" can you
break. th e barrier
of the seewingly
obvious. /And so
also In our case...
With tiwe and
persistence, the
"stupfdrfieation"
began to pay o ff. as
184
,"3ertie Russell!
/Actually, we were at the ofher extreme frow Y
giants: we had becowe chja rfs !And I wean this
quite literally. For, often, the night way to
.vA-?
-------------
- j j —^ philosophize is.., I
...To wake
yourself
artificially
stupid!
r CAN ^
YOU HELP
ME W ITH MY
GEOMETRY,
BERTIE?
^SHUSH,
MY PEAR,
BERTIE IS
WORKING!
We were finally led
to an astounding
discovery.
. . .A T
L A S T 7
WHITEHEAP,
I G O T I T
TH IS TIM E!
I'M FINALLY
P O N E WITH
THE PAMN
. THING! .
A R E YOU,
REALLY?
YES, IT'S PR O V E N ...
185
To achieve this wonui/uenta! task, took us a
were.., 362 pages! T hink o f that-. 362 pages
to prove what every child knows.
I >
DON'T
S E T IT,
BERTIE.
W HY
3 6 2 RACES?
' LET
ME PLAY,
BEETLE
OLD
CHAP!
HURRAH!
MISSED!
r ^ I'LL
REPORT YOU TO
TH E NATIONAL
CROQUET
. BOARD! z '
' BUT >
WHY A L L
TH E S E
PACES... .
...TO PROVE
1
+
1
=
2
?
HM... >
HOW
SHALL I
PUT IT?
^ IT'S >
THE PRICE
YOU PAY
FOR BEING
T R U L Y
C ER TA IN .
TH AT'S US, N
YOUR DAD AND I.
WE ARE DOING
, ANTS' WORK..,
LO O K!
' NOT ^
GLAMOROUS,
BUT V ER Y
CRUCIAL!
Puning th e
last phase
of our work,
wy wife spent
w ostof her
tiw e in
rest hoi/ues.
ANY NEWS OF ALYS?
NO, NOT
THIS WEEK.,.,
I often think back
a t wy stay with
th e Whiteheads,,,
Halcyon
days.
186
Y c-
If took us hem
years fo complete
the first three
volumes of our
grand edifice.
Though, actually,
I dicin' f know
at fhat time that
"first" would
also M&am "last".
In fact, I didn't even
Know), back then,
that we had
completed anything.
I'M
TK EP,
/MAN.
r |'M \ i
T O T A LL Y 1
WRECKEP.J
COME,,,
THERE'S
SOMETHING
I WANT YOU
TO SEE.
Whitehead had found
a penfeot symbol fo r
his argument.
CONPEMNEP BY THE OOPS,
ENPLES5LY TO POUR. WATER
TO FILL A L E A K IN G J A R !
TH E P A N A IP E S ?
187
He led we fo if
fhrough fhe e^upfv/,
resoundi/ig halls.
THERE,..
^ WE1 VE
A LONG, LO N G
WAY TO GO!
W WHAT ^
r are you '
' TALKING
ABOUT? WE
AREN'T YET
FINISHED NOT
. BY H A L F ! i
AS LONG
AS T H E IR S
PO YOU
^ THINK?
BUT...
BUT YOU
MADE A
PROMISE,
r ...YOU SAID 1
AT SOME POINT
W E 'L L R E -E X A M IN E
v THE
BASICS!!! ,
188
I THINK THE
TIME HAS COME...
...THE TIME TO P U B LIS H !
"P U B L IS H "'? ? '?
7 IF THE ^
' " P R IN C IP IA " WAS >
P U R E
PHILOSOPHY,
WE COULD IMPROVE
ITS
PREMISES
AP IN FIN ITUM !
BUT
v IT'S
LOGIC!
7
AND
LOGIC
H A S TO BEGIN
SOMEWHERE.
...SOMETIME,
r ISO
H A T E
LOGIC!
'a" wiade the quest
possible... "g" which
marks wy own entry, is
the major crisis.,.
And we were successful
in our task, in all ways but
one: no waiter bow deep
we went, our too-too-solid
systewi was being built
on sand. Or worse.,,
189
P e a r frie n d s , I Know/
well -th a t, des pite
a n y th in g S ocra te s
ju ia y have believed...
...Lay people often
feel wiles away from a
philosopher's worries.
[So, I want toasK:
Can you at all unalerstand
the state I was in, back then? Poes]
my desperation wake any sense to,
k you? ...What say you, Madavw? i
Wei/, I II 0\cm\T ^
that it's not foo clear
Professor Russell! ^
.Alright,,, Let's 90 over
the stages of -the
journey so fa r: V is
_
______
the need that
/ ^ ^ X s t a r t e d it,,, .
r ,,,/And "4",
the struggle to
. overcome it! .
/ So, what Whitehead and I were >
I really doing, in building a paradox
ic free Logic that could support
>
____
Mathematics, was...^
...F ix in g ih e . h o le
I had exposed in
Frege's ideas!
I've said that the Foundations of '
Mathematics were like a mythical turtle
supporting the Cosmos. Vet, all we did
when we tried to create solid ground for
the beast to stand on,., was,.
,,"A tower
of "turtles", all
the way down!
190
i To wie, a philo sopher, 1
th e fnony o f +h e s itu atio n
o f a "fo u n d atio n a l system
w ith ou t fo u n d a tio n s " I
L was h a rd to bean,,,
,"Too
h a rd !
Yet, despite wy initial n e tlo e n ce '
ho publish, I eventually agreed:
Maybe a book would help us fin d
new associates in our e ffo rts !
r Also, o f c o u rs e , ^
I s u ffe re d fro w i a b a d ]
ca s e o f inte lle ctual J
^ ca b in fe v e r
".And publication
o ffe re d away
out o f wiy prison!
J 111 WEEP
FOR. YOU, THE
WALRUS 5AIP..."
PON'T BE A ^
SPOILSPORT,
BERTIE. THIS IS
E X C IT IN G ! a
I walked with
Whitehead to |
th e publishers.
~ THE N
"P R IN CIP IA "
SHOULP HAVE
LEP US INTO
PARAPISE.,.
2 r , " B i r \
' WITHOUT A >
SOUP FOOTING
IT'S MERELY A
COMPETENT
SURVEY OF j
k . H E L L ! A
X TUSH, \
RUSSELL,
YOU'RE TUST
GETTING
COLP
w FEET. A
ALE A /A C T A
E S T !
I d id n 'tg o in.
A i I vjaiiecl foir
Whifaloe-Qcl, suddenly,
a p ro f o u n d sei'ise o f
loss ove ro a w e we..
'I?'!
OH, IT'S SUCH A U NIQ U E MANUSCRIPT!
' QUITE.
NOW LE T ME
T E LL YOU...
Sowe.+hiing
load died...
But-
what?
WHAT ON EA R TH??? \ \
r IT'S.., ^
IT'S NOT...
N O T THE
BOOK! .
WHAT
IS?T ...A N P 1
I K N O W ...
NOW, I KNOW
V WHY,,, >
YOU'RE
SUCH A
MESS!
W ELL,..
...IT 'S 1
A L L
W R O N G !
WHAT
ARE VOU
T A LK IN G
ABO UT?
skills os 01 cowwunicator had
mot vet reached th e ir present leve.1.
19£
Suddenly,
I'd realized
th a t I was
following
the wrong
track.
EVEL YYYYNNH!
i... i...
SEE IT...
G A S P ...
IT'S...
IT'S.,.
193
IT HAS™.
...A LY S ?
YES, OUR
MARRIAGE.
IT'S A.,.
...A
FA RC E!
...ANP, OF
COURSE, I AM...
I AM...
...HEAP OVER HEELS IN LOVE
WITH... WITH...
| V | | K ; V ER...
... YOU!
/K N O W IT !
I K N OW
YOU LOVE
ME TOO!!!
SO SAY IT...
m
OH B ERTIE.,,
S A Y IT!
I SHALL REMEMBER.
WHAT YOU SAID FOR
AS LONG AS I LIVE...
EVELYN.,,
...SAY IT.
PLEASE!
DEAREST
BERTIE...
MV DARLING,
S A Y IT!
OH
MY
COP!!!
A t t h a t n/iowtemt,
I Knew wy life
Was ab o u t
ta k e a s h a rp
tu r n .
ACTUALLY,
TH E PUBLISHERS1
ANSWER WAS A
P 0 L IT E"N O ".
W H A T ?
r T H E Y
HAVE NO
FAITH IN T H E
" P P IN C IP IA " .
195
IT 'S
NOW OR
N EVER .
But..,
' 50 .,. T E L L ^
U5, P E AR, HOW
PIP IT G O WITH THE
^P U B L IS H E R S ? A
As it turned out,,.
r
ARE..,
ARE THEY IN ^
LO I/E WITH TH E
. BOOK? >
,"The tuw was in a totally
u/i&xpec-ted di/ectioio!
T H E Y 'L L
PUBLISH ONLY...
IF W E PAY FOR
T H E PRINTING! .
r TH EY CO ULDN'T FIN PA ^
SIN G LE REAPER TO EVALUATE
T H E MANUSCRIPT, SO TH EY
FIGUREP: "IF NO ONE WILL ACCEPT
TO REAP T H E 'P P IN C IP IA ' j
k ANP BE P A IP FOR IT..." 4
"...TH E N, OBVIOUSLY,
NO ONE WILL PAY T O
REAP IT E IT H E R !"
Ten years of
daydreams of
th e triumph
o f our opus
Magnuiu h ad
come fo fhis.
STAY AWAY
FROM THE
MUP, KURT!
Y /4nd one final thing on >
this sad day, and its parallel
, emotional misadventures,,.
In the th irty years since ft was
published, I've only wet one. person
who I'm/i convinced has read the two
thousand or so pages of forbidding,
sywibol- pacKed text, cover to cover.
6>ut, co n vin ced t h a t t h e "Principia" should e n te r
th e cow iw unity o f Ideas, we decid e d to a c ce p t
th e Ign om iny o f paying to se e o u r w o rk in p rin t.
T he publishers' thinking was
pretty sound.
196
With hindsight I say th a t
I was wrong In wy self-analysis:
. wy problem was the book!
But he was only a
child back in /I9/10.
E NTRACTE
I THINK RUSSELL'S CRUSH
ON EVELYN WAS REALLY
AN OUTLET FOR HIS
FRUSTRATION, OVER TH E
"P P IN C IP IA " CONSTANTLY
MISSING ITS
ULTIM A TE
AIM!
199
".A N P THIS IS
ABOUT AS FAR
AS WE'VE COME!
HM . W HAT PO YOU M EAN BY
TH E L A S T SEN TENC E :
^ "T HE PROBLEM W AS THE
B O O K"? >
TH A T IT W A S !
THE PEEPER THEY
WENT, TH E MORE
FRUSTRATE? HE
k BECAME! a
I HAVE TO 1
RUN,,, BYE
EVERYONE!
I LIKE? THIS! POES THE LITTLE GUY
AC TU ALLY LO O K LIKE
. 11 L IT T L E K U RT C d P E L " ?
SPITTING
IMAGE!
OK, LE T US ASSUME I ACCEPT YOUR.
VIEWPOINT ON T H E "G U E S T". BU T
WHAT'S TH E B O T T O M L IN S '?
SEE
YOU
IN TUNE,
CHRISTOS!
WHAT ^
B O T T O M
LIN E"? >
£00
SO, VOU SINK Z E
M ATH E M A TIC HAS
ANY MISTAKES?
^ NOT MUCH ^
"M A T H EM A TIC ",
IS THER E?
"M A T H E M A T IC S ANP
COMICS, LIKE
OIL ANP
WATER.
...P O N 'T >
EVER
M IX ! /
5 T IL L , I PO TH IN K IT S
COOP TO EXPLAIN THE LOGIC
A BIT MORE, SO THEN...
...YOU CAN PUT '
IN A FEW THINGS ON
COMPUTERS
S
___
LATER! >
' CHRISTOS >
WOULP LIKE US
IPEALLY TO WRITE
A COMIC BOOK
"TH E O RE T IC A L
COMPUTER
SCIEN CE FOR
' MORONS"! J
TH AT'S
T O T A L L Y
UN FAIR!
WELL,
OBVIOUSLY
OUR AIMS
PIVERGE!
WELL, IF T H E "FOUNPATIONAL OUEST"
IS, AS YOU IMPLY, A "SPIRITUAL TRAOEPY" OF
SOME KINP, THEN IT SHOULP HAVE A MORAL!
r ARISTO TLE SAYS T H E
ACTION OF A TRAGEPY IS
'C O M P LE T E IN IT SE L F 1:.
^ ...I.e. FOR A
MORAL YOU NEEP
AN ENPING.
BUT O U R
STORY IS
N O T FINISHEP
. Y E T !
OK. SO,
WHERE IS IT
H E A P IN G '?
IT 'S X
r NO T T U S T T H E
FINAL PESTINATION
T H A T 'S IMPORTANT..,
k IT 'S T H E R O A P !
' T H E MEANING >
IS IN EVERY T U R N
T H E HEROES M AKE
EVERY STO P, EVERY
C U L -P E \- S A C ,,. .
r IN A X
SENSE T H E
FOUNPATIONAL
Q.UEST IS AN
INCOMPLETE
. O P Y S S E Y ! y
AH, YES! WITH
L O G IC A L
C E R T A IN T Y IN
T H E ROLE OF
ITHACA!
B IT OF
BO TH, I
THINK..
AS W R IT E R ,
w o u l p you SAV
IT 'S M ORE
M O TIV AT E ? BV
CHARACTER
^ OR ACTION?
I f c J r ALSO, I MUS T N I ,
y SAY I'M N O T REALLY y k
' CO M FO R TA BLE W ITH T H E 1
"LO GIC f r o m m a p n e s s "
T H E M E , AS IT KEEPS
POPPING UP IN T H E STORY.
IT 'S N O T T H A T I P O N 'T T H IN K
T H A T IT 'S IN TE R ES T IN G ,
I T O A L S O LOOK A T A
t h a t s ip e ...
T jr L IS TE N : ^
f FORGET FOR A
M O M EN T T H E S E ARE
HISTORICAL PEOPLE,
IN T E N T ON BUILPING
FOUNPATIONS FOR
M ATH E MATIC S...
I T H IN K
IT 'S T H E
C E N T R A L
. ISSUE! .
...ANP T H IN K OF T H E ^
"Q U E S T" AS A FICTIONAL
STORY...
W H Y S HOULP IT B E ?
wHi
^ T O ME IT S lOO'/o "
CHARACTER! NOT TU S T THEIR
ACTIONS B U T TH E IR /P E A S
COME FROM IT: ONLY MEN LIRE
T H E M COULP HAVE T H O U G H T
^ T H E M ! ^
YOU M EA N IF TH E Y
W E R E N 'T NEU RO TIC , OR
W HATEVER , T H E Y W O U LP N 'T
HAVE T H E NEC ESS ARY PASSION
ANP PE R SIS TEN CE T O
^ C R E A T E LOGIC?
r ... OR T H A T ^
T H E IPEAS
T H E M S E L V E S
WERE INSPIREP BY
^ NEUROSIS? A
R E M E M BER
BERTIE T O ERIC
ON "1 +1:2. IN
5 6 2 PAGES"?
r i g u e s s y s
V MY POINT IS \
T H A T L E S S
TO R TU R E P
CHAR ACTERS
WOULP N O T HAVE
FOUNP THIS
k PRICE W O R T H A
X P A Y IN G ! A
YEP! ^
f HE SAIP
SOMETHING
LIRE "T H E PRICE
YOU PAY FOR
ABSOLUTE
N C ER TA IN TY !"/
J *
AN NE!
WHAT'S
WRONG?
..ON T H E "MAPNESS" SIPE? A F T E R ALL, TH ER E
ARE M A N Y PSYCHO T IC S , B U T ONLY O N E ■
TH E PAWN
THING BROKE
POWN! I'M
LATE FOR A
REHEARSAL!
RIGHT!
GOP NO! I PO
TH E VISUAL
RESEARCH.
SEE YOU IN
THE SUMMER,
CHRISTOS!
SO, POES
YOUR BEING
THE RESEARCHER
MEAN YOU HAVE
M A T H E M A TIC A L
TRAINING?
EOH
OK, TH IS BEGINS TO MAKE
SOME SENSE. BUT L E T ME
THINK A BIT. WHAT IF
YOU' RE PUTTING TO O .
L . M UCH E M P H A S IS ...^
I MAKE TH E '
MASKS FOR SOME
FRIENPS POING
AESCHYLUS' j
O P .E S T E IA " l/
f WHAT A ^
PAY FOR YOU: 1
FROM MOPERN
LOGIC TO
ANCIENT
W T R A G E P Y Ii
r in f a c t ;
FROM O N E
TRAG EPY
TO
A N O T H E R !
CAN I COME ANP
WATCH? I SO LOVE
. REHEARSALS...
SURE...
r THOUGH BY \
G ETTING T O KNOW
TH E CHARACTERS
I HOPE I'L L ALSO
UNPERSTANP THEIR
MATHEMATICS .
A A BIT ! y
y A H A ...
f IP E T E C T \
INFLUENCES '
OF TH E "LOGIC
FROM MAPNES5"
THEORY! A
£05
YOU
PON'T
RELIEVE
IT ?
W ELL,
OBVIOUSLY
SOM E OF
T H O S E PEOPLE
WERE CONTROL
FREAKS - OR
C A L L T H EM
OBSESSIVES!
BUT ^
IF THIS WAS
MAPNESS, THE
SYMPTOM WAS
SANITY ITSELF:
MAKING COMPLEX
THINGS SIMPLE!
REALLY
SIMPLE!
SURE,
"SIMPLE",,,
LIKE PROVING
"p| \+ /I = 2 " IN
5 6 2 PAGES!
< W ELL, >
T H A T 'S WHY
I PREFER GOING
ALGORITHMS
TO LOGIC.
SORRY,
WE HAVEN 'T
PONE
ALGORITHMS
. YET ! >
OH, I RATHER POUBT YOU WILL,
IN A "SPIRITUAL TRAGEPY"!
£06
YOU SEE, RATHER. TH A N
BUILDING ABSTRACT
THEORIES, WE CREATE
METHODS TO SOLVE '
. P R O B L E M S ! f
HAVE
YOU G O T
A METRO
T IC K E T ?
NO.
SO YOU USE THIS MACHINE... WHICH WORKS
WITH AN INTERACTIVE ALGORITHM!
T H A T ?
SURE! IT RUNS A PERFECTLY SIMPLE
. METHOD FOR SELLING YOU YOUR J
\
_____
TICKE T.
T O O SIMPLE! I DON'T
HAVE ANY SMALL BILLS.
NOW, AN ALGO R IT H M LIKES TO LIVE IN
A CLEAN-CUT KIND OF ENVIRONMENT...
LIKE...
A M A P !
FOR EXAMPLE, SAY YOU
WANT AN ALGORITHM TO
FIND TH E WAY FROM
"STATION X" TO /
V 1 STATION
...THE ALGORITHM GOES..,
"STEP A, LO CATE X,
YOUR POINT OF
DEPARTURE." /^ | S
"STE P 2 , 5
LOCATE Y...
...YOUR
POINT OF
ARRIVAL".
SIR, DO YOU WANT A
TIC K E T OR P O N 'T YOU?
COME TO THINK. OF IT, YOU
KNOW, THE HEROES OF TH E
"FOUNDATIONAL QUES T"
WERE T U S T T H A T !
£07
r ...STEP 5, IS: "CHECK IF TH ER E 'S \
A LINE GOING FROM X T O Y. >
IF YES, REAP THE NAME OF TH E
LAST STATION OF THE LINE IN TH E
DIRECTION X-Y THEN GO TO STEP V ,
WHICH IS " ENTER TH E TRAIN
. WITH THIS INDICATION 7
V ANP E X IT -" ^ / |
H E R E, GIVE '
ME T H E TICKETS!
WHAT?
M A P
M A K E R S !
THE Y REDUCED MESSY REALITY TO
T H E CLARITY OF MAPS, i.e. S IM P L E R
T H IN G S , WHERE LOGIC COULD >
. APPLY MORE N A T U R A L L Y !^ /8*
"T H E SIMPLER
THE BETTER", IS
TH A T IT ?
RIGHT THE
K.I.S.S. PRINCIPLE...
WHAT
"K IS S " ?
"KEEP
IT SIMPLE
STUPID!"
H allo FiHand,
I a rriv ed a t B e rk e le y th is worm ing
Bu t IW been th in k in g o f A th en s.
O f t h e " Foundational Q uest in C on ie s'
,To whioh I've given th e nickname "Logfcow ix
...A n d its w a rn in g
W hioh I th in k , b ro ugh t w e a b it clo ser
.T o t h e "L o gic an d M adn e ss" th e m e
So I w a n t to teJ I you a little s to ry .
I M U S T FIN P IT
ON T H E MAP... " . . . I H AVE N 'T B EEN '
> >
\--\----\--
T O TH IS REHE A R SAL
SPACE BEFORE .
' PLEASE! I WALKEP
TO SCHOOL THROUOiH
THESE STREETS FOR
v SIX YEARS! A
r THE MAP ^
OF THE AREA
IS ENGRAVE? IN
MV NEURONS!
mm
£09
A n A th e m s s to r y .
T MV HICiH "
s c h o o l w as
FIVE MINUTES
FROM HERE! ,
When we got out,
I thought I'dcowie
book to wy old
neighbourhood.
B u t ha d I?
WHERE IS
TH E PAMN
THINOi...
y COOP T O ^
7 SEE THE
VEG ETABLE
M AR K ET IS STILL
v STANPINCi... a
l/i fact, I have to adwit
I am gulU"'/ o f having
frie d fo show o ff fo
Amine- t/wy superior-
Knowledge of th e area!
£10
/ ft firs t, I couldw't
see too i/uuch
C han ge..,
Well, ro t
really,..
r TH/R.P ^
RIGHT FROM
. HERE! A
Vet as we turneol
frowi th e vegetable
warket,,.
~ EVRIPIPOU \|
STREET IS
THROU GH HERE,
A T E N P OF
T H E S T R E E T ,
RIGHT! A
£11
Sure- e n ough , th e re , h o o f b e a n sow ie
c h a n g e s in t h e d e c a o fe s o f i/i/iy a b s e n c e !
PIP YOU
SEE'
THO SE.,.
...PROSTITUTES?.
WHAT
ABO UT
T H EM ?
CHRISTOS!
I'L L BE
. L A T E !
f SORRY,
I'M J U S T
TRYING TO
REGISTER
T H E NEW
IMAGES!
r NOW,
WE NEEP
T O FINP...
E R ",
MENANPROU
^ STR EET! J
SECONP
RIG HT!
YOU SEE , ^
IMAGES CHANGE,
BU T MAPS STAY
T H E SA M E!
r GOLLY! A ^
BARBER SHOP WITH
A L L T H E SIGNS J
IN HINPI! j U
CHRISTOS,
COME!
SO W HAT
POES YOUR
'ENGRAVEP
MAP" T E L L
YOU NOW?
r TH AT TH ES E ^
INSTRUCTIONS
WERE W RITTEN
BY A PERSON NOT
TRAINEP IN
A L G O R IT H M IC
E X A C T N E S S !
IT 'S A POOR A R EA .! W HAT
PO YOU E X P E C T ? NOW..,
" IN TH E MIPPLE OF
T H E THIRP BLOCK.
TH E R E 'S A GALLERY TO
T H E RIGHT, LEAPING TO
A SMALL SQUARE..."
...BUT
HERE TH E R E
AR E T W O
G A LLERIE S !
2d
L E T ME SEE... W E 'L L APPLY A SIMPLE
SEARCH PROCEPURE: FIRST, T A K E TH E
ONE GALLERY, ANP IF IT PO ESN'T LEAP
WHERE WE W ANT IT T O ...
WE CONGLUPE
BY T H E " R E O U C T IO A O
A & S U R P U M " WE MUST
TAKE TH E O T H E R i
NO T IM E ! YOU T A K E ONE,
/ T A K E TH E OTHER ANP
WHOEVER FINPS T H E "SMALL
SQUARE" GIVES THE OTHER
^ A CALL!
BUT
SIS
W H A T PO YOU FINP SO STRAN G E,
YOU OF A L L PEOPLE, LIVING IN
BERKELEY? ATHENS IS NOW AN
INTERNATIONAL CITY.,,
r I GUESS I'M ^
COMPARING IT
TO A C ITY WHICH
EXISTS ONLY
k IN M E M O R Y .^
r VOU KNOW, ^
A T LE A ST 1 0 %
OF T H E POPULATION
OF GREECE IS NOW
IMMIGRANT.
AROUNP
HERE IT'S
PROBABLY
MORE LIKE
39% !
r PHEW! 1
TH A T WAS
" CLOSE!
A PUNK T U S T STO LE MY W ALLE T !
C OULP I BORROW YOUR PHONE,
T O C A LL MY S O N ?
NO "SMALL
SQUARE"HERE.
B E TT E R C A LL
A N N E ... .
SURE, NO
PROBLEM.
S IP
£13
SPARE
A EURO,
MAN?
TH E R E WAS
NO "SMALL
SS.UARE" T H A T
WAV EITHER...
r WHAT 1
PO WE PO
. N O W ? j
I PON 'T
B E L IE V E T H IS
HAPPENEP
TO ME!
W HAT AM I,
IP IO T ?
AN
£14
YOU W AN T
COMPANY,
BIG PUY?
NO TH A N K S ,
I'M FINE!
ARE
YOU
PONE?
C O M E
B A C K
H E R E !!!
I WAS ^
TRYING
TO C A L L
YOU.
W ELL, TRY AGAIN ANP
YO U 'LL TALK. TO TH E
NICE CON ARTIST...
" ...WHO >
TUST S T O L E
MV PHONE!
EXCUSE ME
SIR, PO YOU
KNOW WHICH
W A Y - .
NO GREEK,
ME NO GREEK
SPEAK. .
r ONE, M AKE ^
SU RE WE TOOK
T H E RIGHT
L . TURNS... A
LISTEN
W ELL, W H AT
ARE OUR v ...TAKE
OPTIONS,,. M IT FROM
.T HE TOP:
£15
W ELL, P O N'T
PESPAIR... IT
MUS T BE
SO M E W H ER E!
OH, T H A T ^
K N 0 W L E P 6 E
IS V E R Y
COMFORTING!
SOMEONE IS
IN PISTRESS!
r ANP I THINK T
I KNOW E X A C T L Y
^ W H O ! a
I sat \- t h r o u g h the rehearsal
■totally enthralled.
I SEE
M U R P E R
IN YOUR EYES,
ANP I FEAR,
MY SON!
r IT IS ^
YO U WHO
IS T H E
MURPERER,
N O T I!
But what d o
we learn,
I wonder ?
B E W A R E !
IF YOU RILL
ME, A M O T H E R 'S
C U R S E WILL
H U N T YOU
POWN!
L o gician s h a te
contrad iction s.,,
But what Is life,
in th e tragic view
which you espouse,
if not a bundle of
contradictions?
£16
Aeschylus' words brought fo rth
a dark proverbial wisdowu,
"We only learn
through suffering,,,'
What does life, in its full
complexity, teach us?
&U T IF l
PON'T, MY
F A T H E R 'S
WILL!
Good g rie f!
",My own puny little. hubris
of earlier that evening,,,
...My thinking
that I knew an
area of Athens
". Just because I had, as I
said, "ifswiap engraved
in wiy neurons."
Strangely, the Vneafeta"'s
conflicts brought to laind...
But waybe
eventually they
confused their
reality with their
wiaps.11
,"And I fe lt this idea
gives your hhewie of
"Logic frowi Madness"
a forn/i I understand.
WHAT A
PERFECT
PEFINITION OF
IN S A N IT Y !
And then, strangely, this brought
back i/uy earlier cowti/uent to Anne
on "wiap-eiakers"...
And I thought: " S u r e \- , Frege,
Russell, Whitehead, were
excellent wap-wiakers,,,"
,"And the heroes of this
"Lqgicowix" we're trying
to iwake.
£17
5. LOGICO-PHILOSOPHICAL WARS
50, CHRISTOS'S CO MMENT ABOUT
MAPS VS. REALITY COMES A T A TIM ELY
MOM ENT IN OUR STORY, T U S T AS THE
REAL WORLP BEGINS TO BARGE INTO
RUSSELL'S CLOISTEREP LIFE.
ANNE, PIP
YOU RESEARCH
BR IG HTO N?
/ TUFR.E AEP N J
SHOME W O N P E R F U L ^
OLP PICTURESH,.. B E L LE
E P O aU E A T TH E
. SHEASHIPE! a
Sitting on Brighton
beach one wintry
cloy, m s / wind went
book to wy early
years,.,
,"The tiwie when
Eidiof saved wie
frowi th e clutches
of Grandwother's
stern religion.
The promise of
certainty in total
rationality was
wiy olreawi of
a perfect c o s m o s .
Z IS IS N
VERY NICE!
Z E COLORS...
SO, HOW POES
TH E SEASIPE
CONCERN US?
WELL, RUSSELL
O FTEN ESCAPEP
IN TIMES OF
CRISIS.
TO BE
ALO N E.,.
TO ^
TH IN K...
M \| ow n v isio n
o f H e a v e n .
A fish in a bowl,..
...Content to chop at the pants
of it that could f it through the
grid protecting my austere
intellectual lair.
Cut off from the world,,,
All in all, I'd spent twenty years stringing
with the Foundations of Mathematics ,"
, My own idealistic foray into the
great Ocean of Truth.
And now the time had come..,
The frirtdpioi MafheMaffca"was about to be ,
published, bringing my labours to the world,,,
Apart from Mathematics, and
a bungled attempt a t marriage,
I was totally isolated!
eee
",0r, to be exact, to the tiny part of
the world that could u^de-rsfo/yd it.
This sense of an ending,
however incomplete,
prompted we to review
my life until then.
And the review brought
home an unwelcome truth:
H e re I was ",
££3
I realized then Thor, at the human
level, I hadn't progressed much from the
sad little boy desperately seeking ways out
of the deadly vortex of uncertainty. ^
r ^ The ^
//O * \- * // ,
r/nncipi4 was
my outstretched
hand to the
S. world... V
,"But
would it
reach its
target?
So! We are now In the
year In it, two
events of momentous
import occurred,,,^
One starteol as a dream\-
come-true but eventually
turned into a nightmare.
The other, on
the contrary—
But let's take the
story in its proper
Order! ^
I was sitting in my rooms at College, one
afternoon.There was a knock at the door,,,
YOU ARE
TH E H E R R
PROFESSOR
RUSSELL?
A v/oung
fo re ig n e r
entered
wy rooiw.
HE IS SAYING NO ONE IS
B E T TE R TH A N YO U TO
T E A C H ME T H E LOGIC!
TH A T
WHICH IS MERELY
EMPIRICAL HAS
N O T PLA CE IN
T H E PISCOURSE
OF TH E
TRUTH!
COME NOW...
I'M SURE YOU'LL
AGREE T O T H E
R E A L ITY OF S O M E
EMPIRICAL FACTS!
T W ON'T YOU ^
A C C EP T AS T R U E
FOR EXAM PLE,
T H E ST A TE M E N T:
" TH E R E IS N O
RHINOCEROS
" IN TH IS ROOM"? ,
£ £ H
r IT IS ^
T H E HERR 1
P R O FESSO R
FREGE WHO
SENP5
S. M E . y
HOW IS PROFESSOR
FR E G E? ^
...Therefore, I had a new student!
Frown t he
sta rt ha
impressed
wte with the
intensity
of his
philosophical
convictions.
r B U T WE ^
CAN ONLV
KNOW F O R S U R E
TH E RESULTS OF
T H E LOGICAL
^OPERATIONS!
SURELY, W E ALSO HAVE AC CESS T O
EMPIRICAL OBSERVATION!
NO.
WHAT A BO U T TH E
INFORMATION GIVEN
BY T H E SENSES?
N O !
Such intensity I'd previously
seen only in iwy younger self.
NO, I
WILL
N O T !
new
students
nawie was
Ludwig
Wittgenstein.
But +loe
rest of the.
ver-y few
who could
u n d e rs ta n d
the book,
were less
enthusiastic!
S TH E Y ARE
OUR S A F E G U A R P
AG AIN ST PARAPOX,
TH E Y ARE ES SEN TIA L
T O LOCtIC IT SE L F !
TYPES MUST BE
V SALVAGE?.,. >
££5
The firs t volume of the Pr'incipia" had
been published Just before he arrived-
/ TH EY ARE \
' SAYING T H A T PESPITE '
OUR HUNPREPS OF PAGES
OF SYMBOLIC CALCULATIONS,
W E'VE N O T M A PE TH E
. FOUNPATIONS ANY y
V LESS SHAKY.
TH IS IS >
M U S IC
RU SSELL.
TH IS IS
M O Z A R T !
Wittgenstein's aesthetic
appreciation was, of course,
heartwarming
A C H ! THEY
ARE SUCH
S l o o p v
FO O L S ! * V
T H E GIST OF IT IS T H AT T H E
P R E M IS E S OF T H E THEORY OF
TYPES PO N'T GO POWN WELI
__
...T U ST
A S I'P
FEAREP!
r B U T P O N 'T
T H E Y UNPERSTANP
T H E S IG N IF IC A N C E
OF T Y P E S ? A
...A T
A L L C O S T S !
( 4nd the reason: I was convinced by^
now that wy new student was a
geniusl Oh, he
certainly exhibited
all the obvious
wiamifestati:
He was,,
ons.
I W ILL NOT A L L O W
YOU TO ABANDON
T H E "P R IN C IP IA 11!
,"Ar\cl
s o w c t iw e s
in fu ria tin g ly
doM \V\0ifi»lQ\
5 T u ^
m & m
££6
Of course., I couldn't agree
More with Wittgenstein. But
Whitehead and I were, sadly,
too intellectually worn-out
to attempt the- rescue. .
' Vet,
I was rather 1
. optiwwstic. J
l o g ic is t o o
IM P O R T A N T
T O BE L E F T
TO T H E
LOGICIANS!
Passionate
WE M US T GO
DOWN T O TH E
B A R E S T
E S S E N T IA L S !
Prof ound
...G E T R IP OF
TH E UNNECESSARY!
Intense...
OF COURSE WE P0 : S E T
TH EO R Y IS E S S E N T IA L
^ T O OUR ARGUMENT.
~ A HELL >
THROUGH WHOSE
^ G ATES... ^
T H E BLOOPY ASS
HILBERT CALLS IT
A "PARAPISE"! BUT
. IT IS H E L L ! A
^ V , . T H E
~ MONSTER ~
INFINITY C R E E P S
INTO
.MATHEMATICS!.
~ "CREEPS IN"? ^
WHAT H O T ! INFINITY
IS ALREAPY T H E R E
FROM TH E START;
k OLP CHAP! U
IT 'S IN TH E CONCEPTUAL
UNIVERSE, P R IO R TO OUR
POKING OUR PUNY L IT T L E
. BRAINS INTO IT ! A
££7
f o th e r w ord s , ^
h e h o d e x a c tly w h a t
was n e ed ed o f a
. repairtw an f o r th e .
^ _
"R 'fa c ip ia " /
As for
w ys elf...
r ...I'd wove on, ^
to e xp lo re — now w ith
a firw te r fo o tin g —,
how w e cowie to know
t h e tro th s o f t h e J
w ia te ria l
w o rld .
Y et we all kn o w
w h a t h a p p e ns to th e
b e s t la id p la n s o f
wiice a n d wen...
RUSSELL, 1
/OU WASTE SO
M A N Y PAGES
TO ESTABLISH
SETS! >
r A C H , ^
RUSSELL, '
I AM IN SUCH
. P A IN ! J
P L E A S E P O N 'T T E L L ~
M E YOU ARE ASSUMING T H E
IN D E P E N D E N T E X IS T E N C E
O F A MATHEMATICAL
^ REALITY?
r OF COURSE ~
I AM. EITHER T H A T
OR WE LIVE IN
.TOTA L CHAOS! A
P O N T
T E L L ME
r P O N T YOU S E E ...
B L A H B L A H ...NO
O B J E C T IV E EXISTENCE
B L A H B L A H
hoped, noiivalv/, th a t ha
would do precisely os asked.
When I'd assigned Wittgenstein to fine-tune I
Oun technical arguments ...
But now he was
questioning wy
Most
basic,
unspoken
premises about the
natune of "Truth!
££8
B U T T H E M A JE ST Y OF TH E
" p p i n c i p i a " is t h a t i t p u ts
ON PAPER EVERYTHING ANP .
. O NL Y W HAT IT SAYS! \ A
SO W H E R E IS
THIS "INFINITY"
. OF YOURS?
W H E R E ???
IT
C A N T
FIT...
...IN A F IN IT E
BOOR!!!
^ GOP ~
P R E V E N T ME
FROM SANITY!
OOP
CERTAINLY
W IL L !
Wittgenstein
b a n g e d in to
w y noowts a t
SAM one nig h t,
in extnewie
ag o n y a b o u t
sow /e f in e
logical p oint.
I wanned
hiw/1 t h a t
he should
bew ane: t h e
way h e w as
d n t/in g hiw iself
h e co u ld well
g o (n sa ne.
B u t h e
Said,..
£ £ 9
THIS GUESTIONING
CREATEP IN RUSSELL
EXTREME ANXIETY,
ATTACKING ONE OF
HIS S T R O N G E S T
PEFENCES, HIS BELIEF
IN OBTECTIVE A
V REALITY! / '
..,ANP THIS A T '
A TIM E WHEN
O T H E R REFUGES
WERE NO MORE
AVAILABLE! A
HE' P ALREAP Y L E F T ALYS.
ANP PEAR EVELYN WAS NOT WILLING
TO SUCCUMB TO HIS CHARM.
A T LE A S T ONE
OF ZE M WAS
REMEMBERING
SHE WAS MARRIEP
TO HIS CLOSEST
v FRIENP! J
I TH IN K TH AT
RUSSELL SAW
WITTGENSTEIN
AS A M I R R O R :
HE HAP SO MANY
ELE M EN TS OF
. HIMSELF! y /
/ BUT >
' ABOVE
A LL THIS
"INTENSITY"
WHICH HE
GRAPUALLY N
REALIZEP WAS B UT
AN EXPRESSION y
S ;. OF... /
F T . AN
UNPERLYING '
INSTABILITY!
RUSSELL W R ITE S :"LIK E ME, HE WAS
CO N STANTLY A N A L Y Z IN G E V E R Y T H IN G ,
A HABIT PEAPENING T O T H E EMOTIONS
£30
HM. HERE RUSSELL SEEMS TO IMPLY
TH A T MAPNESS COMES F R O M LOGIC ANP
N O T T H E OTHER WAY ROUNP, AS YOU SAY!
I P O N 'T
THINK
SO...
IN MANY COMMENTS, HE PESCRIBES
W ITTGENSTEIN AS "VERY LIK E " HIM,
WHICH HE EXPLAINS AS "TYPICAL
v OF LOGICIANS"!
______
^
HERE,
LISTEN ...
RUSSELL'S GHILPHOOP GAVE HIM
G O OP RE A SO N TO WANT TO
S
__
PEAPEN Z E EMOTIONS! .
E X A C T L Y !
IT 'S HIS CHARACTER, HIS >
INSECURITIES, HIS NEUROSES,
WHICH PROVE HIM T O LOGICLj
I! be no
J accident that
wy deep-seated fear
of Madness resurfaced
exactly at the tiwe the
value of w/y work in
. Logic started to .
be questioned.
~ It 5
been said
before:
"The sleep
of reason
p ro du ces
wonsters."
The one a paragon o£ purify,
whose creo/o was Reason.,.
...E>uf the ofher
a disgusting
re p ro b a te ,
always seeking
Unrestrained
voluptuousness!
If was As who Wittgenstein
had brought to the surface, by
underwlining wy logical work.
5eing a true child of the Victorian age, I had learned
to regard every human being as essentially split into
"T H O U S H A LT
N O T LIVE HAPPILY
UNTIL TH O U SO LVE ST
T H E PROBLEMS OF
T H E TH EO R Y OF
. T Y PE S!" >
COME W ITH ME!
L E T 'S HAVE SOME
. F U N ! ^
It was then that the second momentous
event occurred.
£31
I was driven info a ghastly cul-de-sac.
^ T H E HUN
IS IN T E N T ON
GETT IN G HIS WAY,
^ GENTLEMEN!
P o rin g t h i s c ris is , I w a s c lin g in g f o r
s o m e Kind of support to t h e e x te rn a l
ro u tin e s o f a ca d e m ic life .
A n d i t w as
d u rin g a
b o rin g d in n e r
a t C olle g e
th a t It
s ta rte d ,.,
£3£
r ANP IF HE C A N 'T 1
G E T IT PEACEFULLY,
HE WILL RESORT
l T O A R M S ! y
~ I BEG YOUR
PARDON,
GENTLEMEN! |
A short sequence
of events which,,,
YOU ARE ^
U R G E N T L Y
REQUESTED AT
YOUR HOME, SIR!
...S ho w ed
me, to ta lly
unexpectedly,
a b ra n d -n e w
way to fa c e
life .
WAIT FOR M E,
RU S S ELL!
IT 'S AN
EM ERG EN C Y,
M A N ! R U N !!!
AH, WHITEHEAD, AT LAST!
WE HAVE
S E VE R E CHEST
PAIN...
...AND 1
PALPITATIONS...
...INDICATING
H E A R T F A IL U R E !
HEART...
F A IL U R E ?
As I stood there, a duwb
witness to the suffering
of a wowan I'd loved, the
lost footholds of wiy
austere worldview crumbled.
DEAR.
BERTRAND,
r NONSENSE, ^
OLD GIRL! YOU
SHALL LOOK AFTER
L HIM YOURSELF...
PROMISE YOU'LL
LOOK A FTE R POOR
. ALFRED!
S ta rin g in to h e r eyes
I fa c e d , te r r ifie d , w y
own m o rta lity .
£33
EVELYN,
MV PEAK?
MY COP.., '
3HE'S W IN G !
OH! TME PAIN IS
E X C R U C IA T IN G !
Yet.,
r P L E A S E ...
HELP ERIC FACE
MY DEATH LIKE
V A M A N !
...A newfound
S en s e o f
re sp on sib ility
s h o w e d wie a w a y
ou f o f w y d esp a ir.
A n d t h i s M ade.
th e e n c o u n te r
w ith d e a th , th is
^wewem/a M ori.,,
".A n o c ca sio n
f o r a s u rp ris in g
new o u tre a c h
t o life .
LET'S HAVE 1
A LITTLE TALK,
OLD CHAP! .
In Evelyn's eyes, I'd seen
fh a stark, unvarnished
image of our pr-edlic.aMe.trt.
T h e -tragic- lone liness
o f e ve /'/ h u m an bein g.
The f imiteness and
profound futility of life.
T h e t e r r i b le h a rs h n e s s
o f p a in a n d d is e a s e .
The
unmitigated
horror of
death.
But talking to Eric
I understood there is
also an alternative:
Redemption
In compassion
Yet, though the, report
of her Iwiiwinenh dewiise
had been rather grossly
exaggerated, the
transformation if
cou&ad in me was
totally real.
So real, in fact that when
I received a letter- from
Wittgenstein, who had
gone to a God-forsaken
Norwegian fjord "to think
about the weaning of
logical propositions
..I was not so
effected by his
doubts, or so
pe/iu/b&d
by his
criticises.
My naw concern with the, welfare
of wiy fellow human beings hod
iejMper&d wy passion for
the Foundations of Mathematics.
In this new spirit, I also
started giving lectures
— like, the one I'm gluing
to you right now — trying
to apply the Higher
Logic to human affairs.
£35
,"l must tell you that Mrs.
Whitehead, soon after...
...Totally recouered!
.../And is still
a live and well
today!
Apparently, her
"heart failure" was
indigestion, slightly
aggravated by a
nervous disposition!
".so, IN
ORPER TO
AC
,T
AEASONASLW...
This, wtind you, at a time
when any kind of Logic
was in very short Supply!
" RUSSIA TO
FIG HT GERMANS IF TH E Y
A TTAC K. SERBIA!"H E A P
A L L AB>O UT IT U !
OF COURSE, THIS
C O U L P COMPLICATE
. M A TT E RS !
IN T H E WORST CASE, THE Y
WILL INVAPE SERBIA! ^
BU T APPARENTLY T H E GERMANS
ARE A LS O TARING IT PERSONALLY.
£36
You see, wa Houa
wow waoched fine
suwhme y o f '(9'IH.
OH
PEAR!
r COULPN T
THIS C REATE
PROBLEMS
R FO R US'
MERELY A ~
LO CAL AFFAIR,
MV PEAR. ^
£37
I still shudder as
I dwe.ll on the.
next f aw days.
B L A S T !
In b u t a
few weeks'
fiwie....
A se/ies of
illogical
actions,,,
Brought
US +0 th e
b rin k ,,,
...Of a
te r rib le
nightwtare.
.And beyond.
Continental Eunopa
was sick with
nationaliswi,,,
,"And the qerwis of belligerence
flew acnoss the Channel at full speed.
£38
dui they didn't
get to you, Sir!
Well,,, I had
beer partially
immunized.
,"The inclination to
always try and be
logical, against ary
irrational instincts.
Pon't forget "Old Fbrker",
the maimed ve.fe.ram I bad seen
in the cemetery, as a child! ,
This tragic wreck yN
of a main,.,
,"Told me all ^
I needed to know
k about war! >
So, at its "
prospect, I took
action.
I wrote tracts and articles,
spoke at meetings and rallies
whose purpose was to find
. peaceful ways to
resolve the
'TrP ty A crisis.
Always, the appeal was
to Logic.,.
...To ^
Reason.
On August H-, ^9/lif, I was
attending a pro-peace rally
ah Trafalgar Square.
If was there float tine,
mews reached us,.,
...The, Uni fed Kingdom
had just declared
war on (nerwiany!
« jf^ RULE & r ita a a n n ia 6 J
BRITANNIA R u ir T H E W /\V I5 '
But strangest of
all was wy own
first reaction.
HURR A H !!!
COP
SAVE TME
KINS!
At float wiowent, I desired
the defeat of Germany as
wuoh as any retired Colonel!
£39
And then,
I witnessed a
terrifying i/uiracle.
People who were
there united by a
dreaw of peace..,
mWere, suddenly,
feasting the new
reality of war!
GET THE
H U N S !
Thankfully, this strange | started, in ww lectures and My 0wifj Crown Prince
upsutga of wiy deep-buried, ov+ides, to araue against the °f Hogio had now enlisted,
tribal instincts lasted madness that was engulfing as a volunteer in the
but a few hours. Then J s o m e of the cleverest i L Austro-Hungarian Imperial
^reason took over again. J L people I knew, including,,. J ^ ^ _ ^ A n m y ! J
WE WANT 0/G CANNON,
W ITTGENSTEIN : T H E
G E N ERAL IS A BIT
_ SHORT- SIGHTEP! A
,"The military
powers-that-be, no
doubt influenced by
his family's immense
wealth, put him at
Corps Headquarters.
He was now an Assistant
Mechanic, no less!
J 'AW OHL, H ER R
SERG EA NT!
Although Wittgenstein had
his own, eccentric reasons
for becoming a soldier,,,
£40
Vet no occupation could interrupt
his latest train of thought, about
the meaning of logical propositions,,,
",And their
relations to
language.
NOW T H A T IS
INTERESTING!
... SO, OUR, ARTILLERY
WILL COMMENCE FIRING
ON TH E ENEMY'S
^ WESTERN FLANK.
EACH CANNON ^
STANDS FOR ONE 1
BATTERY ME/NE I
HERREN.
7 A T O Y ^ T
f SOLDIER
J REPRESENTS
fC J l AN INFANTRY
BATTALION.
It was Toy
Mode.
Is
that I
ed
hi
<m
+o his
■first1 big idea.
SO, EACH ^
PART OF
REALITY IS
REPRESENTED
BY A SYM BOL/
£41
In the first years
o f
the war, he
got sow/e letters through to
m &."
YOUR
CANNON M E N
GENERAL!
...describing his latest efforts.
r ...AND t h e n t h e ^
FOUR. INFANTRY BRIGADES
WILL ATTACK THEIR
FRONT.
_
__
^
B R IL L IA N T !
IN GOES TH E
INFANTRY,..
...AND TH ES E ^
f RECOMBINE, IN ^
1 ACCORDANCE WITH
TH EIR REAL RELATIONS
AS MEDIATED BY
k L A N G U A G E ! A
I re tu rn e d to Condorid g e o n ly
fo r tine oc casio n a l ta lk .
£ 4£
W IT T G EN S TE IN ,
PUS H TH E
INFANTRY
L IN! A
WHICH,
IN F A C f
M EA N S
TH A T ...
PRIVATE
W IT T —
LANGU AGE 1
IS BUT A \
M O P E L .,.^
IT 'S A ^
V P IC T U R E
OF
1 R E A L IT Y !.
Wittgemstein^'^NBB
was in ve stig atin g
language's p ote ntia l f o r '
tr u th , b u t wiy exp erie nces
w ith it in w a rtiw e J
E n g l a n d A $
...C o nfirm e d wiy opinion
o f i t as an in s tru m en t
o f falsehood!
A s t h e w a r e sc ala ted, I took, a
s ta r 's leave to cam paign a g ain st
tin e im m inen t p ro s pe c t
o f co n s c rip tio n .
I SAY,
RUSSELL...
OH, '
CH E E PS!
£43
MOW MUCH IS TH E KAISER.
PAYING YOU TO UNPERMINE
OUR. YOUNOi M EN'S S P R IT ?
BLOOPV
TRAITOR!
Many encounters with otherw ise
rational wen had now becowe
exercises in irrationality.
A n d r e w s w ere w o re
pa in fu l th a n o th e rs .
WELCOME! WE HAVE
A TOLLY COOP
TU R N O UT! ^
WAIT A
MINUTE.
W HITE H EAP ?
RUSSELL.
' MY >
COOPNESS,
IT 'S ... a
ERIC WHITEHEAP, TRAINEE |
OFFICER IN T H E
ROYAL
A
^ FLYING. CORPS!
BUT...
WHY PIP YOU
ENLIST, BEETLE?
OH, FOR KING. ANP
COUNTRY, RUSSELL
- PO T H E WORP5
^ RING A BELL? >
BUT HE
IS A
k CH ILPI
7 YOU PON'T
WANT TO MISS
YOUR TRAIN,
^ ERIC. A
O H , B E R T IE !
PO N'T LISTEN TO THE OLP MAN!
I PO N'T PO IT FOR THE PATRIOTIC
s^. _ MUMBO TUMBO! .
7 I'M
GOING FOR
THE FUN!
"FUN"?
I PON'T WANT TO SPENP MY
LIFE OVER BOOKS, TOILING
TO PROVE "1+1
...I JUST
WANT T O
LIVE !
£45
In a sense., W ittg e ns te in h a d vo lu n te e re d
f o r tin e saw ie re ason, -though h is se n se
o f "fu n "w a s philosophical!'/ b ia se d !
f H ER R \
/MAJOR., I NAVE A >
PETITIO N CONCERNING
MV TRANSFER TO ,
. T H E FR O NT! S
YOU ARE
NOT THE
ONLY
. ONE. .
HAVE YOU GOT
CHILPREN?
NO, HERR
MAJOR.
BUT YOU ARE
MARRIEP?
I AM
NOT.
YOU PON'T UNPERSTANP! YOU
MUST HAVE VERY SERIOUS
REASO NS NOT TO GOTO >
k THE FRO N T.
W WITH ^
/ RESPECT, \
r SIR, IT IS y o u I
WHO POES NOT
, UNPERSTANP. )
MY PETITION
IS TO G O TO
T H E FRONT!
Wittgenstein believed that before
being a logician he "should beeoi/ue
a huwtan being".
And so he took.
Schopenhauer's
word for it:
there is nothing
like a good
near-death
experience, to
huwianiee you!
I SHALL
RECOMMENP
YOU.
And recowwiend
hiwi he did.
Wittgenstein got Inis wish
He was assigned
+0 a front-bound
artillery regirwent.
Relish In +He comfort of
sleeping In wud.
____
At last he could enjoy the
luxuries of trench life.
No task was too
demeaning,,.
Too arduous
HIaTC-1
Y OF COURSE,
I H E R R STAFF
IV SER G EA N T !
Or too dangerous
H fy / HOW v
V ' A 60U T >!
i l l YOU? i
^ ANY
V O L U N T E ER S ?
Ah, the superior masochism
of the privileged!
None too
harsh for hiwi...
TO PERFORM IT
APEQUATELY, A MAN
MUST POSSESS A
CERTAIN,,. A
VOU P O N T
UNPERSTANP '
TH A T THIS MISSION IS
CRUCIAL FOR TH E
k NEW ATTACK'S ,
u W SUCCESS? A
..S A NG
FRO IP.
I UNPERSTANP,
H E M M ATOR!
Eventually, he found exactly
what he wanted.
But apparently he
didn't understand.
Not at first,
anyway.
A "forward observer
is placed right at the
nucleus of danger.
T o u ^ d e y sta m o l t h is
situ atio n , you ha ve
to live it!
POOR
SOP.
CAN THIS
WORLP HAVE
A M E A N IN G ?
...IF IT POES, IT'S
CERTAINLY N O T
TO O EAGER TO
PISPLAY IT !
This is one
of floe iwojon
disadvantages
of Reality
f now up close,
if looks very
diffenenf f row
onY "piotune".
A difference
which no theory
can explain.
S A N G -F R O / P
S A N G -F R O tP ..
Face to face with death,
Wittgenstein arrived at
his fundamental insight.
T H E . M E A N IN G
O F T H E WORLP
PO ES N O T R E S IP E
/SV T H E W ORLP!
Put a warn on th e brink o f the abyss and — in
th e unlikely event that he doesn't fall into it —
he will become either a mystic or a madman,..
.Which is probably the same thing!
£5 £
r LuoKy ^
Ludwig W ittg ens tein
had a succe ssful bout
With existe ntial
I— e xty e iv iis iM .^ X ^
Not so Eric.
Whitehead.
PEARLY BELOVEP, WE ARE HERE TO
THANK THE LORP TOR TH E LIFE OF
A YOUNG MAN WHOSE LOVE FOR
T-r-r
\----\--
— st ^ ^ C O U N T R Y ...
The confluence of
religious and patriotic
rhetoric would be too
much for wie, so,,.
I didn't attend
his funeral.
I iMourned in
n/y own way.
~ You<MUsr^
take a stance.
Professor
^ Russell! ^
Honour you r
C onvictions!
£53
...Which involved changing
My tack: frow arguing simply
fora peaceful resolution of
,tne conflict, I now urged people
to become conscientious
—— objectors. J
If you'll pardon the
oxymoron, I became,,,
," A '
m ilita n t
p a c if (s t!
Good
Im o io !
So, do it a g ain , y
Professor Russell!
The problem is the
same, a totally
irrational war! A
r r f// ^
Wars a re
irra tio n a l!
Hear,
hear!
r We
didn't come
here to
listen to you,
fellows!
Prove '
you are a
pacifist!
Please m
'l l I ''" " 1
".A N P WE HEREBY
SENTE N C E T H E AGGUSEP
B E R TR A NP R U S SELL TO S IX
M O N T H S ' IM P R IS O N M E N T !
Incidentally, it
will interest you
ho know that
what sent we
to jail was an
article protesting,
precisely, your
own country's
entry in the war!
^ INPEEP,
r WILSON! TH E l
INTROPUGTION TO
T H E "P H IL O S O P H Y O F
M A T H E M A T IC S " is
, ALM O ST FINISHEP! A
PROPUGTIVE
PAY, SIR?
,"l now returned to pure thought,
writing a defence of tine
prewises of iwy logical work.
But I can't cowplain: wy tiwe in
Brixton offered concentration of
the highest quality.
£54
I was expelled frow wy
College, I was prosecuted, I was
taken to court \- do you need
^ wore proof than fhat?
But that's
all history!
What about
. now? .
Be patient, wy story is now
alwost fintshed. /And as it
happens, Its wiost Important
events, in what concerns
you, lie in the ending...
Wly appetite
for woral duty
having been
fully Satiated,,.
The wanus c r ip t
of his opus iMagnuiM, written
in the trenches of the Eastern Front
was, at least in parts, obscure to wie.
But still, I could sense what his
total solution" imiplied,,,
£55
A few wiontHs after fine
end of the war, I received
o totally unexpected gift.
THAN K
GOP.,. HE IS
A L IV E !
r "I HAVE ^
T O T A L L Y SOLVEP 1
ALL TH E PROBLEMS
OF PHILOSOPHY!" J
G o o d old
W ittg e n ste in ,,,
M od esty
w a s ne ver h is
stro ng p o in t!
WAIT W AIT! I TRIEP TO
REAP TH IS " T R A C T A T U S
L O G IC O -P H IL O S O P H IC U S ".
• X S T L L N F
' CALLEP, BACK N
THEN, " L O G IS C H E -
P H IL O S O P H IS C H E
A B H A N P L U N G
...WHATEVER, ANP I GOT
VE RY GONFUSEP!
^ IN T H A T >
YOU ARE N O T
A L O N E . V
' W HAT IS ZE
"NUTSH ELL"
s. OF IT ? J
BUT, IN CRITICIZING THE PREMISES
OF THE " P R IN C IP IA ", WITTGENSTEIN
QUESTIONED THIS VERY SWITCH.
£56
^ FIRST REMEMBER: ^
FUELLING RUSSELUS QUEST
FOR ABSOLUTE CERTAINTY
WAS A PEEP M IS T R U S T OF
-— N EVERYDAY, ORPINARY ,
\ LANGUAGE.
LIRE FREGE, >
HE SAW IT AS A
C O R R U P T IO N OF
PURE THOUGHT..,
^ ...AND SO
SUBSTITUTED FOR
IT A "LOGICALLY
PERFECT" j
■V V E R S IO N .^ /
f ...AW \
EMPLOYED
THE ORDINARY
LANGUAGE
V A G A IN ! a
r THE FIRST SENTENCE OF ~
THE "T R A C T A T U S " REFERS TO
THE REALITY OF THE WORLD.,.
" T H E W O R L P
IS A L L T H A T IS
T H E C A S E ." .
r AND THE WORLD N
IS M O P E L LE P BY
LANGUAGE.., THIS IS
THE GIST OF WHAT
HE CALLS "PICTURE
THEORY". J
LOOK = AS THE TOY
CANNON 15 A M O P E L ...
...OF THE
REAL, SO...
...IS '
ALSO THIS,
THE W ORP
FOR IT!
...Tlnovofn T'oy after
its saok comimot have
looked worse than Ypres!
Feat'.
£57
ANP THE SENTENCE "THE
CANNON FIREP AOiAINST THE
ENEMY" PICTURES THE
SITUATION IN THE REAL
> WORLP!
K A -B O O M !
BUT LET'S
MOVE ONWARPS,TO \
PEGEMBER A3'19 ANP
THE SEMINAL MEETINCi I
^ IN THE HACiUE...
With the wounds
of won still gaping,
on Austrian couldn't
visit England.
So we hod to weet
on neutnol ground.
To describe to you
the state of post\-
W
an BejgiuiM, I'd need
to be on Aeschylus,
o Eunipides,,,
My joy ot the pnospect of our
reunion often seven years,,,
/"Wos tinged with
trepidation.
f THERE YOU ^
GO AGAIN! THERE'S
NO SUCH THING
AS A
"HIGHER
LANGUAGE"! TRUTH
COMES ONLY IN
\ ONE VARIETY! J
^...TH E "PICTURE" THEORY
IS CLEAR ENOUGH. BUT IT GIVES
US
TRUTH
ONLY BECAUSE OF THE
UNPERLYING HIGHER
LANGUAGE
OF
LOGIC! ^
For a
week., we spent
all our waking hours
going over each and
every argument
o f
th e " l? a c M u & '
r ...YOU HAVE TO UNPERSTANP ^
THAT MY CENTRAL IPEA IS THE
EXACT OPPOSITE OF YOURS! FROM
YOU -A N P FREGE OF COURSE -1 ONLY
v TAKE SOME METHOPS! ^
£58
r A "PICTURE ^
LANGUAGE" IS ALL YOU
NEEP TO PESCRIBE
THE WORLP, i.e . ALL
^ THE FA C T S !^ A
...ANP 1
LOGIC?,
LOGIC IS THE FORM OF THE
LANGUAGE, IT'S EMSEPPEP IN IT, LIRE
v THE IRON STRUCTURE THAT A
SUPPORTS A B U IL P IN G .^ ^ ^ ^
BUT TRY \
LIVING IN
THE IRON
STRUCTURE!
r YOUR FAILURE TO
CREATE FOUNPATIONS
FOR LOGIC IS EXPLAINED BY
^ ITS VERY NATURE. a
r ...YOU
CANNOT SPEAK
"OF" LOGIC!
. LOGIC... .
...YOU T
CAN ONLY
SHOW!
ONE THING
AT A TIME,
OLP CHAP!
I f S in o f easY fo
d ig e sf sou/ieotoe's
" to ta l s o lu tio n
o f fine- problem s
o f Philosophy"...
...E s p e c ia lly
i-f
if m/iplies fine, fofal
ainwihilofioin o f your
oun life's work!
...LIKE
SO-CALLEP
"INFINITY"!
IT'S THIS WHICH PROPUCES MONSTERS!
Y BUT WE PON'T 1
NEEP SETS! ANP TO
SAY "X IS TRUE OF
INFINITY" IS AS SLOOPY
INANE AS TO /MAKE
STATEMENTS "OF ,
K THE UNIVERSE"! A
Y YOU'RE WRONG,
HERE'S A NON-INANE
ONE: "AT LEAST THREE
THINGS EXIST IN
V THE UNIVERSE." >
TO
WIT LOOK
AT... .
£59
BY THE WAY, POES YOUR
BOOK ALSO PEMOLISH
. M A TH E M A T IC S ?^
OH,
MATHEMATICS IS
A PECENT ENOUGH
TOOL. BUT TO
GIVE ITS...
r ENTITIES ^
SOME KINP OF
"INPEPENPENT
EXISTENCE",
IS INSANITY!
MJKE "THE
SET OF ALL
.SETS"... .
^ TM E
PARAPOXES SHOULP
HAVE A L E R T E P YOU,
v RUSSELL! n
7 TRY ANP \
MAKE EMPTY '
FORM SPEAK
OF SUBSTANCE
ANP YOU GET
POPPYCOCK! A
S LOGIC ^
IS V A C U O U S ...
IT C ANNO T SPEAK
. REALITY! ^
WHAT ABOUT T H E
S TATE M EN T "TOMORROW IT
WILL EITHER SNOW OR NOT
SNOW"? IT'S "EMPTY FORM"
YET T O T A L L Y T R U E ! >
F YES, BUT IT ^
T E LL S US N O T H IN G
OF T H E W EATHER
TOMORROW! >
T h e r e i t w a s : f o r tw e n ty
ye a rs , I h a d s w e a te d t o
ju s t if y t h e e xis te n ce o f a
M a c h in e f o r M a n u fa c tu rin g
ta uto lo g ie s!
£60
...THIS B RANC H : 1
T H R E E L E A V E S EXIST
ON I f THER EFOR E
'AT LEAST TH R EE .
THINGS EXIST IN /
THE UNIVERSE!" J K
V NO, NO, N O ! N
YOU ARE S O W RO NG !
YOU CAN SAY "AT LEAST
THREE LEAVES EXIST
ON THIS TREE..." >
...b u t you
~ C A N N O T SAY " IN ^
THE UNIVERSE"! LOGIC
MUST N O T ALLOW IT, SINCE
YOU CANNOT PIC T U R E
k THE UNIVERSE! A
£61
r MY BOOK
PELIMITS
LANGUAGE,
TH U S ALSO
THOUGHT.
BUT TH E '
REAL ISSUE IS
BEYONP A LL
. THAT.,, ^
IT'S
HOW TO
L IV E
C . A W OF
T H A T WE
CANNOT
v T A L K !
A LL T H E FACTS OF SCIENCE
A R E N 'T ENOUGH TO
UNPERSTANP T H E W O RLP 'S
M E A N IN G . FOR THIS, YOU
. MUST STEP O U T S IP E A
T H E W ORLP! J D
' W ITHOUT >
LANGUAGE
OR THO UGHT
HOW CAN YOU
UNPERSTANP
A N Y T H IN G ? .
WHO KNOWS, MAYBE
BY W H IS T L IN G ? ^
IT'S TOO
COLP FOR
WHISTLING,
RUSSELL.
TO O >
SLOOPY
COLO. ,
6. INC0MPLETENE55
£65
r so,
WELCOME
SACK,
CHRISTOS!
, ANP OFF WE \
' GO, TOWARPS >
TH E RESOLUTION
, OF TH E STORY, j
' INTERESTINGLY, "
ARISTOTLE'S WORP
FOR IT WAS SIMPLY:
"SOLUTION". .
H IN T ^
HINT...
NOW, WE SAW THE WAY RUSSELL'S PREAM
OF CERTAINTY WAS SHAKEN UP BY THE
" TRACTATUS LOGICO-PHILOSOPHICUS " ...
...WHICH, > r
f FRANKLY, IS >
IN MY "TOP TEN"
LIST OF HUGELY
OVERRATEP
K BOOKS! /
' EVEN >
SO, IT 'S AT
T H E HEART OF
YO UR THEM E
OF "REALITY
v VS. M A P S " ..
...OR, AS I ^
PREFER TO
THINK OF IT,
REALLY, "T H E
CONCRETE
vs. T H E
ABSTRACT".
YOU SHOULPN T KNOCK
TH E ABSTRACT, THOUGH!
IN THIS STORY, IT'S OF THE
UTMOST ESSENCE!
RIGHT.
ANP SO, WE
CONTINUE
WITH —
YES, BUT FOR ONE HOUR
ONLY! ANNE IS EXPECTING
US ALL FOR Z E PRESS
^ — REHEARSAL OF ZE
IK "ORESTEIA". y
M ONSIEU R,
I INVITE YOU T O
T H E C R E ATIO N
OF A POEM!
r T H E Y ^
C A LL THIS
A R T ?
T h e / w o rld
had
cowe*
fopsy-+ury/y.
PRECISELY:
"A RT TH AT
P E N E S
A R T !"
WHAT ON E AR TH ARE YOU GOING?
WE
BEGIN,
SHHH,
The- war
had changed
e ^ e ry fln 'rig -
arid not ju s f
for wie..
£66
/O IL A !
"ELBOW KNEE KIPN AP
EERIE TUBE FIZZLER
WINTRY CURSES TERK
INTERREGNUM F E Z 11.
ART T
COPIES
LIFE,
M ONSIE U R ..,
",A
RANPOM
STEW!
IF THIS IS
ART, I'L L TA K E
M A T H E M A T IC S !
C onclu sion:
Tine/ Old World's
values and the art
that embodied
them should be
destroyed.
Premise-: the Old World had
created a monstrous war.
r A T A LE ^
TO LP BY
AN
IPIOT,
M O N CHER,
SIGNIFYING
V NOTHING.'
J
HISTORY IS
HISTORY IS
HISTORY. . HUMPTY- PUMPTY
SAT ON A PAPA
HUMPTY - PUMPTY
HAP A GREAT PAPA.
I was as critical o[
Ihe Old World as the
anarlest artist.
But I feared
the void
created by
its demise,,,
,"A\r\ open invitation
to the Irrational.
£67
0 4 0 4 /
#/#The argument had
a lot Qo\r\Q for i t !
W.&.Yeats's lines expressed
uwy apprehension perfectly.
'Things fall apart, the centre cannot hold**."
"/"Mere anarchy is loosed upon the world."
THE "TRACTATUS"
PEALS WITH THE
PROBLEMS OF
LOGIC...
FROM ARISTOTLE TO YOURS TRULY,
SAYS WITTGENSTEIN, LOGICIANS
ARE CREATING ELABORATE WAYS
TO "SAY THE SAME THINGS IN
PIFFERENT WORPS"...
Y WHITEHEAP ~
ANP I SPENT OVER
A THOUSAND PAGES
TO BUILP FOUNPATIONS
FOR LOGIC ANP - ...
f TO TRY N
ANP BUILP
FOUNPATIONS,
OLP CHAP!
£68
Wltfge.Kisfe.iWs ^
l? a c ,ia tu s ' was
oublfshed W /192Z.
Though it was not exactly
a bestseller, its influence
started slowly to accrete.
f Those \
to L/hoM '
its Message
Mattered,
paid it
increasing
v attention.
J
At the top of that list was
w\y old friend Moore, wiy
first inductor into Logic. .
YES, LIRE ALEXANPER
"UNTIEP" THE GORPIAN
KNOT — WITH A
SW O RP ! A
...ANP
SOLVES
THEM!
r WHAT TOOK ^
TWENTY-THREE
CENTURIES TO BUILP,
HE PISPOSES OF
v IN A TIFFY! W
| ...TAUTOLOGIES!
WELL, EXCUSE ME, BUT I FEEL WITTGENSTEIN
HAS STACKEP THE PECK IN HIS FAVOUR! THIS
"EVERYTHING IS A TAUTOLOGY" STUFF
SMELLS OF METAPHYSICAL BOSH! ^
Y
0H? anp 1
ARE YOU SURE
YOUR REACTION
POES NOT SMELL
OF "SOUR
^ GRAPES"? ,
BUT WHY TH EM ?
SURELY, HE GOULD FIND
NEEDIER RECIPIENTS!
' MONEY \
CORRUPTS,"
HE SAYS, "SO,
BEST GIVE IT TO
TH E ALREADY
CORRUPTED!"
V HA HA! A
'A / Z
YES, AND HE'S FOUND A NEW
GODFORSAKEN PLACE IN WHICH
TO EXERCISE HIS VOCATION...
A VILLAGE IN T H E ALPS, FOR
GOODNESS'SAKE!
£69
Yet, despite wiy doubts about Wittgenstein's Logic,
I was full of adi/wiration fon his integrity.
WHAT'S
SO FUNNY,
MY LOVE?
THE MAN HAS SUPPASSEP
HIMSELF IN ECCENTRICITY!
...HE G A VE
TH E HUGE
FORTUNE HE
INHERITED
TO HIS
BILLIONAIRE
SISTERS!
AND LISTEN TO T H IS : HAVING "SOLVED
ALL THE PROBLEMS OF PHILOSOPHY,"
HE HAS NOW DECIDED TO BECOME
A TEACHER! A
A ^
SCHOOL
TEACHER?
WELL, '
ISAY,
BERTIE..,
...THAT'S QUITE
COM M ENPA& LE!
YES, LE T'S 1
XU ST HOPE HIS
S T U P E N T S WILL
AGREE WITH
v YOU! A
Ob, by
the way...
I forgot to tell you that, by
now, there had been a
change in miy personal life.
A pleasant change...
...Well, a/f/nst
anyway!
£70
M y new wife/, Pbra, sh a re d wiy in te re s t in
th e . w e lfa re , o f th a t w iost in d is c riw iin a tin g
o f c lu b s : H uw ianbinol.
______
,"7 o w hich, in cid e n ta lly ,
a new wtewtber w a s
a b ou t to be a d d e d .
PROFESSOR
RUSSELL?
CONGRATULATIONS!
IT 'S A BOY!
A... A...
& O Y ?
P h iloso phy 's
con sola tio n s
ha d n o t
pre p a red wie
fo r s u c h jo y.
271
... Whfch, like, all joys, was mot unadulterated!
r COULP 1
HE BE TOO
C O L P ?
...OR, TO 0
WARM?
^ THE ^
TEMPERATURE'S
TU S T RIGHT!i
IS HE
HUNGRY
MAYBE?
r HE A T E
HALF AN
HOUR AGO!
I'M
GOING TO
CHECK!
As always in wiy adult
life., I turned to Reason
for assistance-.
T h e new scie n ce o f
psychology seewted
to o ffe r a way out.
In fact, time seeiwed
propitious fo r or extension
of wiy lo^icist project.
A project to apply the
tools of Logic, Mathematics and
the Physical Sciences to the
study of human matters.
WHO LAIP THE GROUNP FOR A LOGICAL LANGUAGE
MAKING POSSIBLE THE SCIENTIFIC WORLP-VIEW! IT
IS H IS PIONEERING VISION WHICH INSPIRES TH E WORK.
OF OUR CIRCLE...
TOGETHER WITH TH A T
OF THOSE VENERABLE OLP
GENTLEMEN, FREGE ANP
WITTGENSTEIN!
£7£
Y a 1
group o f
vis io na rie s
in Viennoi
had drof+ eo l
a wianifes+0
adv'OCG'flVig
" t h e
s cie n tific
con ception
o f fine.
. w orld".,, ,
K S o < ^
d e s p ite th e
failure, o f wiy
own w o rk in
Logic, I did
^ n o t - ^
VOU TUST CAN'T GO CALLING RUSSELL'S
WORK IN LOGIC A "FAILURE"... r i
\----\----\----\----
\---------
\----\------\-----
V ...N O W A Y! A
' IT 'S >
H IS WORPS
WE ARE
. USING! .
BUT TH E "P R IN C IP IA " IS T H E BASIS OF
E V E R Y T H IN G T H A T FOLLOWEP!
BU T-
M M M",
OK, MAYBE 1
W E'LL PUT
SOME OF THAT
IN BERTIE'S TRIP
. T O VIENNA! j
...IT IS THE GREATEST HONOUR
FOR US, PEAR COLLEAGUES,
TO HAVE LISTENEP TO
. PROFESSOR RUSSELL...
WlnicJn, like, all legends, load only a
fewous neJatfonship wifh fine. frufln!
MAY I INTRODUCE A
YOUNOi COLLEAGUE?
AN HONOUR,
HEAR PROFESSOR.
HERR
KURT
GODEL!
£73
IN VIVA VOCE TOO, HE
HAS INSPIRED US!
EXCELLENT
TALK, HERR
PROFESSOR!
THANK
YOU!
' SAY '
5CHLICK...
WOULD YOU MIND ELABORATING ON YOUR
BRANDING WITTGENSTEIN "O LO "?
To members o f the. V i m Circle,
W l+tgens+ein h a d beeowe. a legend.
^ I AM WORKING
" ON LOGIC FOR MY "
DOCTORATE. FOR
THIS, I WISH TO ASK YOU:
IN THE WHOLE OF THE
. "P R IN C IP IA " YOU DO NOT
V MENTION O N C E - a
THE "WHOLE"?
YOU MEAN YOU
HAVE REAP A L L
OF IT? y
r INDEED, ^
EVERY PAGE!
HOW COULD I
COMMENT ON IT
IF I HAP NOT?
GOODNESS, YOU
MUST BE GIVEN
SOME SORT OF
V MEDAL... A
...THOUGH ^
I'M NOT SURE
EXACTLY F O R
. W H A T ! a
TO THE POINT: IN THE WHOLE
BOOK I DO NOT FIND A CLEAR
STATEMENT OF ITS M O S T
n . S A S IC A S S U M P TIO N ! >
OH? A W
WHAT
WOULD
THAT
v BE?
S OP COURSE, N
THAT THE TRUTH
-O R FALSITY, SHOULD IT
SO BE — OF EVERY
LOGICAL PROPOSITION
CAN, IN THEORY,
V BE P RO VE N ! y
This young wian's questions brought we book
to wy philosophical salad days... And wade
we painfully aware that at the heart of wy
quest was a void. A void I had all wy
tfe tried to fill — but failed!
ANYWAY, I' LL TAKE HIS WHINING, AS LONG AS
T H E POINT IS C LE A R: WITHOUT T H E "P R IN C IP IA "
GOING TH E PONKEVWORK, GOPEL COULPN'T
ASK HIS QUESTIONS!
£7A
SATISFIEP
NOW?
✓ SURELY, ^
THAT'S THE >
B A S IS OF THE
LOGICAL POINT
OF VIEW, IS ,
L IT NOT? /
F THAT ^
SOMETHING
IS TRUE IS
SYNONYM OUS
TO IT BEING
. PROVABLE! ,
SO, YOU
POSIT IT
AS AN
AXIOM?
r N -N O .,. I GUESS IT T
TU ST REFLECTS THE
ESSENCE OF A LOGICAL
SYSTEM. LIKE OLP HILBERT
PUT IT: " IN MATHEMATICS
THERE IS N O .
. 'IPi N O R A SIM U S'! "
T SHOULP X
NOT T H A T
BE AMENABLE
TO PROOF
N THEN? J
' WITH V
RUSSELL>
HAVING
"F A tLE P"?
Y WELL, HE ^
PIPN'T GET 1
MANY A N S W ER S
FROM RUSSELL...
OR A NY ON E ELSE,
FOR TH AT
v MATTER! A
' ANP TH A T'S >
WHY HE HAP T O
IN V E S T IG A T E
. O N H IS OW N!
I sat- out- fo r home
rejuvenated by
floe optiwiswi of
floe Vienna Circle*.
f WONPERFUL N
TO SEE YOU STILL
AT WORK., H ERR
V PROFESSO R! S
...THER E ^
HE IS, A T HIS
"LOGIC",
" ALWAYS.., .
^ JA , TA,
I CANNOT STOP!
TH E PANGER, IS
_ T O O G R E A T !
£75
...M akin g a stop to
v is it an old fr ie n d .
0H '" ...W HIC H ^
H B "P A N G E R "
T | V is t h a t ?
« THE t
JEWISH ONE,
OF COURSE!!!
I PROVE IT
f BY STRICT LOGIC,.. \
INESCAPABLE CONCLUSION.,. '
UNPERMININGi THE NATION'S
FOUNPATIONS... WE MUST
PEFENP.,, EXCISE THEM
FROM THE SOCIAL CORPUS...
ANY MEANS AVAILABLE A
\ FOR A NEW A
V SOCIETY... y f l
Logic. is
a to o l,,,"
% own
words.
Like o knife,
you can use it
to cut bread
with — or kill!
Frege's paranoid
vision played on
a Malignant
variation of an
ancient theMe:
"You cam't Make- a
good OMelet Wifh
bad eggs."
B u t th o u g h I s tro n g ly
d is a g re ed w ith his
ra cis t c rite ria o f
excellence.,,
...Like hiivi, I also dreawied
of a better world.,.
r —
H E 'LL BE FINE
AS SOON AS HE
OVERCOMES HIS
PANIC.
It's the oldest story around:
Instinct, Ewiotion and Habit
get the better of hvwian beings.
£76
' HELP ! \
H E -E E E L P
M E ! .
^ 4R E
YOU SURE HE'S
ALRIGHT, MY
V UOVE? y
Or, in other words: start from
the wrong premises and Logic can
be the executioner's handmaiden
— as in Frege's cruel theories. Or,
L alternatively, a fool's ideal
. accomplice! a
So how to straighten "the
crooKed timber of
humanity"? Mow to annul
the harm done by...
Instinct, Emotion and
Habit?
£77
r lo my mind, there could only
be one answer, a rather obvious
^ one: Education.
But of what
Kind should
. it be? a
M y p h ilo soph ic a l
h e ir w e n t a t it
h is Own way.,.
IT IS CLE A R >
A S PAYLIPHT
WHAT YOU PO!
tK . " '
WELL.,,
^ A C H , ^
"A LIN E ", J A \ BUT
W HIC H LIN E ?
T H E .,,
H E -
HEICtHT...
YOU KNOW THE
TO OLS: COMPASS
ANP RULER.! SO,
. T E L L M E ! >
E R ." l," l,"
PRAW A LINE FROM
...ANGLE " B ".
^ IT 'S NOT G EOMETRY T H A T SHOULP <
STOOP POWN TO YOU, YOU BRAINLESS C R EATU R E!
IT IS YOU WHO MUST RAISE YOUR THIC K HEAP
TO IT S LEVEL!!! <
^ 3
Ac+uoillv, the onl'Y idea
Wittgenstein brought to
education was a new
use for the ruler in
geowetric proofs!
BUT OF
COURSE RUSSELL,
THE INVETERATE
MOPERNIZER, COULP
NOT ACCEPT AN
OLP EPUCATIONAL
SYSTEM!
£78
^ R IG H T ! SO AFTER REPEATEP ^
"EAR* BOXINGS", FREQUENT
HAIR PULLING ANP SOME BEATINGS
THE VILLAGE COUNCIL PECIPEP i
s. TO EXPEL HIM.
PEAR N
FRIENPS, WE
SMALL NOW
INTROPUCE
YOU TO,.,
TOMORROW'S
SCHOOL! .
& y m o w , 1 w as
c o nv in e e d t h a t /
t h e ad van ces
in s c ie n tific
psych ology
o ff e r e d a u
w ay out...
>
& & ' \ !
" . T h e w a y t o a
p e r fe c t tr a n s fo r m e r ,
o f b a d eg g s in to g o o d !
' TOPAY WE BEGIN WORK ON
GEOMETRY WHOSE ACQUAINTANCE
I MAPE, LIKE YOU, WHEN I WAS >
VERY YOUNG...
OH,
I THOUGHT
YOU WERE
A LW AY S
OLP!
In o u r b ra n d -n e w s c h o ol, P o ra
a n d I s h a r e d + h e ta s K s .
£79
NYAH N YAH
NO RULES!
GEOMETRY
IS N O T FUN!
NYAH
. NYAH
PAMN
& R A T 5 !
Sowetiw/ies, the best angi/wient in
favour of tine old.., is the mew!
£80
you SEE.? TWO DIAMETRICALLY
OPPOSED VIEWS OF EPUGATION,
AUTHORITARIAN ANP
RULE-BASEP FOR
WITTGENSTEIN
R U L E R -
RATHER!
_r~ — i —
ANP TOTALLY
/4AT7-AUTHORITARIAN
FOR RUSSELL, YE T BOTH
EQUALLY INEFFECTIVE
IN PRACTICE!
*Z ey are crazy ease logicians!
£81
ts/ori, To E51A6L/514 ^
A PROOF CALCULUS, W %
/W /A/DEPCNDen/T AXIOMATIC
FouNpATiON 6A$ZP ON J/+E
k RV&£U -_u/HlTEH£AD
MINI? VOU, RUSSELL'S
SON WAS EVENTUALLY
PIAGNOSEP WITH
SCHIZOPHRENIA, ANP HIS
ORANPPAUGHTER, LATER
^C O MM ITTEP SUICIPE.
___
MAYBE WHAT BRINGS THEM TO
LOGIC IS FEAR OF AMBIGUITY
ANP EMOTION, FEARS LEAPING
TO BAP PARENTING.
V TRUE? FALSE? S
£ 8£
A t this point, I return to Logic-. For,
while- I was experimenting with education,
logicians, based on our "Prfnc'ipia", reached
the apex of the struggle towards my
youthful dream,.
r ,"To build
Mathematics
on absolute
certainty, to
place the
lowest of the
beastly
things,,
WITH T H E TOOLS OF
T H E NEW LOGIC, WE SHAL L
A T LAST C E M EN T T H E
C O R N E R S T O N E O F OUR
' V SCIENCE... ^
...T H E P R O V A B IL IT Y O F
E V E R Y M A T H E M A T IC A L
S T A T E M E N T - OR ITS
N E G A T IO N ! >
True to the spirit
of his Paris talK
of "1900, that had
also inspired Me so
much, Pavid Hilbert
continued to preach
as the struggle's
High Priest.
H e s p re a d his m essag e by
e v e ry m ea ns ava ilable,,,
Including
t h e n e w est
te c h n o lo g y
of the
radio!
£83
KWiSlid
...A speaker at the, next logical conference,
held right inside the lair of the Vienna Circle.
SAY, I WONPER. HOW YOU CHAPS CAN LIKE ME
A M P WITTGENSTEIN? E S P E C IA L L Y GIVEN OUR.
PIFFER.ENCES ON M A TH EM A TIC S? V
»
----
--------
-------------------------\- —" < j?W Y
£8H
..MY RESEARCH ON TH E
PROVABILITY OF THE
PROPOSITIONS OF
ARITHMETIC.
v '
n
LIK E YOU, N
H E BE LIEVES \
I LOGIC IS AN IMAGE )
A OF T H E H IG H E S T /
FORM OF /
v T R U TH !
T H E POWERFUL METHOPS
OF T H E " P R IN C IP IA " NOW
ALLOW US, FOR T H E FIRST TIME
IN HISTORY, T O SPEAK. OF A
"CORRECTLY FORMULATEP
QUESTION" IN THEORIES
OF MATHEMATICS...
OBVIOUSLY!
,.ANP TH U S ALSO, FURTHER, T O A S K : "IS A CORRECTLY
FORMULATEP MATHEMATICAL QUESTION
N E C E S S A R IL Y A N S W E R A B L E ? »
£85
l/i Model's
lecture, the
audience, had
expected fine
Gonfi/w/iafiom
of f heir wosf
cherished
vision.
50, "Oi&n-n" IS NOT
PROVABLE," FOR, IF IT WERE,
THERE WOULP BE AN
W SUCH THAT,,
% & ) & ) = 3b('
Substi tu tin g X And y
X SK (1 R Gen r) R< ^ L ^ z o o ) ],
x B ^ G a w - ) - .
They got" sowefWng
cowplefely different.
IT'S A L L OVER!
£86
""AII over!"Vo/1
Neumann's comment
perfectly sums-up th e ,
' _ essence o f G to d e l' s
p /o o f .
I K now
1+ w a y b e h a rd
f o r la y p e rs o n s t o
u n d e rs ta n d .
...But fo r a lot of
very intelligent people,
the Incompleteness
Theorem wieant the
end of a Preaw!
The Preawi had
theological ancestry.
Its credo had been
, written in (nreek, two and
half Millennia ago!
H E R R P RO FE S S O R ...
WOULP YOU LIKE US
TO T A K E YOU TO
^ YOUR H O T E L ? "
! M I ■»'!
And now,
suddenly,
th e rug had
been pulled
from under
the feet
of the
dreamers.
There's no
getting round
a proof...
_
...Even if it proves
that something is
unprovable!
That is the
beauty,
th a t is the
terror of
Mathematics,
T H E TO URN EY THROUG H ABSTRACT
THOUGHT, FROM ARISTOTLE, VIA BOOLE,
A L L T H E WAY T O GOPEL'S THEOREM , IN
E FF E C T LE P TO A N E W B E G IN N IN G ,
WHICH-
£87
Now, as if Model's proof wasn't enough
wty Vie,nnese adwiirers soon received a
new blow, which added insult to injury,,,
'll- ~
Completely
Subverting,
as it did,
the iwiage of
wiy arch-rival ^
for the Circle's |
admiration.
.T O C E LE B R A TE
OUR FIRST /MEETING,
WE OFFER YOU OUR
" M A N IF E S T O O F T H E
S C IE N T IF IC
YOUR WORK
V CAVE US T H E MEANS
TO EXPEL RELIGION,
METAPHYSICS, ETHICS,
E T C . FROM RATIONAL
£88
...Ojien tf)e waaaay for tyt
'Brown 'Battalion*... We raise
tlje Swastika, tlje fjape of ft
I t J many million*! j>r
Though arguments With hiwi always
involved sowie awount of sound and fury,
Wittgenstein thankfully never resorfed
to
physical violence.
,"/4t least
not against
his peers!
Oh, if only that were also true
of the acolytes o f Europe's
newest avatar of Irrationality.
WHATSA MATTER, PANSY?
In _ 4955, I I earned
GicJdel was
hospitalized for
Melancholia.
J wasn't surprised.
£89
The\- yvo-Naii newspaper
declared +ha+ the- Code's
rationalistic\- world-view
had "desecrate-d hallowed
fterw/iamio valuer,.."
£90
f A journey o f some jo ys^
and more disappointments,
the latest o f which is
the r e a l iz a t i o n that I Ve
failed — also — as an
Whose twain victims wene,
alas, wiy own children.
f NO "BUTS"! \
IT'S YOUR PUTY
T O PUT YOURSELF
IN T H E PLACE OF
T H E CHILPREN
. T O WHOM I'M
L N O T "PAPPY".
Y YOU M U S T ^
R E A L IZ E , TOHN
T H A T TH IS IS YOUR
S C H O O L , A N P -
Y b u t n
IT 'S ALSO
MY HOME, PAP
A N P - >
Here is a new, and much wore
bitte r "Russell's Paradox",,,
£91
£ 9 £
But today's world has
more serious problems
than my family troubles.
Remember-,
a year ago,
Hitler's troops
entered Austria
to effect the
"Anschluss",,,
,"The
long-expected
"union"of sorts.
N O O O !
The Jews weJIas
as
anyone
sserit
from
the
Nazi
ng
violently
deology
beenha ve
rounded-up
destinations
sent
to
and
off
st
unknown.
^ '-H 't il \- II
o f the firs t acts of the new,
Nazi rulers was to release Schlfck's
murderer.
J finally get to
what I consider
to be the central
question:
£93
£94
£95
Reflect on this, please:
. — if ever in Logic and Alathewiafics,
S' 6 the paragons of certainty, we cannot
have perfect assurances of Reason,
then even /ess can finis be achieved
in the wessy business o f huwian
a f fains - either private,
o r public!
R
Yes, but what
does this tell us
about the War?
Pireatly about the war,,, wiaybe
nothing. 0>ut it tells you a lot about
your stance on it. Or, rather about your
conviction that you are absolutely right
in your views!
Wait! I don't want you to wiisunderstand
wie: even today, I'd def ine wtyself as
a rationalist! Even row, I believe that
Logic is a wiost powerful tool,,,
>h, we understand
you don't think is very
far!
When if cowtes
to talking about humran
life, it certainly isn't! And
when Logic cong&als info
all-encowipassing and
perfect-seeding theories,
then it can actually
becowie a very evil
con trick!
Wittgenstein
£96
Maybe it's
to try another
old triad: Responsibility,
Uusfice... even a sense of
Good us. Evil, i.e. all the
eonoepts wry Viennese
friends considered
"beyond the dignify
of serious
Minds."
Listen: take my stony as a cautionary
tale, a narrative argument against
ready-made solutions. It tells you that
applying formulas is not good enough
— not, th a t is, when you're faced
with really hard problems!
im not e-vadin^. Ar\d
I'm not saying you should join —
or shouldn't. I can't stand in
your shoes and tell you what
to do. /My contribution to
your present dilemma was
my tale. Pe.riod-
h n n n i s l
£97
£98
FI N A L E
.ATREUS
GETS 5 0 0 0 MAP,
T H A T HE SLAUGHTERS
THYESTES' CHILPREN,
ANP SERVES THEM TO
HIM FOR A MEAL!
7 THEN, N£
' THYESTES \
HAS A NEW SON,
AEGISTHUS,
WHO WILL BECOME
THE VEHICLE OF
k HIS REVENGE. .
Y NOW \
ATREUS'S SON,
AGAMEMNON,
AS LEAPER OF THE
EXPEPITION TO
TROY, SACRIFICES
HIS OWN YOUNG
PAUGHTER... J
HOW NICE
ZE S E OLP
BUILPINGS!
AH, YES,
IPANEMA!
...WELL, N l
( "iP H IfiE N IA ",
ACTUALLY!
SO, HIS WIFE
CLYTEMNESTRA,
N PLOTS... >
301
...WITH AEGISTHUS,
NOW HER, LOVER, ANP
TO GETHER THEY RILL
AGAMEMNON AS HE
RETURNS FROM TROY.
IN PLAY TWO, ORESTES,
i.t. AGAMEMNON'S SON,
IS INSTRUOTEP BY T H E
ORESTES IS AWARE OF HIS
TRAGIC PILEMMA: TO
T A K E OR N O T T O T A K E
REVENGE? H E'S AFRAIP
YOU SEE,
HE EVENTUALLY KILLS
CLYTEMNESTRA, HIS MOTHER,
ANP SO T H E F U R IE S , T H E OLP
GOPPESSES OF REVENGE,
a X .a . T H E "S LO O P -TH IR ST Y
H0UNP5", NOW STAR T TO
ASK FOR H IS BLOOP!
WHAT
THEN?
IT IS! APOLLO 'S RITE
'PURIFICATION'
APPEASE T H E FURIES'
WRATH, SO ORESTES ENPS
UP HERE, IN ATHENS, A
SUPPLIANT T O ATHENA,
GOPPESS OF WISPOM.
POESN'T
y NOW, ^
ATHENA MAKES
AN UNPRECEPENTEP
M O V E -F O R A GO P,
ANYWAY: SHE ASKS
T H E CITIZENS OF
ATHENS T O PECIPE T H E
CASE, ESTABLISHING A
COURT OF LAW,
\ WITH J U R Y ! j
...IT'S NOT
THE ENPlNft!
TH E
ENPIN6,
ACTUALLY,
I QUITE
LIKE!
3 0£
IN FACT, I LIKE IT A LOT!
ESPECIALLY RUSSELL
COMING TO W ITTG ENSTEIN 'S
POSITION, THAT ANSWERS TO
REALLY IMPORTANT
QUESTIONS ARE TO BE
CONTEMPLATE? "BEYONP
WORPS"...
BUT
CONTEMPLATING
REALITY - NOT
MAPS!
ISN'T
EXACTLY
A COMEPY,
EH?
"HAPPY" FOR WHOM? CANTOR,
GOING INSANE? GOPEL STARVING
HIMSELF TO PEATH OUT OF PARANOIA?
HILBERT OR RUSSELL AN? THEIR
. PSYCHOTIC SONS? OR FREGE A
WITH -
"THE MEANING ^
IS IN THE ENOINOi!"
YOU SAIP SO YOURSELF!
SO, FOLLOW ^
THE "G UEST"
FOR T EN MORE
. YEARS... >
r ...ANPYOU
GET A BRANP-NEW,
TRIUMPHANT
. FINALE... "
P\..WITH THE N
1 CREATION OF
THE COMPUTER,
WHICH IS THE
"QUEST"'S REAL
\ HERO! y
' YOUR
PROBLEM IS,
SIMPLY, THAT
YOU SEE IT AS
A STORY OF
PEOPLE!
<<
WELL,
STORIES PO
TENP TO BE
"ABOUT
PEOPLE"!
303
SO, CHOOSE T H E R IG H T
PEOPLE! ANP SHOW WHAT
THEY R E A L L Y PIP! ALL
WE LEARN OF THE GREAT
VON NEUMANN IS HE SAIP
" IT 'S OVER" WHEN HE
HEARP GOPEL!
...A L A N
T U R IN G !
r B UT ^
T H E N CAME
TH E "aUEST"'S
T E U N E P R E M IE R
ITS PARSIFAL...
Y HE SAIP, " O K , ^
r WE C A N 'T PROVE '
EVER Y TH ING! SO, LE T'S
SEE W HAT WE C A N
PROVE!" ANP TO PEFINE
PROOF, HE INVENTEP, IN
•1956, A T H E O R E TIC A L
"M ACHINE " WHICH
CONTAINS A L L TH E
L IPEAS OF TH E A
I v COM PUTER!
W ...WHICH, AFTER N J
f TH E WAR, HE ANP
VON NEUMANN, T H E
"G UE S T "'S PROUPEST
l SONS, BROUGHT .
L . TO FU LL LIFE! A
30A
/ P O N 'T AG R EE! TH E
IN TE RN ET IS OUR
PRIME HOPE FOR PEACE,
DEMOCRACY ANP
V FREEPOM! ^ 4
ALSO WEAPONS,
GAMBLING,
AND CHILD
PORNOGRAPHY!
PAMN!
r WHAT
SAY YOU
WE CALL Z E
"ATHENIAN
_ TURY"? .
ANP WHICH
IS M O R E
k R IG H T ? ^
COME HERE
MAN6AAAU!
GRANTED,
TH E R E'S
T W O SIDES
T O IT,
LOOK., T H E "O R E S T E IA " IS
REALLY IN P E R F E C T ANALOGY WITH
T H E "GUEST"! THERE, TYRANNICAL
KINGS RULE — HERE HITLER! THERE
A REVENGE ETH IC ANP O LD-STYLE
GODS - HERE TH E IRRATIONALITY OF
WAR ANP RACIAL HATRED!
ANP IT'S ONLY
ATHENA'S R A T IO N A L IT Y
WITH T H E INNOVATIONS
OF A NEW O E M O C R A T IC
S T A T E T H A T BREAK.
T H E CYCLE OF MURDER
- THERE! ANP HERE,
TURING D EFEATS HITLER
WITH HIS L 0 6 IC A L
M A C H IN E !
305
r Z IS IS A ^
MORE INTERESTING
EPILOGUE...
EE POG EATING ZE
BIRPOF WISPOM,
INSPIREP BY ZE
V FURIES! .
T ...I'M ~
TOO UPSET
FOR A PROPER
GOOPBYE...
HEY, IT'S
STARTING!
...ANP SO,
YOU'LL HEAR
T H E ACTUAL
F IN A L E
FROM
A G REA TE R
A R TIS T ! .
306
NOW,
FURIES,
FINISH W ITH
YOUR CASE!
IF YOU A C & U IT T H IS WAN, WHO
K IL L E P H IS O W N M O T H E R ...
WE APPRESS A PREA P
W A R N IN G , T O A L L YOUR C IT IZ E N S
PIVINE A TH E NA !
...H E R SLO O P
W ILL B E FOREVER
ON YOUR H E A P S !!!
^ AS IF T H E CASE W ASN'T
COMPLICATEP ENOUGH, W ITH O U T
T H E POOR T U R O R S ALSO HAVING
T O T A K E INTO ACCO UNT
T H E FUR IES' RA GE! >
T LIK E T H E ^
AMERICANS IN
WORLP WAR TW O:
IF T H E Y PECIPE TO
H ELP ENGLANP,
T H EY G E T H IT L E R 'S
K RAGE! >
NO SUCH TH ING AS
AN "EASY PROBLEM"!
HAVE BOTH
SIPES SAIP
ENOUGH?
ANP NOW
ATHENIAN TUPGES
CAST YOUR. VOTES!
I'VE SPOKEN:
I KILLEP IN
TUSTIFIEP
REVENGE.
r WE HAVE! "
THE MURPERER
IS CONPEMNEP
BY THE ANCIENT
LAW OF
BLOOP! A
0 PARK M O T H E R ,
ARE YOU WATCHING THIS?
EITHER OUR
HONO UR THRIVES
OR WE ARE
FINISHEP... a
0 NIGHT! ^
GUIPE THE
JURY TOWARP
ANCIENT
^ LAW! a
r OHGOPS...
HOW WILL THEY
VOTE? THEIR
ACT DEGIPES
WHETHER
I LIVE
OR
V p ie ! y
NOW, COUNT
THE BALLOTS WITH
GREAT CARE!
w LET'S SEE
HOW ZE "WISE ATHENIANS'
WILL THINK, TO TUPGE ZE
. MOTHER-KILLER!
r WHAT AN
AMAZINPi WORK
OF ART!
"THINKING"
IS NOT MUCH
HELP HERE.
...YES, WHO KILLEP
A HUSBANP-KILLER
FOR KILLING A
PAUGHTER-KILLER!
divine ath ena
TH E COUNT IS OVER!
THE NUMBERS OF
THE VOTES ARE
. E Q U A L ! >
IT IS TH EN M Y
TA SK T O GIVE A
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T O GIVE M Y VOTE
TO ORESTES!
0 VOU... YOU...
YOUNG GODS!
0 GODDESS,
YOU'VE
SAVED M E !
HE IS
A C Q U IT T E D !
O O O H H H ,
r YOU'VE ^
R A V A G E D
OUR ANCIENT
LAWS! a
A A R R F ... TH EY 'VE
DISHONOURED US!
A A A A R R G G G G H H H !
I SCREAM, IN
V PA IN !!! ^
DAUGHTERS
OF THE NIGHT
THEY HAVE
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GRRR
ATHENIANS,
MY HATE-FILLED
ANGER TURNS
A G A IN ST
k YOU!!!
AND ZIS
ATHENA IS
GODDESS OF
WISDOM?
...AND
ACQUITS Z IS
MURDERER?
\ I V
FURIES, HOLP NOT THIS TRIAL
IN CONTEMPT! YOU'VE N O T BEEN
PISHONOUREP- SEEK N O REVENGE!
WELL, TO MAKE A
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YOU HAVE TO MAKE A
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SOMEWHERE!
SHE'LL
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NEW "AXIOMS",
THOUGH..,
A A R R R F FF W
7 GNNN...
THE
PLAY'S NOT
FINISHEP
YET!
ATHENS M O C K E D U S !
OUR ANCIENT WISPOM
IS D IS G RA C E D !!!
r LL BEAR WITH YOUR ANGER, FOR H
YOU ARE OLPER, ANP THUS WISER.., M
BUT PON'T LET IT M IS L E A D YOU7
■ T H V INSTEAP, HEAR MY OFFER:
■ ' MAKE ATHENS YOUR HOME
I PROMISE, MY CITIZENS
WILL R E V ER E YOU! "
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"R E V E R E " '
US? THEIR JURY
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^ GOOP! ^
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OF GIVING THE
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A VOICE.
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TO MATCH WHAT IT TOOK.?
WHAT CAN REPLACE TH E
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IF THERE'S A MAN OR WOMAN
WHO'S NEVER FELT YOUR ANGER
THEY'RE TOTALLY IG N O R A NT 0 F
LIFE'S TRUE NATURE!
r so, STAY, "
ENRICHING
MV WISPOM WITH
YOURS! STAY ANP
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L WITH ME! .
SHE IS
NOT 2E OWL,
215 ATHENA.,
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YOU ANP I SHALL BE VENERATEP
SIPE BY SIPE... ANP SIPE BY SIPE WE'LL
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O OO HN H...
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I FEEL
MY RAGE
PIMINISH,
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r I SAY WE ^
BETTER TA K E
HER OFFER!
IT IS AGREEP, THEN,
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WE, FURIES, A C C E P T
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TO ^
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. MUST,,.
...ALSO
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OF U NEX PEC TE D L U C K !
REJOICE REJOICE!
REJOICE, YOU HAPPY CITIZENS,
WHO LOVE TRUE WISDOM!!!
p c
Many thanks for their help to our friends
Aliki Chapple, Doukas Kapantais, Avraam Kawa,
Margaret Metzger, Apostolia Papadamaki, Dimitris Sivrikozis,
Chloe Theodoropoulou, Panagiotis Yiannopoulos
final_BOOK_bloomsbury_USA:Layout 3 4/22/09 12:56 PM Page 314
315
Logicomix and reality
Logicomix was inspired by the story of the quest for the foundations of
mathematics, whose most intense phase lasted from the last decades of the
19
th
century to the eruption of the Second World War. Yet, despite the fact that
its characters are mostly real persons, our book is definitely not ― nor does
it want to be ― a work of history. it is ― and wants to be ― a graphic novel.
Particularly in our reconstruction of Bertrand russell's life, we've had to
wander through an immense amount of material, to select, reduce, simplify,
interpret and, very often, invent. Also, though our major characters are
based as closely as possible on their real-life counterparts, we have on more
than one occasion departed from factual detail, in order to give our
narrative greater coherence and depth. most of these deviations consist in
inventing meetings for which there is no historical evidence — or even, in some
cases, where there is evidence that they did not occur. But these imagined
meetings are always based on the actual intellectual interaction of the
thinkers involved, conducted in reality either through correspondence or
publications.
A few examples of such deviations from fact: from the existing evidence, or
lack thereof, it is safe to assume that russell never met Frege or Cantor in
the flesh; there are no indications that he was present in hilbert's seminal
1900 lecture on the "Problems of mathematics", although he was certainly in
Paris a few days earlier, attending the Congress of Philosophy, where he met
Peano; there is no evidence whatsoever that he was in the audience during
go
..
del's "incompleteness" talk ― he probably wasn't and hilbert certainly
wasn't, though Von Neumann certainly was, and did say "it's all over" right
after. Furthermore, russell couldn't have visited Frege right after this talk,
as the latter had been dead for six years. And although the timing of
Frege's rabid anti-Semitic diatribes is incongruous in our book, it is totally
true that he wrote them a few years earlier.
Historically keen readers can have fun locating many more such deviations
from fact. For our part, we take comfort in the words of the painter
domi
΄
nikos Theotoko
΄
poulos (better known as "el greco") explaining
the freedoms he took in his painting "Storm over Toledo":
final_BOOK_bloomsbury_USA:Layout 3 4/22/09 12:56 PM Page 315
316
I found it necessary to reduce the size of the hospital of don Juan
Tavera, not just because it covered the gate of Bisagra, but also
because its dome came up too high, passing the city's skyline.
And so, since i've made it smaller and moved it, i think it is better
to show its fac,ade, rather than its other sides. As for its actual
position in the city, you can see it in the map.
Still, we must add this: apart from the simplification that was necessary to
accommodate it into a narrative work of this kind, we have not taken any
liberties with the content of the great adventure of ideas which forms our
main plot, neither with its central vision, its concepts, nor ― even more
importantly — with the philosophical, existential and emotional struggles
which are inextricably bound with it.
final_BOOK_bloomsbury_USA:Layout 3 4/22/09 12:56 PM Page 316
Notebook
final_BOOK_bloomsbury_USA:Layout 3 4/22/09 12:56 PM Page 317
The following notes are by no means necessary for the enjoyment of Logicomix, but may give
additional information on persons and ideas. A name or term in blue indicates that it also
has its own entry, while italics, when not used for emphasis, indicate technical terms.
Aeschylus One of the three great Greek tragedians,
the precursor of Sophocles and Euripides, Aeschylus
is the creator of tragedy as we know it. He introduced
a second actor into the earlier dramatic form, which
only used a protagonist and the chorus, thus also
inventing the technique of dramatic dialogue. Born
in
525 BCE in Eleusis, near Athens, he fought against
the invading armies of Darius at Marathon (490 BCE)
and Xerxes at Salamis (480 BCE), the latter battle
also providing the subject matter of his earliest extant play, the Persians
(first produced in 472 BCE). The titles of seventy-nine of his plays are known
to us, but only seven of these have survived in their totality, three of which
constitute the Oresteia trilogy.
Algorithm A methodical, step-by-step procedure
described in terms of totally unambiguous instructions,
which starts at a specified initial condition and eventually
terminates with the desired outcome. Though there is no
reason why a well-written cooking recipe, or the instructions
for finding a certain geographical location or address
cannot be called algorithms, the term originated in
mathematics, where it is still mostly used. The word
"algorithm" comes from a European transcription of the
name of the 9
th
century astronomer and mathematician
Al Khwarizmi of Baghdad, who catalogued and championed
these methods, having invented many of them. His compendium
of algorithms, the Hisab al-jabr w'al-muqabala, is generally considered to be the
first algebraic treatise, the very words al-jabr in it also providing the root for
our word "algebra". An example of a simple mathematical algorithm is the
method we learn in elementary school for adding two integers: "write the two
numbers one under the other with their rightmost digits justified to the right;
add their last digits; if the sum is less than 10, write that number right under
the other two; if it is greater than 10,write the second digit of the sum right
under the other two, and add the first digit to the sum of the digits immediately
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to the left ..." and so on. Probably the earliest sophisticated Western algorithm
is the one given in Euclid's Elements for computing the greatest common divisor
of two non-negative numbers. Algorithms gained prominence in the West in the
15
th
century with the introduction of the decimal system, which, in stark contrast
with the roman numerical system, was amenable to fast calculations, such as the
one described above. Numerical algorithms played a central part in the scientific
and technological revolutions. Today, algorithms are usually coded in advanced
notations called programming languages. They are often transmitted over the
internet, and constitute the software that is the workhorse, platform, and
backbone of computers and the internet.
Aristotle Born in 384 BCe, in Stageira, Chalcidice,
Aristotle is, with Plato, the most influential of greek
philosophers. After he left Plato's Academy, Aristotle
developed his own philosophy, which departed from his
teacher's in its emphasis on the systematic observation
of reality and the attempt to shape general, inductive
laws. Perhaps his most lasting contribution is the
systematization and exposition of logic in a series
of works which later commentators edited collectively
as the Organon ("instrument" or "tool"). The books comprising the Organon,
i.e. The Categories, On Interpretation, The Prior Analytics, The Posterior Analytics,
The Topics and the Sophistical Refutations formed the core of the canon
of the study of logic until the 19
th
century. At the heart of Aristotle's logic
is the combination of non-ambiguous statements in syllogisms to create new
statements, different from the original but following necessarily from them.
Aristotle also had a huge and lasting influence on mathematics, mainly through
his emphasis on the notion of first principles from which any logical investigation
must begin. it was this notion that found its mathematical incarnation in
Euclid's concept of the axioms from which every theory has to begin. Aristotle
died in
322 BCe.
Athena The ancient greek goddess of wisdom,
as well as of the arts and the city. Athena sprang
in full armour from the head of Zeus, father of the
gods, whose favourite child she became. Athena was
the patron goddess of ancient Athens and greatly
beloved of the Athenians, to whom, according
to legend, she gave the gift of the olive tree.
The Parthenon, in the centre of the Acropolis,
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is a temple to her ― the word comes from parthenos, meaning "virgin".
Athena's role in Aeschylus' trilogy, the Ore steia, gives her a central role in
the origin myth for the Athenian democratic invention of trial by jury, a
system based on reason, as opposed to the older ones, where juridical
authority emanated from a ruler's absolute power.
Axiom Since the time of Euclid, who was working in the wake of Aristotle's
philosophy of logic, mathematicians agree that a workable theory
must rest on some (few) agreed-upon first principles that don't require
proof. This is a logical necessity if one wants to avoid, on the one hand,
infinite regression (endlessly having to base something on something
else) and, on the other, circuitous thinking (constructing proofs
for statements which, however indirectly, assume the original statement
to be true in the first place). Up to the
19
t
h
century, axioms were
generally considered to be self-evident truths about the world, a view more
or less still valid in Frege's idea of axioms as the reflection of an ulterior
logical reality. After Hilbert, however, and under the influence of
the mathematico-philosophical school of formalism, which developed
from his ideas, axioms came to be seen as existing independently
of any outside reality, the only requirements of an axiomatic system
being: for the individual axioms their grammatical correctness
(in other words, their being well-formed according to the rules
of the logical language in which they are expressed), and independence
(their not being derivable from the other axioms of the particular
theory); and, for the whole set of axioms, its internal consistency (not
containing axioms which contradict one another).
Boole, George Born in 1815, Boole was a largely
self-taught mathematician who later became
a professor of mathematics and logic at Queen's
College in Cork, ireland. his great contribution
to mathematics is in the field of logic. in his book
An Investigation of the Laws of Thought, Boole
developed the idea that logical propositions can be
expressed in a purely symboliclanguage which
allows them to be manipulated by operations,
similar to the operations of elementary
arithmetic. At the heart of Boole's work is the idea
of a propositional calculus, constructed somewhat as Leibniz imagined it. The
"Boolean search" on the internet, involving use of the logical connectives "and",
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"or" and "not", can be traced directly back to his ideas. Yet, despite the
great value of his work in mathematizing logical arguments, Boole did not
offer any great insights into the study of logicitself, having worked wholly
within Aristotle's classical model. in Boole's system, symbols such as x and Y
(essentially they are variables that can take only the two values 0 and 1) are
joined via the three connectives mentioned above, as well as the "implies"
connective envisaged by Aristotle. (interestingly, the Stoic Chrysippus had
already identified these connectives in the 3rd century BCe.) The application
of algebraic identities, such as the three below, allow a logician to simplify
logical expressions and deduce useful conclusions from them:
(x or Y) = (Y or x)
not (not x) = x
not (x and Y) = (not x) or (not Y)
What thislogical formalism is lacking is the ability to express semantic
connections between propositions. So, for example, there is no way
to denotein the above that x and Y may stand for the two propositions
"Plato is older than Socrates" and "Socrates is older than Plato."
This weakness is remedied in the predicate calculus. Boole died in
1864.
Cantor, Georg Born in 1845, Cantor studied
under some of the greatest mathematicians of his
time, including richard dedekind and Karl
Weierstrass. he spent the greatest part of his
career teaching at the University of halle, where he
wrote his seminal papers demonstrating the great
power of the ideas of set theory. his most famous
theorem is that the set of so-called real numbers
(all the numbers on the number line, i.e. the natural
numbers 1, 2, 3... etc., together with the decimals,
including
0 and the negatives) is uncountable, in other
words cannot be put into a one-to-one
correspondence with the whole numbers 1, 2, 3,... etc. on the contrary, as
Cantor had already proved, the set of all rational numbers, i.e. all fractions
of natural numbers, such as 2/3 or 11/476, is countable and can be put in
such a correspondence. As both countable and non-countable sets have an
infinity of elements, Cantor's results essentially proved that there are
various, mutually exclusive kinds of infinity. As his theorems were extremely
counter-intuitive and thus totally unexpected, they created much skepticism
about set theory in the mathematical community.
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One of Cantor's teachers, the great mathematician Leopold Kronecker, as well
as the mathematical giant Henri Poincare
΄
were strongly critical of sets, though
the other mathematical giant of that time, David Hilbert,was one of Cantor's
greatest supporters. The identification of two distinct 'sizes' of infinity in the set
of real numbers, a smaller and a larger one, ushered in the question of whether
there could exist a third kind: could there be a subset of the real numbers that is
neither countable nor can be put in one-to-one correspondence with the reals?
Cantor conjectured that none exists, a guess ever since called "the Continuum
hypothesis" ― the Continuum being another name for the number line. Cantor
worked towards a proof of the Continuum hypothesis for many years, but
never achieved it. in
1940, Kurt Go
..
del proved that the Continuum hypothesis is
consistent with the prevailing axiomatic system of set theory (which does not
amount to a proof of it). in 1963, the young American mathematician Paul Cohen
proved that it is independent of it, i.e. that no real proof of the hypothesis can
be established from it, or, alternatively, that the axioms of set theory are
consistent with the hypothesis being either true or false. This discovery
earned Cohen a Fields medal, a distinction often called "the Nobel Prize of
mathematics". Cantor suffered from severe emotional problems and was
repeatedly hospitalized with a diagnosis of melancholia, which certain historians
of mathematics have ascribed to the hostile reactions of some mathematicians
to set theory, and others to the constant anxiety resulting from his fruitless
attempt to prove the Continuum hypothesis. in the last decades of his life
Cantor did no mathematical work, but wrote extensively trying to substantiate
two strange theories: a) that the plays of Shakespeare were in fact written by
the elizabethan philosopher Sir Francis Bacon, and b) that Christ was the
natural son of Joseph of Arimathea. The second of these is a basic component
of many variations of the holy grail legend, and a standard part of esoteric
lore. Cantor died in a mental asylum, where he had been interned against his
will, in 1918.
Euclid Born around 325 BCe, euclid is the earliest
greek mathematician whose work is extant in the
form in which he actually gave it ― theorems of
earlier mathematicians survive only as transcribed
by others. he lived and worked in Alexandria, where
he was associated with the great Library. his opus
magnum, the Elements, has been a best-seller for
twenty-three centuries, and is the book with the most editions in the Western
world, after the Bible. Though many of the theorems appearing in it are
probably not euclid's own discoveries, the work of compilation, classification
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and presentation of the existing mathematics of his day is totally his own. The
Elements is a majestic conceptual edifice which, inspired by Aristotle and his
work on logic, starts from definitions and first principles, the axioms
(aiteˆmata ― literally "requests"― in euclid's original greek) and then proceeds
to arrive at all the theorems through rigorous proof. Though later students of
logic, especially at the time of the quest for the foundations of mathematics
and after, have criticized euclid for relying too much on geometric insight or
taking many more things for granted than his axioms, the influence of the
Elements has been colossal, and it is rightly considered to be the fountainhead
of the mathematical method. euclid died around 265 BCe.
Foundations of Mathematics Since the time
of Pythagoras, mathematicians have wondered
about the nature of mathematical truth, the
ontology of mathematical entities and the reasons
for the validity of proof and, more generally,
mathematical knowledge. From the enlightenment
until the middle of the 19
th
century, the prevailing
scientific ideology saw mathematics as the only
way of reaching a truth that is final, absolute and
totally independent of the human mind's capacity
to understand it. The basic notions of mathematics were thought to reflect
essential properties of the cosmos and the theorems to be the truths
of a higher reality. This absolute faith in mathematics is reflected in the
crowning of the discipline as the "Queen of the Sciences", a title whose previous
holder, significantly, was theology. This view is usually termed mathematical
Platonism, having its roots in the views of Plato ― and, at least partly,
Pythagoras before him ― on the transcendent ideas (eideˆ ). Yet, in the 19
th
century this traditional belief was undermined in the minds of some people
and eventually led to a serious foundational crisis in mathematics. The first
of the discoveries which caused this loss of faith, dating from the time of
the renaissance, was that of the imaginary numbers (i.e. those involving the
square root of minus one). But in the 19
th
century the appearance of non-
Euclidean geometries strengthened the arguments against the "self-evident"
truth of the axioms. The most troublesome of all mathematical concepts,
though, was that of infinity. Problems concerning the mathematical handling
of the infinite had first been alluded to by Zeno, in his paradoxes, resurfaced
with the invention of the calculus in the 18
th
century and the counterintuitive
and ill-defined concept of an infinitesimal, and peaked in the last two decades
of the 19
th
century, most especiallywith set theory and Georg Cantor's
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results on infinite sets. The problems that came to the surface via set
theory ― chief among them Russell's Paradox ― culminated in severe doubts
about "self-evident" truths and thus, indirectly, about the value of all
mathematical knowledge. It was principally the wish to overcome these doubts
that fuelled the quest for secure foundations. The "Program" proclaimed
by David Hilbert in the early 1920's bearing his name, expresses
the most optimistic version of the foundational dream: the creation
of a formal system for all mathematics, also containing a proof that this
axiomatization is consistent (i.e. can lead to no contradictions), complete
(leaves no unprovable truths) and decidable (one is able to decide in every
occasion whether a formula follows from the axioms or not, through the
application of a set of algorithms.)
Frege, Gottlob Born in 1848, Frege spent the
greatest part of his mathematical life as a professor
at the University of Jena. He is generally considered
to be the father of modern logic, whose notation and
method he expounded first in his Begriffsschrift (which
literally translates from the German as "concept
script"), published in 1879.In it, Frege departed from
the earlier logiciansworking in the wake of Aristotle,
by explicitly introducing the notion of variable in
logical statements. In the place of the older type of
statements like "Socrates is a man", he introduced
propositions like "x is a man", propositions that can
be true or false according to the value given to x ― this particular one,
for example, is true if x is equal to "Alecos" but false if it's "Manga". Frege
also invented the notion of quantifiers, the universal (written )which
makes a statement true "for every x"; and the existential (written )
which says that "there exists an x" which makes a statement true.
He later applied his new logical system to the quest for the foundations
of mathematics. His Grundgesetze der Arithmetik (The Basic Laws of
Arithmetic ) is the first great work of the school of logicism, whose central
tenet is that mathematics is essentially a branch of logic. The first volume
of the Grundgesetze was published in
1893 and the second, containing
the addendum on Russell's Paradox, in 1903. Though Frege's logical
symbolism has been abandoned as particularly cumbersome, most of
the basic concepts and methods he invented still form the backbone of logic.
After the Grundgesetze, Frege didn't do any important foundational work.
In the last decades of his life he became increasingly paranoid, writing
Ε
Α
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a series of rabid treatises attacking parliamentary democracy, labour
unions, foreigners and, especially, the Jews, even suggesting "final solutions"
to the "Jewish problem". he died in 1925.
Go
..
del, Kurt He was born in 1
906
in the town
of Bru
..
nn, moravia, then a part of the Austro-
hungarian empire (the city now called Brno,
in the Czech republic). Go
..
del studied mathematics
in Vienna, where he became fascinated with
mathematical logic and the question of the
foundations of mathematics. in his doctoral
dissertation, he advanced Hilbert's Program
by proving his Completeness Theorem,
a result establishing that all valid statements
in Frege's first-order logic can be proved from
a set of simple axioms. in
1931, however, he proved
the Incompleteness Theorem for second-order
logic, i.e. for a logic powerful enough to support arithmetic and equally or
more complex mathematical theories. Go
..
del became one of the youngest
members of the Vienna Circle, though his deeply-ingrained, idealist belief in
the independent, Platonic existence of mathematical reality eventually
alienated him from the other members, who embraced a materialist-
empirical worldview. during the late thirties, Go
..
del was hospitalized twice
for severe melancholia. in 1940, after the Anschluss, i.e. the annexation
of Austria to Nazi germany, he managed to escape the country with his
wife and took the trans-Siberian route to the United States. he became
one of the first members of the institute for Advanced Study
at Princeton, where he spent the rest of his life. his most important
mathematical result from this period is the proof that Cantor's Continuum
hypothesis is consistent with the axioms of set theory (i.e. that it would not
be in contradiction with them, if true). At Princeton, Go
..
del developed a close
friendship with Albert einstein and worked for a while on the theory
of relativity, establishing the mathematical possibility of a non-expanding,
rotating universe, in which time travel can be a physical reality. in later life,
Go
..
del became increasingly paranoid. he died in January
1978, at the
Princeton hospital, where he had been admitted for the treatment
of a non-life-threatening urinary tract problem. The cause of his death
was malnutrition: he refused to eat for fear that the hospital staff
was attempting to poison him.
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Hilbert, David Hilbert was born in 1862
in Ko
..
nigsberg, Prussia (now Kaliningrad, Russia)
and spent the greatest part of his life at the
University of Go
..
ttingen, the world's most
renowned mathematical centre at that time.
He is one of the greatest mathematicians in
history and, with Henri Poincare
΄
, the greatest
of his era. He made important contributions
to many branches of mathematics including
invariant theory, algebraic number theory, functional analysis, the calculus
of variations, the theory of differential equations and more, also pioneering
new methods of proof. In 1899 he published Grundlagen der Geometrie
(Foundations of Geometry), a book which gave geometry a firm basis, with
new axioms, thereinimproving on the work of Euclid. In his famous 1900 talk
at the international Congress of mathematicians, in Paris, he attempted
to give a bird's-eye view of the mathematics of the twentieth century,
by way of twenty-three great open questions. Of these, now renowned
"hilbert's problems", eleven have been fully solved, seven partly, while the
rest ― the eighth, also known as "the riemann hypothesis" is the most
famous of these ― are still unsolved. The Second Problem is the one
demanding a proof of the consistency (the completeness was considered
more or less obvious) of arithmetic ― and it was this that spurred on a lot
of the work on the foundations and logical structure of arithmetic, including
Go
..
del's. In the 1920s, hisideas of the previous decades related to the
foundations of mathematics culminatedin what became known as "hilbert's
Program", i.e. a project to formalize all mathematics on an axiomatic basis,
including a proof that this axiomatization is consistent. Hilbert's battle
cries of "in mathematics there is no ignorabimus" (i.e. no "we shall not know")
and "we must know, we shall know" ― thelatter spoken only a few days before
go
..
del's first announcement of his Incompleteness Theorem — encapsulate
the quintessence of foundational optimism. Though the results of go
..
del,
Alan Turing and Alonzo Church put an end to hilbert's grand ambition,
the Program continued to exert a great influence on logic and foundational
matters, and especially the development of proof theory. Though in outward
appearance and behaviour hilbert gave the impression of a paragon
of normality and mental health, the way he treated his only son, Franz,
raises questions. When the boy was diagnosed with schizophrenia, at age
15,
his father sent him off to an asylum, where he spent the rest of his life.
Hilbert never visited his son. He died in 1943.
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Incompleteness Theorem In 1931, the 25 year-old Kurt Go
..
del proved
two theorems that are sometimes referred to as "the" incompleteness
Theorem ― though occasionally this form is used to denote the first of
these. The completeness of a logical system is the property that every
well-formed (i.e. grammatically correct by the rules of the system)
proposition in it can be proved or disproved from the system's axioms.
Go
..
del's earlier Completeness Theorem shows that there is a simple
such axiomatic system for first-order logic.However, the holy grail
of Hilbert's Program was to create a complete and consistent axiomatic
system that can support arithmetic, i.e. the mathematics of whole numbers.
Such a system would require second-order logic, i.e. a system that is also
able to accept sets as values of variables. Go
..
del shocked the mathematical
world by proving, in his famous paper "on Undecidable Propositions in the
Principia Mathematica and related Systems", that any consistent axiomatic
system for arithmetic, in the form developed in the Principia, must
of necessity be incomplete. More precisely, the first of the two
incompleteness Theorems establishes that in a logical axiomatic system
rich enough to describe properties of the whole numbers and ordinary
arithmetic operations, there will always be propositions that are
grammatically correct by the rules of the system, and moreover true,
but cannot be proven within the system. The second incompleteness
Theorem states that if such a system were to prove its own consistency
it would be inconsistent. This was a new, devastating blow to hilbert's
Program, with its goal that a strong axiomatic system should be equipped
with a proof of its own consistency.
Intuitionism This is the philosophy
of mathematics created by the great dutch
mathematician Luitzen egbertus Jan Brouwer
(1881-1966), though some consider Henri Poincare
΄
,
with his strong belief in the role of intuition in
mathematics, a clear precursor. Intuitionism is
based on the belief that intuition and time are
fundamental to mathematics, which cannot be
made a-temporal or formal in the sense of
Hilbert. Contrary to what logicists like Frege
and Russell thought, Brouwer was convinced
that logic is founded upon mathematics rather
than the other way round. Also, he was totally
against the theorems of Georg Cantor in the theory of sets, considering
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them ill-formed. Time-hallowed logical laws, such as that of the excluded
middle, and mathematical techniques in use since the time of the ancient
greeks, such as the reductio ad absurdum, were put on trial and their
use condemned. In fact, Brouwer believed that allthe theorems making
use of these in their proofs ―where infinite sets of mathematical objects
were concerned — shouldbeexcised from the body of mathematics,
a view that made the brilliant British logician and mathematician
Frank ramsey callintuitionism "mathematical Bolshevism". Although his
logic and mathematics were formalized by his student Arend heyting,
Brouwer remained skeptical towards any such attempt to the end
of his life.
Leibniz, Gottfried This great german
philosopher, mathematician, scientist and
student of logic was born in 1646. He served
in the courts of several german rulers
as diplomat, political advisor and historian,
all the while pursuing his theoretical studies.
He invented the infinitesimal calculus concurrently
with, but independently from, isaac Newton,
also proposing the notation for its operations that is still in use today.
He was a strong proponent of philosophical optimism,with his theory
that our world is the "best of all possible worlds", having been created
by a god who is both loving and almighty. He is considered the most
important logician after Aristotle and before Boole, having envisioned
the calculus ratiocinator. This was a kind of computational propositional
logicthat would enable completely rigorous and rational decision-making
which could eliminate all disagreement among rational (as Leibniz thought
them) human beings. Sadly, Leibniz did not manage to realize this most
coveted of his many projects. He died in 1716.
Logic The term covers a broad spectrum of disciplines ― not unexpectedly,
as it derives from one of the semantically richest greek words, logos,
some of whose meanings are word, speech, thought, reason, ratio, rationality,
and/or concept ― but can perhaps be best described as the study
of methodical thinking, deduction and demonstration. The books of
Aristotle's Organon present an extensive study of the deductive patterns
called syllogisms, which for over two millennia were considered practically
synonymous with logical thinking. Until the middle of the 19th century, logic
was considered a branch of philosophy. But with the advent of Boole and his
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algebra of propositions and, more importantly, Frege and his "concept
script" whichled to a predicate calculus, it increasingly came within
the province of mathematics. The new logic revealed both the underlying
mathematical nature of the subject and its potential role in the creation
of solid foundations of mathematics. The basic claim of the school
in the philosophy of mathematics known as logicism ― the school founded
by Frege, of which Bertrand Russell was one of the primary exponents ―
was that all of mathematics can be reduced to logic or, in other words,
that mathematics is essentiallyabranch of logic. After the years
of the foundational quest, however, and especiallyafter Go
..
del's results,
logic became a well-developed, diversified fieldin the interface between
philosophy and mathematics. In the second half of the 20
t
h
century
it also found unexpected applications in computer science, where it provides
solid foundations for the design and verification of software and hardware,
as well as for databases and artificial intelligence.
Oresteia Written by Aeschylus and first
performed in the theatre of dionysus, in Athens,
two years before the poet's death, in 458 BCe,
it is the only extant trilogy of greek dramas
— although the satirical play Proteus, intended
to be performed after the trilogy, is missing.
In the trilogy's first play, the Agamemnon,
the eponymous hero and leader of the greek
forces in Troy returns a victor to his hometown
of Argos, with the captive prophetess, Cassandra. Though his wife,
Clytemnestra, at first appears to rejoice at his return, she has other plans.
She and her lover, Agamemnon's cousin Aegisthus, murder Agamemnon
and become the new sovereigns of Argos.In the Libation Bearers, the second
play, the chorus of women accompanies Agamemnon's daughter electra
to her father's tomb. The forlorn electra is hoping for revenge, which
she can only carry out with the help of her brother, orestes, who is in exile.
When orestes clandestinely returns to Argos, he and electra plan and
execute the murder of Aegisthus and then, in a highly dramatic scene
in which Clytemnestra bares her breasts before his naked sword, orestes
also kills her, his own mother. The third play, the Eumenides, or "beneficent
ones", is one of the most unusual in the history of drama: all its speaking
parts, apart from that of orestes himself, are taken up by gods or other
supernatural entities. The chorus consists of the Erinyes or Furies, archaic
goddesses of revenge, who chase orestes from the temple at delphi, where
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he has been ritually purified by the god Apollo, to Athens.
In a totally unprecedented move ― for a god anyway ― Athena, the patron
god of Athens, decides to let the citizens of Athens judge orestes' case,
thus giving a mythological origin-story for the democratic innovation
of a court of law, with citizen jury. The trial and its aftermath develop
as shown in our book's finale, though our text is only an approximate
translation, slightly adapted, of Aeschylus' original words.
Peano, Giuseppe Born in 1858, this great
italian mathematician and logician spent the
greatest part of his creative life as a professor
at the University of Turin. Though his ideas were
not as influential as Frege's in the search for the
foundations of mathematics, Peano, like Frege,
created a notation for first-order logic and
a system of axioms for arithmetic, that is still
in use ― in fact, our arithmetic is formally called
Peano arithmetic. He influenced Bertrand Russell
greatly, especially with his logical notation, which was much more user-friendly
than Frege's. Peano believed that all mathematics could be formalized
and expressed in a common, minimal language that originates from just
a few axioms. But when he tried to present his own version of this universal
mathematics in textbook form and use it for teaching, his students
revolted, eventually causing the book's withdrawal. Inspired by his attempts
to unify all mathematics by use of a common logical language, Peano later
created an international auxiliary natural language, for use among people
of different linguistic backgrounds, based on a simplified form of Latin
which he called Latino sine flexione. However, like so many other artificial
international languages, such as Esperanto, Volapu
..
k, Ido ― all of them the
offspring of an overoptimistic age ― Peano's brainchild proved to be
a mere pipe dream. Peano died in 1932.
Poincare
΄
, Henri Born in 1854 in Nancy,
France. Although he studied engineering
at the e
΄
cole Polytechnique and the e
΄
cole de
mines, Poincare
΄
was to become, with David
Hilbert, the greatest mathematician of his
time. He has been called the "last universal
mathematician", i.e. the last one to have
profound knowledge of all the mathematics
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of his time. He made important contributions to many diverse fields
of mathematics, among them differential equations, automorphic functions,
the theory of several complex variables, probability and statistics.
With his Analysis situs he essentially created the major
20
th
century field
of algebraic topology, and his work on the 3-body problem laid the groundwork
for what is now called chaos theory. Despite his many great innovations,
Poincare
΄
was an extremely practical man, involved to the end of his life ―
alongside of his mathematical research — with the most down-to-earth
of affairs, as for example the inspection of mines and an engineering
project to make the Eiffel tower function as a huge antenna broadcasting
time signals to navigators. He was probably the last of the great
mathematicians to adhere to an older conception of mathematics, which
championed a romantic faith in intuition over rigour and formalism.
This stance was made famous by his reaction to the set theory of Georg
Cantor as a "disease, from which mathematics will eventually be cured."
His views on mathematical creation, encapsulated in his saying that "logic
is barren, unless fertilized by intuition," are seen by many as the precursor
of Luitzen Brouwer's school of intuitionism, a theory at the antipodes
of Hilbert's strict formalism. Poincare
΄
died in
1912.
Predicate calculus Often used synonymously with predicate logic
and first-order logic, the predicate calculus is Frege's extension of the
propositional logic developed by Boole.In the predicate calculus, elementary
propositions (or predicates) are composite objects of the form P(a, b, c,...),
where P is a symbol in the language, and a,b,c, etc. are constants
or variables. For example, if "older" is a propositional symbol, "Plato"
is a constant and "x" is a variable, then "older(Plato, x)" is a well-formed
proposition, describing that Plato is older than x. Propositions of this
type can then be combined by Boole's connectives "and", "or", "not" and
"implies" and prefixed by Frege's quantifiers, such as "for all x" (written )
and "there exists y" (written ). Thus, "there exists x older(x, Plato)"
means that there is (at least) one individual who is older than Plato.
Evidently, this is a much more ambitious attempt at creating Leibniz's
calculus ratiocinator than Boole's simpler formal logic. By employing
symbols from various fields of mathematics (such as "
<", "+", and so on)
one can create predicates expressing mathematical statements in this
formal, logically exact language. For example, the theorem in arithmetic
stating that every integer is either odd or even can be written thus:
x y (x=y+y or x=y+y+1)
Ε
Α
Ε
Α
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Rigorously defined, the version of the predicate calculus called
first-order logic employs simple mathematical objects as variables,
whereas in second-order logic variables can also be sets, making possible
statements like "there is a set S". This, more powerful language, can
express all known mathematics. Whether a sentence in the predicate
calculus, first- or second-order, is true or false depends on the model
whereby the sentence is interpreted. Thus, for example, the simple
arithmetical theorem given above is true of the whole numbers in the
ordinary interpretation of "
+", but becomes false if we interpret the
symbol "+" as multiplication. However, some sentences ― called valid ―
are true independently of interpretation, because they embody basic
properties of Boolean connectives and quantifiers. Kurt Go
..
del's
Completene ss Theorem provides a simple, complete axiomatic system
for proving validity in first-order logic.
Principia Mathematica The extremely influential, but highly
controversial, essentiallyunfinished work in which Alfred North Whitehead
and Bertrand Russell attempted to rescue Frege's grand project
to create foundations of mathematics built on logic, in the wake
of the crisis brought on by Russell's Paradox. The title Prin cipia
Mathematica (i.e. "Principles of Mathematics") in itself provoked
controversy, as it is the exact same as that of Newton's greatest work;
many in the British mathematical community thought this choice
to be in bad taste, if not actually blasphemous. The three volumes
of the Principia, published in 1910, 1912 and 1913, were based on a developed
version of Russell's theory of types, the so-called "ramified", which
imposed a hierarchical structure on the objects of set theory. This could
not be made to yield the required results, however, without the addition
of what Russell called an axiom of reducibility, which eventually became
one of the main reasons for negative criticism of the whole work.
Logicians found this axiom extremely counter-intuitive, a far-fetched
and basically artificial method to sweep the very problem it was trying
to solve under the rug. Despite the fact that the Principia fell short of its
authors' immense ambition, it had a huge influence on the shaping
of modern logic, its greatest effect possibly being the inspiration
and context it provided Kurt Go
..
del for his groundbreaking discovery,
the Incompleteness Theorem.
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Proof The process of arriving at the logical
verification of a mathematical or logical statement,
starting from a set of agreed-upon first principles
(these could be either axioms or already proven
statements, deriving from these axioms), and
proceeding by totally unambiguous and unabridged
logical steps or rules of inference. The demonstrations
of geometric propositions in Euclid's Elements were
considered for over two millennia to set the standard
of excellence to which mathematical proof should
aspire. Yet, towards the end of the
19th century
his method came under logical and philosophical
scrutiny and was found to lack, principally, in two
directions: a) in its sense of the logical "obviousness"
of the axioms, and b) in its logical gaps, where
intuition ― which, in euclid's case was mostly visual-
geometric ― took over from strict application
of a formal system of rules. In a sense, Frege's and Russell and Whitehead's
logicist project was developed as a reaction to the imperfections found
in euclid's proofs, as well as all those developed in his wake. The logicists,
as well as the formalists working on the foundations of mathematics,
aimed at a fully developed theory and practice of rigorous proof,
by which arithmetic (as the basis of all mathematics) would begin
from a small number of consistent axioms, and eventuallylead, via proof,
to the full range of truth. Hilbert's seminal question, which he called
the Entscheidungsproblem ("decision problem"), posed in 1928 and answered
seven years later by Alan Turing, is equivalent to the demand for
a totally powerful apparatus of proof, which can provide a provable
or unprovable response to any mathematical statement by virtue
of a rigorous algorithm.
Russell, Bertrand Born in Wales, in 1872, Bertrand
Arthur William, the Third earl russell ― this is
his full name, by virtue of his noble descent — was
the grandson of the important politician Lord John
russell, whose title he eventually inherited. An orphan
at the age of four, russell was raised by his paternal
grandparents, and after his grandfather's death
two years later, exclusively by his grandmother,
Lady russell. He grew up at the family home of
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Pembroke Lodge, in richmond Park, to the west of London. Russell
is now perhaps best known to a wider public for his work in philosophical
exposition. His History of Western Philosophy, published in 1945, remains to
this day a classic of idiosyncratic, yet intelligent and highly readable
exposition of complex ideas. And while his later work as a pro-peace
and anti-nuclear activist also earned him international fame, russell's
greatest contribution is in mathematical logic, ranking him, along with
Aristotle, Boole, Frege and Go
..
del, with history's greatest logicians.
Despite the momentous importance of his work in the establishment
of a scientific logic, its direct influence on Go
..
del's great discoveries,
and the indirect on the Vienna Circle's "scientific worldview" and the
philosophies of logical positivism and logical empiricism, russell's work
in logic essentially ends with the Princip ia Mathematica, the book he
co-authored with Alfred North Whitehead, completed just before he
turned forty. Russell considered the Principia essentially a failure,
as it fell short of his —and the other logicists'— grand ambition,
of founding mathematics securely on logic. Russell married four times
and fathered three children. His first son, John, as well as John's
daughter, were diagnosed as schizophrenics, and the latter committed
suicide. This pathology was very possibly another instance of the streak
of mental illness running in the family, manifest both in russell's uncle
William and his aunt Agatha. During the last decades of his life,
russell gave all his energy to the struggle for nuclear disarmament,
becoming an emblematic figure of pacifism. He died in
1970.
Russell's Paradox discovered in 1901,
as Russell was working on his first book on the
foundations of mathematics, the Principles
of Mathematics (published in 1903), the Paradox,
in the form originally expressed, shows an
essential flaw in Cantor's set theory, developed
from Bolzano's simple concept of a "collection
of elements with a common property".
By the generality of this definition, which Frege
extended to the realm of logic, one can speak
of a "sets of sets" and thus, eventually of the "set of all sets".
Of the elements of this all-encompassing set one defines the property
of "self-inclusiveness", i.e. of a set containing itself as an element. Thus,
for example, the set of all sets is a set (and thus contained in itself),
as is the set of all entries in a list (it can appear as an entry in a list),
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but the set of all numbers is not a number and thus not contained
in itself. By virtue of this property, we can define the "set of all sets
which don't contain themselves", and ask, with the young russell, the
question: "does this set contain itself or not?" See what happens:
if it does contain itself, it follows that it is one of the sets which don't
contain themselves (as this is the property that characterizes elements
of this set) and thus cannot contain itself. But if it doesn't contain itself,
then it does not have the property of not containing itself, and thus
does contain itself. This situation, in which assuming something implies
its negation, and vice versa, is called a paradox. When a paradox,
such as russell's, arises in a theory, it is a sign that one of its basic
premises, definitions or axioms is faulty. Though historically developed
within the context of the theory of sets, russell himself later viewed
his paradox as essentially having to do with self-reference, i.e. with
statements referring to themselves, such as euboulides' "i am now
lying to you."
Self-reference Literally, the quality
of a statement of referring to itself.
However, it is also used more generally
in logic to characterize statements which
include themselves within their scope
of reference, as in the "barber" story used
to explain Russell's Paradox. The barber
lives in a town wherein a law decrees that
"all residents of the town must either shave themselves or be shaved
by the barber." This law is self-referential as the barber, apart from being
"the barber" referred to, is also one of the "residents of the town".
Self-reference has played a seminal role in logic and mathematics,
already from the time of the greeks. From euboulides' self-referential
statements, to Cantor, whose proof of the non-denumerability of the real
numbers relies heavily on a numerical variant of self-reference, to Russell
and his paradox, and to Go
..
del. In fact, Go
..
del proved his Incompleteness
Theorem by creating, in the context of modern logic, a statement that
is quite similar in spirit to that of euboulides, with one crucial difference:
while euboulides states "this statement is false", Go
..
del's ingenious variant
essentially says, in the language of arithmetic, "this statement
is unprovable." Any consistent axiomatic theory in which one can formulate
such a statement must be necessarily incomplete: for either this
statement is false, in which case it is both false and provable,
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contradicting the consistency of the axiomatic system, or true, in which
case it is both true and unprovable, establishing its incompleteness.
Set theory The study of collections of objects united by a common
property ― in some cases this property can be nothing more than the
fact that they are defined to be members of the same set, as for
example in the arbitrarily defined set whose elements are the numbers
2, 3, 8, 134, 579. Sets were first studied by the Czech mathematician Bernard
Bolzano (1781-1848), who also introduced the term Menge ('set') and defined the
notion of a set's cardinality, i.e. of its "size" in a way not directly involving
measurement. Thus, one can speak of two sets having the same cardinality if
their elements can be put in a one-to-one correspondence – without ever needing
to know via the precise number of these elements. This has the great advantage
that it also works for infinite sets, where the notion of number does not apply:
mathematicians don't think of "infinity" as a number. However, some seeming
paradoxes, such as the fact that the whole and the even numbers can be put
into a one-to-one correspondence (just by multiplying each whole number by
2, or dividing each even number by 2), thus showing a subset to have the
same cardinality as the containing set, prevented Bolzano from developing
the theory further. The advanced mathematical discipline of set theory was
arguably born on december 7, 1873, when Georg Cantor wrote to his teacher,
richard dedekind describing his proof of the non-denumerability of the real
numbers (the set of the whole numbers, decimals, zero and the negative
numbers), as opposed to the denumerability of the rationals (all fractions),
which Cantor also proved ― denumerability is defined as a one-to-one
correspondence with the natural numbers 1, 2, 3... etc. The concept of a set
is almost too primitive to merit a mathematical definition, and is practically
impossible to define informally without the use of some synonym (here we
used the word "collection"). It is precisely this "naturalness" of the concept in
Bolzano's and Cantor's work that led to Russell's Paradox. To overcome it,
and to rule out the flawed concept of "the set of all sets" it allowed for, one
has to come up with bottom-up constructions and axioms for sets, as in the
Principi a Mathematica and, later, the system called "
ZFC", from the names
of its two creators, ernst Zermelo and Abraham Fraenkel, and the Axiom of
Choice, a necessary additional axiom that allows the theory to deal with
infinite sets. Set theory is considered by some the most basic branch of
mathematics, as all others can be defined in terms of it. This was the gist
of an over-ambitious project undertaken, from the 1930s onwards, by the
group of brilliant French mathematicians writing under the pen name of
"Nicolas Bourbaki".
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Tractatus Logico-Philosophicus Ludwig Wittgenstein wrote
his seminal philosophical work during W
Wi
, building on his pre-war
notebooks and ideas on logic. It contains his solution of (in his own words)
"all the problems of philosophy", dealing with the world, representation,
and language. Originallycalled Logische-Philosophische Abhandlung
("Logical-Philosophical Treatise"), it was renamed for its english
publication under the influence of g. e. moore, with his predilection for
Latin titles. In the Tractatus, Wittgenstein uses many techniques and ideas
from logic, especially those of Frege and Russell, as well as insights from
totally different philosophical positions, mostly that of Arthur
Schopenhauer. Though publication by the then totally unknown
Wittgenstein was only made possible when russell accepted to write an
introduction, calling the book "an important event in the philosophical
world", the Tractatus was the cause of the two men's falling-out.
Wittgenstein considered russell's ― not altogether appreciative ―
introduction to his work to be fraught with misunderstandings and
philosophical errors, while russell saw in the Tractatus the first signs of
Wittgenstein's decline ― as he saw it ― into mysticism. The tight structure
of the book proceeds with seven main propositions, each developed in a
chapter, which are further developed in propositions arranged by a
rather pedantic, and often somewhat confusing, system of numbering.
The first two propositions (1 and 2) expand the positions that "the world
is all that is the case", and that "what is the case" are facts, and
combinations of facts. This is a departure from classical philosophy and
the metaphysics of Aristotle in particular, according to which the world
consists of objects. In the logical language of the Tractatus, objects do figure
within states of affairs, but in complex combinations and relationships with
each other, and not as elementary units. The next two propositions (3 and
4) develop mostly what has been called the picture theory of language,
whereby a "thought is a proposition with sense." Passing here to
representation and language, Wittgenstein delimits thoughts to logical
propositions, but within a context and in reference to the world. This is
perhaps the most subtle part of the book, and also the one which relates
to Wittgenstein's idea of mathematics and logic as machines for
producing tautologies. Propositions
5 and 6 develop the idea that
"propositions are truth functions of elementary propositions", in which
mathematical-symbolic notation is used to explain precisely what a truth
function is. Here Wittgenstein uses logic to define propositions (and thus
language and thought) as the combinations of atomic, or elementary
propositions, combined through Boole's laws of composition. This part of
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the book actually contains the first mention of what is now known as the
"truth table method" for dealing with Boolean functions. The book's final
clause, proposition 7, is: "What we cannot speak of, we must pass over in
silence." (This and other quotes are from the d. F. Pears and B.F.
mcguinness translation.) This last proposition was given two highly
divergent interpretations, the extreme positivist one of the Vienna Circle,
by which what one "cannot speak of" (logically) is, quite literally, non-
sense, and the one that Wittgenstein and others himself later gave, which
russell termed "mystical", according to which what "one cannot speak of"
is the truly important. The Tractatus is one of the most influential and
closely-studied books in Western philosophy. Its influences are legion and
it may have also influenced ― and certainly was vindicated by ― the way
in which computers and databases model the world today.
Turing, Alan Born in London in 1912, this great
British mathematician is generally considered
to be the father of computer science. Turing
contributed to many areas of mathematics,
but is mostly remembered for one of his earliest
results in logic. While a student at Cambridge,
he became fascinated by the foundations
of mathematics and especially the Incompleteness
Theorem of Kurt Go
..
del, which inspired him to
study Hilbert's Entscheidungsproblem ("decision
problem"), a question that had survived Go
..
del's
analysis. The Entscheidungsproblem asks
whether, given a logical system, there is an
algorithm for deciding whether a proposition is
provable within the system or not. Turing's answer was a devastating
"no". To reach this, he first had to define rigorously the notion of
algorithm. his ingenious definition in terms of a theoretical "machine"
with a central control and a tape for memory, input and output,
anticipated in important ways the digital computer and has had, since
then, an enormous influence on computational practice and thought.
Turing machines ― as they are now called — share with today's computers
the key property of universality, in that a machine can carry out any
computational task, provided it is supplied with an appropriate program
for it. Two other mathematicians, Alonzo Church (later Turing's thesis
advisor at Princeton) and emil Post, came up independently, and at about
the same time, with algorithm formalisms that were ultimately shown
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equivalent to Turing's. Yet his formalization had the greatest impact,
mainly because of the extreme simplicity of its basic construction, which
can, nevertheless, achieve extremely complex results. The work of Turing
― as well as that of the others mentioned — on algorithms and methods
for the general solvability of problems, is an obvious outgrowth of the
foundational quest and thus, in a sense, its culmination. During World
War ii, Turing presided over the design and construction of two series of
electronic computers, the "Bombe" and the "Colossus". These were used
successfully ― and crucially for the war effort ― for breaking several
german cryptographic codes, including the notoriously hard "enigma"
of the german navy. After the war, Turing worked in the fledgling British
computer industry, did important work in biology and founded the field
of artificial intelligence by proposing what became known as the Turing
test, a method for determining whether an artifact "can think". Always
interested in sports and games — he was an accomplished long-distance
runner — Turing was the first to develop ideas for a chess-playing program,
making mastery in the game one of the goals towards which the designers of
intelligent machines should strive. In 1952 he was prosecuted on account of
his homosexuality, then a punishable offense in Britain. As an alternative
to a jail sentence, he agreed to undergo an experimental "treatment" with
estrogens, which probably caused the severe depression which led him to
take his own life, in 1954.
Vienna Circle A group of philosophers
and philosophically-minded scientists,
who met in Vienna between 1924 and 1936.
Their main aim was two-fold: to build
a strong empiricist philosophy using the
insights into scientific methodology
garnered from recent advances in logic,
mathematics and physics, and to apply
the methodology of the physical sciences to the social. The scientifically-
trained philosopher of science moritz Schlick is generally recognized
to be the group's leader. Some of the most prominent members were:
the mathematicians hans hahn, olga hahn-Neurath, gustav Bergmann,
Karl menger, and Kurt Go
..
del for a short period of time; the physicist
Philipp Frank; the social scientist otto Neurath and the philosophers
Viktor Kraft and rudolf Carnap. The group met informally Thursday
evenings at Vienna's "Cafe
΄
Central", but was later constituted as
a society with public meetings. Despite the group's informal nature,
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the members had a common core of philosophical beliefs, expressed
in a sort of manifesto, titled "The Scientific Conception of the World".
The members of the Circle declared that the work of Frege, Russell
and einstein provided their first inspiration, while the Tractatus
L ogico-Philosophicus of Ludwig Wittgenstein functioned as their direct
model. The philosophies of logical positivism and logical empiricism,
expressing the worldview of the members of the Circle, state that
knowledge comes from experience ― and thus, basically, from scientific
observation and experiment ― developed into theory through logical
analysis and synthesis. Still, following the Tractatus, members of the
Vienna Circle held that logic and mathematics only deal in tautologies,
and thus do not provide knowledge as such, but only one of the tools
for the elaboration of empirical knowledge. According to the worldview
of the Circle, statements that cannot be reduced to experience (such as
theological or ethical pronouncements) cannot be right or wrong, as they
are ― quite literally ― non-sense, having no meaning. The most extreme
version of this tenet, due to Carnap, actually required that for a
statement to be meaningful, its truth or falsity must be verifiable by
an algorithm reducing it to observable truths ― a new incarnation of
Leibniz's "calculemus". Carnap later tried to reconcile this view with
the Incompleteness Theorem. Though the Vienna Circle, in its original
form, was dissolved in 1936, after Schlick's murder by a paranoid
ex-student and Nazi sympathizer, its spirit continued to live on.
Most of its members managed to flee Austria and emigrate to england
and the United States, where they had a major influence on
the development of post-war philosophy.
Von Neumann, John Born in Budapest in 1903
("John" is the anglicized form of the hungarian
"Janos"), von Neumann showed very early signs
of unusual intellectual prowess, being able to do
mental division of
8-digit numbers and converse
in ancient greek by the age of six. He studied
mathematics in Budapest, obtaining a Phd at 22,
meanwhile also working towards a degree in
chemical engineering at the renowned Technical
University of Zu
..
rich, to please his father.
He rapidly became the star mathematician of his generation, legendary
for his penetrating and rapid-fire mathematical genius. Upon attending
the lecture where Go
..
del announced the first Incompleteness Theorem,
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von Neumann was the first to realize the result's import, and did indeed
proclaim "it's all over" after the talk. But he made crucial suggestions
to Go
..
del right after it, and went on to prove the second incompleteness
Theorem ― which however Go
..
del himself had also proven independently
in the meantime. Von Neumann never worked on the foundations of
mathematics again. Possessing a wide-ranging mathematical genius, he
made contributions to many different branches, he has been called "the
last of the great mathematicians", having made great contributions to
many different branches of mathematics, among them set theory,
operator algebras, ergodic theory and statistics. He also did important
work in quantum theory, fluid mechanics and mathematical economics,
being the co-founder (with economist oscar morgenstern) of the field
of game theory. During
WWII, he was one of the brains behind the atomic
bomb, and after it headed the U.S. government committee in charge
of the construction of the hydrogen bomb. Perhaps most important
of all his work, however, was his contribution to the creation of computers.
While he was working as a consultant in the design of one of the first
electronic computers, in 1946, and influenced by Alan Turing's ideas,
von Neumann developed an array of fundamental design principles,
postulating, among others, a central processing unit and separate
memory devices where both data and programs are both stored.
Practically all subsequent computer designs have been based on this basic
model, now known as the von Neumann architecture. Von Neumann went on
to become one of the first great computer scientists, especially excelling
in what now would be called scientific computing, i.e. the use of computers
for scientific research. He died of cancer — possibly the result of his
attendance of thermonuclear tests — in 1957.
Whitehead, Alfred North english
mathematician and philosopher. Born in 1861,
he studied mathematics at Cambridge, where
he also taught for many decades. In
1891
he married evelyn Wade, an irish woman much
younger than himself. Before his intense,
decade-long collaboration with Bertrand Russell
on the Principia Mathem atica, Whitehead
published his book Universal Algebra, an attempt
to study the types of symbolic reasoning
in various algebraic systems from a very modern ― for its time — formal
viewpoint. After russell's abandonment of the Principia, in 1913, Whitehead
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tried to write a fourth volume, on geometry, but never completed it.
The two men had very little interaction after the publication of the
Principia, and Whitehead did not contribute to the
1925, second edition
of the book, having moved on to mathematical physics and later
philosophy. He died in 1
947
.
Wittgenstein, Ludwig Wittgenstein is
considered by many to be the greatest
philosopher of the 20
t
h
century. He was one of the
eight children of industrialist Karl Wittgenstein,
one of Austria's wealthiest and most powerful
men, and a great patron of the arts. Of his
four brothers, three committed suicide in early
manhood, while the fourth, Paul, went on
to become a renowned concert pianist. After
two years of engineering studies, Wittgenstein
developed a strong interest in logic and the foundations of mathematics.
He went to see Frege, who suggested that he go to Cambridge to study
with Russell, a piece of advice Wittgenstein followed. The association deeply
influenced both men, but probably the teacher more than the student.
During his service with the Austro-hungarian army in
WWi, Wittgenstein
won several medals for his valour, his citations underlining his "sang-froid
under fire". He was eventually captured by the enemy and completed his
magnum opus, the Tractatus Logico-P hilosophicus, in an italian prisoners'
camp. After the war he donated the huge fortune left to him by his
father to his three sisters and, having, as he believed and declared,
"solved all the problems of philosophy" with the Tractatus, he worked
as a gardener, architect, and eventually as a teacher in a small village
in Lower Austria. In 1929, possibly inspired by interactions with members
of the Vienna Circle, as well as attending a lecture on the philosophy
of mathematics by Luitzen Brouwer, on intuitionism, Wittgenstein returned
to Cambridge and philosophy. He retracted his earlier work as dogmatic
and went on to create a new, extremely influential philosophical stance
often referred to as "the late Wittgenstein". Unlike the ideas in the
Tractatus,Wittgenstein did not attempt to put his later philosophy
in a systematic treatise, but presented them in a series of more or less
independent remarks. Many of these he saw as forming a book, which
was posthumously published as Philosophical Investigations ― this,
as well as a few books based on his notebooks, or transcripts of lectures
or discussions, are all that we have of his later thought. This is a
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philosophical position of an extreme anti-dogmatic nature, focusing on
language and psychology (what we now call cognitive psychology ), instead
of logic and objective truth, and on fuzzy concepts such as"family
resemblance" and "language games" instead of clear definitions
and propositions. In this later phase, Wittgenstein's thinking is
characterized by a vicious criticism of philosophy as it had been practiced
until then, by others but also himself ― it was for this criticism more than
anything else, that russell was dismissive of his later work, referring
to Wittgenstein's decision to "become a mystic". Most of his negative
criticism of mathematics ― which he increasingly came to view as a purely
practical activity, a craft legitimized only by its use in application —
is contained in transcriptions of his lecture notes at Cambridge.
Of particular interest is the dialogue with one of the attendees
at these lectures, Alan Turing, who strongly disagreed with his ideas
on mathematics. Wittgenstein died in
1951.
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Bibliography
final_BOOK_bloomsbury_USA:Layout 3 4/22/09 12:56 PM Page 345
In preparation for Logicomix we read many books ― in addition to those we had read
earlier, before the idea for the project was even born ― and consulted many more, and
even more articles. Of all these, we mention here very few, chosen either for the wealth
of the information they contain, for their astuteness, profundity and/or synthetic
ability. Clearly, this list represents a personal choice, and nothing more: these are
t
he books that we most liked and found most useful.
Andersson, Stefan. In Quest of Certainty: Bertrand Russell's Search for Certainty
in Religion and Mathematics Up to the Principles of Mathematics (1903).
Stockholm: Almqvist & Wiksell International, 1994.
Davis, Martin. The Universal Computer: The Road from Leibniz to Turing. New
York: W. W. Norton & Company, 2000.
Gray, Jeremy J. The Hilbert Challenge. Oxford: Oxford University Press, 2000.
Janik, Allan, and Stephen Toulmin. Wittgenstein's Vienna. New York: Simon and
Schuster, 1973.
Monk, Ray. Ludwig Wittgenstein: the Duty of Genius. London: Jonathan Cape,
1990.
— Bertrand Russell: the Spirit of Solitude. London: Jonathan Cape, 1996.
— Bertrand Russell: the Ghost of Madness, 1921-1970. London: Jonathan Cape,
2000.
Reid, Constance. Hilbert. Berlin: Springer-Verlag, 1970.
Rota, Gian-Carlo. 1997. "Fine Hall in its Golden Age". In Indiscrete Thoughts, ed.
Fabrizio Palombi, 4-20. Boston: Birkhauser Verlag AG.
Russell, Bertrand. My Philosophical Development. London: George Allen & Unwin,
1959.
— The Autobiography of Bertrand Russell, 3 vols. London: George Allen &
Unwin, 1967-1969.
— Griffin, Nicholas, ed. The Selected Letters of Bertrand Russell. London:
Routledge,
2002.
Scharfstein, Ben-Ami. The Philosophers. Oxford: Oxford University Press,
1980.
Stadler, Friedrich. The Vienna Circle, Studies in the Origins, Development, and
Influence of Logical Empiricism. English translation by Camilla Nielsen.
Vienna: Springer-Verlag,
2001.
Van Heijenoort, Jean. From Frege to Go
..
del. Cambridge: Harvard University
Press,
1967.
Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. (English translation: D. F.
Pears and B. F. McGuinness. London: Routledge and Kegan Paul, 1961.)
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1. Cover
2. Title page
3. Copyright
4. Overture
5. 1. Pembroke Lodge
6. 2. The Sorcerer's Apprentice
7. 3. Wanderjahre
8. 4. Paradoxes
9. Entracte
10. 5. Logico-Philosophical Wars
11. 6. Incompleteness
12. Finale
13. Logicomix and reality
14. Notebook
15. Bibliography
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,225 |
\section{INTRODUCTION}\label{introduction}
Long duration gamma-ray bursts are generally believed to result from
the death of massive stars, and their association with
core-collapse supernovae (SNe of Type Ib/c) has been observed over
the last decade. The first hint for such a connection came with the
discovery of a nearby SN 1998bw in the error circle of GRB 980425
(Galama et al. 1998; Iwamoto et al. 1998) at distance of only about
$40 {\rm Mpc}$. The isotropic gamma-ray energy release is of the
order of only $10^{48} {\rm erg}$ (Galama et al. 1998) and the radio
afterglow modelling suggests an energy of $10^{49}-10^{50} {\rm
erg}$ in a mildly relativistic ejecta (Kulkarni et al. 1998;
Chevalier \& Li 1999). Recently {\it Swift} discovered GRB 060218,
which is the second nearest GRB identified so far (Campana et al.
2006; Cusumano et al. 2006; Mirabal \& Halpern 2006; Sakamoto 2006).
It is also an under-energetic burst with energy in prompt
$\gamma$/X-rays about $5\times10^{49} {\rm erg}$ and is associated
with SN 2006aj. { Another burst, GRB 031203, is a third example of
this group (Sazonov et al. 2004)}.
{If one assumes that GRB 060218-like bursts follow the logN-logP
relationship of high-luminosity GRBs then, as argued by Guetta et
al. (2004), no such burst with redshifts $z < 0.17$ should be
observed by a HETE-like instrument within the next 20 years.
Therefore, the unexpected discovery of GRB 060218 may suggest that
these objects form} a different new class of GRBs from the
conventional high-luminosity GRBs (Soderberg et al. 2006; Pian et
al. 2006; Cobb et al. 2006; Liang et al. 2006; Guetta \& Della
Valle 2007; Dai 2008), although what distinguishes such
low-luminosity GRBs (LL-GRBs) from the conventional
high-luminosity GRBs (HL-GRBs) remains unknown. Their rate of
occurrence may be one order of magnitude higher than that of the
typical ones (e.g. Soderberg et al. 2008). They are rarely
recorded because such { intrinsically dim GRBs can only be
detected from relatively short distances} with present gamma-ray
instruments.
The "compactness problem" of GRBs requires that the outflow of
normal GRBs should be highly relativistic with a bulk Lorentz factor
of $\Gamma_0\ga 100$ (Baring \& Harding 1997; Lithwick \& Sari
2001). The low-luminosity and softer spectra of low-luminosity GRBs
relax this constraint on the bulk Lorentz factor. It was suggested
that the softer spectrum and low energetics of GRB060218 (or
classified as X-ray flash 060218) may indicate a somewhat lower
Lorentz factor of the order of $\sim 10$ (e.g. Mazzali 2006; Fan et
al. 2006; Toma et al. 2006).
An alternative possibility is that low-luminosity GRBs are driven
by a trans-relativistic outflow with $\Gamma_0\simeq2$ (Waxman 2004;
Waxman, M\'esz\'aros \& Campana, 2007; Wang, Li, Waxman \&
M\'esz\'aros 2007; Ando \& M\'esz\'aros 2008). The flat light
curve of the X-ray afterglow of the nearest GRB, GRB 980425/SN
1998bw, up to 100 days after the burst, has been argued to
result from the coasting phase of a mildly relativistic shell
with an energy of a few times $10^{49}$ erg (Waxman 2004). From the
thermal energy density in the prompt emission of
GRB060218/SN2006aj, Campana et al. (2006) inferred that the shell
driving the radiation-dominated shock in GRB 060218/SN 2006aj must
be mildly relativistic. This trans-relativistic shock could be
driven by the outermost parts of the envelope that get accelerated
to a mildly relativistic velocity when the supernova shock
accelerates in the density gradient of the envelope of the
supernova progenitor (Colgate 1974; Matzner \& McKee 1999; Tan et
al. 2001), or it could be due to a choked relativistic jet propagating
through the progenitor (Wang et al. 2007).
In this paper, we investigate the high energy afterglow emission
from low luminosity GRBs for both {relativistic and
trans-relativistic} models and explore whether the Fermi Gamma-ray
Space Telescope can distinguish between these two models with future
observations of low-luminosity GRBs. Since photons from the
underlying supernova are an important seed photon source for
inverse-Compton (IC) scattering, we consider both synchrotron
self-inverse Compton (SSC) and external IC scattering due to
supernova photons (denoted by SNIC hereafter). At early times
(within a few days after the burst), a UV-optical SN component was
recently detected from SN2006aj and SN2008D (Campana et al. 2006;
Soderberg et al. 2008), which has been interpreted as the cooling SN
envelope emission after being heated by the radiation-dominated
shock\footnote{Although there is a disagreement on the origin of the
early UV-optical emission from SN2006aj, an agreement has been
reached for that of SN2008D (Waxman et al. 2007; Soderberg et al.
2008; Chevalier \& Fransson 2008).} (Waxman et al. 2007; Soderberg
et al. 2008; Chevalier \& Fransson 2008). In our calculation, we
take into account this seed photon source in addition to the
late-time supernova emission, which peaks after ten days.
Recently, Ando \& Meszaros (2008) discussed the broadband emission
from SSC and SNIC for a trans-relativistic ejecta in a
low-luminosity GRB at a particular time-- the ejecta deceleration
time. Here we study the time evolution of the high-energy
gamma-ray emission resulted from such IC processes and consider
both the trans-relativistic ejecta and the highly relativistic
ejecta scenarios for low-luminosity GRBs.
The paper is organized as follows. First, we describe the dynamics
of shock evolution in the two models in $\S$ \ref{models}. In
$\S$3, we present the formula for the calculation of the
inverse-Compton emissivity. For the SNIC emission, we take into
account the anisotropic scattering effect. Then we present the
results of the spectra and light curves of SSC and SNIC emission
for the two different models and explore the detectability of
these components by Fermi Large Area Telescope (LAT) in $\S$ 4.
Finally, we give the conclusions and discussions.
\section{Dynamics and Electron Energy Distribution}\label{models}
{We assume that the two models have the same parameters except for
the initial Lorentz factor of the ejecta. For the latter, we adopt
nominal values of} $\Gamma_0=10$ in the conventional
relativistic ejecta model and $\Gamma_0=2$ in the
trans-relativistic ejecta model, respectively. Note that in the
conventional relativistic ejecta model, even if $\Gamma_0\gg 10$
the dynamics of the blast wave is identical to the case of
$\Gamma_0= 10$ from the time tens of seconds after the burst,
because the blast wave has entered the Blandford-McKee
self-similar phase since then.
We consider a spherical GRB ejecta carrying a total energy of
$E=10^{50}E_{50}\rm{erg}$ expanding into a surrounding wind medium
with density profile $n=Kr^{-2}$, where $K\equiv\dot{M}/(4\pi
m_pv_{\rm w})=3\times10^{35}{\rm cm}^{-1}\dot{m}$ with
$\dot{m}\equiv(\dot{M}/10^{-5}M_{\odot}\rm{yr^{-1}})/(v_{\rm
w}/10^3\rm{kms^{-1}})$. As the circumburst medium is swept up by
the blast wave, the total kinetic energy of the fireball is
(Panaitescu et al. 1998)
\begin{equation}
E_{\rm k}=(\gamma-1)(m_{\rm ej}+m)c^{2}+\gamma U'\label{energy}
\end{equation}
where $\gamma$ is the bulk Lorentz factor of the shell, $m_{\rm ej}$
the ejecta mass, $m$ is the mass of the swept-up medium, and the
comoving internal energy $U'$ can be expressed by
$U'=(\gamma-1)mc^2$, which is suitable for both ultrarelativistic
and Newtonian shocks (Huang et al. 1999). Hereafter superscript
prime represents that the quantities are measured in the comoving
frame of the shell. As usual, we assume that the magnetic field and
the electrons have a fraction $\epsilon_B$ and $\epsilon_e$ of the
internal energy, respectively. Following Eq. (\ref{energy}), the
differential dynamic equation can be derived as (Huang et al. 2000)
\begin{equation}
\frac{d\gamma}{dm}=-\frac{\gamma^{2}-1}{m_{\rm ej}+2\gamma m},
\end{equation}
which describes the overall evolution of the shell from relativistic
phase to non-relativistic phase. The initial value of $\gamma$ is
$\Gamma_0=E/(m_{\rm ej}c^2)$. To obtain the time-dependence of
$\gamma$ one makes use of
\begin{equation}
\frac{dm}{dR}=4\pi R^{2}nm_{\rm p},
\end{equation}
\begin{equation}
\frac{dR}{dt}=\beta c\gamma(\gamma+\sqrt{\gamma^{2}-1}),
\end{equation}
where $t$ is the observer time, $R$ and
$\beta=\sqrt{1-\gamma^{-2}}$ are the radius and velocity of the
shell, respectively.
Solving Eqs. 2, 3, and 4, three dynamic phases can be found: (i)
Coasting phase. The shell does not decelerate significantly until it
arrives at the deceleration radius, $R_{\rm dec}=E/(4\pi
K\Gamma_{0}^{2}m_{\rm p}c^{2})$, where the mass of the swept-up
medium $m$ is comparable to $m_{\rm ej}/\Gamma_{0}$ (Sari \& Piran
1995). The corresponding deceleration time can be calculated from
$t_{\rm dec}\simeq R_{\rm dec}/(2\Gamma_0^2c)$. For representative
parameter values $E_{50}=1$ and $\dot{m}=1$, $R_{\rm
dec}\simeq4.4\times10^{15}\rm{cm}$ and $t_{\rm
dec}\simeq1.8\times10^4\rm{s}$ for a trans-relativistic ejecta with
$\Gamma_0=2$, while $R_{\rm dec}\simeq1.8\times10^{14}\rm{cm}$ and
$t_{\rm dec}\simeq29\rm{s}$ for a conventional highly relativistic
ejecta with $\Gamma_0=10$; (ii) Blandford \& McKee self-similar
phase ($t>t_{\rm dec}$ and $\gamma\ga 2 $), where $\gamma\propto
t^{-1/4}$ and $R\propto t^{1/2}$; (iii) Non-relativistic phase
($\gamma\rightarrow1$), where $\beta\propto t^{-1/3}$ and $R\propto
t^{2/3}$, i.e., the Sedov-von Neumann-Taylor solution applies
(Zel'dovich $\&$ Raizer 2002, p.93, Waxman, 2004).
In the absence of radiation losses, the energy distribution of
shock-accelerated electrons behind the shock is usually assumed to
be a power-law as $dN_{e}/d\gamma_{e}\propto\gamma_{e}^{-p}$. As
the electrons are cooled by synchrotron and IC radiation, the
electron energy distribution becomes a broken power-law, given by
(1) for $\gamma_{e,\rm c}\leq\gamma_{e,\rm m}$,
\begin{equation}
\frac{dN_{e}}{d\gamma_{e}}\propto\left\{\begin{array}{ll}
\gamma_{e}^{-2},&\gamma_{e,\rm c}\leq\gamma_{e}\leq\gamma_{e,\rm m}\\
\gamma_{e}^{-p-1},&\gamma_{e,\rm m}<\gamma_{e}\leq\gamma_{e,\rm
max}\end{array}\right.
\end{equation}
(2) for $\gamma_{e,\rm m}<\gamma_{e,\rm c}\leq\gamma_{e,\rm max}$,
\begin{equation}
\frac{dN_{e}}{d\gamma_{e}}\propto\left\{\begin{array}{ll}
\gamma_{e}^{-p},&\gamma_{e,\rm m}\leq\gamma_{e}\leq\gamma_{e,\rm c}\\
\gamma_{e}^{-p-1},&\gamma_{e,\rm c}<\gamma_{e}\leq\gamma_{e,\rm
max}\end{array}\right.
\end{equation}
which are normalized by the total number of the electrons solved
from the dynamic equations. The minimum, cooling, and maximum
Lorentz factors of electrons are, respectively, given by
\begin{eqnarray}
\gamma_{e,\rm m}&=&\epsilon_{e}\frac{p-2}{p-1}\frac{m_{\rm
p}}{m_{e}}(\gamma-1)=92f_{p1}\epsilon_{e,-0.5}(\gamma-1)
\\
\gamma_{e,\rm c}&=&\frac{6\pi m_{e}c}{(1+Y)\sigma_{T}{B'}^{2}
(\gamma+\sqrt{\gamma^{2}-1})t}\nonumber\\
&&\simeq\frac{1.7\times10^3R_{15}^2} {\epsilon_{B,-3}\dot{m} t_{4} Y
(\gamma+\sqrt{\gamma^2-1})\gamma (\gamma-1)}
\\
\gamma_{e,\rm max}&=&\sqrt{6\pi q_{e}\over\sigma_{T}B'(1+Y)}\\
&&\simeq4.5\times10^7R_{15}\gamma^{-1/4}(\gamma-1)^{-1/4}Y^{-1/2}
\end{eqnarray}
where $f_{p1}=6(p-2)/(p-1)$, $B'$ is the comoving magnetic field
strength and $Y$ is the Compton parameter that is defined as the
ratio of the IC luminosity (including SSC and SNIC) to the
synchrotron luminosity\footnote{At very late times, when
$\gamma_{e,\rm m}$ decreases to be close to a few, $\gamma_{e,\rm
m}=\epsilon_{e}\frac{p-2}{p-1}\frac{m_{\rm p}}{m_{e}}(\gamma-1)+1$
is used (Huang \& Cheng 2003).}.
\section{Inverse-Compton Emission}
The accelerated electrons can be cooled by synchrotron radiation and
inverse Compton (IC) scattering of seed photons (including
synchrotron photons and blackbody photons emitted by the supernova).
Since the IC emissivity on the basis of the Thomson cross section is
inaccurate for high-energy $\gamma$-rays, we use the full
Klein-Nishina cross section instead. Once the electron distribution
and the flux of seed photons ($f'_{\nu'_{\rm s}}$) (the distribution
of which is isotropic) are known, the IC emissivity (at frequency
$\nu'$) of electrons can be calculated by(Blumenthal $\&$ Gould
1970; Yu et al. 2007)
\begin{equation}\label{SSCiso}
{\varepsilon'}^{\rm IC}_{\rm iso}(\nu')=3\sigma_{\rm
T}\int^{\gamma_{e, \rm max}}_{\gamma_{e, \rm min}}d\gamma_{e}
{dN_{e}\over d\gamma_e}\int^{\infty}_{\nu'_{\rm s,\min}}d\nu'_{\rm
s}\frac{\nu'f'_{\nu'_{\rm s}}}{4\gamma_{e}{\nu'}_{\rm
s}^{2}}g(x,y),
\end{equation}
where $\gamma_{\rm e,\min}=\max[\gamma_{\rm e,c}, \gamma_{\rm e,m},
{h\nu'/( m_{\rm e}c^2)}]$, $\nu'_{\rm s,\min}={\nu'm_{\rm
e}c^2/4[\gamma_{\rm e}(\gamma_{\rm e} m_{\rm e}c^2-h\nu')]}$,
$x=4\gamma_{\rm e} h\nu'_{\rm s}/m_{\rm e}c^{2}$,
$y=h\nu'/[x(\gamma_{\rm e} m_{\rm e}c^{2}-h\nu')]$, and
\begin{equation}
g(x,y)=2y\ln y+(1+2y)(1-y)+\frac{1}{2}{x^{2}y^{2}\over
(1+xy)}(1-y).
\end{equation}
In our case, because the radius of the supernova photosphere is
much smaller than that of the GRB shock, supernova seed photons
can be regarded as a point photon source locating at the center in
the comoving frame of the GRB shock. These soft photons from
supernova impinge on the shock region basically along the radial
direction in the rest frame of the shock, so the scatterings
between these photons and the isotropically-distributed electrons
in the shock are anisotropic. For a photon beam penetrating into
the shock region where the electrons are moving isotropically, the
inverse Compton scattering emissivity of the radiation scattered
at an angle $\theta_{\rm SC}$ relative to the direction of the
photon beam in the shock comoving frame is (Aharonian $\&$ Atoyan
1981, Brunetti 2000, Fan et. al. 2008):
\begin{equation}\label{AIC}
\begin{array}{ll}\varepsilon'^{\rm AIC}(\nu',{\rm cos}\theta_{\rm SC})=\frac{3\sigma_{\rm
T}c}{16\pi} \int^{\gamma_{e,\rm max}}_{\gamma_{e,\rm
min}}d\gamma_{\rm
e}\frac{dN_{\rm e}}{d\gamma_{e}}\\
\int^{\infty}_{\nu'_{\rm s,min}}\frac{f'^{\rm SN}_{\nu'_{\rm
s}}d\nu'_{\rm s}}{\gamma_{\rm e}^2\nu'_{\rm s}}
[1+\frac{\xi^{2}}{2(1-\xi)}-\frac{2\xi}{b_{\theta}(1-\xi)}+\frac{2\xi^2}{b_{\theta}^2(1-\xi)^2}]\end{array},
\end{equation}
where $\xi\equiv h\nu'/(\gamma_em_{e}c^2)$, $b_{\theta}=2(1-{\rm
cos}\theta_{\rm SC})\gamma_eh\nu'_{\rm s}/(m_ec^2)$ and
$h\nu'_{\rm s}\ll h\nu'\ll \gamma_e
m_ec^2b_{\theta}/(1+b_{\theta})$. On integration over $\theta_{\rm
SC}$ for whole solid angle (i.e. in the case that the photon
distribution is also isotropic), Eq.(\ref{AIC}) reduces to
Eq.(\ref{SSCiso}), i.e. the usual isotropic inverse Compton
scattering emissivity. In the observer frame, the angle $\theta$
between the injecting photons and scattered photons relates with
the angle $\theta_{\rm SC}$ in the comoving frame by ${\rm
cos}\theta_{\rm SC}=({\rm cos}\theta-\beta)/(1-\beta {\rm
cos}\theta)$, where $\beta$ is the velocity of the GRB shock.
The observed SSC and SNIC flux densities at a frequency $\nu$ are
given respectively by (e.g. Huang et al. 2000, Yu et al. 2007)
\begin{equation}
F_{\nu}^{\rm SSC}=\int^{\pi}_{0}\frac{\varepsilon'^{\rm IC}_{\rm
iso}(\nu/D)}{4\pi D_{\rm L}^2}D^3{\rm sin}\theta d\theta,
\end{equation}
and
\begin{equation}
F_{\nu}^{\rm SNIC}=\int^{\pi}_{0}\frac{\varepsilon'^{\rm
AIC}(\nu/D, {\rm cos}\theta_{\rm SC})}{4\pi D_{\rm L}^2}D^3{\rm
sin}\theta d\theta,
\end{equation}
where $D_{\rm L}$ is the luminosity distance and
$D\equiv[\gamma(1-\beta {\rm cos}\theta)]^{-1}$ is the Doppler
factor. Note that ${\rm cos}\theta_{\rm SC}$ in Eq.(15) can be
transformed to ${\rm cos}\theta$ through ${\rm cos}\theta_{\rm
SC}=({\rm cos}\theta-\beta)/(1-\beta {\rm cos}\theta)$.
\subsection{Seed Photons from the Supernova}\label{seed photons}
For seed photons from the supernova, we consider the contributions
from two components. One is the early thermal UV-optical emission
from the cooling supernova envelope after being heated by the
radiation-dominated shock (Waxman et al. 2007; Chevalier $\&$
Fransson 2008). Such UV-optical emission has been observed recently
by {{\it Swift}} UVOT from SN2006aj (Campana et al. 2006) and
SN2008D (Soderberg et al. 2008). Another component is the supernova
optical emission at later time, powered by the radioactive elements
synthesized in supernovae.
The characterization of the emission from the cooling supernova
envelope is expressed as follows, which is correct at
$t\ga10^2\rm{s}$ (Waxman, M\'{e}sz\'{a}ros, $\&$ Campana 2007):
\begin{equation}\label{rph}
R_{\rm ph}(t)= 3.2\times10^{14}\frac{E_{\rm snej,51}^{0.4}}{(M_{\rm
snej}/M_{\odot})^{0.3}}t_{\rm day}^{0.8}\rm{cm},
\end{equation}
\begin{equation}\label{Tenv}
T_{\rm ph}(t)=2.2\frac{E_{\rm snej,51}^{0.02}}{(M_{\rm
snej}/M_{\odot})^{0.03}}R_{0,12}^{1/4}t_{\rm day}^{-0.5}\rm{eV},
\end{equation}
\begin{equation}\label{Lenv}
L_{\rm ph}(t)=4\pi R_{\rm ph}(t)^{2}\sigma T_{\rm ph}(t)^{4},
\end{equation}
where $R_{\rm ph}(t)$ and $T_{\rm ph}(t)$ are the radius and
temperature of the envelope, $L_{\rm ph}(t)$ is the luminosity from
the photosphere of the cooling envelope, $M_{\rm snej}$ and $E_{\rm
snej}$ are the supernova ejecta mass and energy, and $R_0$ is the
initial stellar radius of the SN progenitor. We take the following
values for supernovae like SN2006aj and SN2008D: $E_{\rm
snej,51}=E_{\rm snej}/(10^{51}\rm{erg})=2$, $M_{\rm
snej}=2M_{\odot}$ and $R_{0,12}=R_{0}/(10^{12}\mbox{cm})=0.3$
(Mazzali et al. 2006; Soderberg et al. 2008). For hypernovae such
as SN1998bw associated with GRB980425, modelling of the SN optical
emission gives a larger kinetic energy and ejecta mass such as
$E_{\rm snej,51}\simeq22$ and $M_{\rm snej}=6M_{\odot}$ for SN1998bw
(Woosley et al. 1999).
The luminosity of the late component is rising before $\sim \rm 10
days$ and then decaying exponentially. For SN2008D, it is found
that the rising is roughly in proportion to $t^{1.6}$ (Soderberg
et. al. 2008), so we can describe the luminosity as
\begin{equation}\label{Lrad}
L_{\rm rad}(t)=\left\{\begin{array}{ll}3\times10^{42}(\frac{t}{10\rm{days}})^{1.6}\rm{erg~ s^{-1}}, &t<10\rm{days},\\
3\times10^{42}\exp(1-\frac{t}{10\rm{days}})\rm{erg~ s^{-1}},
&t\geq10\rm{days}\end{array}\right .
\end{equation}
We assumed that the radiation temperature is approximately
constant, $T_{\rm rad} \simeq 1\rm{eV}$. The bolometric luminosity
of the cooling supernova envelop is dominant over that of the late
component before $t=7.8\times10^5\rm s$.
Through the calculation of the shock dynamic evolution, the radii
of the GRB shock are $R=1.9\times10^{14}$ $\rm{cm}$ at $t=10^3
\rm{s}$, and $R=1.5\times10^{15}\rm cm$ at $t=10^4\rm s$ for the
trans-relativistic ejecta model, and for the conventional ejecta
model $R=1.0\times 10^{15}\rm cm$ at $t=10^3\rm s$ and
$R=4.1\times 10^{15} \rm cm$ at $t=10^4\rm s$. According to
Eq.\ref{rph}, the shell radii for supernova are $9.6\times
10^{12}\rm cm$ at $t=10^3\rm s$ and $6.1\times 10^{13}\rm cm$ at
$t=10^4\rm s$, and the one for hypernova are $1.8\times 10^{13}\rm
cm$ at $t=10^3\rm s$ and $1.1\times 10^{14}\rm cm$ at $t=10^4\rm
s$. Consequently the shell radius is less than ten percents of the
GRB ejecta height, so it is reasonable to approximate that
supernova photons come from a point source at the center and
impinge onto the electrons in the shock from behind.
\subsection{Compton Parameter $Y$} \label{Y_f}
Following Moderski, Sikora $\&$ Bulik (2000) and Sari $\&$ Esin
(2001), we define the Compton parameter $Y$ as
\begin{equation}\label{Y}
Y\equiv\frac{L_{\rm IC}}{L_{\rm
SYN}}=\frac{{u'}_{\gamma}}{{u'}_{B}}=\frac{{u'}_{\rm SYN}+f_{\rm
a}{u'}_{\rm SN}}{{u'}_{B}},
\end{equation}
where
\begin{eqnarray}
{u'}_{\rm SN}&=&\gamma^{-2}\frac{L_{\rm SN}}{4\pi
cR^2}\nonumber\\
&&\simeq65{\rm~erg~cm^{-3}}t_4^{-0.4}R_{15}^{-2}\gamma^{-2},\\
{u'}_{\rm SYN}&=&{\eta\epsilon_e{u'}\over(1+Y)}\nonumber\\
&&\simeq5.4\times10^2{\rm~erg~cm^{-3}}\epsilon_{e,-0.5}\dot{m}R_{15}^{-2}\gamma(\gamma-1)\eta Y^{-1},\\
{u'}_{B}&=&\epsilon_B{u'}\nonumber\\
&&\simeq1.8{\rm~erg~cm^{-3}}\epsilon_{B,-3}\dot{m}R_{15}^{-2}\gamma(\gamma-1),
\end{eqnarray}
are, respectively, the comoving energy densities of the blackbody
supernova seed photons, synchrotron seed photons and magnetic
fields, and $f_{\rm a}$ is the factor accounting for the
suppression of the photon energy density due to the anisotropic
inverse Compton scattering effect ($f_{\rm a}=1$ corresponds to
the isotropic scattering case). Here $L_{\rm SN}=L_{\rm ph}+L_{\rm
rad}$, which is dominated by the luminosity of the cooling envelop
$L_{\rm ph}$, $u'$ is the comoving internal energy density,
$\eta=\eta_{\rm rad}\eta_{\rm KN}$ is the radiation efficiency
where $\eta_{\rm rad}$ is the fraction that the electron's energy
radiated, and $\eta_{\rm KN}$ is the fraction of synchrotron
photons below the KN limit frequency (Nakar 2007). For slow
cooling, $\eta_{\rm rad}=(\gamma_{e,\rm c}/\gamma_{e,\rm
m})^{2-p}$ (Moderski, Sikora $\&$ Bulik 2000), and
\begin{equation}\label{eta_KN_slow}
\eta_{\rm KN}=\left\{
\begin{array}{ll}
0,&\nu'_{\rm KN}(\gamma_{e,\rm c})\leq \nu'_{\rm m}\\
(\frac{\nu'_{\rm KN}(\gamma_{e,\rm c})}{\nu'_{\rm c}})^{(3-p)/2},&\nu'_{\rm m}< \nu'_{\rm KN}(\gamma_{e,\rm c})<\nu'_{\rm c}\\
1,&\nu'_{\rm c}\leq\nu'_{\rm KN}(\gamma_{e,\rm c})\\
\end{array}\right.
\end{equation}
For fast cooling, $\eta_{\rm rad}=1$ and
\begin{equation}\label{eta_KN_fast}
\eta_{\rm KN}=\left\{
\begin{array}{ll}
0,&\nu'_{\rm KN}(\gamma_{e,\rm m})\leq \nu'_{\rm c}\\
(\frac{\nu'_{\rm KN}(\gamma_{e,\rm m})}{\nu'_{\rm m}})^{1/2},&\nu'_{\rm c}< \nu'_{\rm KN}(\gamma_{e,\rm m})<\nu'_{\rm m}\\
1,&\nu'_{\rm m}\leq\nu'_{\rm KN}(\gamma_{e,\rm m})\\
\end{array}\right.
\end{equation}
Solving eq. (\ref{Y}), we get
\begin{equation}\label{Y_eta}
Y={1\over2}\left[\sqrt{4\frac{\eta\epsilon_{e}}{\epsilon_{B}}+\left(1+\frac{f_{\rm
a}{u'}_{\rm SN}}{\epsilon_{B}{u'}}\right)^{2}} +\left(\frac{f_{\rm
a}{u'}_{\rm SN}}{\epsilon_{B}{u'}}-1\right)\right].
\end{equation}
Roughly, the above expression can be simplified in three limiting
cases as follows
\begin{equation}\label{Y_eta}
Y=\left\{
\begin{array}{ll}
f_{\rm a}u'_{\rm SN}/(\epsilon_Bu'),&f_{\rm
a}{u'}_{\rm SN}\gg u'_{\rm SYN}\\
\sqrt{{\eta\epsilon_{e}/\epsilon_{B}}},&u'_{B}\ll f_{\rm a}{u'}_{\rm
SN}\ll u'_{\rm SYN} \\
(\sqrt{4{\eta\epsilon_{e}/\epsilon_{B}}+1}-1)/2,&f_{\rm a}{u'}_{\rm
SN}\ll {\rm min}[{u'}_{\rm SYN},u'_{B}]\\
\end{array}\right.
\end{equation}
In the latter two cases, $Y$ can be treated as a constant when the
electrons are in the fast-cooling regime and $\nu'_{\rm
m}\leq\nu'_{\rm KN}(\gamma_{e,\rm m})$ with $\eta=1$.
\subsection{Pair Production Opacity For High Energy Photons}\label{tao}
High-energy gamma-rays can be attenuated due to interaction with
low-energy photons through the pair production effect. We consider
the pair production opacity in the shock frame due to the absorption
by low-energy photons, which include thermal photons from the
supernova, synchrotron photons, SSC photons and SNIC photons. A high
energy photon of energy $E'_{\gamma,1}$ in the shock frame will
annihilate with a low energy photon of $E'_{\gamma,2}$, provided
that
$E'_{\gamma,1}E'_{\gamma,2}(1-\cos\theta_{12})\geq2(m_ec^{2})^{2}$,
where $\theta_{12}$ is the collision angle of the two annihilation
photons. The pair creation cross section is given by
\begin{equation}
\sigma(E'_{\gamma,1},E'_{\gamma,2})=\frac{1}{2}\pi
r_{0}^{2}(1-\tilde{\beta}^{2})[(3-\tilde{\beta}^{4}){\rm
ln}\frac{1+\tilde{\beta}}{1-\tilde{\beta}}-2\tilde{\beta}(2-\tilde{\beta}^2)],
\end{equation}
where $\tilde{\beta}\equiv
v/c=\sqrt{1-2(m_{e}c^{2})^{2}/[E'_{\gamma,1}E'_{\gamma,2}(1-\cos\theta_{12})}]$
is the velocity of electrons in the center-of-mass frame (Heitler
1954, Stecker, De Jager, $\&$ Salamon, 1992) and $r_0=e^2/(m_ec^2)$
is the classic electron radius. By using the photon distribution in
the shock frame, we obtain the optical depth for high-energy photons
of energy,
\begin{equation}
\tau(E'_{\gamma,1})=\int_{E'_{thr}}^{\infty}\sigma(E'_{\gamma,1},E'_{\gamma,2})n_{\gamma}(E'_{\gamma,2})\frac{R}{\eta_{\rm
R}}dE'_{\gamma,2},
\end{equation}
where $\eta_{\rm R}$ is the shock compressed ratio, which is
$\eta_{\rm R}=4\gamma+3$, and the threshold energy of low energy
photons is $E'_{\rm thr}=2(m_{e}c^{2})^{2}/E'_{\gamma,1}(1-
\cos\theta_{12})$. In the calculation, we estimate the cutoff energy
conservatively by assuming that colliding photons are moving
isotropically in the shock frame. Such a treatment may overestimate
the pair-production opacity because in reality the supernova seed
photons move anisotropically (i.e. moving outward in radial
direction as seen by high-energy photons emitted from the shock at
much larger radii).
The result of the cutoff energy in the observer frame
$E_{\gamma,{\rm cut}}=\gamma E'_{\gamma}(\tau=1)$ is given in Figure
\ref{nucut}. Since the luminosity and peak energy of the supernova
envelope emission decreases with time in general and the shock
radius increases with time, the cutoff energy of the high energy
spectrum increases with time, which is clearly seen in Fig.
\ref{nucut}. From this figure, we can see that the cutoff energy is
above $1\rm{GeV}$ after the starting time of our calculation
($10^{2.5}\rm{s}$) , so we can calculate the light curves at energy
$\sim 1\rm{GeV}$ without considering the opacity. Since the cutoff
energy is large enough, it does not affect the detectability of
Fermi LAT, whose sensitive energy band is
$20\rm{MeV}\sim300\rm{GeV}$.
\section{Results}
\subsection{Spectra and Light Curves: Numerical Results}\label{spectra and light curves}
We first compare the spectra of different IC components for the two
models. The spectra at $t=10^3\rm{s}$ are shown in Figure
\ref{spectra_13}, including the synchrotron emission, the thermal
emission from the supernova or hypernova, the synchrotron
self-Compton emission and the SNIC emission, for parameters
$\epsilon_e=0.1$, $\epsilon_B=0.001$, $p=2.2$, $E=10^{50}\rm{erg}$
and burst distance $D_L=100\rm{Mpc}$. From the spectra we can see
that for the supernova case the SSC emission dominates over the SNIC
emission at energies from $\rm 1 MeV$ to $\rm 100 GeV $ for
conventional relativistic ejecta model, and for trans-relativistic
ejecta model the SSC emission is higher than SNIC emission and the
two components are both important at high energies below the cutoff
energy. For the hypernova case the SSC emission is dominant in the
conventional relativistic ejecta model while the SNIC emission is
dominant in the trans-relativistic ejecta model at the high energy
band. This is because in the case of hypernovae and $\Gamma_0=2$,
the energy density of hupernova photons $u'_{\rm HN}$ (multiplied by
the suppressed factor $f_{\rm a}$ due to the anisotropic scattering)
is significantly higher than the synchrotron radiation density
$u'_{\rm SYN}$.
Figure \ref{lc_sup} shows light curves of SSC and SNIC afterglow
emission at $h\nu=\rm{1 GeV}$ in the two models for
$E=10^{50}\rm{ergs}$, $p=2.2$ and four different sets of parameters
of $\epsilon_e$ and $\epsilon_B$ for the supernova case. In the
conventional relativistic model with $\Gamma_0=10$, a sharp decay
phase of a GeV afterglow is produced during the early hours, which
is mildly dominated by the SSC emission, and a slightly flatter
decay phase dominated by SNIC emission takes over at late times
$t>10^6\rm s$. On the other hand, for the trans-relativistic ejecta
model with $\Gamma_0=2$, a plateau of SSC emission, due to the
presence of a coasting phase in the ejecta dynamic, dominates in the
early time, which transits to a faster decay at later time and SNIC
emission become dominant after the time $10^5\sim10^6\rm s$.
Figure \ref{lc_hyp} show light curves for the hypernovae case. Light
curves in the conventional relativistic ejecta model are similar
with those for the supernova case except that SNIC emission become
dominant at earlier time; for the trans-relativistic ejecta model,
SNIC emission is always dominated with parameters $\epsilon_e=0.1$
and $\epsilon_B=0.001$, a plateau is seen in the early time, which
transits to a faster decay at later time.
The two models also predict different flux levels at
high-energies. In early hours, the total flux from the conventional
relativistic ejecta model is more than one order of magnitude higher
than that in the trans-relativistic ejecta model. This can be
explained by the different amount of energy in shocked electrons.
For the conventional relativistic ejecta model, the ejecta has been
decelerated at this time and a great part of its energy has been
converted into shocked electrons, while in the trans-relativistic
ejecta model, only a small fraction of the ejecta kinetic energy has
been converted to shocked electrons at this early time. A lower
amount of energy in shocked electrons results in a lower flux level
in the trans-relativistic ejecta model.
In addition, a comparison among four panels in Figure \ref{lc_sup}
and Figure \ref{lc_hyp} indicates that the flux decreases as
$\epsilon_e$ decreases, which is obvious since the energy of
radiating electrons $E_e\propto \epsilon_e$. As $\epsilon_B$
decreases, the SSC flux changes little and the SNIC flux increases
for the case that SSC emission is dominant at early time. This can
be understood from the following analysis. Since the SSC emission
dominates and $f_{\rm a}u'_{\rm SN}\gg u'_{B}$ at the early times,
$Y\simeq \sqrt{\eta
\epsilon_e/\epsilon_B}\propto\epsilon_B^{-\frac{1}{2}}\epsilon_e^{\frac{1}{2}}$.
The SSC and SNIC flux at $\nu>\nu_{\rm min}>\nu_{\rm c}$ scale as
\begin{equation}
\nu F_{\nu}^{\rm SSC}=\nu F_{\rm max}^{\rm SSC}(\frac{\nu_{\rm
min}^{\rm SSC}}{\nu_{c}^{\rm
SSC}})^{-\frac{1}{2}}(\frac{\nu}{\nu_{\rm min}^{\rm
SSC}})^{-\frac{p}{2}}\propto\epsilon_B^{\frac{p}{4}-\frac{1}{2}}\epsilon_e^{2p-3}
\end{equation}
and
\begin{equation}
\nu F_{\nu}^{\rm SNIC}=\nu F_{\rm max}^{\rm SNIC}(\frac{\nu_{\rm
min}^{\rm SNIC}}{\nu_{c}^{\rm
SNIC}})^{-\frac{1}{2}}(\frac{\nu}{\nu_{\rm min}^{\rm
SNIC}})^{-\frac{p}{2}}\propto\epsilon_B^{-\frac{1}{2}}\epsilon_e^{p-\frac{3}{2}}.
\end{equation}
\subsection{The effect of the anisotropic scattering on the SNIC emission}\label{aiciso}
The incoming supernova photons are anisotropic as seen by the
isotropically distributed electrons in the GRB shock, so the IC
scatterings are anisotropic. In order to see how the anisotropic
inverse-Compton scattering (AIC) affects the SNIC flux, we
compare the light curves of the SNIC emission obtained by using
the isotropic scattering formula Eq.11 and using the AIC
scattering formula Eq.13 in Figure \ref{lc_iso}. The thinner lines
show the SNIC light curves obtained using the usual isotropic
scattering formula, while the thicker ones correspond to the
calculations with the AIC scattering effect taken into account.
One can see that the flux of the SNIC emission with the AIC effect
correction is reduced by a factor of about $\sim0.4$ compared to
the isotropic scattering case. This is consistent with the
calculation result obtained by Fan $\&$ Piran (2006), who studied
the anisotropic inverse Compton scattering between inner
optical/X-ray flare photons and electrons in the outer GRB forward
shock.
Fig.5 shows that the AIC effect suppresses the SNIC flux only
slightly. The anisotropic photon distribution results in more
head-on scatterings, i.e. the photon beam scatter preferentially
with those electrons that move in the direction antiparallel to
the photon beam, so one can expect that the scattered IC emission
power has a maximum at $\theta_{\rm SC}=\pi$ and goes to zero for
small scattering angles (e.g. Brunetti 2000). The photons
scattered into the angles $0\la\theta_{\rm SC}\la\pi/2$ relative
to the shock moving direction in the shock comoving frame will
fall into the cone of angle $1/\Gamma$ in the observer frame,
according to the transformation formula ${\rm cos}\theta_{\rm
SC}=({\rm cos}\theta-\beta)/(1-\beta {\rm cos}\theta)$. Therefore
the AIC scatterings decrease the IC emission in the $1/\Gamma$
cone along the direction of the photon beam, but meanwhile they
enhance the emission at larger angles (about half of the emission
falling into angles between $1/\Gamma$ and $2/\Gamma$, see Wang \&
M\'esz\'aros~ 2006). For a spherical outflow as we consider here, the
IC emission after integration over angles should have the same
flux in every direction in the observer frame, with a flux level
only slightly reduced comparable to the isotropic scattering case.
\subsection{Analytical Light Curves}\label{light curves}
As a comparison, we derive here approximate analytical expressions
for afterglow light curves, which provide an explanation for the
physical origin of the behavior. Since the anisotropic SNIC emission
are depressed by a factor of $~0.4$, which is almost constant,
relative to the isotropic seed photons case, we can consider the
isotropic seed photons case for the approximate analytic treatment
of the afterglow light curves. The blackbody photons from the
supernova can be approximated as mono-energetic photons with
$h\nu_{\rm SN}=2.7 \rm{K}T_{\rm SN}$. Thus, similar to the
description for synchrotron emission of a single electron (Sari et
al. 1998), the radiation power and characteristic frequencies of
SNIC from a single electron scattering supernova photons in the
observer frame can be described by
\begin{equation}
P(\gamma_{e})=\frac{4}{3}\sigma_Tc\gamma^2\gamma_e^2\frac{L_{\rm{SN}}}{\gamma^2\pi
R^2c},
\end{equation}
and
\begin{equation}
\nu(\gamma_{e})=2\gamma\gamma_e^2\nu_{\rm{SN}}/\gamma=2\gamma_e^2\nu_{\rm{SN}},
\end{equation}
respectively. Similar to the analysis in Sari and Esin (2001), the
maximum flux of the SNIC spectrum is
\begin{equation}
F_{\rm max}^{\rm SNIC}=\frac{N_e}{4\pi
D_{\rm L}^2}\frac{P(\gamma_{e})}{\nu(\gamma_{e})}
\end{equation}
Characteristic SNIC frequencies are
\begin{equation}\label{numinsnic}
\nu_{\rm min}^{\rm SNIC}=2\gamma_{e,\rm m}^2\nu_{\rm SN}
\end{equation}
and
\begin{equation}
\nu_{c}^{\rm SNIC}=2\gamma_{e,\rm c}^2\nu_{\rm SN}
\end{equation}
respectively.
By adopting the broken power-law approximation for the IC spectral
component ( Sari \& Esin 2001) and the dynamics of the shock
discussed in $\S$ \ref{models}, one can derive the analytic light
curves in an approximate way.
\subsubsection{Light Curves In The Conventional Relativistic Ejecta
Model} In the conventional relativistic ejecta model, at time
$t_{\rm dec}<t<t_{\rm tran}$, where $t_{\rm tran}$ is defined as the
transition time when $\gamma_{e,\rm m}=\gamma_{e,\rm c}$, we have
$\eta=1$ and $u'_{B}\ll f_{\rm a}u'_{\rm SN}\ll u'_{\rm SYN}$, so
$Y\simeq\sqrt{\eta\epsilon_e/\epsilon_B}\propto t^{0}$.
The shock
dynamic follows the Blandford \& McKee self-similar solution in the
wind medium, i.e. $R\propto t^{\frac{1}{2}}$ and $\gamma\propto
t^{-\frac{1}{4}}$. Then we can obtain the evolution of the break
frequencies of the SSC and SNIC spectral components and their peak
flux in the following way:
\begin{equation}
\nu_{\rm min}^{\rm SSC}=2\gamma_{e,\rm m}^2\nu_{\rm min}\propto
t^{-2},
\nu_{\rm c}^{\rm SSC}=2\gamma_{e,\rm c}^2\nu_{\rm c}\propto t^{2},\\
\end{equation}
\begin{equation}
F_{\nu,\rm max}^{\rm SSC}=\frac{\sigma_TN_e}{4\pi R^2}F_{\rm
max}^{\rm SYN}\propto t^{-1}.
\end{equation}
and
\begin{equation}
\nu_{\rm min}^{\rm SNIC}\propto t^{-1}, \nu_{\rm c}^{\rm
SNIC}\propto t, F_{\nu,\rm max}^{\rm SNIC}\propto t^{-0.4}.
\end{equation}
where $\nu_{\rm SN}\propto T_{\rm SN}\propto t^{-1/2}$ has been used
for the cooling envelope emission (see Eq. 13). The SSC and SNIC
flux at an observed frequency $\nu$ higher than characteristic
frequencies vary as $\nu F_{\nu}^{\rm SSC}\propto t^{-p+1}$ and $\nu
F_{\nu}^{\rm SNIC}\propto t^{-\frac{p}{2}+0.6}$ respectively for
$t<t_{\rm tran}$.
At $t>t_{\rm SNIC}$, where $t_{\rm SNIC}$ is the time when $f_{\rm
a}u'_{\rm SN}=u'_{\rm SYN}$, we take $Y\simeq f_{\rm a}{u'}_{\rm
SN}/(\epsilon_{B}{u'})$ because the SNIC emission becomes dominated.
At such time, the shock is likely to enter the non-relativistic
phase, so we take the Sedov-von Neumann-Taylor solutions $R\propto
t^{\frac{2}{3}}$ and $\beta\propto t^{-\frac{1}{3}}$, which induce
that $Y\propto t^{\frac{2}{3}-0.4}$. So the minimum and cooling
Lorentz factors of electrons vary as $\gamma_{e,\rm m}\propto
\beta^{2}\propto t^{-\frac{2}{3}}$ and $\gamma_{e,\rm
c}\propto\beta^{-2}R^2t^{-1}Y^{-1}\propto t^{0.4+\frac{1}{3}}$.
Thus, the break frequencies of the SSC and SNIC spectral components
and their peak fluxes evolve with time in the following way:
\begin{equation}
\nu_{\rm min}^{\rm SSC}\propto t^{-\frac{11}{3}}, \nu_{\rm c}^{\rm
SSC}\propto t^{\frac{1}{3}+1.6}, F_{\nu,\rm max}^{\rm SSC}\propto
t^{-1},
\end{equation}
and
\begin{equation}
\nu_{\rm min}^{\rm SNIC}\propto t^{-\frac{11}{6}}, \nu_{\rm c}^{\rm
SNIC}\propto t^{\frac{1}{6}+0.8}, F_{\nu,\rm max}^{\rm SNIC}\propto
t^{-\frac{2}{3}+0.1}.
\end{equation}
Then the SSC and SNIC fluxes at high energy $\nu$ vary as $\nu
F_{\nu}^{\rm SSC}\propto t^{1.8-\frac{11}{6}p}$ and $\nu
F_{\nu}^{\rm SNIC}\propto t^{\frac{5}{6}-\frac{11}{12}p}$ at
$t>t_{\rm SNIC}$.
To summarize, the temporal evolution of the SSC and SNIC afterglow
emission at high energies are
\begin{equation} \nu
F_{\nu}^{\rm SSC}\propto\left \{\begin{array}{ll}
t^{-1.2}&t\la t_{\rm tran}\\
t^{-2.2}&t>t_{\rm SNIC}\end{array}\right.
\end{equation}
and
\begin{equation}
\nu F_{\nu}^{\rm SNIC}\propto\left \{\begin{array}{ll}
t^{-0.5}&t\la t_{\rm tran}\\
t^{-1.2}&t>t_{\rm SNIC}\end{array}\right.
\end{equation}
in the two asymptotic phases for $p=2.2$.
\subsubsection{Light Curves In The Trans-relativistic Ejecta Model}
In the trans-relativistic ejecta model, one would expect a much
flatter light curve of high-energy gamma-ray emission at early
times, which could be dominated by both the SSC emission and SNIC
emission, depending on the properties of the underlying supernova
and the shock parameter $\epsilon_e$ and $\epsilon_B$. For the
supernova case, the SSC emission is dominant before the deceleration
time and the transition time for most parameters, while for the
hypernova case, the SSC emission is dominant at earlier time with
$\epsilon_e=0.3$ and $\epsilon_B=0.01$ and the SNIC emission is
always dominant for $\epsilon_e=0.1$ and $\epsilon_B=0.001$.
For cases where SSC emission dominated in trans-relativistic ejecta
model, we have $\eta=1$ in fast cooling regime and $u'_{B}\ll f_{\rm
a}u'_{\rm SN}\ll u'_{\rm SYN}$, so
$Y\simeq\sqrt{\eta\epsilon_e/\epsilon_B}\propto t^{0}$. Since
$t<t_{\rm dec}$, we adopt the approximation $R\propto t$,
$\gamma\propto t^0$. So the SSC and SNIC flux vary as $\nu
F_{\nu}^{\rm SSC}\propto t^{1-\frac{p}{2}}$ and $\nu F_{\nu}^{\rm
SNIC}\propto t^{0.1-\frac{p}{4}}$ respectively at $t<{\rm
min}(t_{\rm dec},t_{\rm tran})$. At later time when $t>t_{\rm dec}$
and $\gamma\rightarrow1$, the SNIC emission become dominant, the
evolution of light curves is the same as that in conventional
relativistic ejecta model, i.e. the SSC and SNIC flux vary as $\nu
F_{\nu}^{\rm SSC}\propto t^{1.8-\frac{11}{6}p}$ and $\nu
F_{\nu}^{\rm SNIC}\propto t^{\frac{5}{6}-\frac{11}{12}p}$ at
$t>t_{\rm dec}$. Therefore, for the supernova case and the hypernova
case with early dominated SSC emission, the temporal evolution of
the SSC and SNIC emission at high frequency $\nu$ for $p=2.2$ are
given by
\begin{equation}
\nu F_{\nu}^{\rm SSC}\propto\left \{\begin{array}{ll}
t^{-0.1}&t\la{\rm min}(t_{\rm dec}, t_{\rm tran})\\
t^{-2.2}&t\gg t_{\rm dec}\end{array}\right.
\end{equation}
and
\begin{equation}
\nu F_{\nu}^{\rm SNIC}\propto\left \{\begin{array}{ll}
t^{-0.45}&t\la{\rm min}(t_{\rm dec}, t_{\rm tran})\\
t^{-1.2}&t\gg t_{\rm dec}\end{array}\right.
\end{equation}.
For cases where SNIC emission dominated in trans-relativistic ejecta
model, We have $f_{\rm a}u'_{\rm SN}\gg u'_{\rm SYN}$, so $Y\simeq
f_{\rm a}u'_{\rm SN}/(\epsilon_Bu')$. At the time $t<t_{\rm dec}$,
we adopt the approximation $R\propto t$, $\gamma\propto t^0$ to
yield $Y\propto t^{-0.4}$. So the SSC and SNIC flux vary as $\nu
F_{\nu}^{\rm SSC}\propto t^{1.8-\frac{p}{2}}$ and $\nu F_{\nu}^{\rm
SNIC}\propto t^{0.5-\frac{p}{4}}$ respectively at $t<{\rm
min}(t_{\rm dec},t_{\rm tran})$. At later time when $t>t_{\rm dec}$
and $\gamma\rightarrow1$, the evolution of light curves is the same
as that in conventional relativistic ejecta model, i.e. the SSC and
SNIC flux vary as $\nu F_{\nu}^{\rm SSC}\propto
t^{1.8-\frac{11}{6}p}$ and $\nu F_{\nu}^{\rm SNIC}\propto
t^{\frac{5}{6}-\frac{11}{12}p}$ at $t>t_{\rm dec}$. Therefore, for
the supernova case and the hypernova case with early dominated SSC
emission, the temporal evolution of the SSC and SNIC emission at
high frequency $\nu$ for $p=2.2$ are given by
\begin{equation}
\nu F_{\nu}^{\rm SSC}\propto\left \{\begin{array}{ll}
t^{0.7}&t\la{\rm min}(t_{\rm dec}, t_{\rm tran})\\
t^{-2.2}&t\gg t_{\rm dec}\end{array}\right.
\end{equation}
and
\begin{equation}
\nu F_{\nu}^{\rm SNIC}\propto\left \{\begin{array}{ll}
t^{-0.05}&t\la{\rm min}(t_{\rm dec}, t_{\rm tran})\\
t^{-1.2}&t\gg t_{\rm dec}\end{array}\right.
\end{equation}.
\subsection{Detectability by the Fermi LAT}\label{detectability of Fermi LAT}
We explore here whether Fermi LAT can detect the high energy
gamma-ray emission from low luminosity GRBs in the two models
considered above. Following Zhang $\&$ M\'{e}sz\'{a}ros (2001),
Gou $\&$ M\'{e}sz\'{a}ros (2007) and Yu, Liu $\&$ Dai (2007), the
fluence threshold for long-duration observations is $F_{\rm
thr}=[\phi_0(t/t_{\rm eff})^{1/2}]E_{\rm ph}t_{\rm eff}$ which is
in proportional to $t^{1/2}$ due to the limitation by the
background, where we take the average energy of the detected
photons as $E_{\rm ph}=400\rm MeV$ and the effective time as
$t_{\rm eff}=0.5\rm yr$. $\phi_0$ is the integral sensitivity
above 100 $\rm MeV$ for LAT for a steady source after a year sky
survey, which is $\phi_0\sim3\times10^{-9}\rm phs cm^{-2}s^{-1}$
(atwood et. al. 2009) and is improved by a factor of $3$ by
keeping the GRB position at the center of the LAT field of view as
long as possible (Gou $\&$ M\'{e}sz\'{a}ros 2007). For short-time
observation, the fluence threshold is calculated by $F_{\rm
thr}=5E_{\rm ph}/A_{\rm eff}$ under the assumption that at least
$5$ photons are collected. Taking the effective area $A_{\rm
eff}=6000\rm cm^2$, we can obtain the fluence threshold of Fermi
LAT,
\begin{equation}
F_{\rm thr}=\left\{\begin{array}{ll}
5.3\times10^{-7} \rm{erg~cm^{-2}},&t\leq4.4\times10^{4}\rm{s},\\
2.5\times10^{-9}t^{1/2} \rm{erg~cm^{-2}},&t>4.4\times10^{4}\rm{s}
.\end{array}\right .
\end{equation}
With this fluence threshold, the detectability of high energy
emission (with supernova seed photons luminosity given above and a
total energy $E=10^{50}\rm{erg}$) by the Fermi LAT is shown in
Figure 6. The time-integrated fluence shown in the plot is defined
as an integration of the flux density ($F_{\nu}$) over the Fermi
LAT energy band $[20\rm MeV, 300\rm GeV]$ and the time interval
$[0.5t,t]$ as used in Gou $\&$ Meszaros (2007), which is
$\int_{0.5t}^{t}\int_{\nu_{1}}^{\nu_{2}}F_{\nu}d\nu dt$.
For $\epsilon_e=0.3$ and other representative parameter values,
high-energy gamma-ray emission in the conventional relativistic
ejecta model stays detectable up to $\sim10^6\rm{s}$, while the
high-energy gamma-ray emission in the trans-relativistic ejecta
model can only be detected in a short period around
$10^{4.5}\rm{s}$. For a lower value such as $\epsilon_e=0.1$, the
high-energy gamma-ray emission can still be detected in the
conventional relativistic ejecta model, while it becomes
undetectable for the trans-relativistic ejecta model.
In order to compare with earlier results in Ando \&
M\'{e}sz\'{a}ros (2008), we increase the total energy to
$E=2\times10^{50}\rm{erg}$, which yields a kinetic energy
$E_{\rm k}=(\Gamma_0-1)/\Gamma_0 E=10^{50}\rm{erg}$ in the
trans-relativistic ejecta model, the same as that used in Ando \&
M\'{e}sz\'{a}ros (2008). This will increase the fluence by a
factor of $2$. We also increase the SN luminosity from that of a
normal SN (shown in $\S$ \ref{seed photons}) to SN1998bw-like
hypernovae. In Figure 7, we show the detectability of high energy
emission by Fermi LAT in this case. By comparing the flux of light
curves between the supernova case and the hypernova case, which is
shown in Fig. \ref{lc_sup} and Fig. \ref{lc_hyp}, we can see that
increasing the SN luminosity can hardly enhance the total IC flux
for the two models. Our IC flux is still lower than that obtained
by Ando \& M\'esz\'aros~ (2008). The main difference between our
calculation and that of Ando \& M\'{e}sz\'{a}ros (2008) is the
different minimum Lorentz factors used in the calculations. Ando
\& M\'{e}sz\'{a}ros (2008) may overestimate the minimum Lorentz
factor of electrons by taking $\gamma_{e,\rm
m}=\epsilon_e(m_p/m_e)\gamma$, which is a factor of $(p-1)/(p-2)$
larger than ours. From the formula describing the high energy flux
$\nu F_{\nu}^{\rm SNIC}\propto\nu_{\rm min,\rm
SNIC}^{(p-1)/2}\propto\gamma_{e,\rm m}^{p-1}$, one expects that
the flux is increased by a factor of $(\frac{p-1}{p-2})^{p-1}$,
which is about $8$ for $p=2.2$.
\section{Discussions and Conclusions}
The external stellar wind provides a source of Thomson opacity to
scatter the supernova emission, thus a quasi-isotropic,
back-scattered SN radiation field is present. Let's study whether
this component is important. For a GRB shock locating at radius
$R$ and moving with a Lorentz factor $\gamma$, the Thomson
scattering optical depth of the wind is $\tau_{\rm
w}=\sigma_TRn=\sigma_TKR^{-1}=2.0\times10^{-4}\dot{m}R_{15}^{-1}$.
The scattered SN energy density by the wind in the comoving frame
of the blast wave is $u'^{\rm w}_{\rm SN}=(L_{SN}/{4\pi c R^2})
\tau_{\rm w} \gamma^2$ due to the relativistic boosting effect. On
the other hand, the energy density of the supernova photons
impinging the shock from behind is $u'_{\rm
SN}=\gamma^{-2}(L_{SN}/{4\pi c R^2})$ (see Eq.21). The ratio
between these two energy densities is $u'^{\rm w}_{\rm SN}/u'_{\rm
SN}=2.0\times10^{-4}\dot{m}R_{15}^{-1} \gamma^4$. For the
trans-relativistic ejecta model with $\Gamma_0=2$, the radius of
the GRB ejecta is $R=1.9\times10^{14}$ $\rm{cm}$ at $t=10^3
\rm{s}$, so $u'^{\rm w}_{\rm SN}\ll u'_{\rm SN}$ for typical wind
parameters. For the conventional relativistic ejecta model, the
Lorentz factor and radius of the GRB ejecta are respectively
$\gamma\sim4$ and $R=1.0\times10^{15}$ $\rm{cm}$ at $t=10^3
\rm{s}$, so we also have $u'^{\rm w}_{\rm SN}\ll u'_{\rm SN}$.
This means that the wind-scattered supernova radiation field is
subdominant compared to the direct impinging supernova photon
field and hence we neglect its contribution to the high-energy
gamma-ray emission.
Our estimate of the pair-production opacity for high-energy
photons in $\S$ \ref{tao} is based on the common assumption that
the colliding photons are isotropic in the rest frame. However, as
we have shown above, the low-energy photons from the supernova
essentially move radially outward before colliding with
high-energy photons (e.g. Wang, Li \& M\'esz\'aros~ 2006). Therefore,
the collision process between the high-energy photons and soft
supernova photons is anisotropic, which would decrease the
pair-production opacity. This will be the subject of a more
detailed future calculation and would be useful to explore whether
$\rm{TeV}$ photons can escape from the source, which is important
for checking the detectability by ground-based Cherenkov detectors
such as Magic, VERITAS, Milago, HESS, ARGO etc. Additionally,
besides the high-energy gamma-ray emission discussed in this work,
the high-energy neutrino emission arising from $p\gamma$
interactions between shock-accelerated protons and photons from
the supernova may also provide a constraint on the model for
low-luminosity GRBs (e.g. Yu et al. 2008).
Trans-relativistic ejecta may also exist in the usual high
luminosity long GRBs, besides in low luminosity ones, since
{accelerating shocks are} expected to accompany the supernova.
Berger et al. (2003) and Sheth et al. (2003) found that the radio
and optical afterglow indicates a low velocity component more than
$1.5 \rm{days}$ after the explosion in GRB030329/SN2003dh.
However, since the highly relativistic ejecta is much more
energetic than the trans-relativistic component, high-energy
gamma-ray emission from the latter component {could easily remain
hidden}.
In summary, we have calculated the spectra and light curves of the
high-energy gamma-ray afterglow emission from low luminosity GRBs
for the two main models in the literature, i.e. the
trans-relativistic ejecta model ($\Gamma_0\simeq2$) and the
conventional highly relativistic ejecta model ($\Gamma_0\ga 10$),
considering both synchrotron self inverse-Compton (SSC) and the
external inverse-Compton due to photons from the underlying
supernova/hypernova. Our analysis takes into account a full
Klein-Nishina cross section for inverse Compton scatterings, the
anisotropic scattering of supernova photons and the opacity for
high energy photons due to annihilation with low-energy photons
(mainly from the supernova/hypernova).
We find that for the supernova case the conventional relativistic
outflow model predicts a relatively high gamma-ray flux from SSC
at early times ($<10^5-10^6 {\rm s}$ for typical parameters) with
a rapidly decreasing flux, while in the trans-relativistic outflow
model, a much flatter light curve of high-energy gamma-ray
emission dominated by the SSC emission, is expected at early
times. For the hypernova case, the SSC emission also dominates and
decays sharply at early time in the conventional relativistic
ejecta model, while for the trans-relativistic ejecta model, both
the SSC emission and the SNIC emission could be dominant at early
time, depending on the shock parameters $\epsilon_e$ and
$\epsilon_B$. The main difference between these two models arises
from their different initial Lorentz factors, which induces a
different dynamical evolution of the shock at early times. As a
result, different observational features arise, such as different
light curve shapes, different flux levels and different dominant
components (as detailed in $\S$ \ref{spectra and light curves}).
As shown in $\S$ \ref{detectability of Fermi LAT}, high-energy
gamma-ray emission can be detected in both models as long as
$\epsilon_e$ is large enough, although detection from the
conventional relativistic ejecta is much easier. Thus, with
future high energy gamma-ray observations by Fermi LAT, one can
expect to be able to distinguish between the two models based on
the above differences in observational features.
{\acknowledgments We would like to thank Z. G. Dai, Y. F. Huang
and D. M. Wei for useful discussions. This work is supported by
the National Natural Science Foundation of China under grants
10973008 and 10403002, the National Basic Research Program of
China (973 program) under grants No. 2009CB824800, the Foundation
for the Authors of National Excellent Doctoral Dissertations of
China, the Qing Lan Project, and NASA grant NNX 08AL40G.}
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,447 |
Fan Becky helps Coppinger celebrate 38th birthday
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Rovers midfielder James Coppinger has thanked residents of a local residential care home who helped him celebrate his 38th birthday in style.
Coppinger met residents of the Quarryfields care home who were invited down the club's Cantley Park training ground to deliver a cake they had baked for the occasion.
Rovers' No.26 said: "I'd like to thank everyone who came down to help me celebrate my birthday, especially Becky who made the cake for me.
"The lads were all pleased to see them, and even happier when they got to try some of the cake afterwards! I can confirm it was delicious!"
The cake was made by the home's 'The Platform', a pop-up coffee shop on Balby Road run by residents who live at Exemplar Health Care's Quarryfields care home.
Heather Johnson, education facilitator at The Platform, said: "We are so pleased that we were able to present James with a birthday cake. Becky did a great job in making it and we were all so excited to give it to him.
"James and the team have been fantastic and we can't thank them enough for taking the time to meet with us. We hope James and the team enjoy the cake. You're always welcome to come and visit The Platform for a coffee."
Becky who lives at Quarryfields, said: "I am really pleased with the cake and I am happy James likes it. It's full of chocolate so I'm sure he will like it!"
One of Exemplar's 26-care homes across the country, Quarryfields on Woodfield Road in Balby specialises in providing specialist nursing-led care for younger adults who live with complex nursing needs including autism, learning disabilities and mental health conditions.
The video above is free to watch for anyone who has registered for a free account with iFollow Rovers. Click here to sign up or to enhance your iFollow Rovers package with a paid subscription which provides even more exclusive videos as well as live match commentary.
James Coppinger
Quarryfields | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,029 |
#include "CompositeTurboModuleManagerDelegate.h"
namespace facebook {
namespace react {
jni::local_ref<CompositeTurboModuleManagerDelegate::jhybriddata>
CompositeTurboModuleManagerDelegate::initHybrid(jni::alias_ref<jhybridobject>) {
return makeCxxInstance();
}
void CompositeTurboModuleManagerDelegate::registerNatives() {
registerHybrid({
makeNativeMethod(
"initHybrid", CompositeTurboModuleManagerDelegate::initHybrid),
makeNativeMethod(
"addTurboModuleManagerDelegate",
CompositeTurboModuleManagerDelegate::addTurboModuleManagerDelegate),
});
}
std::shared_ptr<TurboModule>
CompositeTurboModuleManagerDelegate::getTurboModule(
const std::string &moduleName,
const std::shared_ptr<CallInvoker> &jsInvoker) {
for (auto delegate : mDelegates_) {
if (auto turboModule = delegate->getTurboModule(moduleName, jsInvoker)) {
return turboModule;
}
}
return nullptr;
}
std::shared_ptr<TurboModule>
CompositeTurboModuleManagerDelegate::getTurboModule(
const std::string &moduleName,
const JavaTurboModule::InitParams ¶ms) {
for (auto delegate : mDelegates_) {
if (auto turboModule = delegate->getTurboModule(moduleName, params)) {
return turboModule;
}
}
return nullptr;
}
void CompositeTurboModuleManagerDelegate::addTurboModuleManagerDelegate(
jni::alias_ref<TurboModuleManagerDelegate::javaobject>
turboModuleManagerDelegate) {
mDelegates_.insert(turboModuleManagerDelegate->cthis());
}
} // namespace react
} // namespace facebook
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,898 |
North Africa has a plethora of wild animals including the leopard, the dama gazelle and the striped hyena. Whether you go to Morocco, Egypt, Sudan, Tunisia, Libya, Algeria or Western Sahara, you are likely to encounter any of these animals at the local zoo or on a safari. Guide books such as Lonely Planet and the Rough Guide will give you more information on the different kinds of elephants and gazelles. Although the "big seven" African animals--such as the rhino, elephant and lion--tend to congregate in eastern and southern Africa, the impressive animals of the northern part of the continent are worth a visit.
One of the most sighted animals in North Africa, the striped hyena resembles a large dog with its long face and skinny body. These animals weigh less than 100 pounds and typically average four feet long. They are nomadic and tend to hunt alone rather than in packs. Often they can be found in savannas, grasslands and the woods.
Interesting fact: In North Africa and some parts of the Middle East, the striped hyena is often seen as an embodiment of a supernatural creature, often known as "jinn." They supposedly lure humans to perform magic tricks or miracles.
Part of the antelope family, the addax is becoming an endangered species in North Africa and sightings are rare. Often they are hunted by locals for their lustrous hide, even though technically the African government has outlawed this practice.
The addax, which stands about five feet tall, is found all over North Africa but tends to congregate in the Sahara desert. They hunt at night and in small packs.
Interesting fact: the addax does not need to take in large quantities of water; they get much of their water from the dew on local flora.
The dama gazelle resembles something of a dancer. With its long, thin legs, graceful neck and small stature, the average gazelle stands just three feet tall and weighs 140 pounds. One of the most striking creatures in all of North Africa, the dama gazelles were blessed with a burnt red hide and a white underbelly and head.
These creatures prefer to live alone or small groups and they are vegetarians, feeding off the local plants and trees.
Interesting fact: Due to high drought conditions in North Africa, the dama gazelles have moved out of their comfortable environment into more populated habitats, thus allowing more contact with humans. These animals are least likely to flee if they come in contact with a tourist in the wild or safari park.
More than 20 species of leopard live in Africa, and many of these species can be found in the deserts of Algeria and the western Sahara. The average leopard ranges from four to six feet and weighs anywhere from 60 to 160 pounds.
A versatile mammal, the leopard can live in almost any environment, from wet grasslands to arid deserts. Commonly, they hide in forests and trees to avoid being eaten by lions and hyenas. The leopards, who hunt only at night and from trees, tend to feast on monkeys, fish, rodents, dogs, pigs and deer or gazelle. After killing their prey, the leopard will drag the animal into a tree to avoid other predators taking their food. They are fast, adaptable, and tend to travel in small packs.
Interesting fact: Most people think of leopards as having a light brown or yellow coat, but some species are midnight black with occasional dark brown patches on their underbelly or face.
Wagner, Kathryn. "Animals From North Africa." Sciencing, https://sciencing.com/animals-north-africa-6700788.html. 24 April 2017. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,479 |
Crime sur Mégahertz Crime sur Mégahertz est un épisode des cinq dernières minutes réalisé par Johannick Desclers, d'après un scénario de Jean-Pierre Amette.
Synopsis
Après le meurtre d'une animatrice de radio, le commissaire Cabrol enquête dans les couloirs des radios libres...
Fiche technique
Source : IMDb, sauf mention contraire
Scénario : Jean-Pierre Amette
Réalisateur : Johannick Desclercs
Musique : Marc Lanjean
Sociétés de production : ORTF et Antenne 2
Format : Couleur
Création :
Genre : Policier
Durée : 1h30'
Date de première diffusion :
Distribution
Jacques Debary : Commissaire Cabrol
Marc Eyraud : L'inspecteur Ménardeau
Roland Bertin : Raoul Meinhart
Florence Giorgetti : Nina Meinhart
Michel Boujenah : Luc Henno
François Gamard
Bernard Pinet
Christian Rauth
Annie Noël
Voir aussi
Les Cinq Dernières Minutes
Lien externe
Crime sur Mégahertz sur www.imdb.com
Téléfilm diffusé en 1985
Téléfilm de France Télévisions | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,960 |
{"url":"https:\/\/epg.modot.org\/index.php\/Category:403_Asphaltic_Concrete_Pavement","text":"# Category:403 Asphaltic Concrete Pavement\n\n Forms QA\/QC Checklist Profiloqraph Report Figure QA\/QC Questions and Answers QRG SuperPave Adjustments\n AC Price Index Adjustments Guidance for Asphalt Cement Price Index Adjustments Related Information Guidance for Administering and Planning Pavement Maintenance Responsibilities during Construction\n\n## 403.1 Construction Inspection for Sec 403\n\n### 403.1.1 Description (Sec 403.1)\n\nThere will be no commentary for sections in which the intent of the specifications, as written, is clear.\n\nDesign Levels (Sec 403.1.2)\n\nAt the contractor\u2019s expense, a mix with the same size aggregate and one design level higher can be substituted for the mix required by the contract. Substitutions typically require a change order to pay for the higher quality mix at the price of the lower mix. Care should be taken to assure that the material product codes reflect the mix actually placed on the roadway. The substitutions must be done uniformly and various design levels in the same lift will not be allowed.\n\n### 403.1.2 Material (Sec 403.2)\n\nFine Aggregate Angularity (FAA) (Sec 403.2.1)\n\nFine Aggregate Angularity (FAA) ensures a high degree of fine aggregate internal friction and rutting resistance. FAA provides an indication of the particle shape and is defined as the percent voids in loose, uncompacted fine aggregates. More fractured faces results in a higher void content in the aggregate. FAA is determined on the fine portion of the blended aggregate (passing the #8 sieve) in accordance with AASHTO T304 (Level 2 Aggregate Training). The minimum requirements, based on the design level of the mix, are given in Standard Specification Section 403.2.1.\n\nCoarse Aggregate Angularity (CAA) (Sec 403.2.2)\n\nCoarse Aggregate Angularity (CAA) ensures a high degree of coarse aggregate internal friction and rutting resistance. CAA is defined as the percent of coarse aggregates with one or more fractured faces. CAA is determined on the coarse portion of the blended aggregate (retained on the #4 sieve) in accordance with ASTM D5821 (Level 2 Aggregate Training). The minimum requirements, based on the design level of the mix, are given in Standard Specification Section 403.2.2.\n\nClay Content (Sec 403.2.3)\n\nClay content, or sand equivalency, is the percentage of clay-like material present in the aggregate. The higher the sand equivalent value, the less clay-like material present in the aggregate. Clay-like material can coat the aggregate surfaces and prevent the binder from adhering to the aggregate particles. Sand equivalency is determined on the fine portion of the blended aggregate (passing the #4 sieve) in accordance with AASHTO T176 (Level 2 Aggregate Training). The minimum requirements, based on the design level of the mix, are given in Standard Specification Section 403.2.3.\n\nThin, Elongated Particles (Sec 403.2.4)\n\nThis property, also known as flat and elongated, is the percentage of coarse aggregates that have a maximum to minimum dimension ratio of 5:1 or greater. Flat and elongated particles are undesirable because they have a tendency to break more easily than other aggregate particles. When an aggregate particle breaks, it creates a face that is not coated with binder, increasing the potential of the mix to strip or ravel. Also, the gradation of the mix becomes finer, which may be detrimental to the mix properties. Finally, a mix that contains flat and elongated particles may be difficult to place and compact. The percentage of flat and elongated particles is determined on the coarse portion of the blended aggregate (retained on the #4 sieve) in accordance with ASTM D4791 (Level 2 Aggregate Training). The maximum allowable percentage of flat and elongated particles for all mixes other than SP125xSM is given in Standard Specification Section 403.2.4.\n\nSP125xSM Requirements (Sec 403.2.5)\n\nIn a Stone Matrix Asphalt (SMA) mix, the coarse aggregate will consist of crushed limestone and a hard durable aggregate, i.e. low Los Angeles Abrasion and absorption. Durable aggregate is generally either porphyry or steel slag but may be aggregates such as crushed gravel or quartzite. Mixtures designated as SMR, for rural interstates, may use 100% dolomite aggregates. SMA mixes have flat and elongated requirements for ratios of 5:1 and 3:1. The maximum allowable percentages of flat and elongated particles based on these ratios are given in Standard Specification Section 403.2.5.\n\nFiller Restriction (Sec 403.2.5.1)\n\nSee Mineral Filler, Hydrated Lime, and Baghouse Fines in Plant Inspection.\n\nFibers (403.2.5.2)\n\nReclaimed Asphalt (Sec 403.2.6)\n\nBoth reclaimed asphalt pavement (RAP) and reclaimed asphalt shingles (RAS) are allowed in some mix types by specification but not all mix types. When RAP or RAS are allowed, the contractor chooses when and how much recycled material to utilize within the specification limits. Depending on how much RAP or RAS the contractor chooses to use, there may be additional requirements placed on the virgin binder by the specifications. Effective virgin binder is used to account for binder absorbed by the aggregates and is not available for blending with the reclaimed binder.\n\nReclaimed binder contribution is calculated using a spreadsheet provided on MoDOT\u2019s website. When a mix design approval is based off of a blend chart binder grade or extracted binder grade, substitution of a different virgin binder may require additional testing to prove the specification requirements are met.\n\n### 403.1.3 Composition of Mixtures (Sec 403.3)\n\nThe master range for the combined aggregate gradation of each mix type is given in Standard Specification Section 403.3.1. These master ranges apply to the final gradation of the aggregate, including filler materials, before the binder is added. During production, the combined aggregate gradation may be outside of the master range when the tolerances of Standard Specification Section 403.5.1 are applied.\n\nAnti-Strip Agent (403.3.2)\n\nSee Liquid Anti-Strip Additives in Plant Inspection.\n\n### 403.1.4 Job Mix Formula\n\nThe mix design procedure will be in accordance with Materials Inspection.\n\nApproval (Sec 403.4.3)\n\nNo mix shall be produced or placed by the contractor or accepted for use by an inspector without an approved JMF. This includes mix transfers. The Materials Field Office (MFO) will give written approval. In some extreme cases, approval may be verbal with written approval to follow. Occasionally, a contractor may elect to place mix while the request is still pending. In this situation, the contractor is proceeding at his own risk and should be so advised by an order record.\n\nJob Mix Formula Modifications (Sec 403.4.4)\n\nA new JMF will be required if a material source is changed or if unsatisfactory results are obtained. The exception for the new JMF requirement will be when a binder source change has been made to a supplier, previously provided by the contractor in the original JMF, for which an alternate JMF number has already been created. Unsatisfactory results may include a mix that fails to meet specifications (binder content, volumetrics, and\/or density) or if the visual appearance of the mix is unacceptable. If a new JMF is required, the procedures outlined in Standard Specification Section 403.11 should be followed.\n\n### 403.1.5 Mixture Production Specification Limits (Sec 403.5)\n\nIntentional deviations from the JMF will not be permitted. The plant shall be operated in such a manner that the mix is produced as shown on the JMF. The specification tolerances are developed in an attempt to keep the mix as consistent as possible and to allow for some variation during production. However, these tolerances are not production limits. For example, if the target binder content is 5.0%, the binder content of the mix can range from 4.7% to 5.3% when the tolerances are applied. The contractor will not be allowed to produce the mix at 4.7% to save money.\n\nBoth QC and QA will use the following procedures to determine volumetrics of the mix and compliance with Standard Specification Sections 403.5.3 through 403.5.5. These procedures are discussed in greater detail in the Levels 1 and 2 Bituminous Training.\n\nA loose mix sample consisting of roughly 100 lbs. will be taken from the roadway behind the paver, in accordance with AASHTO T168, at the required frequency. The sample will be thoroughly mixed and quartered in accordance with AASHTO R47, or with an approved splitting\/quartering device. Two opposite quarters will be retained for testing during the dispute resolution process, if necessary. The remaining two quarters will be mixed together and quartered again.\n\nThe required weight of mix, as listed on the JMF, will be taken from one quarter and used to compact a specimen in accordance with AASHTO T312. The mix will be compacted to Ndes gyrations while the mix temperature is within the molding range listed on the JMF. Using the opposite quarter, follow the same procedure for the second specimen. The Gmb of each specimen will be determined and the average will be used to calculate the air voids Va and the voids in the mineral aggregate (VMA). By specification, a minimum of two compacted specimens must be used to calculate these properties.\n\nA third quarter will be used to determine the Gmm of the mix in accordance with AASHTO T209. The minimum sample size for each type of mix can be found in the training manual. This property is used to calculate the Va and density. The volume of the sample, which is needed in the calculation, can be determined by either the weigh-in-air method or the weigh-in-water method. The weigh-in-air method consists of weighing the sample and container (with the lid) completely filled with water in air. The weigh-in-water method consists of weighing the sample and container (without the lid) completely submerged in water.\n\nThe remaining mix should be mixed together and quartered again. To determine the binder content using the nuclear gauge, enough mix should be taken from opposite quarters. The required weight of mix is listed on the JMF. A moisture content sample should be taken from the same quarters. To determine the binder content using the binder ignition oven, enough mix should be taken from one quarter. The minimum sample size for each type of mix can be found in the training manual. A moisture content sample should be taken from the same quarter. Sometimes the ignition oven may not shut itself off. The oven may be shut off manually as long as 3 consecutive readings show less than 0.01% loss. The sample should be examined to assure that a complete burn has been achieved. This will be considered a valid test.\n\nIn situations where a retained sample must be tested, the following procedure should be used to reheat the sample. Heat the sample in an oven until the mix is workable. Take the mix out of the sample container (box, bucket, etc.) and spread it in a large pan or several smaller pans. Using this procedure, the mix will reach the molding temperature much quicker than it would if it were left in a mass in the sample container. Also, less aging of the mix occurs since the mix is in the oven for a shorter period of time. Once the mix has reached an acceptable temperature, the sample must be quartered using the procedures discussed above. The entire suite of tests must be performed on a retained sample.\n\nSee Sieve Analysis in Plant Inspection. The gradation of the mix is not a pay factor item. However, it does have a significant influence on the volumetrics of the mix. Samples may be taken from the hot bins at a batch plant or from the combined cold feed at a drum plant. It is acceptable to determine gradation from the binder ignition sample according to AASHTO Standard Test Method T 308. Contractors should be allowed the option provided that the chosen method is spelled out in the Quality Control Plan. Gradations of extracted samples would be satisfactory as well. QC is required to sample the aggregate and perform a sieve analysis twice per lot. QA is required to independently sample the aggregate and perform a sieve analysis once per lot. These testing requirements are minimums and should be increased as necessary. Minor deviations outside the tolerances given in Standard Specification Sections 403.5.1.1 or 403.5.1.2, whichever is applicable, may be allowed if the test results indicate that the binder content, volumetrics, and density of the mix are satisfactory. If the test results are unsatisfactory, adjustments of the JMF, in accordance with Standard Specification Section 403.11, are necessary.\n\nStone Matrix Asphalt Tolerances (Sec 403.5.1.1)\n\nThe tolerances from the JMF for SMA mixes are given in Standard Specification Section 403.5.1.1.\n\nMixture Tolerance (Sec 403.5.1.2)\n\nDuring production, the combined aggregate gradation must be within the following limits:\n\nPercent Passing by Weight\nSieve Size SP250 SP190 SP125\n1 \u00bd in. 100 -- --\n1 in. 90-100 100 --\n\u00be in. 92 Max. 90-100 100\n\u00bd in. -- 92 Max. 90-100\n3\/8 in. -- -- 92 Max.\n#4 -- -- --\n#8 17-47 21-51 26-60\n#16 -- -- --\n#30 -- -- --\n#50 -- -- --\n#100 -- -- --\n#200 1-7 2-8 2-10\n\nDensity (Sec 403.5.2)\n\nSee also Density in Plant Inspection Density Samples in Paving Operations. One sample per sublot will be taken for QC testing. QA will randomly test one of the samples from each lot to verify that a favorable comparison is obtained. These testing requirements are minimums and should be increased as necessary. SMA mixes shall have a minimum density of 94.0% with no upper limit. All other mixes shall have a density of 94.0 \u00b12.0%.\n\nShoulder Density (Sec 403.5.2.1) and Integral Shoulder (Sec 403.5.2.2)\n\nIf the shoulders and the traveled way are placed in the same pass (integrally), the cores will be taken on the traveled way. No cores will be taken on the shoulder. For example, if the paving width is 16\u2019 with a 12\u2019 travel lane and a 4\u2019 shoulder, the shoulder will not be subject to density testing.\n\nAsphalt Content (Sec 403.5.3)\n\nQC is required to sample and test the mix for the binder content once per sublot and QA is required to independently sample and test the mix once per lot. These testing requirements are minimums and should be increased as necessary. During production, the binder content of the mix, as determined by sampling and testing, shall be within \u00b10.3% of the target listed on the JMF.\n\nVoids in the Mineral Aggregate (VMA) (Sec 403.5.4)\n\nQC is required to sample and test the mix for the VMA once per sublot and QA is required to independently sample and test the mix once per lot. These testing requirements are minimums and should be increased as necessary. The VMA of the mix shall be within \u20130.5% and +2.0% of the minimum required for the corresponding mix type (see Standard Specification Section 403.4.6.2).\n\nThe following table gives the ranges for each mix type:\n\nMix Type VMA Limits (percent)\nSP250 11.5-14.0\nSP190 12.5-15.0\nSP125 13.5-16.0\nSP095 14.5-17.0\nSP048 15.5-18.0\nSMA 16.5-19.0\n\nAir Voids (Va) (Sec 403.5.5)\n\nQC is required to sample and test the mix for the air voids once per sublot and QA is required to independently sample and test the mix once per lot. These testing requirements are minimums and should be increased as necessary. The Va for all mixes shall be 4.0 \u00b11.0%.\n\nTensile Strength Ratio (TSR) (Sec 403.5.6)\n\nThe TSR is used to evaluate the impact that water saturation and freeze-thaw cycles have on the strength of an asphalt mix. It can also be used to predict the susceptibility of the mix to stripping.\n\nDuring production, loose mix samples will be taken and quartered as described in Mixture Production Specification Limits. TSR samples do not need to be located by random numbers. However, they should be taken whenever it is convenient to production, such as during a big gap between QC volumetric tests. QC has the option of taking the loose mix samples from any point in the production process. The recommended locations are from the roadway behind the paver or from the plant. The QA sample(s) should be taken from the same point as the QC sample(s). If QC takes their sample from the plant, QA should take their sample from the plant also. This does not mean that QA should be taking their samples at the same time as QC. Two opposite quarters will be retained and the remaining two quarters will be mixed together and tested in accordance with AASHTO T283.\n\nQC should obtain enough mix to retain a sample. QC will sample and test each mix at a minimum of once every 10,000 tons, or fraction thereof. QA will independently sample and test each mix at a minimum of once every 50,000 tons. The TSR sampling requirements are best described with an example. Suppose that 112,960 tons of SP190 are to be placed on a project. By specification, QC is required to take twelve samples and QA is required to take three samples. There are two possible scenarios for sampling this mix. QC may take eleven samples representing 10,000 tons each and a twelfth sample that represents the remaining 2,960 tons. Or QC may take ten samples that represent 10,000 tons each and two samples that represent the remaining 12,960 tons (6,480 tons each). Either scenario is acceptable. Likewise, QA may take two samples representing 50,000 tons each and a third sample that represents the remaining 12,960 tons. Or QA may take one sample that represents 50,000 tons and two samples that represent the remaining 62,960 tons (31,480 tons each). The contract quantity may be used to approximate sample 1 locations.\n\nQA will send approximately 125 lbs. of loose mix (approximately 4 \u2013 13\u201d x 13\u201d x 4.5\u201d boxes) to the Central Laboratory for testing. Each box must be labeled with the SiteManager ID Mix Type VMA Limits (percent) number and the mix number. A SiteManager record must be created for each sample, which must include all required information, the mix number, lot, sublot, and the represented tonnage. The represented tonnage is explained in the example in the preceding paragraph.\n\nAdditional information that may be included in the SiteManager record is the Gmm from the sublot that the sample was taken in (QC or QA) and the specimen weight that QC has been using. The specimen weight may be different from that shown on the JMF because of bin percent changes, etc. This information is helpful because it results in less trial-and-error for the Central Laboratory.\n\nIn the laboratory, a minimum of six specimens are compacted to a height of approximately 95 mm. The air voids of the specimens are calculated. For all mixes other than SMA, the air voids must be within 7.0 \u00b10.5%. For SMA mixes, the air voids must be within 6.0 \u00b10.5%. Half of these specimens are saturated, frozen, and thawed. These are the conditioned specimens. The degree of saturation of the conditioned specimens is also calculated. The remaining specimens are unconditioned. Then, the indirect-tensile strength of all of the specimens is determined. Therefore, the TSR is the ratio of the average tensile strength of the conditioned specimens to the average tensile strength of the unconditioned specimens.\n\nA favorable comparison will be obtained if the QC and QA test results are within 10% of each other. The contractor\u2019s pay will be adjusted in accordance with Standard Specification Section 403.23.5 based on the QC test results. For example, if the QC TSR is 95% and the QA TSR is 93%, a favorable comparison has been obtained and the contractor will receive a 3% bonus. However, if the difference is greater than 10%, the field office should be consulted. The field office will evaluate the air voids and saturation levels. The raw data should be collected from QC and forwarded to the field office for comparison in order to determine whether it will be necessary to proceed with 3rd party testing. QC and QA retained samples should be kept for an extended period of time so that they may be used during dispute resolution, if necessary.\n\nThe QC data should be reported in SiteManager (Test - SAA402AB). Contractors may report their own test results using the TSR Contractor Reporting Excel to Oracle Spreadsheet available on the MoDOT Quality Management website. Furthermore, this information is quarried regularly and, provided that a favorable comparison is reached, used to signal the appropriate time for disposal of the remaining TSR sample at the Central Lab.\n\nAggregate Properties (Sec 403.5.7)\n\nThe aggregate consensus tests (Fine and Coarse Aggregate Angularity, Clay Content, and Thin, Elongated Particles) are performed on the blended aggregate. The aggregate will be sampled from the combined cold feed whether dealing with a drum-mix plant or a batch plant.\n\nFor each mix that is produced, QC shall sample the aggregate and perform the consensus tests once every 10,000 tons with a minimum of one per mix per project. QA will independently sample the aggregate and perform the consensus tests once per project. QA should also test a minimum of one QC retained sample per project. For large projects, enough QC retained samples should be tested to ensure that QC is performing the tests correctly. These testing requirements are minimums and should be increased as necessary. During production, the following tolerances are applied (see Standard Specification Sections 403.2.1 through 403.2.5 and Consensus Testing).\n\nProperty Tolerance\nFAA 2% below the minimum\nCAA 5% below the minimum\nClay Content 5% below the minimum\nThin, Elongated Particles 2% below the minimum\n\nMoisture Content (Sec 403.5.9)\n\nContamination (Sec 403.5.10)\n\nSee Material Acceptance in Paving Operations.\n\n### 403.1.6 Field Laboratory (Sec 403.6)\n\nSee Field Laboratory in Plant Inspection.\n\n### 403.1.7 Bituminous Mixing Plants (Sec 403.7)\n\nSee Batch Plants and Drum-mix Plants in Plant Inspection.\n\n### 403.1.8 Hauling Equipment (Sec 403.8)\n\nSee Haul Trucks in Paving Equipment.\n\n### 403.1.9 Pavers (Sec 403.9)\n\nSee Pavers in Paving Equipment.\n\n### 403.1.10 Construction Requirements (Sec 403.10)\n\nWeather Limitations (Sec 403.10.1)\n\nSee Weather Conditions in Paving Operations.\n\nSubstitutions (Sec 403.10.2)\n\nThe intent of this specification is that there be no additional cost to MoDOT as a result of the allowed substitution. Payment should be made for the mixture originally set up in the contract. Material codes for the substitute mixture should be entered in SiteManager on the line for which payment is being made. For example: Assume that the contractor wishes to use SP125 in lieu of the SP190 that is set up in the plans and that the SP125 has a higher contract unit price. Payment for the substitute mix should be paid as SP190. Material codes for SP125 should be added to the line for SP190 so that material quantities can be tracked and documented.\n\n### 403.1.11 Field Adjustments of Job Mix Formulas (Sec 403.11)\n\nWhen test results indicate that the mixture does not meet the specification requirements, the contractor may adjust the JMF in the field. The total binder content may be adjusted by a maximum of 0.3% from the original JMF. Virgin aggregate fractions may be adjusted as necessary except that they may not be eliminated entirely unless they are 5% or less of the original JMF. Consult the Field Office before eliminating virgin aggregate fractions greater than 5%. The addition of any new fraction will require a new mix design. The RAS fraction may be adjusted by a maximum of 3% from the original JMF. The RAP fraction may be adjusted by a maximum of 15% from the original JMF.\n\nField Mix Redesign (Sec 403.11.1)\n\nIf a new mix design is required, the contractor may redesign the mix in the field. All requirements of Standard Specification Section 403.4 will apply. A representative sample of a minimum of 50 lbs. shall be submitted with the new mix design to the Central Laboratory for verification testing.\n\nApproval (Sec 403.11.1.1)\n\nConstruction and Materials will grant approval and assign a new mix number to the mix upon successful verification.\n\nResume Production (Sec 403.11.1.2)\n\nNo mix shall be produced or placed by the contractor or accepted for use by an inspector without approval of the new field mix design from the Materials Field Office. Once the mix design has been approved, production can resume.\n\n### 403.1.12 Application of Prime or Tack (Sec 403.12)\n\nSee Surface Preparation in Paving Operations\n\n### 403.1.13 Spreading and Finishing (Sec 403.13)\n\nPaving Widths (Sec 403.13.1)\n\nStandard Specification Section 403.13.1 puts restrictions on the paving widths and lengths if the pavement is constructed under traffic.\n\nSegregation (Sec 403.13.2)\n\nSee Material Acceptance in Paving Operations.\n\nRelease to Traffic (Sec 403.13.3)\n\nTraffic must not be allowed on the pavement until its surface temperature is 140\u00b0F or less. Otherwise, the traffic will overconsolidate the mat while it is still hot and cause the pavement to be more susceptible to rutting during its early life.\n\nDraindown (Sec 403.13.4)\n\nSee Material Acceptance in Paving Operations.\n\nShoulder Substitutions (Sec 403.13.5)\n\nThe same Superpave mix that was used on the travel lanes may also be used on the shoulders. The density shall be in accordance with Standard Specification Section 403.5.2.1 if nonintegral shoulders are placed or 403.5.2.2 if integral shoulders are placed.\n\n### 403.1.14 Spot Wedging and Leveling Course (Sec 403.14)\n\nSee Surface Preparation in Paving Operations.\n\n### 403.1.15 Compaction (Sec 403.15)\n\nVibratory rollers shall be operated in static mode when the mix temperature is below 225\u00b0F. Pneumatic tire rollers shall not be used on SMA mixes. See Compaction in Paving Operations.\n\nRolling (Sec 403.15.1)\n\nDefective Mixture (Sec 403.15.2)\n\nSee Material Acceptance in Paving Operations.\n\nNon-traffic Areas (Sec 403.15.3)\n\nMixes used for non-traffic areas (medians, shoulders, and similar areas) shall be compacted to the required density. Density testing for Superpave mixes placed on the shoulders may be waived, at the RE\u2019s discretion, once the contractor has established a roller pattern that has been shown to produce the required density. This means that cores must be taken until the RE is confident that density will be obtained consistently with this roller pattern. If testing has been waived, density must still be obtained and coring may be necessary to ensure that it is. Density testing will again be required at any time that changes in the material, mix temperatures, or roller pattern are made. The intent of the specification is to attain the required density on the shoulders, particularly on full depth pavements. On resurfacing projects, the existing shoulders may not be able to withstand the compactive effort needed to attain density. In this situation, the RE can relax the density requirements, but only to the point that conditions will allow. In other words, get the most density possible without tearing up the shoulders.\n\nDensity Measurement (Sec 403.15.4)\n\nSee Density in Plant Inspection and Density Samples in Paving Operations.\n\n### 403.1.16 Joints (Sec 403.16)\n\nSee Transverse Joints and Longitudinal Joints in Paving Operations.\n\nJoint Composition (Sec 403.16.1)\n\nThe density requirements in this section apply to the traveled way pavement within 6 in. of the longitudinal joint, including the pavement on the traveled way side of the shoulder joint. All mixes, except for SMA, shall have a minimum unconfined joint density of 90.0%. SMA mixes shall have a minimum unconfined joint density of 92.0%. Confined joint densities will be evaluated with the remainder of the mat and must meet the density requirements of Standard Specification Section 403.5.2.\n\n### 403.1.17 Quality Control (Sec 403.17)\n\nQuality Control Operations (Sec 403.17.1)\n\nAsphalt Test Results (Sec 403.17.1.1)\n\nA copy of all QC test results shall be furnished to the QA inspector no later than the beginning of the day after testing has been performed. All raw data and printouts must be included with the testing records. Raw data consists of all weights, measurements,etc. used to arrive at the final test results. Printouts include the gyration\/height data from the gyratory compactor and the asphalt content ticket from the binder ignition oven or nuclear gauge. The testing records must be available to the QA inspector at all times. A self-test is a test that QC may perform between random testing to determine whether or not the mix is within specifications. Self-testing is not required and may be performed at any time and at any frequency. Generally, self-testing will be performed early in the production period. The self-test may not be completed in full. For example, QC may only compact the gyratory specimens. Doing so will yield specimen heights and the contractor may or may not make production adjustments based on these heights. Self-test samples must be clearly marked as such if they are tested and stored in the field laboratory. Self-test data may be used to determine removal limits if it is adequately documented. It should not be used for QLA under any circumstances. To be considered adequately documented the following criteria should be met:\n\n\u2022 The gyratory pucks should be clearly identified and labeled and made available for verification.\n\u2022 The gyratory printout should be available.\n\u2022 The printout from the AC test should be available.\n\nIf the preceding conditions are met and the gyratory specimens are used to troubleshoot the placement, the specimens can then be weighed and bulked to determine the volumetric properties. Data from self-tests is approximate. Its only legitimate use to the QA inspector is to help determine the point on the roadway where the mixture transitioned either above or below the removal limits. We don\u2019t want to remove acceptable mix or leave unacceptable mix in place.\n\nSee the Figure Appropriate Use of Self Tests.\n\nIt is QC\u2019s responsibility to take appropriate action if unsatisfactory mix is being produced. This may include making adjustments to the plant to bring the mix back into specification, sampling the mix from the roadway and performing complete testing, removing mix from the roadway, etc. QC is not required to provide the QA inspector with self-test results. Self-test results will never be used to determine pay factors. However, if the self-test is well documented, the results may be used to determine removal limits, if necessary. A self-test is considered well documented if the gyratory specimen(s), gyration\/height printout, and asphalt content ticket are available for QA\u2019s review. The compacted specimens should be clearly marked as self-test specimens and may be tested if necessary.\n\nInertial Profiler Test Results (Sec 610)\n\nSurface of the pavement should be thoroughly tested with an inertial profiler or straightedge as required by Sec 610. The procedures for testing with an inertial profiler and analyzing the results with the ProVAL software program are set forth in EPG 106.3.2.59 TM-59, Determination of the International Roughness Index.\n\nBituminous Quality Control Plan (Sec 403.17.2)\n\nPlant Calibration (Sec 403.17.2.2)\n\nRetained Samples (Sec 403.17.2.3)\n\nQC must retain the portion of each sample that is not tested after the sample has been reduced to testing size. This includes gradation, consensus, TSR, and volumetrics samples. The retained samples must be clearly identified in accordance with Standard Specification Section 403.17.2.3 and stored in the field laboratory for a minimum of 7 days. Also, all cores must be retained for a minimum of 7 days.\n\nQC will retain the portion of their gradation sample that is not tested. This includes the sample of the combined cold feed from a drum plant and all hot bin samples from a batch plant.\n\nLoose Mix Sample (Sec 403.17.2.3.2)\n\nA companion sample for all loose mix samples shall be taken and retained. However, the contractor is encouraged to sample a large amount of mix from the roadway, thoroughly blend the mix together, and then reduce the sample down to the necessary testing size. The portion that is not tested will be retained for possible use in the dispute resolution process. This is the preferred method because both halves should yield similar results.\n\nQuality Control Laboratory (Sec 403.17.3)\n\nCalibration Schedule (Sec 403.17.3.1)\n\nCalibrations and verifications of the testing equipment are very important. If the equipment has not been calibrated or verified as required, false test results may be obtained. The maximum intervals are given in Standard Specification Section 403.17.3.1. These frequencies are taken from the AASHTO test methods and\/or the manufacturer\u2019s recommendations.\n\nCalibration Records (Sec 403.17.3.1.2)\n\nPeriodically, the QA inspector should check the QC calibration records to ensure that the equipment has been calibrated or verified in accordance with Standard Specification Section 403.17.3.1.\n\n### 403.1.18 Quality Assurance (Sec 403.18)\n\nAssurance Testing (Sec 403.18.1)\n\nAll QA samples will be independent from QC. QA must sample enough material to retain a sample. This retained sample, as with the QC retained sample, may be used during dispute resolution. QA will randomly sample the mix from the roadway once per lot and perform volumetric testing. At the beginning of the project, QC and QA should be given the opportunity to witness each other\u2019s sampling and testing procedures. Any discrepancies should be immediately resolved at the project level, if possible. QA should test a QC retained volumetric sample once per day to ensure that both QC and QA are testing correctly. These samples should also be chosen at random (do not consistently test the retained sample from the same sublot every lot or develop a pattern).\n\nWhen both QC and QA are confident in each other\u2019s testing procedures and favorable comparisons have been obtained on the retained samples, testing of the QC retained volumetric samples may be performed on days that an independent sample is not taken. QA should test a QC retained gradation sample at a minimum of once per week. A minimum of one QC retained consensus sample should be tested per project. Again, all of the testing requirements previously mentioned are minimums and should be increased as necessary. QA test results will be furnished to the contractor no later than the day after testing has been performed. A QA\/QC Checklist is attached. For additional information see QA\/QC Questions and Answers.\n\nAggregate Comparison (Sec 403.18.2)\n\nA favorable comparison will be obtained when the independent QA sample(s) meets specifications. In addition, the QA test results of a QC retained sample must be within the following tolerances from the QC test results:\n\nProperty Percentage Points\n\u00be in. sieve and larger \u00b15.0\n\u00bd in. sieve \u00b15.0\n3\/8 in. sieve \u00b14.0\n#4 sieve \u00b14.0\n#8 sieve \u00b13.0\n#16 sieve \u00b13.0\n#30 sieve \u00b13.0\n#50 sieve \u00b12.0\n#100 sieve \u00b12.0\n#200 sieve \u00b11.0\nCAA \u00b15.0\nFAA \u00b12.0\nClay Content \u00b15.0\nThin, Elongated Particles \u00b11.0\n\nIf a favorable comparison is not obtained, dispute resolution procedures should be initiated.\n\n### 403.1.19 Acceptance of Material (Sec 403.19)\n\nRandom Numbers (Sec 403.19.1)\n\nAll random numbers will be generated by QA at least one lot in advance. This includes the random numbers for the core locations and loose mix sample locations. A copy of the random numbers will be sealed in an envelope and given to the contractor upon completion of the lot. QC samples that are used to determine the pay factors must be taken at the locations designated by the random numbers unless circumstances warrant relocation. This could include close proximity to another QC sample location in the same production period, areas where mix must be placed by hand, etc. If necessary, the random samples may be separated by 200 tons. QC should be notified of the core location after rolling has been completed. QC should be notified of the loose mix sample location approximately 100 to 150 tons before the test. The independent QA sample must be taken at the location designated by the random number unless circumstances warrant relocation. This could include close proximity to a QC sample location in the same production period, areas where mix must be placed by hand, etc. If necessary the random samples may be separated by 200 tons. The QA inspector should maintain possession of the QA density core from the time of extraction till it\u2019s tested, however; in the event that it\u2019s not possible, the inspector shall place and seal the QA core in a tamper-proof bag immediately after extraction and mark the bag label with the project number, lot number, location and inspector signature. The test results from the independent QA sample will be compared to the QC test results to determine whether or not the QC test results adequately define the characteristics of the entire lot. However, QA may take additional samples to determine if an area of concern complies with the specifications. The test results of these additional samples will not be compared to any QC test results.\n\nLots (Sec 403.19.2)\n\nFor the purposes of pay factor determination, the mat will be divided into lots with a minimum of 4 sublots per lot. The maximum sublot size is 1000 tons. If a full lot cannot be completed, the extra sublots will be added to the previous full lot and the pay factors will be determined on the large lot. If there is no previous lot, the mix will be treated as small quantities and Standard Specification Section 403.23.7.4.1 will apply.\n\nIf the target binder content is adjusted from the original JMF, a new lot shall begin. This will ensure that the binder content pay factor will represent the population of the adjusted mix. If the cold feed settings are adjusted from the original JMF alone, a new lot is not required. Adjusting the cold feed settings will change the Gsb and, therefore, the VMA of the mix. However, the VMA specification limits are based on the type of mix (see Voids in the Mineral Aggregate (VMA) (Sec 403.5.4) and do not change. The VMA is required to be within this range, even if changes are made to the JMF. A new lot sequence shall begin when a new mix design is established. The limits of adjustment can be found in Standard Specification Section 403.11.\n\nTest and Pay Factor Items (Sec 403.19.3)\n\nThe minimum sampling and testing requirements for both QC and QA, as shown in the table in Standard Specification Section 403.19.3, have been modified as a result of the QC\/QA Process Team. The guidelines set forth in this document should be followed. In regards to note b, one core equals one sample and the results will be used to determine the density pay factor for the corresponding sublot. However, if stated in the QC Plan, a maximum of two additional cores may be taken per sublot. This gives a maximum total of three cores per sublot. One core must be taken at the location selected by random numbers. The remaining cores must be taken at the same transverse offset within one foot longitudinally of the location selected by the random numbers. If more than one core is taken per sublot, all of the cores will be combined into one sample. This means that the average density of the cores will be used to determine the density pay factor for the corresponding sublot.\n\nTest Method Modification (Sec 403.19.3.1)\n\nBinder Ignition Modification (Sec 403.19.3.1.1)\n\nThis specification adjusts the temperature of the binder ignition oven due to the breakdown of certain aggregate formations as a result of intense heat.\n\nRice Test (Sec 403.19.3.1.2)\n\nIf the absorption of any aggregate fraction used in the mix is greater than 2.0%, AASHTO T209 must be modified in accordance with Standard Specification Section 403.19.3.1.2. This procedure is called the dry-back method. The final surface-dry weight will be recorded in the APIW as \u201cA2\u201d. If necessary, the dry-back method should be performed on all samples taken in the first lot of mix produced. If the initial Gmm and the dry-back Gmm of a sample are within 0.002 of each other in all sublots of the first lot, the dry-back may be reduced to every fourth sublot. Otherwise, the dry-back will be required every sublot.\n\nMiscellaneous Applications (Sec 403.19.3.2)\n\nSmall Quantities (Sec 403.19.3.2.1)\n\nA mix that requires less than 3000 tons on a project is referred to as small quantities. Testing frequencies will be as stated in Standard Specification Section 403.19.3.2.1(b). If a project is initially setup with less than 3000 tons, pay factors will not be determined unless an adjustment is made to the contract to before production begins. If a project is initially setup with more than 3000 tons but less than 3000 tons are placed, pay factor determination is not required and Standard Specification Section 403.23.7.4.1 will apply.\n\nDispute Resolution (Sec 403.19.4)\n\n### 403.1.20 Method of Measurement (Sec 403.22)\n\nWeight Determination (Sec 403.22.1)\n\nIf a batch plant is used to produce the mix, the weight of the load will be determined by the batch weights. If the mix is produced in a drum plant, the weight of the load will be determined by weighing each load of mix. This can be accomplished with either a silo scale or a truck scale. These individual load weights will be added together for the total tonnage accepted for the project and rounded to the nearest 0.1 ton.\n\nFull Depth (Sec 403.22.2)\n\n(Sec 403.22.2.1)\n\nThe final driving surface area (length multiplied by width) of the pavement will be used as the area of all underlying lifts and courses. Any mix that is placed outside of this area, including the mix used to construct the 1:1 slope, will not be directly paid for.\n\n(Sec 403.22.2.2)\n\nFull depth pavements will be paid for by the square yard. If authorized changes are made to the contract quantity during construction or if errors are found in the contract quantity, the applicable completed pavement will be measured to the nearest 0.1 yd2. The revision or correction will be added to or deducted from the contract quantity. If no changes are made or errors found, the pavement will not be measured and the contractor will be paid for the quantity of mix as shown in the contract.\n\nAlternate Overlay (Sec 403.22.3)\n\nAn overlay project may be bid as Portland cement concrete or asphalt.\n\nField Established Quantity (Sec 403.22.3.1)\n\nThe field established plan quantity is the tonnage of mix that is determined from the set or adjusted profile. This will be the contract quantity for an asphalt overlay.\n\nOverlay Measurement (Sec 403.22.3.2)\n\nOverlays will be paid for by the ton. If authorized changes are made to the contract quantity during construction, the applicable completed pavement will be measured to the nearest 0.1 ton. The revision will be added to or deducted from the contract quantity. If no changes are made, the pavement will not be measured and the contractor will be paid for the quantity of mix as shown in the contract.\n\nPavement Testing (Sec 403.22.4)\n\nSee Density Samples in Paving Operations.\n\n### 403.1.21 Basis of Payment (Sec 403.23)\n\nAggregate Variation (Sec 403.23.1)\n\nThe specific gravity of the aggregates used in the mix may fluctuate because of a variation in the quality of the rock within the quarry ledge. The gradation of the aggregate may also cause some fluctuation. However, this contribution is usually negligible. Because of such fluctuations, the quantity of aggregate used in the mix may vary from the quantity specified in the contract. Since this is expected and unavoidable, the contract unit price will not be adjusted.\n\nCompacted Samples (Sec 403.23.2)\n\nThe cost of cutting QC cores is included in the contract. Therefore, no direct payment will be made. QA samples will be paid for at $75.00 per sample, per Standard Specification Section 109.15. If one QA core is cut per location, that core is equal to one sample. If more than one QA core is cut per location, the test results will be averaged and those cores will equal one sample. Smoothness Adjustment (Sec 610.5) Diamond Grinding Diamond Grinding (Sec 403.23.4.1) Areas of the final driving surface that must be corrected by diamond grinding will be considered as a marred surface (Sec 610.5.3). A tack coat will not be applied to these areas. No direct payment will be made for diamond grinding. Tensile Strength Retained Adjustment (Sec 403.23.5) The tonnage represented by each QC TSR sample is subject to a pay adjustment that depends on the test results. The adjustments to the contract unit price are given in Standard Specification Section 403.23.5. Continuing with the sampling example in Tensile Strength Ratio (TSR) (Sec 403.5.6), the contractor takes ten samples that represent 10,000 tons each. The last two samples represent 6,480 tons each. The contractor\u2019s test results are shown in order in the table below. The price per ton is$35.00. The contract adjustment is calculated as follows:\n\nContract Adjustment = ((Percent of Contract Price-100)\/100) * Price\/ton * Tons\nTSR Tonnage Percent of Contract Price Contract Adjustment (Bonus\/Deduct)\\$\n84 10,000 100 0\n87 10,000 102 7,000\n88 10,000 102 7,000\n92 10,000 103 10,500\n86 10,000 102 7,000\n83 10,000 100 0\n81 10,000 100 0\n76 10,000 100 0\n74 10,000 98 -7,000\n80 10,000 100 0\n78 6,480 100 0\n85 6,480 102 4,536\nTotal 112,960 20,036\n\nThe Pay Factor Worksheet will automatically calculate the contract adjustment once the appropriate information has been entered. The contractor\u2019s TSR results should be recorded in the Pay Factor Worksheet that corresponds with the lot that the sample was taken in.\n\nQC will take one unconfined longitudinal joint core per sublot, if applicable. These cores will be taken within 6 in. of the unconfined longitudinal joint. Unconfined joint cores can either be located at the same longitudinal location as the corresponding mat density cores or separate random numbers can be generated. The test results for each lot will be averaged to determine compliance with the specifications. Pay adjustments will be in accordance with the following table and will be applied to the corresponding tonnage represented by the core(s):\n\nLongitudinal Joint Density (Percent of Gmm) Pay Factor (Percent of Contract Unit Price)\nFor all SP mixtures other than SP125xSM\n90.0 to 96.0 includsive 100\n96.1 to 96.5 or 89.5 to 89.9 inclusive 90\n96.6 to 97.0 or 89.0 to 89.4 inclusive 85\n97.1 to 97.5 or 88.5 to 88.9 inclusive 80\n97.6 to 98.0 or 88.0 to 88.4 75\nAbove 98.0 or Below 88.0 Remove and Replace\nFor SP125xSM mixtures:\n\u226592.0 100\n91.5 to 91.9 inclusive 90\n91.0 to 91.4 inclusive 85\n90.5 to 90.9 inclusive 80\n90.0 to 90.4 inclusive 75\nBelow 90.0 Remove and Replace\n\nIf pay reductions are necessary, the lower adjusted contract unit price of the PWL or the unconfined joint density adjustment will apply to the corresponding tonnage. For example,assume that the lot size is 4000 tons and that 1000 tons in the lot has an unconfined joint. The total pay factor for the lot due to volumetric testing is 105%. A longitudinal joint core is taken as required and the pay factor due to the unconfined joint density is 90%. As a result, a 10% reduction to the contract unit price will be applied to the 1000 tons represented by the unconfined joint and a 5% bonus will be paid for the remaining tonnage in the lot (3000 tons). On the other hand, if the pay factor due to the unconfined joint density were 100%, the 5% bonus would be paid for the entire lot. Longitudinal joint density is very important and this is an attempt to ensure that density is achieved. If it is not, the joint will ravel.\n\nPercent Within Limits (PWL) (Sec 403.23.7)\n\nThe mean (xa), standard deviation (s), Upper Quality Index (Qu), Lower Quality Index(Ql), and total percent within limits (PWLt) are calculated for each pay factor item in each lot using the equations given in Standard Specification Section 403.23.7. The PWL for an item can be determined using Table III in Standard Specification Section 502.15.8. To use this table, calculate the Qu of the item and round the result to two digits (X.XX). Find the result in the left hand column of the table and move along the row to the right until reaching the column with the corresponding n-value. The n-value is the number of test results for the item in the lot. This process yields the upper percent within limits (PWLu) of the item. Repeat this process to determine the lower percent within limits (PWLl) of the item using the Ql. Finally, calculate the PWLt. If a Q-value is negative, subtract the PWL-value from 100. The Pay Factor Worksheet will automatically calculate the PWLt for each pay factor item in each lot.\n\nQuality Level Analysis (Sec 403.23.7.1)\n\nAcceptance (Sec 403.23.7.1.1)\n\n(Sec 403.23.7.1.1.1) The QC test results will be used to determine the PWL as long as QC and QA compare favorably. If a favorable comparison is not obtained, dispute resolution procedures should be initiated. If dispute resolution is carried out to independent third party testing and the QC test results have been determined to be correct by the third party, the QC test results will be used to calculate the PWL. If the QA test results have been determined to be correct by the third party, the QA test results will be included in the PWL calculation.\n\n(Sec 403.23.7.1.1.2) A favorable comparison is obtained when the QA test results of a random, independent sample are within two standard deviations of the average of the QC test results. This determination cannot be made until all random testing for the lot has been completed. If the QC test results vary within the specification tolerances, the standard deviation will be large. In fact, as the variability in the QC test results increases, the standard deviation also increases. This results in a wide comparison range and low pay factors. On the other hand, if there is little variability in the QC test results, the standard deviation will be small. The comparison range will be narrow and the pay factors will increase. In this case, a favorable comparison is obtained when the QA test results are within one-half of the specification tolerances of the QC average. For example, the specification tolerances for VMA are \u20130.5% to +2.0%. One-half of this range is 1.25%. Therefore, a favorable comparison is obtained if the QA test result is within \u00b10.6% of the QC average.\n\nIf the comparison is not favorable, the first step is to review both QC and QA test results to see if there is any noticeable error. If no errors are found, testing of the retained samples may be performed. Judgment must be used in determining which retained sample(s) to test. When testing a retained sample, the entire suite of tests (%AC, Va, and VMA) should be performed to verify the validity of the original test results. If the test results of the retained sample confirm the original test results, the original test results are used to determine the PWL. If the test results of the retained sample verify that the original test results were incorrect, the test results of the retained sample are used to determine the PWL.\n\nIf the QC and QA test results have been determined to be valid and the comparison is still unfavorable, the test results from the random, independent QA sample will be included in the PWL calculation. The QA test results of QC retained samples or the test results from any additional QA samples will not be used in the PWL calculation. As an example, lot 3 has been completed and consists of 4 sublots. A favorable comparison was not obtained but it was determined that the QC and QA test results are valid. Therefore, the PWL calculation will include the QC test results from all 4 of the sublots and the test results of the random, independent QA sample (n = 5).\n\nA favorable comparison is obtained when the QA test results of a QC retained volumetric sample are within 0.005 of the QC Gmm test results, within 0.010 of the QC Gmb test results, and within 0.1% of the QC asphalt content test results. If larger variances occur, both QC and QA should investigate the sampling and testing procedures to identify and rectify the cause of the discrepancy.\n\nOutliers (Sec 403.23.7.1.2)\n\nIf it is suspected that an individual QC test result is an outlier, the entire lot of QC test results may be checked in accordance with Standard Specification Section 403.23.7.1.2. The eligible measured test results are Gmb, Gmc, Gmm, and %AC. Gmb, Gmc, and Gmm shall be carried out to three decimal places (X.XXX) and the %AC shall be carried out to two decimal places (X.XX). On the other hand, Va, VMA, and density are not eligible because these are calculated volumetric properties.\n\nIf an outlier is found, QC may test the retained sample from the corresponding sublot.Again, the entire suite of tests (%AC, Gmb, and Gmm) must be performed. If the test results from the retained sample confirm the original test results, the original test results will be used to calculate the PWL. If the test results from the retained sample do not confirm the original test results,the test results from the retained sample will be used to calculate the PWL.\n\nWhen any change is made in the JMF, the previous test results cannot be used for future outlier calculations since the mix has changed. For example, if the contractor has made a change in sublot 2B and wants to check for an outlier in sublot 2D, the results from sublot 2A cannot be used since the mix is not the same.\n\nRandom Sampling (Sec 403.23.7.1.4)\n\nSee Random Numbers in EPG 403.2.19 Acceptance of Material (Sec 403.19).\n\nPay Factors (Sec 403.23.7.2)\n\nThe density (PFdensity), asphalt content (PFAC), VMA (PFVMA), and air voids (PFVa) pay factors are calculated for each lot using the corresponding PWLt and the equations in Standard Specification Section 403.23.7.2. The total pay factor (PFT) is then calculated for each lot using the average of the individual pay factors. If coring is not required, such as on a leveling course or non-integral shoulders, the PFT will be calculated for each lot using the average of the PFAC, PFVMA, and PFVa.\n\nThe contract adjustment is used to adjust the contractor\u2019s pay to reflect the quality of the mix. The contractor may receive a bonus if the quality of the mix is good. On the other hand, if the quality of the mix is poor, a deduction will be applied. The contract adjustment is calculated by subtracting 100% from the PFT. The dollar amount of the bonus or deduction is determined by multiplying the unit bid price, the quantity of mix in the lot, and the contract adjustment (in decimal form) together.\n\nMix is typically produced and measured by the ton. Therefore, in order to eliminate confusion and excessive conversions on square yard projects (full depth pavements), the lots will be tracked by tonnage. When the pay factors are calculated at the end of the lot, the \u201cSquare Yard Calculator\u201d in the Pay Factor Worksheet can be used to determine the square yards in the lot. This is best explained with an example:\n\nOn a full-depth paving project, the total thickness of the pavement is 12 in. and the contractor is placing two lifts of SP190, one 6 in. lift and one 4.25 in. lift. The final lift of SP125 is 1.75 in. thick. The lot size is 3000 tons. Suppose that one lot of SP190 has been completed. The total thickness of the pavement and the lift thicknesses are entered in the appropriate cells in the \u201cSquare Yard Calculator\u201d. The length and width of the lot must be measured manually. The width of the lot is 12 ft., the length of the first lift is 4650 ft., and the length of the second lift is 3300 ft. Therefore, the area of the first lift is 6200.0 yd2 (12 ft. * 4650 ft. = 55800 ft2 * (1 yd2\/9 ft2) = 6200 yd2) and is entered in the appropriate cell. The area of the second lift is 4400.0 yd2(12 ft. * 3300 ft. = 39600 ft2 * (1 yd2\/9ft2) = 4400 yd2) and is entered in the appropriate cell. The square yardage represented by each lift is calculated by multiplying the square yards by the lift thickness divided by the total pavement thickness. Therefore, the square yardage of the first lift is 3100.0 yd2 (6200 yd2 * (6 in.\/12 in.) = 3100 yd2) and the square yardage of the second lift is 1558.3 yd2 (4400 yd2 * (4.25 in.\/12 in.) = 1558.3 yd2). This lot represents 4658.3 square yards. This procedure is followed for the remaining lots.\n\nDensity Pay Factor (Sec 403.23.7.2.1)\n\nDensity is calculated using the Gmc of the core and the Gmm of the mix. The PFdensity for each lot is calculated using the density test results of all of the sublots. Cores that are cut in half, as required by Standard Specification Section 403.15.4, will double the number of test results used to determine PFdensity. For example, suppose that the contractor is placing SP190 in 8\u201d lifts and 4 cores are taken per lot, 1 per sublot. The lift is being placed thicker than 6 times the nominal maximum size aggregate used in the mix. By specification, the cores are to be cut in half and the density of each half determined separately. Therefore, 8 test results (as opposed to 4) will be used to determine the PFdensity for the lot.\n\nAsphalt Content Pay Factor (Sec 403.23.7.2.2)\n\nThe PFAC for each lot is calculated using the binder content test results of all of the sublots.\n\nVoids in the Mineral Aggregate and Air Voids Pay Factor (Sec 403.23.7.2.3)\n\nThe Va, VMA, and VFA are calculated using the average Gmb of the compacted gyratory specimens, the Gmm of the mix, the percent stone (Ps) of the mix, and the Gsb of the combined aggregate. The Ps is determined by subtracting the percent binder (Pb) from 100%. The Gsb will be that listed on the JMF. The PFVa and PFVMA for each lot are calculated using the Va and VMA test results of all of the sublots.\n\nRemoval of Material (Sec 403.23.7.3)\n\nIf the PFT for a lot is less than 50.0, the entire lot must be removed and replaced at the contractor\u2019s expense. If the QC test results for density and\/or air voids fall below the removal limits in any sublot, the affected mix must be removed and replaced at the contractor\u2019s expense. The specifications state that the entire sublot must be removed. However, in some cases only a portion of the affected sublot(s) may require removal. Therefore, the limits of removal will be left up to the Resident Engineer's discretion. QC self-test results may be used to help define the limits of removal as long as the self-test(s) are well documented (see Asphalt Test Results (Sec 403.17.1.1) for the documentation requirements). The replacement mix will be sampled and tested as required. These test results will be used to calculate the PWL for the lot.\n\nIf the QA test results fall below the removal limits for density and\/or air voids, the mix should stay in place if a favorable comparison has been obtained with the QC test results. Again, a favorable comparison signifies that the QC test results adequately define the characteristics of the lot and are, therefore, acceptable. If the QA test results fall below the removal limits and a favorable comparison has not been obtained, dispute resolution should be initiated to determine whether or not the mix should stay in place.\n\nMiscellaneous Applications (Sec 403.23.7.4)\n\nSmall Quantities (Sec 403.23.7.4.1)\n\nFor small quantity projects consisting of less than 3000 tons, the statistical analysis of the mix is not required. Therefore, pay factors will not be determined. However, the mix must meet density, binder content, VMA, and Va specifications. The testing frequencies are stated in Standard Specification Section 403.19.3.2.1(b). Density will be adjusted in accordance with the table in Standard Specification Section 403.23.7.4.1(b). TSR testing is also required.\n\nBase Widening and Entrances (Sec 403.23.7.4.2)\n\nSingle Lift or Leveling Course Work (Sec 403.23.7.4.3)\n\nThis specification does not apply to \u201cmill and fill\u201d projects.\n\n## 403.2 Materials Inspection for Sec 403\n\n### 403.2.1 Scope\n\nTo establish procedures for mix design of asphaltic concrete pavement, inspection and acceptance of bituminous mixture. Ingredients for use in asphaltic concrete pavement are to be inspected in accordance with the applicable sections of this Manual. Plant calibrations, if requested, will be performed in accordance with EPG 106.4 Plant Inspections.\n\n### 403.2.2 Mix Design Procedure\n\nIn order for an asphaltic concrete mix formula to be approved, the contractor\u2019s proposed job mix formula (JMF) shall be submitted as required in Specification Section 403.3.1. The time for approval starts when the completed design is delivered to the District. This time restarts when the District receives information omitted from the original JMF or corrected by the contractor. Review times include District and Central Office processing, therefore, each mix should be processed as soon as possible. When the contractor is not accepted as a participant in the AASHTO Proficiency Sampling Program, material sampling is required for mixture verification. Trial mix samples must be obtained and submitted to the Central Laboratory in accordance with Section 1001 of the Materials Manual. When possible, the JMF and correspondence should be transmitted electronically. The Materials Field Office e-mail address is MFO.\n\nDistrict Procedure\n\nWhen the District receives a proposed trial mix formula, as required by the Standard Specifications, the mixture properties, components and proportions should be checked to ensure compliance with Specifications and that they are approved for the intended use. It may be necessary for the District to advise the contractor to make changes in the proposed mixture in order to comply with Department policies. The District shall provide SiteManager ID\u2019s and all pertinent information (gradation, deleterious, etc.) for each fraction of aggregate used in the mixture. A QC plan in accordance with Section 1001 of the Materials Manual covering each aggregate fraction should be on file in the District Office or received with the JMF. The target gradations shown on the QC plan and JMF must match. When the District is satisfied that the proposed mixture is acceptable, a copy of the JMF and the contractor's letter shall be submitted to the Materials Field Office, accompanied by a letter of transmittal with comments, any corrections made and recommendations. The transmittal letter shall contain the following information:\n\nProject information \u2013 Job Number, Route, County, Contract Number.\nMixture Types\nGrade and All Possible Sources of Asphalt Binder Intended for Use\nLetting Date\nProposed Work \u2013 Type of Work, Job Location and Length\nTotal Trucks\nTotal Combination Trucks\nMix Use \u2013 Mainline, Shoulders, Outer Roads, Entrances, etc.\nQuantity of Mix\n\nIncluded in the letter should be information regarding the approximate date on which the contractor intends to begin placing the mixture on the roadway, the type of mixture needed first and whether the JMF is submitted for a 7-day review or verification. Information concerning plant location, type of plant to be used, etc., is beneficial.\n\nIf the mixture design is performed by a laboratory participating in the AASHTO Proficiency Sampling Program with a rating of 3 or more on the applicable test methods, trial mix material does not need to be submitted to the Laboratory unless one or more of the following conditions apply:\n\na. Nuclear calibration for MoDOT asphalt content gauges is needed.\nb. Material for full verification of the mixture is requested by the Field Office.\nc. District personnel have concerns over any aspect of the mix design.\n\nRequests for previously approved mixes shall be submitted and will be approved in accordance with applicable portions of EPG 403 Asphaltic Concrete Pavement. Approved mixtures may be transferred within the time limit of 3 years from the approval date. Transfer requests for mixtures that exhibited poor performance or field problems should be denied.\n\nUpon request by the contractor, the District has authority to change the source of mineral filler, hydrated lime, natural sand from the Missouri and Mississippi Rivers or asphalt binder. However, constant changing throughout a project should not be allowed. The contractor must provide reasonable justification for changing sources during the course of a project. Any adjustments should be made to the JMF to reflect changed properties caused by the new source. (e.g. Change in Gsb, gradation, asphalt content, etc.)\n\nApproval of a new mix design shall be obtained prior to changing the source of aggregates used in a mixture.\n\nField Office Procedure\n\nThe Materials Field Office is charged with the responsibility of processing the mix formula. General procedures for processing an asphaltic concrete mix formula are as follows:\n\na. A letter from a District requesting a mix with a copy of the contractor's JMF and letter is received.\nb. Contract Special Provisions for the project are checked for necessary items.\nc. The urgency of the mix and the status of trial mix samples in the Central Laboratory are reviewed.\nd. Grade of asphalt, possible sources intended for use and the percent asphalt recommended are reviewed.\ne. Gradations of the individual aggregates are checked for specification compliance and compared with the gradations determined by the Laboratory.\nf. All calculations on the proposed JMF are checked.\ng. For verification, a one-point trial is prepared and submitted to the Laboratory.\nh. When Central Laboratory tests are completed, the results are compared to the contractor\u2019s and against the specifications. If the mixes tested cannot be used, the mixture will be rejected.\ni. Formulas to check aggregate and mixture properties are shown in LS 403.\nj. Absorption values obtained by AASHTO T85 or T85 Combined will be shown on the JMF for determining which mixtures require the optional dry-back procedure of AASHTO T209.\nk. Upon JMF approval, a unique JMF number will be created and entered into SiteManager for each possible asphalt source. Each JMF will have a two letter suffix abbreviation for the binder supplier name and location.\n\n### 403.2.3 Field Adjustments of Superpave Mix Design\n\nThe specification criteria are to be used to determine whether or not the mixture meets the specifications. When a mixture is field adjusted, the contractor is to notify the inspector prior to making the adjustment. A new Gsb is required when cold feed adjustments are made. A new lot will begin with any change in asphalt content. Adjustments beyond the limits set in the specifications will require a new mix design. Field adjusted mixture changes are not required to be sent to the Field Office, however, the District will track the changes to ensure proper material quantities are inspected.\n\n### 403.2.4 Field Superpave Mix Design\n\nWhen a field mix design is needed, the contractor must first notify the engineer (the Materials Field Office is to be notified immediately). During the design and verification process, no mixture is to be placed on the project. A plan for producing, sampling and verifying the proposed field mix design is to be agreed on between the contractor and the engineer. One hundred (100) pounds of loose mixture will be required in the Central Laboratory and the Materials Field Office will approve or deny the field mix design. In order to be accepted for use, the test results must meet all of the following:\n\na. Minimum VMA for the mixture type, i.e., 12.0 minimum for 250 mixes, 13.0 for 190 mixes, 14.0 for 125 mixes, and 17.0 for 125 SMA mixes.\nb. Asphalt content within 0.3\u00a0% of the adjusted target. For example, if the contractor chose to lower the asphalt content from 5.0 percent to 4.8 percent for a field adjustment, the initial test results must be within 0.3\u00a0% of 4.8 percent.\nc. Air voids of 4.0 +\/- 0.5\nd. TSR result is equal to or greater than 80%.\n\nThe contractor and Central Laboratory may run the moisture sensitivity test simultaneously. If the contractor\u2019s test results meet the above criteria, and the results are verified by the Central Laboratory, the target VMA will be set at the contractor's test result, the target AC content will be set at the target set for the field mix design, and the air voids target will be set at 4.0.\n\n### 403.2.5 Report\n\nA letter of transmittal will accompany the approved mixture to the District Construction and Materials Engineer with distribution as follows:\n\nTitle (e-mail address) Copy of Letter of\n\nTransmittal & Approved Mix\n\nDistrict Construction and Materials Engineer (D#MaContacts) 1\nConstruction and Materials Clerk 1\nResident Engineer (POorg) 1\nPhysical Laboratory Director (PLO) 1\nChemical Laboratory Director (Extraction) 1\nDesign Representative 1\nField Office File 1\nContractor 1\n\nThe letter of transmittal and the approved mixture will be sent by electronic mail to the individuals listed above.\n\nA copy of the approved formula accompanied by a letter of transmittal from the District Construction and Materials Engineer is to be forwarded to the contractor when an electronic mail address for the contractor has not been provided.\n\n## 403.3 Laboratory Procedures for Sec 403\n\nThis establishes procedures for Laboratory testing and reporting of hot mix asphalt trial mixtures and field compacted density samples.\n\n### 403.3.1 Procedure\n\n#### 403.3.1.1 Trial Mixtures\n\nTest results and calculations required for asphaltic concrete trial mixtures are as shown in Batching below. Test results and calculations shall be recorded through SiteManager. Standard test procedures are as follows:\n\nMixture Property Test Method\nTheoretical Maximum Specific Gravity (Rice) AASHTO T 209\nSpecific Gravity of Compacted Mixture AASHTO T 166, AASHTO T 275\nCompaction and Stability Using Marshall Method AASHTO T 245\nCompaction Using Gyratory Compactor AASHTO T 312\nMoisture Sensitivity AASHTO T 283\nDraindown of Mixture AASHTO T 305\nAsphalt Content AASHTO T 308, MoDOT TM-54\n\n403.3.1.1.1 Batching\n\nAggregates shall be divided into the individual aggregate sizes down to either the minus No. 8 or minus No. 16 sieve except when inconsistencies occur during testing where it may be advantageous to divide to the minus No. 200 material.\n\nSP BB BP\n1 1\/2 in.\n1 to 3\/4 in. 1 to 3\/4 in. 1 to 3\/4 in.\n3\/4 to 1\/2 in. 3\/4 to 1\/2 in. 3\/4 to 1\/2 in.\n1\/2 to 3\/8 in. 1\/2 in. to No. 4 1\/2 in. to No. 4\n3\/8 in. to No. 4\nNo. 4 to No. 8 No. 4 to No. 8 No. 4 to No. 8\nNo. 8 or No. 8 to No. 16 No. 8 No. 8\nNo. 16\nAdditional sizes for batching to No. 200\n(No. 16 to No. 30) (No. 8 to No. 30) (No. 8 to No. 30)\n(No. 30 to No. 50) (No. 30 to No. 200) (No. 30 to No. 200)\n(No. 50 to No. 100)\n(No. 100 to No. 200)\n(No. 200) (No. 200) (No. 200)\nSizes shown in parenthesis () are used when it is desired to batch a partcular fraction down to the minus No. 200.\n\n403.3.1.1.2 Weigh each size aggregate according to the batch sheet in a tared pan and place into a larger pan for combining sizes. Repeat process for number of samples desired. The combined weight of a batch containing 3\/4 in. plus material must be within 5 g of the target weight and a batch with minus 3\/4 in. material must be within 3 g of the target weight. Otherwise, the sample shall be rebatched. If only one sample is out of tolerance, it may be swapped with the butter sample. The butter does not have to meet these tolerances.\n\n403.3.1.1.3 Place samples and asphalt binder in an oven set at the mixing temperature. Heat in the oven a minimum of 2 hours. Aggregates may be heated no more than 28\u00b0 C (50\u00b0 F) above the mixing temperature in order to maintain the desired temperature throughout mixing.\n\n403.3.1.1.4 Use a Hobart mixer with a wire whip to mix samples unless otherwise instructed to use a bucket mixer. Remove heated aggregate from oven and put in heated mixing bowl (bucket) that has been tared and weigh. Dry-mix thoroughly, then make a crater in the center of the aggregate to pool the binder. Calculate the proper binder weight for the actual weight of aggregate and add the binder to the aggregate to attain the proper weight. Recalculate the percent binder based on actual weights. If it is not within 0.02% of the target percent, add or remove binder to get within this tolerance. Mix until all aggregate is uniformly coated. The first batch shall be the butter. Scrape the mixing bowl with a spatula to remove as much of the mixture as possible. For each successive batch, the bowl should be scraped to the same level of removal as the butter. Calculation of the binder weight shall be as follows:\n\n${\\displaystyle BinderWeight={\\frac {Wt.Agg.}{100-P_{b}}}x100}$\n\n403.3.1.1.5 Split the mixtures batched in combined batches into individual specimens. Mixtures shall be placed in pans, at a depth of 1 to 2 in., then placed in a force-draft oven at the compaction temperature for the mixture and short-term aged for 2 hours. At one hour, stir Superpave mixtures making sure free binder and fines are scraped from the bottom and stirred evenly back into the mixture and place back in the oven. Other mixtures are aged with no stirring during the 2 hours.\n\n403.3.1.1.6 After aging, stir mixtures, scraping binder and fines from the pan, and ensure all components are uniformly incorporated into the mixture.\n\n403.3.1.1.7 The mixture shall be tested according to the appropriate test method.\n\n403.3.1.1.8 Percent voids of the mixture (Va) shall be calculated as follows:\n\n${\\displaystyle V_{a}=\\left(1-{\\frac {G_{mb}}{G_{mm}}}\\right)x100}$\n\n403.3.1.1.9 Percent voids in mineral aggregate (VMA) shall be calculated as follows:\n\n${\\displaystyle VMA=100-\\left({\\frac {G_{mb}xP_{s}}{G_{sb}}}\\right)x100}$\nWhere:\nPs = Percent stone in mixture = 100 \u2013 Percent AC\nGsb = Bulk specific gravity of combined aggregate fractions\n${\\displaystyle G_{sb}={\\frac {P_{1}+P_{2}+...P_{n}}{{\\frac {P_{1}}{G_{1}}}+{\\frac {P_{2}}{G_{2}}}+...{\\frac {P_{n}}{G_{n}}}}}}$\nWhere:\nP1, P2, Pn = Bin percentages by mass of aggregate\nG1, G2, Gn = Individual bulk specific gravities of aggregate\n\n403.3.1.1.10 Percent aggregate voids filled with asphalt binder (VFA) shall be calculated as follows:\n\n${\\displaystyle VFA=\\left(1-{\\frac {V_{a}}{VMA}}\\right)x100}$\n\n403.3.1.1.11 Limestone-Porphyry (LP) and SMA mixtures with porphyry requirements by volume of the plus No. 8 material shall be calculated as follows:\n\n${\\displaystyle \\%Porphyry_{+No.8}=\\left[{\\frac {{\\frac {\\%R_{P1+No.8}}{G_{P1}}}+{\\frac {\\%R_{P2+No.8}}{G_{P2}}}...+{\\frac {\\%R_{Pn+No.8}}{G_{Pn}}}}{{\\frac {\\%R_{1+No.8}\\times \\;P_{1}}{G_{1}}}...+{\\frac {\\%R_{n+No.8}\\times \\;P_{n}}{G_{n}}}+{\\frac {\\%R_{P1+No.8}\\times \\;P_{P1}}{G_{P1}}}...+{\\frac {\\%R_{Pn+No.8}\\times \\;P_{Pn}}{G_{Pn}}}}}\\right\\rbrack \\times \\;100}$\nWhere:\n%R1+No.8, %R2+No.8, %Rn+No.8 = Percent of plus #8 non-porphyry aggregate.(100-P#8)\nG1, G2, Gn = Bulk specific gravity of non-porphyry material by AASHTO T 85.\n%RP1+No.8, %RP2+No.8, %RPn+No.8 = Percent of plus #8 porphyry aggregate. (100-P#8)\nGP1, GP2, GPn = Bulk specific gravity of porphyry material by AASHTO T 85.\n\n#### 403.3.1.2 Field Compacted Density Samples\n\nSamples, as submitted from the field, may be routine or Independent Assurance Samples. Tests and calculations shall consist of specific gravity and percent density. Test results and calculations shall be recorded on work Forms \"Asphaltic Concrete (Field Sample)\" and \"Report of Tests on Sample of Asphaltic Concrete\".\n\nSpecific gravity of the field compacted sample shall be determined in accordance with AASHTO T166 except cores to be averaged may be tested as one sample or individually.\n\nPercent density of the compacted sample shall be calculated as follows:\n\n${\\displaystyle PercentDensity={\\frac {G_{mc}}{G_{mm}}}\\times \\;100}$\nWhere:\nGmc=Specific Gravity of Sample\n\n### 403.3.2 Sample Record\n\n#### 403.3.2.1 Trial Mixtures\n\nThe sample record shall be completed in SiteManager, as described in Automation Section, and shall indicate acceptance, qualified acceptance, or rejection. Appropriate remarks, as described in EPG 106.20 Reporting, are to be included in the remarks to clarify conditions of acceptance or rejection. Test results shall be reported on the appropriate templates under the Tests tab.\n\n#### 403.3.2.2 Field Compacted Density Samples\n\nThe sample record shall be completed in SiteManager, as described in Automation Section, and shall include a notation in the remarks, \"Material submitted for the determinations indicated\". Test results shall be reported on the appropriate templates under the Tests tab.","date":"2021-01-27 08:05:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 7, \"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\": 0.5357901453971863, \"perplexity\": 3278.0513947023683}, \"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-2021-04\/segments\/1610704821253.82\/warc\/CC-MAIN-20210127055122-20210127085122-00533.warc.gz\"}"} | null | null |
{"url":"https:\/\/physicsoverflow.org\/16891\/can-we-treat-%24-psi-c-%24-as-a-field-independent-from-%24-psi%24","text":"# Can we treat $\\psi^{c}$ as a field independent from $\\psi$?\n\n+ 5 like - 0 dislike\n87 views\n\nWhen we derive the Dirac equation from the Lagrangian, $$\\mathcal{L}=\\overline{\\psi}i\\gamma^{\\mu}\\partial_{\\mu}\\psi-m\\overline{\\psi}\\psi,$$ we assume $\\psi$ and $\\overline{\\psi}=\\psi^{*^{T}}\\gamma^{0}$ are independent. So when we take the derivative of the Lagrangian with respect to $\\overline{\\psi}$, we get the Dirac equation $$0=\\partial_{\\mu}\\frac{\\partial\\mathcal{L}}{\\partial\\left(\\partial_{\\mu}\\overline{\\psi}\\right)}=\\frac{\\partial\\mathcal{L}}{\\partial\\overline{\\psi}}=\\left(i\\gamma^{\\mu}\\partial_{\\mu}-m\\right)\\psi.$$\n\nNow if we include a term with charge conjugation, $\\psi^{c}=-i\\gamma^{2}\\psi^{*}$, into the Lagrangian (like $\\Delta\\mathcal{L}=\\overline{\\psi}\\psi^{c}$), does this $\\psi^c$ depend on $\\overline{\\psi}$ or $\\psi$? Why or why not?\n\nIf $\\psi^{c}$ depends on $\\psi$, why wouldn't the reason that $\\overline{\\psi}$ and $\\psi$ are independent apply for $\\psi^{c}$ and $\\psi$?\n\nIf $\\psi^{c}$ depends on $\\overline{\\psi}$, how should we take derivative of $\\Delta\\mathcal{L}$ with respect to $\\overline{\\psi}$?\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user Louis Yang\nPossible related? physics.stackexchange.com\/q\/89002\/29216\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user BMS\n\n+ 1 like - 0 dislike\n\nThe Dirac spinor $\\psi$ and its complex conjugate $\\psi^*$ are not independent variables, but in some calculations one can treat them as such.\n\nFor the similar question about a complex scalar field $\\phi$ and its complex conjugate $\\phi^*$, see e.g. this Phys.SE post.\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user Qmechanic\nanswered Apr 26, 2014 by (2,860 points)\nThanks for your brief answer. I am confused. For a complex variable $z$ one can always write it as real and imaginary parts $z=x+iy$. If you compute $\\frac{\\partial z^{*}}{\\partial z}$ or $\\frac{\\partial z}{\\partial z^{*}}$, they are both zero. So this is the same reason why one should take $\\frac{\\partial\\phi^{*}}{\\partial\\phi}=0$, right?\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user Louis Yang\nRecalling the precise definition of $\\frac{\\partial z^{*}}{\\partial z}=0=\\frac{\\partial z}{\\partial z^{*}}$, it does not necessarily imply that $z$ and $z^{*}$ are independent variables. On one hand, if $z^{*}$ denotes the complex conjugate of $z$ (so that $z$ and $z^{*}$ are not independent variables), then $\\frac{\\partial z^{*}}{\\partial z}=0=\\frac{\\partial z}{\\partial z^{*}}$ are merely consequences of pertinent definitions. On the other hand, if $z$ and $z^{*}$ are truly indep. complex variables, then $\\frac{\\partial z^{*}}{\\partial z}=0=\\frac{\\partial z}{\\partial z^{*}}$ is automatic.\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user Qmechanic\nOne can always write $x=\\frac{z+z^{*}}{2}$ and $y=\\frac{z-z^{*}}{2i}$. Then one can express the derivative as $\\partial_{z}=\\frac{\\partial x}{\\partial z}\\partial_{x}+\\frac{\\partial y}{\\partial z}\\partial_{y}=\\frac{\\partial_{x}-i\\partial_{y}}{2}$ So one get $\\frac{\\partial z^{*}}{\\partial z}=\\frac{\\partial z}{\\partial z^{*}}=0$. Maybe \"independent\" is a not a good word to describe it, but at least it is the derivative that enter the derivation of Euler-Lagrange equation.\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user Louis Yang\n+ 0 like - 0 dislike\n\nYes, when we want to obtain the equation of motion using Euler-Lagrange equation, we should treat $\\psi$ and $\\psi^c$ independent, but $\\overline{\\psi}$ and $\\psi^c$ dependent. The reason for this is that we can simply expressed $\\psi^c$ in terms of $\\overline{\\psi}$ by $$\\psi^{c}=C\\overline{\\psi}^{T},$$ where $C=-i\\gamma^{2}\\gamma^{0}$ is the charge conjugation matrix. So $\\overline{\\psi}$ and $\\psi^c$ are the same degree of freedom.\n\nFor the derivative of $\\overline{\\psi}\\psi^{c}$ with respect to $\\overline{\\psi}$, one should be really careful because $\\psi$ is anticommuting. Since the derivative in Euler-Lagrange equation actually comes from the variation of Lagrangian, We should start from the variation \\begin{eqnarray} \\delta\\left(\\overline{\\psi}\\psi^{c}\\right) & = & \\delta\\left(\\overline{\\psi}C\\overline{\\psi}^{T}\\right)=\\delta\\left(\\overline{\\psi_{i}}C_{ij}\\overline{\\psi_{j}}\\right)=\\delta\\left(\\overline{\\psi_{i}}\\right)C_{ij}\\overline{\\psi_{j}}+\\overline{\\psi_{i}}C_{ij}\\delta\\overline{\\psi_{j}}\\\\ & = & \\delta\\left(\\overline{\\psi_{i}}\\right)C_{ij}\\overline{\\psi_{j}}-\\delta\\left(\\overline{\\psi_{j}}\\right)C_{ij}\\overline{\\psi_{i}}, \\end{eqnarray} where I use the anticommutation of the fields to get the minus sign for the last step. Now notice that $C^{T}=C^{+}=-C$. So the last term is $$-\\delta\\left(\\overline{\\psi_{j}}\\right)C_{ij}\\overline{\\psi_{i}}=\\delta\\left(\\overline{\\psi_{j}}\\right)C_{ji}\\overline{\\psi_{i}}=\\delta\\left(\\overline{\\psi_{i}}\\right)C_{ij}\\overline{\\psi_{j}}.$$ and we get $\\delta\\left(\\overline{\\psi}\\psi^{c}\\right)=2\\delta\\left(\\overline{\\psi}\\right)C\\overline{\\psi}^{T}.$ Therefore, the equation of motion from this term is $$\\frac{\\partial}{\\partial\\overline{\\psi}}\\left(\\overline{\\psi}\\psi^{c}\\right)=2C\\overline{\\psi}^{T}=2\\psi^{c}.$$\n\nThis post imported from StackExchange Physics at 2014-05-04 11:36 (UCT), posted by SE-user Louis Yang\nanswered Apr 27, 2014 by (90 points)\n\n Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the \"link\" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)\u00a0\u00a0 Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\\varnothing$ in the following word:p$\\hbar$ysi$\\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.","date":"2019-11-18 21:24:06","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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.88763028383255, \"perplexity\": 457.4180181110561}, \"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-2019-47\/segments\/1573496669847.1\/warc\/CC-MAIN-20191118205402-20191118233402-00331.warc.gz\"}"} | null | null |
{"url":"http:\/\/mathhelpforum.com\/advanced-algebra\/177793-eigenvalue-multiplicity-2-a.html","text":"Math Help - Eigenvalue with multiplicity of 2?\n\n1. Eigenvalue with multiplicity of 2?\n\nhaha im an idiot, i got it figured out","date":"2014-07-22 10:13:19","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.938775897026062, \"perplexity\": 1914.0679612934086}, \"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-2014-23\/segments\/1405997857714.64\/warc\/CC-MAIN-20140722025737-00145-ip-10-33-131-23.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/hpmuseum.org\/forum\/showthread.php?tid=8327&pid=73284&mode=threaded","text":"Deltadays and date functions.\n05-13-2017, 08:57 PM\nPost: #27\n Dieter Senior Member Posts: 2,397 Joined: Dec 2013\nRE: Deltadays and date functions.\n(05-13-2017 07:15 PM)Vtile Wrote: \u00a0....much...\n\nI have read this a few times but I am still not sure if I understand you correctly.\n\nSo here is a short description of the way the JDN formula works. Forget about everything else and give it a try:\n\nFirst of all we define that a year starts with 1 March. This way the irregularity in February appears at the very end of a year where it does not matter if for the last day of the year 28 or 29 days are finally added. So January and February are considered months 13 and 14 of the previous year (m:=m+12, y:=y-1).\n\nSince 1 March of \"year 0\" there were y full years with 365 days each, i.e. 365*y days. Plus one additional day every 4 years, i.e. int(y\/4) more days. Remember that Jan and Feb belong to the previous year, so if y is a leap year the one more day is only added for dates in March and later: if y is divisible by 4, y\u20131 is not and one less day is added.\n\nThis way we get the number of days between 1 March 0 and 1 March of year y.\n\nNext we add the number of days from 1 March in year y to the first day of month m.\nThis is the tricky part - we need a function that returns the accumulated number of days in the previous months since March, i.e. something like this:\n\nCode:\nmonth\u00a0\u00a0#days \u00a0\u00a03\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a00 \u00a0\u00a04\u00a0\u00a0\u00a0\u00a0\u00a0\u00a031 \u00a0\u00a05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a061 \u00a0\u00a06\u00a0\u00a0\u00a0\u00a0\u00a0\u00a092 \u00a0\u00a07\u00a0\u00a0\u00a0\u00a0\u00a0122 \u00a0\u00a08\u00a0\u00a0\u00a0\u00a0\u00a0153 \u00a0\u00a09\u00a0\u00a0\u00a0\u00a0\u00a0184 \u00a010\u00a0\u00a0\u00a0\u00a0\u00a0214 \u00a011\u00a0\u00a0\u00a0\u00a0\u00a0245 \u00a012\u00a0\u00a0\u00a0\u00a0\u00a0275 \u00a013\u00a0\u00a0\u00a0\u00a0\u00a0306 \u00a014\u00a0\u00a0\u00a0\u00a0\u00a0337\n\nExample: at the beginning of month 5 (May) there were 61 days in previous months (31 in March plus 30 in April).\n\nNote that #days adds up only months with 30 or 31 days because February is the last month of the year, so the last added days (306 to 337) are are the 31 in January.\n\nThis is a nearly linear function, so a straight line can be interpolated:\n\nCode:\nmonth\u00a0\u00a030,6*m\u00a0-\u00a091,4 \u00a0\u00a03\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a00,4 \u00a0\u00a04\u00a0\u00a0\u00a0\u00a0\u00a0\u00a031 \u00a0\u00a05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a061,6 \u00a0\u00a06\u00a0\u00a0\u00a0\u00a0\u00a0\u00a092,2 \u00a0\u00a07\u00a0\u00a0\u00a0\u00a0\u00a0122,8 \u00a0\u00a08\u00a0\u00a0\u00a0\u00a0\u00a0153,4 \u00a0\u00a09\u00a0\u00a0\u00a0\u00a0\u00a0184 \u00a010\u00a0\u00a0\u00a0\u00a0\u00a0214,6 \u00a011\u00a0\u00a0\u00a0\u00a0\u00a0245,2 \u00a012\u00a0\u00a0\u00a0\u00a0\u00a0275,8 \u00a013\u00a0\u00a0\u00a0\u00a0\u00a0306,4 \u00a014\u00a0\u00a0\u00a0\u00a0\u00a0337\n\nThe interpolation line's slope of 30,6 equals 153\/5. This is because the number of days sequence consists of two groups of five months with 31\u201330\u201331\u201330\u201331 days, i.e. 153\u00a0altogether. Which means an average of 30,6\u00a0days per month. Note the 153 and 306\u00a0days in the table.\n\nNow simply take the integer part of the right column and the result is exactly what we want.\nSo the days from 1 March to the first day of month m are\n\nint(30,6*m - 91,4) = int(30,6*m + 30,6 \u2013 122) = int(30,6*(m+1)) \u2013 122.\n\nNow we got the number of days in previous years plus the day count until the first of month m. Finally add the day d and we're done.\n\nSo the complete formula is\n\nday number = 365*y + int(y\/4) + int(30,6*(m+1)) \u2013 122 + d\n\nSince the constant 122 only determines from which day in the past we are counting, it can also be omitted:\n\nday number = 365*y + int(y\/4) + int(30,6*(m+1)) + d\nor\nday number = int(365,25*y) + int(30,6*(m+1)) + d\n\nThis is the simplified method that assumes every fourth year is a leap year. As it is true for the Julian calendar, or the Gregorian calendar between 1\u00a0Mar 1900 and 28\u00a0Feb\u00a02100.\n\nFor a correct implementation of the complete Gregorian calendar we have to add the 100-year- and 400-year-exceptions: Subtract the century years that are no leap years, and finally add back the 400-year cases that are. So the result is\n\nday number = 365*y + int(y\/4) \u2013 int(y\/100) + int(y\/400) + int(30,6*(m+1)) + d\nor\nday number = int(365,25*y) \u2013 int(y\/100) + int(y\/400) + int(30,6*(m+1)) + d\n\nOr with integer arithmetics, where \"div\" denotes an integer division:\n\nday number = 365*y + y div 4 \u2013 y div 100 + y div 400 + (153*(m+1)) div 5 + d\n\nThat's all. No magic, just simple math. ;-)\n\nDieter\n \u00ab Next Oldest | Next Newest \u00bb","date":"2022-09-27 01:50:08","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\": 0.272587388753891, \"perplexity\": 1621.786232238242}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030334974.57\/warc\/CC-MAIN-20220927002241-20220927032241-00438.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.biostars.org\/p\/157232\/","text":"Identify point mutations from each read in sam files\n1\n0\nEntering edit mode\n5.6 years ago\nDVA \u25b4 550\n\nHello,\n\nI wonder if there is an easy way to look at point mutations from SAM files. I understand there are mutation calling software such as GATK, but I need to look at each individual read, not the sequence region as a whole.\n\nThe CIGAR column in SAM does not have any information about point mutations, and the TAG field does not seem to help either. My current method is just to align each read back to the reference using the coordinates given, but I would appreciate any other better method.\n\nThanks everyone!\n\nsam mutations \u2022 4.1k views\n1\nEntering edit mode\n\nYou can use the MD field in the BAM file\u00a0that stores information about mismatching positions. If your bam files doesn't have MD tag, you can generate it using samtools calmd function. Check these relevant posts:\u00a0Position Of Mismatches Per Read From A Sam\/Bam FileHow To Find Out The Mismatchs Of An Alignment Entry In The Sam File?\n\n0\nEntering edit mode\n\nThank you! I didn't know samtools could do that. I really appreciate your help!\n\n4\nEntering edit mode\n5.6 years ago\n\nuse my tool sam2tsv https:\/\/github.com\/lindenb\/jvarkit\/wiki\/SAM2Tsv\n\n\\$ java -jar dist\/sam2tsv.jar -A \\\n-r samtools-0.1.18\/examples\/toy.fa\nsamtools-0.1.18\/examples\/toy.sam\nr001 163 ref 0 T . 7 T M\nr001 163 ref 1 T . 8 T M\nr001 163 ref 2 A . 9 A M\nr001 163 ref 3 G . 10 G M\nr001 163 ref 4 A . 11 A M\nr001 163 ref 5 T . 12 T M\nr001 163 ref 6 A . 13 A M\nr001 163 ref 7 A . 14 A M\nr001 163 ref 8 A . . . I\nr001 163 ref 9 G . . . I\nr001 163 ref 10 A . . . I\nr001 163 ref 11 G . . . I\nr001 163 ref 12 G . 15 G M\nr001 163 ref 13 A . 16 A M\nr001 163 ref 14 T . 17 T M\nr001 163 ref 15 A . 18 A M\nr001 163 ref . . . 19 G D\n\n\n(...)\n\n0\nEntering edit mode\n\nThanks so much! Just one more question: the output for single mismatches is marked by \"M\", just like exact matches. (e.g.: In your output example of sam2tsv website:\u00a0r003 0 ref 2 C . 11 A M) - does your program offer a way to distinguish these from exact matches?\n\n0\nEntering edit mode\n\nThe M is the 'M' of the cigar string. You can use awk to test if the column (REF-base)==column(READ-base)\u00a0 . Or you can use https:\/\/github.com\/lindenb\/jvarkit\/wiki\/SamFixCigar to changed the 'M' to 'X' or '='\n\n0\nEntering edit mode\n\nThank you. The java version used is 1.7.0_65, and other requirements on your websites are also fulfilled. However, make sam2tsv gave me the following errors:\n\n### COMPILING sam2tsv ######\nmkdir -p \/dir\/jvarkit\/jvarkit\/_tmp-1.133\/META-INF \/dir\/jvarkit\/jvarkit\/dist-1.133 galaxy-bundle\/jvarkit\n#create galaxy\nrm -f galaxy-bundle\/sam2tsv.xml\nxsltproc --path \/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xml --output galaxy-bundle\/jvarkit\/sam2tsv.xml --stringparam name \"sam2tsv\" --stringparam class \"com.github.lindenb.jvarkit.tools.sam2tsv.Sam2Tsv\" --stringparam classpath \"commons-jexl-2.1.1.jar commons-logging-1.1.1.jar htsjdk-1.133.jar snappy-java-1.0.3-rc3.jar sam2tsv.jar\" --stringparam version cat \/dir\/jvarkit\/jvarkit\/.git\/refs\/heads\/master \/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xsl\/tools2galaxy.xsl \/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xml\/tools.xml || echo \"XSLT failed (ignored)\"\nXSLT failed (ignored)\ncp galaxy-bundle\/jvarkit\/sam2tsv.xml \/dir\/jvarkit\/jvarkit\/_tmp-1.133\/META-INF\/galaxy.xml\ncp: cannot stat galaxy-bundle\/jvarkit\/sam2tsv.xml': No such file or directory\nmake: [sam2tsv] Error 1 (ignored)\n#generate java code if needed = a file with .xml exists, requires xsltproc\nif [ -f \"\/dir\/jvarkit\/jvarkit\/src\/main\/java\/com\/github\/lindenb\/jvarkit\/tools\/sam2tsv\/Sam2Tsv.xml\" ] ; then mkdir -p \/dir\/jvarkit\/jvarkit\/src\/main\/generated-sources\/java\/com\/github\/lindenb\/jvarkit\/tools\/sam2tsv\/ && xsltproc --xinclude --stringparam githash cat \/dir\/jvarkit\/jvarkit\/.git\/refs\/heads\/master --path \"\/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xml\" -o \/dir\/jvarkit\/jvarkit\/src\/main\/generated-sources\/java\/com\/github\/lindenb\/jvarkit\/tools\/sam2tsv\/AbstractSam2Tsv.java \/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xsl\/command2java.xsl \"\/dir\/jvarkit\/jvarkit\/src\/main\/java\/com\/github\/lindenb\/jvarkit\/tools\/sam2tsv\/Sam2Tsv.xml\" ; fi\n\/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xsl\/command2java.xsl:304: parser error : Opening and ending tag mismatch: template line 279 and if\n<\/xsl:if>\n^\n\/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xsl\/command2java.xsl:325: parser error : Opening and ending tag mismatch: stylesheet line 2 and template\n<\/xsl:template>\n^\n\/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xsl\/command2java.xsl:328: parser error : Extra content at the end of the document\n<xsl:template match=\"c:option[@type='outFile' and not(@multiple='true')]\" mode=\"\n^\ncannot parse \/dir\/jvarkit\/jvarkit\/src\/main\/resources\/xsl\/command2java.xsl\nmake: *** [sam2tsv] Error 4\n\n\nI also tried re-download and install, but that doesn't help. Could you please let me know what I missed here? Thank you.\n\n0\nEntering edit mode\n\nHi Pierre, Thanks again for your help - but I'm having issues with \"make sam2tsv\" and wonder if you could please give me some hints (see the previous comment). I'm sorry to bother you and I hope it's not a very silly question to waste your time... Thank you very much in advance.\n\n0\nEntering edit mode\n\nopps ! thank you this is a file I recently modified. It's Fixed now : https:\/\/github.com\/lindenb\/jvarkit\/commit\/b1ddf8ca859f85caec9c29b4ee62f2364b56902f\u00a0 . Sorry, download or pull again .\n\n0\nEntering edit mode\n\nThanks a lot for the clarification. I'm going to try it out tomorrow:)\n\n##Update:\n\nIt works~ Yay! Thanks a lot.\n\n0\nEntering edit mode\n\nHi Pierre,\n\nmany thanks for this tool! It's very useful also for me.\n\nI have just one question. Is there a way how to get records only for selected locations\/snps, not for the whole read? I have quite long list of snps for which I would like to get a record in tsv file. As the tsv file contains all positions of every read covering these snps, it gets really big (hundreds of G). I can post-process the file to select only positions that I am interested in, but it would be much easier if I could limit the positions at the beginning.\n\nAs an alternative, I am now splitting my original sam\/bam file into a number of small files and I will try to process it in pieces.\n\nI am sorry if my question is naive, I don't have any real bioinformatic background.\n\nThank you!\n\n0\nEntering edit mode\nsamtools -b view your.bam your.bed | java -jar sam2tsv.jar\n`\n0\nEntering edit mode\n\nHi Pierre,\n\nthanks for reply. I tried it already but evidently the amount of snps I want to process is way too big. This approach reduces the original bam file to approximately 78% of it's original size.\n\nBut I can proceed with genome split in parts.\n\nThanks","date":"2021-04-19 11:38:26","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\": 0.4644446074962616, \"perplexity\": 6180.029226897469}, \"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-2021-17\/segments\/1618038879374.66\/warc\/CC-MAIN-20210419111510-20210419141510-00377.warc.gz\"}"} | null | null |
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